Local Scale Models of Coral Reef Ecosystems for Scenario Testing and Decision Support
Tak Ching Fung
UCL
PhD Modelling Biological Complexity
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I, Tak Ching Fung, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis.
Signed: …………………………………………
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Acknowledgement First and foremost, I would like to thank my supervisor Professor Robert M. Seymour for his indispensable and generous guidance throughout my PhD, on technical and nontechnical matters. I would also like to extend my thanks to researchers of the Modelling and Decision Support Working Group (MDSWG) of the Coral Reef Targeted Research (CRTR) Project, in particular Professor Craig R. Johnson, for invaluable advice and indepth discussions during the progression of my PhD project. In addition, I am grateful to colleagues at UCL’s CoMPLEX and Maths departments who have supported me during my PhD. I am indebted to the Engineering and Physical Sciences Research Council (EPSRC) for funding my PhD and making the completion of it possible.
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Abstract The world’s coral reefs have been severely degraded over the past four decades and better management is urgently required to reverse this trend. Mathematical models are important tools for hypothesis and scenario testing, and are thus essential for better management. In this thesis, models of a coral reef ecosystem are constructed that aim to identify the key ecological processes responsible for reef degradation. The models operate at a ‘local’ intra-reef scale and they are dynamic, deterministic and non-spatial. A modelling strategy is used which adds complexity in a step-wise fashion, and this aids in the interpretation of results. The models are analyzed using mathematical equilibrium theory and are then parameterised using empirical data from reefs worldwide. Afterwards, the potential of reefs to undergo phase shifts from coral- to algaldominance under the effects of three anthropogenic stressors – fishing, nutrification and sedimentation – are investigated. This involves the application of a novel type of genetic algorithm, systematic sweeps of the parameter ranges and two types of sensitivity analysis. A key result is that the presence of macroalgae can give rise to multiple equilibria and hence discontinuous phase shifts under each of the three stressors investigated, by strengthening positive feedback to an algal-dominated state. Another key result is that reef benthic covers show a high degree of non-linearity under the three stressors. Overall, the results show that reefs are inherently prone to phase shifts under anthropogenic stress. These results suggest that in order to maximise resilience to reef degradation, there is an urgent need to effectively manage different anthropogenic stressors simultaneously. The models in this thesis can be parameterised for local reef areas at specific locations and then used for scenario testing, and they also serve as a solid foundation on which to add and investigate more complexity.
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Table of Contents Glossary..........................................................................................................................15 Part I: Introduction ......................................................................................................21 Chapter 1. General Overview of Coral Reef Ecosystems......................................23 1.1 Genesis and Distribution of Coral Reefs...........................................................24 1.2 Coral Reef Ecosystem Goods and Services ......................................................27 1.3 Worldwide Degradation of Coral Reefs............................................................30 1.3.1 Over-harvesting..........................................................................................32 1.3.2 Increased Nutrients.....................................................................................32 1.3.3 Increased Sediments...................................................................................33 1.3.4 Increased Atmospheric CO2 Concentration................................................33 1.3.5 Underlying Causes of Reef Degradation....................................................34 Chapter 2. Review of Coral Reef Modelling Literature ........................................36 2.1 Non-Spatial Models ..........................................................................................36 2.1.1 Network Models.........................................................................................36 2.1.2 Differential Equations Models ...................................................................38 2.1.2.1 Ecosystem-fisheries Model of McClanahan (1995)............................38 2.1.2.2 Benthic Model of McCook et al. (2001b) ...........................................41 2.1.2.3 ECOSIM Models.................................................................................41 2.1.2.4 Benthic Model of Mumby et al. (2007a).............................................42 2.1.2.5 Social-ecological Model of Kramer (2007) ........................................43 2.2 Spatial Models...................................................................................................45 2.2.1 Local Spatial Models with No Consideration of Fish or Urchins..............46 2.2.2 Local Spatial Models with Consideration of Fish and/or Urchins.............47 2.2.3 Regional Spatial Models ............................................................................48 2.3 Summary ...........................................................................................................49 Chapter 3. Thesis Contribution ...............................................................................51 3.1 Overview of Thesis ...........................................................................................51 3.2 Significance and Novelty of Thesis ..................................................................53 3.2.1 Systematic Investigation of Phase Shifts ...................................................54 3.2.2 Determining the Potential for Non-linearity ..............................................56 3.3 Further Comparison with Existing Coral Reef Modelling Studies ...................57 Part II: Local Models of Coral Reef Ecosystems .......................................................61 5
Chapter 4. Benthic Model without Macroalgae .....................................................61 4.1 Modelling Methodology for Benthic Models ...................................................61 4.2 The Coral-Turf Model (CTm)...........................................................................62 4.2.1 Details of Functional Groups Modelled in the CTm..................................63 4.2.2 Details of Interactions Modelled in the CTm.............................................65 4.2.3 Equations for the CTm ...............................................................................67 4.2.4 Mathematical Analysis of the CTm ...........................................................70 Appendix A4 ..............................................................................................................71 A4.1 Theorems and Conditions Used .....................................................................71 A4.2 The Coral-Turf Model (CTm) ........................................................................75 A4.2.1 Showing that 0 ≤ C (t ), T (t ), S (t ) ≤ 1 for the CTm....................................75 A4.2.2 Finding the Number of Equilibria for the CTm ......................................76 A4.2.3 Finding the Local Stability of Equilibria for the CTm............................81 Chapter 5. Benthic Models with Macroalgae .........................................................84 5.1 The Coral-Algae Model (CAm) ........................................................................84 5.1.1 Details of the Macroalgae Functional Group .............................................85 5.1.2 Details of Extra Interactions Modelled in the CAm...................................85 5.1.3 Equations for the CAm...............................................................................88 5.1.4 Mathematical Analysis of the CAm...........................................................90 5.2 The Coral-Turf-Macroalgae Model (CTMm) ...................................................90 5.2.1 Equations for the CTMm ...........................................................................91 5.2.2 Mathematical Analysis of the CTMm........................................................92 Appendix A5 ..............................................................................................................96 A5.1 The Coral-Algae Model (CAm) .....................................................................96 A5.1.1 Showing that 0 ≤ C (t ), A(t ), S (t ) ≤ 1 for the CAm...................................96 A5.1.2 Finding the Number of Equilibria for the CAm......................................96 A5.1.3 Finding the Local Stability of Equilibria for the CAm ...........................98 A5.2 The Coral-Turf-Macroalgae Model (CTMm) ..............................................100 A5.2.1 Showing that 0 ≤ C (t ), T (t ), M (t ), S (t ) ≤ 1 for the CTMm.....................100 A5.2.2 Equilibrium Analysis for the CTMm ....................................................100 A5.2.3 Finding the Local Stability of Equilibria for the CTMm ......................101 Chapter 6. Fish and Urchin Models ......................................................................104 6.1 Modelling Methodology for Fish and Urchin Models ....................................104 6.2 The Herbivorous Fish-Piscivorous Fish-Urchins Model (HPUm)..................104 6.2.1 Details of Functional Groups Modelled in the HPUm.............................105 6
6.2.2 Details of Interactions Modelled in the HPUm........................................107 6.2.3 Equations for the HPUm ..........................................................................112 6.2.4 Mathematical Analysis of the HPUm ......................................................115 6.3 The Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous Fish-Urchins Model (HSLUm) .........................................................117 6.3.1 Equations for the HSLUm........................................................................119 6.3.2 Mathematical Analysis of the HSLUm ....................................................120 Appendix A6 ............................................................................................................128 A6.1 The Herbivorous Fish-Piscivorous Fish-Urchins Model (HPUm)...............128 A6.1.1 Proving the Existence of at Least One Equilibrium for the HPUm ......128 A6.2 The Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous Fish-Urchins Model (HSLUm) .........................................................129 A6.2.1 Derivation of the φ Ps Term ...................................................................129 A6.2.2 Proving the Existence of at Least One Equilibrium for the HSLUm....131 Chapter 7. Integrated Models ................................................................................134 7.1 Mathematical Analysis of Integrated Models .................................................135 7.1.1 General Analysis for All Integrated Models ............................................135 7.1.2 Example Analysis for the Most Complex Integrated Model ...................137 Chapter 8. Discussion for Part II...........................................................................141 Part III: Quantifying the Parameters .......................................................................146 Chapter 9. Parameter Ranges for the Benthic Groups .......................................146 9.1 Hard Coral Parameters ....................................................................................147 9.2 Algae Parameters ............................................................................................151 Chapter 10. Parameter Ranges for the Fish Groups ...........................................161 10.1 Fish Recruitment Parameters ........................................................................163 10.2 Fish Feeding and Growth Parameters ...........................................................167 10.3 Fish Mortality Parameters .............................................................................173 Chapter 11. Parameter Ranges for the Sea Urchin Group .................................185 11.1 Urchin Recruitment Parameters ....................................................................185 11.2 Urchin Feeding and Growth Parameters .......................................................187 11.3 Urchin Mortality Parameter ..........................................................................190 Chapter 12. Discussion for Part III .......................................................................193 Part IV: Exploring the Parameter Space..................................................................196 Chapter 13. Parameterising and Investigating Fishing, Nutrification and Sedimentation ..........................................................................................................196 7
13.1 Continuous and Discontinuous Phase Shifts under Anthropogenic Stress ...197 13.1.1 Defining Phase Shifts.............................................................................197 13.1.2 Stressors Investigated.............................................................................198 13.1.2.1 Fishing.............................................................................................198 13.1.2.2 Nutrification ....................................................................................199 13.1.2.3 Sedimentation..................................................................................200 13.1.2.4 Caveats ............................................................................................200 13.1.3 Searching for Phase Shifts under Anthropogenic Stress........................202 13.2 Parameterising Fishing, Nutrification and Sedimentation ............................202 13.2.1 Fishing....................................................................................................202 13.2.2 Nutrification ...........................................................................................203 13.2.3 Sedimentation.........................................................................................205 13.3 Detecting Phase Shifts...................................................................................206 13.3.1 Finding Continuous Phase Shifts ...........................................................206 13.3.2 The Immune-Inspired Algorithm ...........................................................207 13.3.3 Finding Discontinuous Phase Shifts ......................................................208 13.4 Phase Shift Results........................................................................................210 13.4.1 Continuous Phase Shifts.........................................................................210 13.4.2 Discontinuous Phase Shifts ....................................................................212 13.5 Investigating Synergy between Fishing, Nutrification and Sedimentation...215 13.5.1 Finding Synergy.....................................................................................215 13.5.2 Synergy Results......................................................................................219 Appendix A13 ..........................................................................................................224 A13.1 Immune-Inspired Algorithm Details ..........................................................224 A13.2 Applying the Immune-Inspired Algorithm to the Coral-Turf Model (CTm) ...............................................................................................................................225 A13.2.1 Deriving and Computing the Fitness Function for the CTm...............228 A13.3 Applying the Immune-Inspired Algorithm to the Coral-Algae Model (CAm) ...............................................................................................................................230 A13.4 Applying the Immune-Inspired Algorithm to the Coral-Turf-Macroalgae Model (CTMm).....................................................................................................231 A13.4.1 Deriving and Computing the Fitness Function for the CTMm ...........235 Chapter 14. Parameter Sweeps..............................................................................238 14.1 Mathematics of Parameter Sweeps ...............................................................238 14.1.1 Defining Different Types of Continuous Phase Shifts...........................239 8
14.1.2 Defining Different Types of Discontinuous Phase Shifts ......................245 14.1.3 Sweeps Performed for the Benthic Models............................................246 14.2 Parameter Sweep Results ..............................................................................249 14.2.1 CTm Results...........................................................................................249 14.2.2 CAm Results ..........................................................................................255 Appendix A14 ..........................................................................................................266 A14.1 Derivation of the Explicit Forms of θ (C ) .................................................266 Chapter 15. Sensitivity Analyses............................................................................269 15.1 Mathematics of Sensitivity Analysis Techniques .........................................269 15.1.1 Elementary Effects Method....................................................................270 15.1.2 Sobol’ Variance Decomposition ............................................................272 15.1.3 Convergence of Results .........................................................................273 15.2 Sensitivity Analysis Results..........................................................................276 15.2.1 Elementary Effects Method Results for Benthic Models.......................276 15.2.2 Elementary Effects Method Results for Fish and Urchin Models .........288 15.2.3 Elementary Effects Method Results for Integrated Model ....................298 15.2.4 Sobol’ Variance Decomposition Results ...............................................301 Chapter 16. Discussion for Part IV .......................................................................311 Part V: Conclusions ....................................................................................................320 Chapter 17. General Discussion for Whole Thesis...............................................320 17.1 Thesis Results in Relation to Existing Studies and Management .................320 17.1.1 Construction and Parameterisation of Models .......................................320 17.1.2 Phase Shifts under Anthropogenic Stress ..............................................323 17.2 Limitations and Further Research .................................................................332 17.3 Conclusion ....................................................................................................337 Bibliography ............................................................................................................340
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List of Figures Figure 1.1. Cross-section of hard coral polyp .................................................................25 Figure 1.2. Worldwide distribution of coral reefs...........................................................25 Figure 1.3. The four major biogeographic regions in which coral reefs occur...............26 Figure 1.4. Hard coral cover in the Caribbean from 1977-2001.....................................31 Figure 1.5. Hard coral cover in the Indo-Pacific from 1968-2004..................................31 Figure 3.1. Conceptual graph of model complexity against model ‘payoff’ ..................52 Figure 4.1. Schematic diagram showing the functional groups and interactions in the CTm ........................................................................................................................69 Figure 5.1. Schematic diagram showing the functional groups and interactions in the CAm ........................................................................................................................89 Figure 5.2. Schematic diagram showing the functional groups and interactions in the CTMm .....................................................................................................................92 Figure 6.1. Graphs showing how (a) θ H changes with H for different values of iH and (b) θU changes with U for different values of iU .................................................109
(
)
2 Figure 6.2. Graph showing how X 2 i PX + X 2 changes with X for different values of
i PX .........................................................................................................................111 Figure 6.3. Schematic diagram showing the functional groups and interactions in the HPUm....................................................................................................................115 Figure 6.4. Schematic diagram showing the functional groups and interactions in the HSLUm .................................................................................................................121 Figure 7.1. Graphs of C B (θ ) and θ (C B ) for the CTM-HSLUm, for (a) a system with one stable equilibrium that has no macroalgae, (b) a system with one stable equilibrium that has macroalgae and (c) a system with two stable equilibria (one with and one without macroalgae) and one unstable equilibrium.........................140 Figure 10.1. Graph showing how X (i FX + X ) changes with X ∈ {H , P, Ps , Pl } for different values of i FX ...........................................................................................178 Figure 13.1. Conceptual diagram of (a) a continuous phase shift and (b) a discontinuous phase shift .............................................................................................................199 Figure 13.2. For the CTm, continuous phase shifts under (a) fishing, (b) nutrification and (c) sedimentation ............................................................................................213
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Figure 13.3. For the CAm, discontinuous phase shifts under (a) fishing, (b) nutrification and (c) sedimentation ............................................................................................216 Figure 13.4. For the CTMm, discontinuous phase shifts under (a) fishing, (b) nutrification and (c) sedimentation .......................................................................218 Figure 13.5. Graphs showing synergy between fishing, nutrification and sedimentation on coral and algal covers at equilibrium, for (a) the CTm, (b) the CAm and (c) the CTMm ...................................................................................................................221 Figure 13.6. Graphs showing how nutrification increases the value of θ at which a discontinuous phase shift occurs with fishing, for (a) the CAm and (b) the CTMm ...............................................................................................................................223 Figure A13.1. For the CTm, conceptual diagrams showing the cases where (a)
θ ′(T ; v ) < 0 for 0 ≤ T ≤ 1 and (b) θ ′(T ; v ) > 0 for some T in 0 ≤ T ≤ 1 ..............229 Figure A13.2. For the CTMm, conceptual diagrams showing the cases where (a)
g ′M (M ; v ) < 0 for 0 ≤ M ≤ 1 and (b) g ′M (M ; v ) > 0 for some M in 0 ≤ M ≤ 1 ..237 Figure 14.1. Graphs of (a) θ (C ) , (b) E (C ) , (c) θ (T ) , and (d) G (T ) for an arbitrary set of parameters for the CTm ....................................................................................244 Figure 14.2. Graphs of (a) θ (C ) and (b) θ (A ) for an arbitrary set of parameters for the CAm ......................................................................................................................247 Figure 14.3. For the CTm, graphs showing the % of sampled parameter sets giving (a) an RC* -continuous phase shift, (b) an S *A -continuous phase shift and (c) an
(R
* C
)
, S A* -continuous phase shift ...........................................................................252
Figure 14.4. Graphs showing the frequency distributions of C * for continuous phase shifts found for the CTm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s ...............................................................................................................255 Figure 14.5. Graphs showing the frequency distributions of T * for continuous phase shifts found for the CTm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s ...............................................................................................................257 Figure 14.6. For the CAm, graphs showing the % of sampled parameter sets giving (a) an RC* -continuous phase shift, (b) an S *A -continuous phase shift and (c) an
(R
* C
)
, S A* -continuous phase shift ...........................................................................258
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Figure 14.7. For the CAm, graphs showing the % of sampled parameter sets giving (a) an RC* -discontinuous phase shift, (b) an S *A - discontinuous phase shift and (c) an
(R
* C
)
, S A* -discontinuous phase shift ......................................................................260
Figure 14.8. Graphs showing the frequency distributions of C * for continuous phase shifts found for the CAm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s ...............................................................................................................263 Figure 14.9. Graphs showing the frequency distributions of A * for continuous phase shifts found for the CAm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s ...............................................................................................................265 Figure 15.1. Graphs showing (a)-(c) M (d ij ) and (d)-(f) s.d. (d ij ) for each of the 10 inputs for the CTm. The output is C at (a), (d) t = 5 yrs , (b), (e), t = 25 yrs and (c), (f) t = 50 yrs ..........................................................................................................279 Figure 15.2. Graphs showing (a)-(c) M (d ij ) and (d)-(f) s.d. (d ij ) for each of the 10 inputs for the CTm. The output is T at (a), (d) t = 5 yrs , (b), (e), t = 25 yrs and (c), (f) t = 50 yrs ..........................................................................................................281 Figure 15.3. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CAm. The output is C at t = 25 yrs .................................................................283 Figure 15.4. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CAm. The output is A at t = 25 yrs .................................................................284 Figure 15.5. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CTMm. The output is C at t = 25 yrs ..............................................................285 Figure 15.6. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CTMm. The output is T at t = 25 yrs ..............................................................286 Figure 15.7. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CTMm. The output is M at t = 25 yrs .............................................................287 Figure 15.8. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 18 inputs for the HPUm. The output is H at t = 25 yrs ..............................................................290 Figure 15.9. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 18 inputs for the HPUm. The output is P at t = 25 yrs...............................................................291 Figure 15.10. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 18 inputs for the HPUm. The output is U at t = 25 yrs ..............................................................292 12
Figure 15.11. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is H at t = 25 yrs ............................................................294 Figure 15.12. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is Ps at t = 25 yrs ..........................................................295 Figure 15.13. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is Pl at t = 25 yrs ..........................................................296 Figure 15.14. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is U at t = 25 yrs ............................................................297 Figure 15.15. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is C at t = 25 yrs ............................................300 Figure 15.16. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is T at t = 25 yrs.............................................301 Figure 15.17. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is M at t = 25 yrs............................................302 Figure 15.18. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is H at t = 25 yrs ............................................303 Figure 15.19. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is Ps at t = 25 yrs...........................................304 Figure 15.20. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is Pl at t = 25 yrs...........................................305 Figure 15.21. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm. The output is U at t = 25 yrs ............................................306 Figure 15.22. Graphs showing sij and sijT for each of the 10 inputs for the CTm, for the output C at (a) t = 5 yrs, (b) t = 25 yrs and (c) t = 50 yrs .....................................308 Figure 15.23. Graphs showing sij and sijT for each of the 10 inputs for the CTm, for the output T at (a) t = 5 yrs, (b) t = 25 yrs and (c) t = 50 yrs .....................................309
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List of Tables Table 1.1. The main coral reef ecosystem goods and services .......................................27 Table 5.1. Parameters for the CTm, the CAm and the CTMm, with their ecological meanings .................................................................................................................95 Table 6.1. Parameters for the HPUm and the HSLUm, with their ecological meanings ...............................................................................................................................127 Table 7.1. Table showing the six integrated models.....................................................135 Table 9.1. Parameters for the CTM, the CAm and the CTMm, with their ecological meanings and empirically derived ranges .............................................................160 Table 10.1. Fish parameters for the HPUm and the HSLUm, with their ecological meanings and empirically derived ranges .............................................................184 Table 11.1. Urchin parameters for the HPUm and the HSLUm, with their ecological meanings and empirically derived ranges .............................................................192 Table 13.1. Parameters for the benthic models that are affected by fishing, nutrification and sedimentation, together with the pristine ranges and the ranges extended by the stressors .................................................................................................................207 Table 13.2. Summary of sub-regions of parameter spaces searched using the IIA, and which models they apply for .................................................................................211 Table 13.3. Part of the IIA results for the CAm and CTMm ........................................215 Table 14.1. Summary of the 12 parameter sweeps performed for each of the CTm and the CAm ................................................................................................................250
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Glossary The following table gives definitions of abbreviated expressions used in this thesis, for ease of reference. Terms are listed in alphabetical order, with Greek letters following all the Latin letters.
Expression A
Definition Algal proportional cover (sum of turf algal and macroalgal proportional covers)
ASS B BCA C CAm
Alternative stable states Biomass B-Cell Algorithm Hard coral proportional cover Coral-Algae model
CA-HPUm
Coral-Algae-Herbivorous Fish-Piscivorous Fish-Urchins model
CA-HSLUm
Coral-Algae-Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous Fish-Urchins model
CCA
Crustose coralline algae
CLS
Characteristic length scale
CoT
Crown-of-Thorns
CPCe
Coral Point Count with Excel extensions (program)
CRTR-
Coral Reef Targeted Research and Capacity Building for
CBMP
Management Program
CTm CTMm CT-HPUm CT-HSLUm
Coral-Turf model Coral-Turf-Macroalgae model Coral-Turf-Herbivorous Fish-Piscivorous Fish-Urchins model Coral-Turf-Herbivorous Fish-Small to Intermediate Piscivorous FishLarge Piscivorous Fish-Urchins model
CTM-HPUm Coral-Turf-Macroalgae-Herbivorous Fish-Piscivorous Fish-Urchins model CTM-
Coral-Turf-Macroalgae-Herbivorous Fish-Small to Intermediate
HSLUm
Piscivorous Fish-Large Piscivorous Fish-Urchins model
[CO2]atm
Atmospheric carbon dioxide concentration
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d
Day
dC
Coral mortality rate
dH
The mortality rate of herbivorous fish from all factors other than predation and fishing
dP
The mortality rate of piscivorous fish from all factors other than predation and fishing
d Pl
The mortality rate of LP fish from all factors other than predation and fishing
d Ps
The mortality rate of SIP fish from all factors other than predation and fishing
dU
The mortality rate of sea urchins
EA
Eastern Atlantic
ECLSP
Exuma Cays Land and Sea Park
EP
Eastern Pacific
eqn.
Equation
eqns.
Equations
f
The maximum catch rate due to fishing
GBR
Great Barrier Reef
GDP
Gross Domestic Product
GNP
Gross National Product
gM
Maximum rate at which macroalgae is grazed
gP
The maximum predation rate of piscivorous fish on herbivorous fish
g Pl
The maximum predation rate of LP fish on herbivorous fish
g Ps
The maximum predation rate of SIP fish on herbivorous fish
gT
Maximum rate at which turf algae is grazed
H
Herbivorous fish biomass
ha
Hectares (1 hectare = 10,000 m3)
HPUm HSLUm
Herbivorous Fish-Piscivorous Fish-Urchins model Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous Fish-Urchins model
IIA
Immune-Inspired Algorithm
IWP
Indo-West Pacific or Indo-Pacific
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i FH
A parameter which measures the inaccessibility of herbivorous fish to fishermen
i FP
A parameter which measures the inaccessibility of piscivorous fish to fishermen
i FPl
A parameter which measures the inaccessibility of LP fish to fishermen
i FPs
A parameter which measures the inaccessibility of SIP fish to fishermen
iH
A parameter which measures the inaccessibility of algae (turf and macroalgae) to herbivorous fish grazing
i PH
A parameter which measures the inaccessibility of herbivorous fish to predation by piscivorous fish
i PlPs
A parameter which measures the inaccessibility of SIP fish to predation by LP fish
i PP
A parameter which measures the inaccessibility of piscivorous fish to predation by other piscivorous fish
iU
A parameter which measures the inaccessibility of algae (turf and macroalgae) to sea urchin grazing
KCl
Potassium chloride
LAS
Locally asymptotically stable
LHS
Latin hypercube sampling
LP
Large piscivorous
l Cb
Rate at which coral larvae, produced by local established brooding corals, recruit onto space
l Cs
Rate at which exogenous spawning coral larvae recruit onto space
l Hen
The endogenous recruitment rate of herbivorous fish
l Hex
The exogenous recruitment rate of herbivorous fish
l Pen
The endogenous recruitment rate of piscivorous fish
l Pex
The exogenous recruitment rate of piscivorous fish
l Plen
The endogenous recruitment rate of SIP fish due to reproduction by LP fish
en l Ps
The endogenous recruitment rate of SIP fish due to reproduction by SIP fish
17
ex l Ps
The exogenous recruitment rate of SIP fish
lUen
The endogenous recruitment rate of sea urchins
lUex
The exogenous recruitment rate of sea urchins
M
Macroalgal proportional cover
MBRS MDSWG
Mesoamerican Barrier Reef System Modelling and Decision Support Working Group
MM
Morris Method
ODE
Ordinary differential equations
P
Piscivorous fish biomass (except in ECOPATH models, where it is production)
PC
Principal component
PCA
Principal Components Analysis
PLD
Pelagic larval duration
ppm
Parts per million
PG
Gamete production
Pl
Large piscivorous fish biomass
Ps
Small to intermediate piscivorous fish biomass
PS
Somatic production
PSA
Somatic production due to algal consumption
Q
Consumption
QH
Consumption of herbivorous fish
QP
Consumption of piscivorous fish
QPs
Consumption of small to intermediate piscivorous fish
RS
Random sampling
rC
Lateral growth rate of corals over space
rM
Lateral growth rate of macroalgae over space
rP
The proportion of biomass consumed by piscivorous fish used for growth
rPl
The proportion of biomass consumed by LP fish used for growth
rPs
The proportion of biomass consumed by SIP fish used for growth
S
Space proportional cover 18
SIP
Small to intermediate piscivorous
SST
Sea surface temperatures
t
Time
T
Turf algal proportional cover (except in model of McClanahan (1995), where it is the invertivorous fish biomass)
U
Sea urchin biomass
USVI
U.S. Virgin Islands
WA
Western Atlantic
αC
Growth rate of corals over turf, relative to the rate over space
βM
Coral growth is inhibited by the presence of nearby macroalgae and this is represented as depression of rC by the factor (1 − β M M ) , where M is the macroalgal cover
γ MC
Lateral growth rate of macroalgae over corals, relative to the rate over space
γ MT
Lateral growth rate of macroalgae over turf, relative to the rate over space
εC
Recruitment rate of corals onto turf, relative to the rate onto space
ζT
Growth rate of fine turf (occupying space)
θ
Grazing pressure
κU
A parameter which measures the biomass accumulated by urchin grazing that contributes to growth, relative to that for herbivorous fish grazing
λH
Competitiveness of herbivorous fish relative to sea urchins
λU
Competitiveness of sea urchins relative to herbivorous fish
µ M , µT , µ S
The herbivorous fish biomass accumulated through growth from grazing on 100% cover of macroalgae, turf or space (the turf within space is what is consumed) respectively
νM
The proportion of total algal cover that is macroalgal cover
ρH
The proportion of the total fishing pressure f which acts on herbivorous fish
ρP
The proportion of the total fishing pressure f which acts on piscivorous fish
ρ Pl
The proportion of the total fishing pressure f which acts on LP fish 19
ρ Ps
The proportion of the total fishing pressure f which acts on SIP fish
φ Ps
A constant parameter that determines how quickly SIP fish biomass becomes LP fish biomass due to predation and subsequent growth
ψP
The predation rate on piscivorous fish by other piscivorous fish, relative to that on herbivorous fish
ψ Pl
The predation rate on SIP fish by LP fish, relative to that on herbivorous fish
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Part I: Introduction The world’s coral reefs are being degraded by anthropogenic activities (Hughes et al. 2003), threatening the livelihoods of tens of millions of people (Birkeland 1997a). There is thus an urgent need for more effective management of coral reefs based upon improved scientific knowledge (Bellwood et al. 2004). This thesis aims to improve the theoretical basis for effective reef management by constructing dynamical coral reef models and using them to investigate reef dynamics under anthropogenic stress. Following a stepwise-refinement modelling procedure that adds complexity incrementally, three differential equations models of a coral reef benthos are constructed. For each model, functional groups that are important indicators of reef health are included, together with key interactions between these groups. The state variable for each group is the proportional cover. In the simplest benthic model, hard corals and turf algae compete for space, whereas the two other models also include macroalgae, which are structurally more complex than turf algae. One model represents macroalgae amalgamated with turf algae as one dynamic group and the other model represents turf algae and macroalgae as two separate dynamic groups. Again following a stepwiserefinement procedure, two models of a coral reef fish and urchin community are constructed. In both models, the two main grazing groups, herbivorous fish and herbivorous sea urchins, and their key interactions are modelled. Piscivorous fish and their interactions are also modelled, with the more complex model splitting this group up into two size classes, thus allowing different demographic rates for the two groups to be included. The state variable for each group is the biomass. Integrated models are then constructed which combine the dynamics of a benthic model with that for a fish and urchin model. All models are for a local scale of approximately tens of metres to a few kilometres. Various techniques are used to analyze the equilibrium properties of each model. Generic regularity assumptions together with the Poincaré-Hopf Theorem (Milnor 1965) are used to show that each of the two simplest benthic models have an odd number of isolated equilibria in the biologically realistic domain (a simplex), and the RouthHurwitz conditions (Murray 2002) are used to show which of these equilibria are locally asymptotically stable (LAS). Direct calculations then show that both models can theoretically exhibit multiple stable (LAS) equilibria. Bendixson’s Negative Criterion (Jordan and Smith 2005) is used to show that, when parameters take values from the 21
empirically derived ranges (see next paragraph), the simplest benthic model cannot exhibit periodic orbits. Also, results for the simplest benthic model are used to show that the most complex benthic model (with turf algae and macroalgae treated as separate dynamic variables) must have an odd number of equilibria without macroalgae in the biological domain. However, there may be additional equilibria with macroalgae present. For the fish and urchin models and the integrated models, the Richeson-Wiseman Theorem (Richeson and Wiseman 2002) is used to show that each model has at least one equilibrium in the biological domain. The parameters for each model are quantified using a variety of techniques and using data derived from many sources for reefs worldwide, resulting in empirically derived parameter ranges, and the effects of three key anthropogenic stressors – fishing, nutrification and sedimentation – on model parameters are also quantified. Thus, ‘generic’ reefs are investigated that have functional groups and interactions typical of a coral reef regardless of location, and which have parameter ranges that incorporate values from reefs worldwide. Using the parameterised benthic models, the likelihood of phase shifts on a generic reef is determined. These phase shifts involve a sizeable decrease in coral cover and/or a sizeable increase in algal cover, and are indicative of reef degradation. Two broad types of phase shifts are investigated: continuous and discontinuous. Continuous shifts are phase shifts where the model reef system has a single LAS equilibrium that changes continuously as a parameter (or parameters) affected by stress changes (change) continuously. In contrast, discontinuous shifts are phase shifts where the system passes through a region of multiple LAS equilibria. Discontinuous shifts thus involve a saddle-node bifurcation as two equilibria collide and cease to exist in the biological domain as the system passes the region of multiple LAS equilibria. The Immune-Inspired Algorithm (Kelsey and Timmis 2003), a form of genetic algorithm, is used to search the parameter spaces of each benthic model to determine regions where multiple LAS equilibria and hence discontinuous phase shifts are possible. In addition, for each of the two simplest benthic models, the entire parameter space is randomly sampled, and for different categories of continuous and discontinuous phase shifts, the proportion of parameter sets in the parameter space that can give a phase shift is estimated using the randomly sampled parameter sets. For each of the benthic models, each of the fish and urchin models and the most complex integrated model, the Elementary Effects method (Morris 1991) is applied as a type of sensitivity analysis to determine which parameters have the biggest effect on the state variables and the degree of non-linearity the state variables exhibit with changes in 22
the parameters. Sobol’ Variance Decomposition (Helton and Davis 2003) is applied, as another type of sensitivity analysis, to the simplest benthic model to determine the degree of interaction between parameters in changing the benthic covers. It is important to recognise that the model reefs analyzed are generic in the sense described above. This means that although the results derived further the knowledge of the general processes behind and trends in reef degradation and recovery, the results do not necessarily apply to a reef at a specific location because model reefs, by representing reefs spanning more than one location, are not necessarily typical of reefs at a specific location. Reefs from different locations can exhibit wide variability in reef dynamics, such as in algal and coral growth rates that can each vary over an order of magnitude. These differences arise because of different species compositions of organisms such as corals, and also physical factors such as depth, wave exposure and distance from the coast. This means that reefs at different locations respond differently to anthropogenic stress and a reef at a specific location may respond differently when compared to the generic model reefs analyzed in this thesis. For example, a reef system with a coral community that is composed predominantly of sediment tolerant species would experience less coral morality due to sedimentation than implied by the range found for the generic reefs. Another example would be a reef system with an algal community exposed to high natural levels of nutrient fluxes (this may be the case for an inshore reef (Hughes et al. 1999)), which would experience less algal growth due to nutrification than implied by the range for generic reefs. Thus, for the purposes of management of specific reefs, the results in this thesis must be interpreted with caution and in conjunction with knowledge on specific reef conditions. To test scenarios for reefs at a specific location, the generic models in this thesis should ideally be parameterised using data from that location, or locations which are similar to as far an extent as possible.
Chapter 1. General Overview of Coral Reef Ecosystems Over geological time, corals have created immense limestone structures over 2,000 km long (Great Barrier Reef) and 1,300 m thick (Eniwetok atoll) (Birkeland 1997a). These structures are called coral reefs and cover 284,000 km2 of the Earth’s surface (Spalding et al. 2001). No definition of coral reef ecosystems is universally accepted (Wilkinson 23
1999). However, a working definition is that they are complex ecosystems of marine plants, animals and minerals, with the primary reef builders being hard (or stony) corals and coralline algae; these primary reef-builders secrete calcium carbonate to produce the physical substructure of the reef, on and in the vicinity of which the biotic components of the system live (Wilkinson 1999). This definition shows that the study of coral reefs lies squarely in the domain of biological complexity, and is used in this thesis.
1.1 Genesis and Distribution of Coral Reefs Coral reefs are biogenic, that is, they are formed by biological processes in addition to geological ones. They have a rigid skeletal structure higher than the surrounding sediments, which influences the deposition of these sediments (Hallock 1997). Coral reefs are formed primarily by the accumulation of calcium carbonate (CaCO3) by reef organisms, in particular by hard corals. In total, there are 845 reef-building hard coral species (Carpenter et al. 2008). Reef-building corals are colonial organisms with each colony consisting of individual coral polyps connected by a common gastrovascular system. This system circulates and digests food particles derived from zooplankton that are captured by tentacles lining the polyp mouths (Muller-Parker and D’Elia 1997, Spalding et al. 2001). However, the majority of the corals’ energy requirements come from translocated photosynthetic products from symbiotic algae living in its cells. These algae are called ‘zooxanthellae’ and can provide over 95% of the metabolic requirements of the coral host (Muller-Parker and D’Elia 1997, Hoegh-Guldberg et al. 2007) (see Figure 1.1). Hereafter, ‘coral’ is used to denote both the coral host and its symbiotic algae. Physiological requirements for reef-building corals restrict the geographical range within which coral reefs are formed. Reef-building corals usually live in regions with average seawater temperatures in the range 24.8-30.2oC (Kleypas et al. 1999). Temperatures outside this range can affect the stability of the animal host-zooxanthellae symbiotic relationship, possibly leading to expulsion of the algae and death of the coral host (Muller-Parker and D’Elia 1997). At the Earth’s equator, incoming solar radiation makes a large angle with the Earth’s surface throughout the year. As a result, the heat energy per unit surface area is relatively high at the equator and decreases moving away from it. Thus, average temperatures decrease away from the equator (Birkeland 1997b). 24
The temperature requirements for reef-building corals mean that they usually occur between the latitudes 30oN and 30oS (Wood 1983). Figure 1.2 shows the distribution of coral reefs around the world.
Figure 1.1. A cross-section of a hard coral polyp and the surrounding skeleton (theca). The inset is a magnified view of the gastrodermis (inner cell layer) and the epidermis (outer cell layer), showing the location of a zooxanthella (ZOOX). The mesenterial filaments digest food particles. Diagram adapted from that in Muller-Parker and D’Elia (1997).
Figure 1.2. Worldwide distribution of coral reefs, which are represented as orange patches. The numbers on the right are the latitudes north and south of the equator (0). Diagram from Spalding et al. (2001).
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Since reef-building corals rely primarily on photosynthesis for their metabolic requirements, they need sufficient exposure to sunlight. Each location on Earth receives 12 hrs of sunlight a day on average, but this is more evenly distributed yearly near the equator (the tropics), where there is no annual cooling (Birkeland 1997b). Thus, another reason why coral reefs tend to form in a zone around the equator is because of the availability of sunlight all year round. There are four major biogeographic regions in the world in which coral reefs exist: the Indo-West Pacific or Indo-Pacific (IWP), the eastern Pacific (EP), the western Atlantic (WA) and the eastern Atlantic (EA) – see Figure 1.3. Reef communities in the IWP and WA are extensive and diverse, whereas those in the EP and EA are limited. Each region is characterised mainly by its endemic organisms (Paulay 1997). The IWP is the area between roughly 30oN and 30oS latitudes which covers the area from the Red Sea and eastern African coast to eastern Polynesia. The EP is the tropical western American coastline together with nearby islands such as the Galápagos Islands, the WA consists of the tropical eastern American coast, Bermuda, and nearby islands in the Caribbean and off Brazil, and the EA consists of the tropical western African coast and nearby islands such as the Cape Verde Islands (Paulay 1997).
Figure 1.3. The four major biogeographic regions in the world in which coral reefs occur: the Indo-West Pacific, the eastern Pacific, the western Atlantic and the eastern Atlantic. Diagram from Paulay (1997).
Coral reef communities are very diverse. In total, coral reefs have the greatest number of phyla and classes per hectare of any marine ecosystem, and the number of species is in the millions (Birkeland 1997a, Knowlton 2001).The vast number of species in coral reefs is not evenly spread geographically. Species diversity for corals and fishes on reefs in the IWP far exceeds that on reefs in the EP and Atlantic (Birkeland 1997a). There are approximately 700 hard coral species in the IWP compared to approximately 26
40 in the EP, 65 in the WA and 14 in the EA. For reef fish, there are approximately 3,000 species in the IWP, 300 in the EP and 750 in the WA (Paulay 1997). The greater diversity in the IWP is likely to result in greater functional redundancy for corals and fish. For example, the Great Barrier Reef (GBR) in Australia has more species in all coral and fish functional groups than Caribbean reefs (Bellwood et al. 2004).
1.2 Coral Reef Ecosystem Goods and Services Coral reefs provide a wide variety of ecosystem goods and services. Ecosystem goods are those goods that help to sustain human life, whereas ecosystem services are defined as the natural processes and conditions of the system that help to sustain human life. Ecosystem services include the production of ecosystem goods (Chapin et al. 2000). Table 1.1 gives a summary of the main ecosystem goods and services provided by coral reefs. Coastal societies benefit directly from these goods and services, whereas people from other societies can benefit if these goods and services are traded in regional and global markets. Over 100 countries have coral reefs along their coastlines and tens of millions of people depend on reefs as a source of protein and income (Birkeland 1997a, Moberg and Folke 1999). Koop et al. (2001) estimate that reefs contribute to the livelihoods of over 100 million people. Thus, coral reefs are very important for the survival of a large number of people.
Coral Reef Ecosystem Goods
Coral Reef Ecosystem Services
Dead and live fish
Production of all coral reef goods
Crustaceans
Protection of shorelines
Molluscs
Contribution towards formation of mangrove and seagrass habitats
Holothurians
Generation of sand for beaches
Dead and live corals
Provision of tourism opportunities
Sponges
Recording of climate and pollution Storage of genetic diversity Strengthening of social and cultural values
Table 1.1. The main coral reef ecosystem goods and services 27
Marine organisms caught from coral reef areas are a key coral reef good and most of these organisms are used as food. Currently, fish catch from coral reefs is in the range 1.4-4.2 million t yr-1, which is about 2-5 % of global fish catch (Pauly et al. 2002). This is a very high figure considering the low spatial area of coral reefs, which is only about 0.1% of the world’s oceans (Spalding et al. 2001). One quarter of total fish catch in developing countries originate from coral reefs (Bryant et al. 1998). Other organisms regularly caught are crustaceans (e.g., lobsters and prawns), molluscs (e.g., conches, oysters and clams) and holothurians (mainly sea cucumbers) (Birkeland 1997a, Moberg and Folke 1999). Some coral reef organisms caught, or parts of these organisms, are not consumed. For example, corals, dried fishes, Mother-of-pearl and giant clam shells are all sold as souvenirs. Substantial amounts of live corals and fish are also taken for the aquarium trade. For example, in 1991, the U.S. imported 250,000 live corals and the live aquarium industry is valued at US$ 28-44 million yr-1 (Birkeland 1997a, Pomeroy et al. 2008). Globally, about 2,000 species of coral are traded for aquarium use (Cesar et al. 2003), and hundreds of thousands of fish are caught each year for the aquarium trade (Kolm and Berglund 2003). Furthermore, live and dead hard corals are extracted and used for construction, for example, in the production of lime and for use as building materials. The Maldives is one country which does this, extracting ≈ 20,000 m3 of corals every year (Birkeland 1997a, Moberg and Folke 1999). Coral reef organisms also have medical uses. Soft corals, sponges, molluscs, seaweeds and sea anemones possess substances with anti-viral, anti-tumour, anti-microbial, anti-inflammatory and anticoagulating properties which are useful for making pharmaceutical products (Moberg and Folke 1999, Sipkema et al. 2005). Coral reefs provide many ecosystem services, one of which is the production of all the goods mentioned earlier. They also help to protect shorelines and coastal communities from the impact of storms and waves (Birkeland 1997a). In addition, by absorbing the energy of waves, coral reefs favour the creation of lagoons (Moberg and Folke 1999). This in turn is conducive to the formation of mangroves and seagrass beds, which are important nursery habitats for juvenile fish, some of which could be commercially important (Williams 1991, Moberg and Folke 1999, Mumby et al. 2004). Reefs also serve to protect these mangroves and seagrass beds once established (Birkeland 1997a). Coral reef ecosystems generate the sand seen on tropical beaches through erosion of the reefs by abiotic and biotic agents (such as storms and bioeroding 28
parrotfish respectively) (Moberg and Folke 1999). Tropical beaches are a component of the aesthetic value that reefs provide, important for attracting tourists. Tourists are also attracted by the diverse coral reef communities which form the basis of scuba diving and snorkelling activities. Thus, coral reefs contribute to the provision of tourism (Moberg and Folke 1999). Tourism is a major source of income for many countries and is the world’s largest industry. It provided US$ 3.5 trillion to the world GDP in 2002 and employed 199 million people. For 83% of the world’s countries, it is a top-five source of foreign exchange (Cesar et al. 2003). A lesser known service rendered by coral reef ecosystems is that they can act as records for climate and pollution. It is possible to track changes in salinity and historical sea surface temperatures (SST) using the chemical compositions of coral skeletons. The skeletons can be used to deduce the level of metals in the surrounding seawater through time as well (Moberg and Folke 1999). Owing to their vast biodiversity, coral reefs also serve as genetic libraries. These libraries can be used to create products directly useful to humanity, such as anti-cancer drugs (Moberg and Folke 1999), and can also be used in aquaculture (Birkeland 1997a). Coral reefs have a social and cultural significance too. Traditional coastal communities have developed religious rituals around reefs, such as in southern Kenya, and their cultures can be linked to nearby coral reefs, such as in the traditional reef management schemes employed in Pacific islands (Moberg and Folke 1999). Since these communities tend to fish cooperatively on reef waters, an activity where roles are defined, reefs can have a stabilising effect on society. Without such stabilisation, crime and suicide rates can increase (Birkeland 1997a). The total economic potential of coral reefs is enormous. A study by Costanza et al. (1997) found that the total worth of coral reef goods and services is US$ 375 billion yr-1. This was in comparison to a figure of US$ 33,300 billion yr-1 for all ecosystems and a figure of US$ 18,000 billion yr-1 for the sum of the GNPs of all countries in the world (Costanza et al. 1997). The figure for coral reefs is proportionately very high, considering they only cover about 0.1% of the world’s oceans (Spalding et al. 2001). Cesar et al. (2003) calculated a more conservative figure of US$ 30 billion yr-1 for the total net benefits provided by coral reefs – this is because they did not include as many different services as Costanza et al. (1997). In conclusion, there is a wealth of evidence that coral reefs are important to human survival, through the goods and services that they provide. They are very important economically, but also in intangible ways such as by providing aesthetic and cultural values. Thus, it is in the best interests of humanity to manage for the 29
sustainability of coral reefs, sustainability being “the ability to meet the needs of today without jeopardising the ability of future generations to meet their own needs” (Giller and O’Donovan 2002).
1.3 Worldwide Degradation of Coral Reefs The central problem in coral reef management is the worldwide degradation of coral reefs. Gardner et al. (2003) did a meta-analysis of 263 sites from 65 separate studies and found that average hard coral cover for reefs in the Caribbean basin decreased from about 50% to 10% over 24 yrs, starting from 1977 (see Figure 1.4). Similarly, Bruno and Selig (2007) performed a meta-analysis of 2,667 Indo-Pacific reefs and found that average hard coral cover nearly halved from 42.5% in the early 1980s to 22.1% by 2004 (see Figure 1.5). About 20% of the world’s coral reefs have lost 90% of their corals (Wilkinson 2004) and no pristine reefs, unaffected by humans, currently exist (Hughes et al. 2003). This declining trend has been attributed to anthropogenic and anthropogenically-enhanced causes, with the most common ones being over-harvesting, pollution and climate change (e.g., Wilkinson 1999, Bellwood et al. 2004, HoeghGuldberg et al. 2007). Bryant et al. (1998) reported results from the ‘Reefs at Risk’ project, which assessed 800 coral reef sites from all over the world. For each reef site, the risk of degradation from overexploitation, coastal development, inland pollution and erosion, and marine pollution was assessed as ‘low’, ‘medium’ or ‘high’. A reef site is then considered to be at high or medium risk if it is at high or medium risk respectively from one or more of these four types of threats. Otherwise, it is considered to be at low risk. It was found that 58% of the sites were at medium or high risk. Globally, coral reefs are threatened because of the increase in frequency, magnitude and diversity of human stressors (Hughes et al. 2003). Reef assemblages are usually resilient to just natural disturbances, such as hurricanes, which are neither too frequent nor too severe. Here, resilience refers to the ability to resist and recover from disturbances (Wilkinson 1999, Bellwood et al. 2004). However, when anthropogenic disturbances are added to the natural disturbances, reefs can fail to recover. In particular, there is often no recovery in the presence of chronic human stressors (Connell 1997). Reef degradation often involves a healthy coraldominated reef changing to a state which is algal-dominated. This can happen rapidly over ecological time-scales of years to decades and is referred to as a ‘phase shift’ or 30
Figure 1.4. Hard coral cover in the Caribbean from 1977-2001. The filled triangles represent the weighted mean percentage coral covers with 95% bootstrap confidence intervals, from a meta-analysis of 263 sites. The filled circles and the crosses both represent unweighted mean percentage coral covers, with the difference that data from the Florida Keys Coral Monitoring Project are left out in calculating the means represented by the crosses. The unfilled circles represent the number of studies. Graph is from Gardner et al. (2003).
Figure 1.5. Hard coral cover in the Indo-Pacific from 1968-2004. The bars and lines represent the means ± standard errors of the percentage coral covers and the unfilled circles represent the number of studies. Graph is from Bruno and Selig (2007).
31
‘regime shift’ (Done 1992, Hughes 1994, Bellwood et al. 2004). The new state can be stable over ecological time scales and is usually associated with a decline in the quality and/or quantity of ecosystem goods and services provided by the coral reef. For example, there could be fewer fish due to fishing, less tourism appeal due to the loss of corals and fish, and lower coral biodiversity due to the loss of coral species (McCook 1999). Thus, coral reef degradation has a societal and an economic loss, and it is important for management to prevent such degradation by maintaining the resilience of coral reefs and to initiate recovery of degraded reefs. To do this, it is necessary to understand the anthropogenic and anthropogenically-enhanced disturbances that affect coral reefs, which are now described in greater detail.
1.3.1 Over-harvesting
Over-harvesting is one of the biggest anthropogenic stressors responsible for the degradation of coral reefs and a common type of over-harvesting is overfishing of reef organisms such as commercially viable fish and invertebrates. The declines in coral cover seen in the western Atlantic from the 1980s is in part due to historical overfishing of herbivorous fishes, without which there was insufficient grazing pressure to suppress the proliferation of algae (Jackson et al. 2001). A factor contributing to overfishing on coral reefs is increasing globalisation of countries such as China, which has led to the fish market becoming increasingly global (Hughes et al. 2003). For example, the international live reef fish trade has led to depletion of stocks of large reef fish such as Cheilinus undulates (humphead wrasse) and is estimated to be worth US$ 0.4-1 billion annually (Pauly et al. 2002, Pomeroy et al. 2008). Another factor is the fierce competition that has resulted from the entry of non-traditional (and non-local) fishers into fisheries (Pauly et al. 2002).
1.3.2 Increased Nutrients
Coastal communities near coral reefs can pollute them by introducing waste directly into coral reef waters or into water systems that flow into reef waters (such as rivers connected to the reefs). The major pollutants are nutrients and sediments. Coral reefs are exposed to nutrients from natural external sources, such as seabirds (Smith and Johnson 1995), but these have been augmented by nutrients from anthropogenic sources. 32
Nutrient additions from human activities usually originate from land (Szmant 2002, McClanahan et al. 2002b). For example, nutrient run-off increases as a result of conversion of indigenous forest to permanent agriculture, and in Queensland, eastern Australia, flood plumes from agricultural lands are the biggest external nutrient source for the GBR lagoon (McClanahan et al. 2002b, Fabricius et al. 2003). There is also significant run-off from sewage plants and septic systems (Smith and Johnson 1995). An increase in nutrient availability can enhance algal growth in circumstances where nutrients are limiting, thus favouring reef degradation (Done 1992).
1.3.3 Increased Sediments
Sedimentation is a persistent and widespread problem affecting reefs near coasts and high islands (Richmond 1997, Brown 1997, Wilkinson 1999). Sediments found within coral reef ecosystems may arise naturally; for example, rivers may carry naturallyderived sediments from inland to coral reefs near their mouths (Wilkinson 1999). However, as with nutrients, this natural source of sedimentation has been augmented by sediments from anthropogenic sources. Increased sediment load is a prominent early effect of coastal development, and activities such as clear-cutting of forests can increase both sediment and nutrient input to reefs (Muller-Parker and D’Elia 1997, Bryant et al. 1998). Sediments can smother corals, causing stress due to the increased energy required to produce mucus to remove the sediment particles, and reduce light levels, inhibiting photosynthesis (Rogers 1990, Brown 1997). This can lead to increased coral mortality and decreased coral growth rates (e.g., Rogers 1983, Cortes and Risk 1985). Increased sediments can also have a detrimental effect on coral settlement and the survival of settled coral juveniles through the same mechanisms (e.g., Babcock and Smith 2000). Thus, sediments can lower the recovery potential of a degraded reef.
1.3.4 Increased Atmospheric CO2 Concentration
There are also global threats to coral reefs superimposed upon the more localised stressors. A main cause of these global threats is anthropogenically-enhanced levels of atmospheric carbon dioxide concentration ([CO2]atm), which gives rise to the two main threats of increased SST and ocean acidification. Since the beginning of the Industrial 33
Revolution in the late 18th century, human activities such as the burning of fossil fuels have contributed significantly to a 40% increase in [CO2]atm, which in turn has contributed to an increase in the global average temperature of ≈ 0.7oC since the late 19th century (Lough 2008). [CO2]atm is predicted to surpass 500 ppm by 2050-2100 and global mean temperature is predicted to rise by at least 2oC (Hoegh-Guldberg et al. 2007). This increase in atmospheric temperature would drive increases in SST; for example, increased SST has been predicted for sites in the Indian Ocean and the Caribbean throughout the 21st century (Sheppard 2003, Sheppard and Rioja-Nieto 2005). This would lead to an increased frequency of coral bleaching. Coral bleaching is the phenomenon whereby corals expel their symbiotic zooxanthellae as a stress response to above-normal water temperature. Because the zooxanthellae provide the pigment for the corals, the corals subsequently become white, as if bleached. The bleached coral populations exhibit higher mortality and reduced growth and fecundity rates (Hughes et al. 2003, Hoegh-Guldberg et al. 2007). Apart from coral bleaching, the increasingly warm waters are expected to increase the intensity of tropical storms such as hurricanes (Wilkinson and Souter 2008). A second major effect of increased [CO2]atm, ocean acidification, arises because the increased [CO2]atm leads to a greater oceanic concentration of CO2. Water reacts with CO2 to produce carbonic acid and this acid dissociates to form bicarbonate ions and protons; the protons produce more bicarbonate ions by reacting with carbonate ions. The net effect is to decrease the concentration of carbonate ions, which are required for the calcification of corals and other calcifying organisms. Thus, increasing [CO2]atm is predicted to lead to reduced coral reef growth, with carbonate ion concentrations possibly dropping below that required for reef accretion by 2050 (Hoegh-Guldberg et al. 2007).
1.3.5 Underlying Causes of Reef Degradation
It is important to note that the underlying basis behind all the anthropogenic and anthropogenically-enhanced stressors mentioned in section 1.3 is the increasing global human population and concomitant economic growth (Birkeland 2004). The world population has increased from 2.5 billion in 1950 to 6.7 billion in 2007, and is projected to increase to 7.8-11.9 billion by 2050 (Department of Economic and Social Affairs of the United Nations 2007). Currently, ≈ 0.5 billion people live within 100 km of a coral reef, some 8% of the world population (Bryant et al. 1998). The number of people 34
living near reefs is expected to increase because the world population is increasing and more than half of this population is expected to reside in urban areas by 2030, with most of these areas being near the coast (Palmer et al. 2004). Similarly, the world economy has increased at least three-fold during the last three decades of the 20th century or so (Wilkinson 1999). There is a need to tackle the ultimate causes of human population growth and associated economic growth in order to stop and prevent reef degradation. Associated with this problem is the need to develop a global human consciousness that values environmental sustainability (Jameson 2008).
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Chapter 2. Review of Coral Reef Modelling Literature Many initiatives have been taken to help conserve coral reefs. Every year, governments spend millions of dollars on reef conservation and a total of 691 organizations in 2004 were involved in conserving reefs, including 451 non-governmental organizations (Birkeland 2004). One of the most famous management initiatives is the Great Barrier Reef (GBR) Marine Park, which has no-take areas (prohibiting fishing) covering 33.4% of the GBR (Hoegh-Guldberg 2006). There is also a global network of volunteer divers, led by scientists, who use the ‘Reef Check’ protocol to monitor reefs (Hodgson 1999). Despite existing efforts to manage reefs, management practices need to be reassessed because reefs are still being degraded worldwide. In particular, there is a need to improve understanding of the underlying biological processes which confer resilience and sustainability to reefs in the context of human activities (Bellwood et al. 2004). Improved knowledge of these processes would allow for more effective management. Modelling is a tool that has the potential to aid reef management by allowing reef dynamics to be investigated and relevant predictions to be made (McCook et al. 2001b, Lirman 2003). Many models of coral reefs exist and the different types of models range from deterministic, non-spatial models to stochastic, spatial models. In this chapter, existing models are reviewed, with a focus on non-spatial models that operate at a local (within-reef) scale and which do not consider just corals, because these are the type of models that are constructed in this thesis (see Chapter 3 for more details).
2.1 Non-Spatial Models 2.1.1 Network Models
Network models usually consist of stocks of energy or nutrients, corresponding to groups of organisms, and flows of energy between them, representing biological flows. In these models, the state of an ecosystem (which could be represented by the biomasses of modelled groups) is fixed, and in this sense, the models are ‘static’. Network software is used to implement these models and a popular program is ECOPATH, which assumes mass balance of all groups. ECOPATH works by balancing two key equations for each modelled group i: 36
Pi = M 1i Bi + Yi + Ei + Ai + M 2i ,
(2.1a)
Qi = Pi + Ri + U i .
(2.1b)
In (2.1a), Pi is the production of the biomass of group i, Bi is the group’s biomass, M 1i is the proportion of Bi that is predated over the time interval considered, Yi is the biomass lost to fishing, Ei is the net migration rate, Ai is the biomass accumulated or lost during the time interval and M 2i is the biomass lost due to mortality processes other than predation. In (2.1b), Qi is the biomass consumed by group i, Ri is the biomass lost due to respiration and U i is the biomass of unassimilated food (for example, some biomass consumed is lost as waste products). If there are missing parameters, ECOPATH attempts to estimate these parameters by solving a set of simultaneous linear equations based on (2.1a), with M 1i = ∑ j Q j p ij , where the sum is over all the groups and pij is the proportion of group j’s diet derived from group i (Christensen et al. 2005). These equations are linear in the sense that each term is a known constant or a known constant multiplied by an unknown that is raised to the first power (Christensen et al. 2005). Network models of coral reefs have been constructed by many authors to investigate flow properties between coral reef groups, and how these may change under different scenarios. For example, Johnson et al. (1995) created a tropho-dynamic network model of a mid-shelf GBR reef slope with 19 groups and found that a degraded algal-dominated reef had a lower efficiency in energy flow up the food chain, compared to a coral-dominated reef. This is detrimental to fish yield. Alino et al. (1993) created an ECOPATH model for a reef flat in Bolinao in the Philippines, with 24 groups. They found that the trophic transfer efficiency, which is the proportion of energy entering a trophic level that is passed to the next trophic level, was low at 9-13%. Opitz (1996) also found low trophic transfer efficiencies using ECOPATH models of a generic Caribbean reef, and identified efficient recycling of organic matter through detritus as a reason. Arias-Gonzalez et al. (1997) used ECOPATH models of reefs in French Polynesia and found efficient recycling as well. Arias-Gonzalez and Morand (2006) used an ECOPATH model of a French Polynesian barrier reef, and found that parasites increased the mean trophic transfer efficiency from 5.2% to 7.7%, which led to a longer 37
food chain, and also increased the number of pathways in the network. ECOPATH models can also be used to directly investigate fishing. For example, Arias-Gonzalez et al. (2004) used ECOPATH models of three reef slopes in the Mexican Caribbean, based on the ECOPATH models of Arias-Gonzalez (1998), and found that the catch at an unprotected site was 3-8 times greater than two semi-protected sites and had a lower trophic level. Gribble (2003) used an ECOPATH model of an inner lagoon and inter-reef area in the GBR and found that trawling had minor impacts on the penaeid prawn community. Tsehaye and Nagelkerke (2008) constructed an ECOPATH model for the artisanal reef fisheries of Eritrea, which operate in the Red Sea. The authors found that, due to the low trophic transfer efficiencies of low trophic levels, fishery yields were also low despite the high primary productivity (Tsehaye and Nagelkerke 2008). The advantage of network models is that they can be used to assess how network properties, such as the number of pathways, change under different scenarios. However, their main limitation is that the ecosystem state is constant. This means that the models cannot be used to investigate coral reef dynamics, where the model state changes over time (possibly reaching an equilibrium). To investigate dynamic properties, dynamic models need to be used, and are considered next.
2.1.2 Differential Equations Models
Non-spatial dynamic models of coral reefs typically use continuous, ordinary differential equations (ODE) and are usually deterministic (i.e. non-stochastic). In these models, the variables are averaged spatially, giving a mean-field representation. The main models of this type are now reviewed.
2.1.2.1 Ecosystem-fisheries Model of McClanahan (1995)
McClanahan (1995) constructed an ecosystem-fisheries model for a coral reef using ODE. The model was designed for a generic reef system, but was parameterised for a Kenyan reef lagoon. In the model, the benthos is comprised of algae and coral, and there are three fish groups (herbivores, piscivores and invertivores – invertivores feed on invertebrates) together with one invertebrate group (sea urchins). The equations for the model are: 38
dA = k 0 ARi ( A)S a ( N ) − k1 A − k 5 AF − k 6 AU , dt
(2.2a)
dC = k 2 CR f ( A, C )S a (N ) − k 3C − k 24 CU − k 25 CM , dt
(2.2b)
dF = k 7 FA − k 8 F − k 21 PF − k16 MF , dt
(2.2c)
dU = k 9UA − k10U − k13TUS w , dt
(2.2d)
dT = k14TUS w + k19TJ r (1 − S w ) − k17 T − k16 MT , dt
(2.2e)
dP = k 22 PF − k 23 P − k16 MP , dt
(2.2f)
where A, C, F, U, T, and P are the biomasses of algae, coral, herbivorous fish, urchins, invertivorous fish and piscivorous fish respectively, J r is the size of invertivorous fish prey other than urchins, Ri and R f are the sizes of environmental resources used by algae and coral respectively, S a is the reef surface area, S w measures the proportion of the diet of invertivorous fish which is urchins, M is the number of fishermen and k i ,
i ∈ {0,1,2,3,5,6,7,8,9,10,13,14,16,17,19,21,22,23,24,25} are coefficients that determine the strength of interactions between the groups of organisms. Ri and R f are both positive functions of the fixed light intensity I. Also, Ri is a negative function of the amount of competition among algae for light and hence is a negative function of A; similarly, R f is a negative function of C due to competition among corals for light. In addition, R f is a negative function of A because it is assumed that algae are superior competitors to coral, since they can shade or overgrow corals. S a changes dynamically as a positive function of the amount of reef accretion N, which is also modelled dynamically as the rate of calcium carbonate (CaCO3) production by A and C. Fishing intensity is fixed for each simulation by M. 39
Using the model, McClanahan investigated three different fishing regimes, viz. fishing all types of fish, fishing only herbivores and piscivores, and fishing only piscivores. Fishing all types of fish was found to lead to increased algae and lower coral, due to sea urchins out-competing the declining herbivorous fish population. Although the maximum fish yield is ≈ 300 kg ha-1 yr-1, yield eventually becomes zero when urchins dominate. Fishing only piscivores releases herbivorous fish, as predicted from theory (Steneck 1998), and this cascades down the trophic levels to cause a reduction in algae and an increase in coral growth. However, in this scenario maximum fish yield is low at ≈ 3 kg ha-1 yr-1. Fishing only herbivores and piscivores gives the highest maximum yield of ≈ 400 kg ha-1 yr-1. Overall, the results suggest that the best fishing regime, in terms of maximising yield, is the one where only herbivores and piscivores are fished. Since simulations were run until equilibrium was reached, McClanahan concluded that different fishing regimes can give different equilibrium states. This suggests the possibility of a ‘phase shift’ from a healthy to a degraded reef as fishing regimes change. The importance of whether fish or urchins graze the algae was also shown – since urchins erode the CaCO3 substratum more, their presence can significantly depress net CaCO3 production. The strengths of McClanahan’s model are that it includes major interactions between the benthos, fish and invertebrates, and also allows the effects of different levels and types of fishing activity to be deduced. Also, it incorporates reef erosion dynamically. However, a weakness of the model is that there is only one algal group. Different types of algae may experience different processes or the same processes at different rates. For example, herbivorous fish prefer eating filamentous turfing algae rather than macroalgae, which may have chemical repellents (Russ and St. John 1988, Hay 1991). Using different algal groups may therefore give different benthic dynamics which are not captured in McClanahan’s model. In addition, his model does not include exogenous recruitment dynamics for corals, fish or urchins. Omitting these dynamics could lead to a misrepresentation of the equilibrium biomass of each group, and could lead to recovery times of coral, fish and/or urchin populations after a disturbance to be underestimated. This could result in unrealistic yield and resilience predictions. Furthermore, direct competition between corals and algae for space is not modelled; rather, competition for light is modelled directly and it is assumed that algae are always the better competitors, which is not always the case (e.g., Jompa and McCook (2003a) for turf algae). Also, the study did not address other anthropogenic factors such as pollution (e.g., increased nutrients and sediments) arising from land40
based human activities, instead focusing solely on fishing.
2.1.2.2 Benthic Model of McCook et al. (2001b)
McCook et al. (2001b) used Logistic/Lotka-Volterra type differential equations to construct a model of the dynamics of corals and algae competing for bare space on a coral reef, with a fixed herbivorous fish population grazing the algae. Corals are split into two age groups, adults and juveniles, to reflect different life-history characteristics. The authors used the model to investigate the effects of cyclones, an acute disturbance, and nutrients and sediments, chronic disturbances. These disturbances are represented as external forcings. Each cyclone is assumed to kill adult corals instantly, reducing their cover by 70% of their pre-cyclone value, whereas nutrients increase algal growth. Sedimentation in the model decreases coral recruitment and survival and also inhibits algal growth and fish grazing. It was found that eutrophication, i.e. increased nutrients and sediments, led to increased algal cover and decreased coral cover at equilibrium, in the absence of acute disturbances. When the model was applied to an inshore-offshore gradient on the GBR, it was found that moving offshore, the average final coral cover increased from 13% inshore to 41% on mid-shelf reefs and to 60% on offshore reefs. Since the same acute disturbance regime was used across the gradient, McCook et al. concluded that eutrophication lowered the resilience to acute natural disturbances, to an extent that phase shifts to algal-dominance can occur. The model results thus demonstrate the importance of the interaction between acute and chronic disturbances in causing reef degradation.
2.1.2.3 ECOSIM Models
The ECOPATH models considered in section 2.1.1 can be converted into dynamic models by changing the mass-balanced equations into dynamic differential equations. The resulting model, called an ECOSIM model, uses parameters derived from the ECOPATH model and can be used to study dynamics through time (Christensen et al. 2005). Gribble (2003) created an ECOSIM model of his GBR model, detailed in section 2.1.1. Using the model, he found that reducing trawling effort by a half from 1997 levels decreased the biomass of two species of prawns by 59% and 64%, because of an 41
increase in their predators and competitors. Tsehaye and Nagelkerke (2008) used an ECOSIM version of their ECOPATH model (see section 2.1.1) to retrospectively model the artisanal reef fisheries of Eritrea. They found that by using official fisheries statistics, fishing has little effect on the sustainability of fish biomasses and subsequent yield. However, if the fishing intensity was increased by five-fold, which may reflect the actual intensity based on anecdotal evidence, there would be long-term declines in the biomass of large demersal fish.
2.1.2.4 Benthic Model of Mumby et al. (2007a)
The analytical model of Mumby et al. (2007a) used differential equations to model the benthic dynamics of coral reefs. This model has three dynamic groups: corals, turf algae and the structurally more complex macroalgae. The differential equations for the model are:
dC = rTC − dC − aMC , dt
(2.3a)
dM gM = aMC − + γMT , dt M +T
(2.3b)
dT dC dM =− − , dt dt dt
(2.3c)
where C, M and T are the proportional covers of coral, macroalgae and turf algae respectively, a is the rate of growth of macroalgae over coral, d is the natural mortality rate of coral, r is the rate of coral recruitment onto and coral overgrowth over turf algae, g is the grazing rate on macroalgae and γ is the rate of growth of macroalgae over turf algae. Mumby et al. parameterised this model for a Caribbean fore-reef and found that the system exhibited multiple stable equilibria across a range of grazing rates, which resulted in a hysteresis effect. This showed that an acute disturbance on a reef with high coral cover, such as a coral bleaching event, can decrease coral cover to the extent that the reef system is attracted to another stable equilibrium with lower coral cover. Also, if the grazing rate is decreased by overfishing, there could be a discontinuous, 42
‘catastrophic’ phase shift to a stable state of lower coral cover, with subsequent recovery inhibited by the hysteresis effect. An increase in the rate of overgrowth of coral by algae had the effect of increasing the threshold grazing rate at which such a phase shift can occur. The authors concluded that reefs are able to show multiple stable equilibria because of ecological feedback mechanisms that favour the path towards an equilibrium with low coral cover. They also concluded that it is vital to protect herbivorous fish stocks to improve the resilience of reefs to phase shifts.
2.1.2.5 Social-ecological Model of Kramer (2007)
Kramer (2007) created a social-ecological model that combines the dynamics of corals, algae, herbivorous fish and piscivorous fish with the dynamics of two fisher populations targeting the two different types of fish. In this model, corals and algae compete with each other for the same resources, herbivorous fish graze on algae and piscivorous fish predate upon herbivorous fish. The equations describing these biophysical components are:
K − A − a AC C dA − a AH HA , = rA A A dt KA
(2.4a)
AS K C − C − a CA S S dC A +B = rC C dt KC
(2.4b)
,
dH = a AH HA − a HH H − a HP HP − FS H a HM E H H , dt
(2.4c)
dP = a HP HP − a PP P − FS P a PM E P P , dt
(2.4d)
where A and C are the algal and coral covers (respectively), H and P are the herbivorous and piscivorous fish biomasses, rA and rC are the maximum (intrinsic) growth rates for A and C, K A and K C are the carrying capacities for A and C, a AC and aCA measure the
competitive strengths of C on A and A on C, a AH is the grazing rate of H on A, a HH and 43
a PP are the mortality rates of H and P due to causes other than predation and fishing, a HP is the predation rate of P on H, F is the total number of fishermen, S H and S P are the proportions of F that fish H and P, aHM and a PM measure the catch efficiencies on H and P, and EH and E P are the fishing efforts on H and P in terms of hours spent
fishing a day. aCA is weighted by a non-linear factor determined by A and the parameters S and B. The justification for this non-linear term given was that as A increases, the proportion of macroalgae to turf algae increases, which causes a rapid reduction in C above a threshold of algae because of the negative effects of macroalgae on coral recruitment and growth. Unlike previous models reviewed, the dynamics of the fisher populations are modelled using two extra differential equations representing S H and S P , which can thus change with time. The fishers show adaptive behaviour, such that they switch between fishing the two fish groups by comparing the payoff from fishing each fish group to the overall mean. If fishing a particular fish group yields an above average payoff, then more fishers will start to target that fish group. The speed at which fishers switch is determined by a parameter, which represents factors such as the speed at which information is disseminated among fishers and the willingness and capability of fishers to change fishing strategies. Kramer’s model was parameterised using data from different reefs worldwide and Kramer found that by considering just the coral and algal dynamics, there could be two stable equilibria for a given set of parameter values. He also found that by starting from a system with alternative stable states, increasing the coral growth rate increases the basin of attraction of the coral-dominated state and increasing herbivory leads to a single coral-dominated state. Thus, both bottom-up and top-down processes are important in determining reef health. Kramer also measured the resilience of the system with changing fisher parameters. Resilience was defined by the size of the basin of attraction of the coral-dominated state as a proportion of the coral/algal phase plane and the minimum distance between a coral-dominated state and the boundary of its basin of attraction in the coral/algal phase plane. Kramer found that fishing only herbivores changes a baseline system with alternative stable states into one that only has an algaldominated state. He also found that, starting from the baseline system, increasing the number of fishers from 250 km-2 to 500 km-2 results in a change from a coral-dominated to an algal-dominated state. Resilience decreased non-linearly as the number of fishers increased and this was partly due to the adaptive behaviour of fishers, which made large shifts in fishing effort between the two fish groups possible. Also, increasing the fishing 44
effort on herbivores from 2.5 hrs/d to 3.5 hrs/d resulted in a transition to an algaldominated state and the resilience of the system decreased as the speed at which fishers switch between fishing the two different fish groups decreased. In contrast, resilience increased as fishing effort on piscivores increased due to release of herbivores from predation; this indicates a trophic cascade effect similar to that found in the model by McClanahan (1995). A significant advantage of the Kramer (2007) model is that it can be used to demonstrate how resilience can be eroded by ecological and socioeconomic factors. By doing so, policy recommendations can be made based directly on changes in socioeconomic factors. However, a significant disadvantage of the model is that a nonlinear competitive effect of algae on coral is explicitly modelled through the non-linear
(
)
factor A S A S + B S (see eqn. 2.4b), and Kramer noted that this was important for giving the multiple stable equilibria found in the study. Thus, multiple stable equilibria is not an emergent property of the model, but is hardwired into the model through a non-linear factor. The reason given for including this non-linear term does not always hold, because the proportion of macroalgae to turf algae does not always increase as the amount of algae increases (e.g., McClanahan et al. 2002a, McClanahan et al. 2007). In addition, exogenous recruitment of corals, which could be important in the recovery of coral cover by structuring the benthic reef community (e.g., Abelson et al. 2005, Vermeij 2006), is not modelled and neither is the recruitment of fish, which could limit the size of reef fish populations (Ault and Johnson 1998) and therefore affect reef fish dynamics significantly.
2.2 Spatial Models An intrinsic weakness shared by all non-spatial models is that they neglect spatial effects and instead average dynamics over space. The implicit assumption is that any individuals modelled exhibit equal mixing, such that they have an equal chance of interacting with other individuals of the same or different group. However, if there is a patchy distribution of individuals, then rates of interaction may differ from the equal mixing case because there may be unequal rates of interaction between individuals in different patches. In this respect, it may be advantageous to explicitly include a spatial dimension. Several authors have attempted to create spatial models of coral reefs, reviewed below. 45
2.2.1 Local Spatial Models with No Consideration of Fish or Urchins
Existing local spatial coral reef models with no consideration of fish or urchins focus just on coral dynamics. Wakeford et al. (2008) constructed a cellular automaton model of 17 physiognomic groups of hard and soft corals at Lizard Island on the GBR, using a 23 yr data set from 1981-2003. The authors found that coral cover dynamics derived from the model without background coral mortality best matched the empirical data up until 1996. However, to match the post-1996 dynamics, background mortality had to be introduced. The authors inferred from this that the post-1996 coral community seemed to exhibit a loss of resilience, possibly due to chronic coral mortality inflicted by Crown-of-Thorns (CoT) starfish and low levels of bleaching and disease. Spatial models of coral dynamics have also been used to find out which types of coral species are favoured under different disturbance regimes. Maguire and Porter (1977) developed a spatial cellular automaton model of six eastern Pacific coral species and found that with a high disturbance frequency, the dominant processes favouring the survival of coral species were high recruitment and fast growth. In contrast, when there is a low disturbance frequency, the ability to out-compete other coral species by overgrowth or extracoelenteric digestion is more important. Langmead and Sheppard (2004) created a cellular automaton model for ten coral species at a Caribbean fore-reef slope and investigated the effects of disturbances of different spatial extent, but which create the same patch sizes, and disturbances which create different patch sizes, but of the same spatial extent. The authors found that with disturbances of a low spatial extent, the superior competitors Montastraea annularis and M. cavernosa dominated coral cover. With disturbances of a high spatial extent, high turnover species with high recruitment, such as Meandrina meandrites, dominated whereas disturbances of an intermediate extent resulted in a mosaic of different species. Similarly, at small patch sizes, competitive dominants were the most prominent because they could overgrow the small patch sizes quickly, preventing establishment of other coral species. At large patch sizes, species with high recruitment and growth dominated. This resulted in low diversity at the extreme patch sizes and the highest diversity at intermediate sizes. These results thus support the intermediate disturbance diversity hypothesis.
46
2.2.2 Local Spatial Models with Consideration of Fish and/or Urchins
Mumby (2006) and Mumby et al. (2006a) used a cellular automaton coral reef model which included stochasticity. The model has a torus shape to represent the reef substratum and 2,500 cells, each representing 0.25 m2 of living and/or dead substrata, and was parameterised for a mid-depth (5-15 m) Caribbean fore-reef. The benthos was modelled in detail, with two types of coral (brooding and spawning) and four types of algae (cropped algae 0-6 months old, cropped algae 6-12 months old, macroalgae 0-6 months old and macroalgae > 6 months old). A cell can also have ungrazeable substratum such as sand, which fills a cell if present. This allowed Mumby et al. to model many benthic interactions: corals recruit onto cropped algae at a constant rate, macroalgae grow over cropped algae with a probability determined by the cropped algae’s von Neumann neighbourhood, macroalgae can grow over corals with a certain probability, corals can reduce the growth of macroalgae, larger coral colonies outcompete smaller coral colonies and corals always overgrow cropped algae. Grazers are modelled as an external forcing – an unfished herbivorous fish community grazes 30% of the substratum every 6 months, whereas an intact urchin (Diadema) community grazes 40%. Competition between the two groups of grazers means that the total percentage of the substratum grazed is less than the sum of the two figures for each group. There is a constant rate of background partial-colony and whole-colony mortality for corals, and dead corals are assumed to be immediately covered by cropped algae. The effects of different levels of hurricanes, nutrients, grazing by herbivorous fish (as a proxy for different levels of fishing) and grazing by urchins were investigated (Mumby 2006, Mumby et al. 2006a). In the model, hurricane frequency follows a Poisson distribution and coral colonies of different sizes experience differential mortalities due to these hurricanes. Mumby et al. (2006a) investigated four levels of nutrients and four combinations of fish and urchin grazing levels. Simulations were run for 50 yrs under different scenarios of hurricane frequency and grazing levels, and for 10 yrs under different scenarios of initial coral cover, nutrient level and grazing level. Each simulation started with random initial coral colony sizes and spatial distributions. The authors found that the presence of urchins permitted reefs to maintain high levels of coral cover even under the most severe hurricane frequency of an average of once per decade (this frequency is found in the northern Caribbean). Urchins also compensated for the lowest herbivorous fish grazing level by providing adequate grazing for coral cover to persist. In the absence of urchins, reefs with a healthy biomass of herbivorous 47
fish were able to maintain coral cover provided that the expected periodicity of hurricanes was 20 yrs or greater. However, heavy exploitation of herbivorous fish resulted in declining coral cover even under the mildest disturbance regime simulated. With urchins, coral cover increased under all but one of the 10 yr scenarios considered, the one scenario being severe levels of nutrients with an initially unhealthy 10% coral cover. Mumby (2006) simulated 10 yr scenarios with different hurricane frequencies and grazing levels and also found that herbivorous fish grazing alone cannot maintain high coral covers, ≥ 30 %, under severe decadal hurricane regimes. However, reefs can withstand such a hurricane regime if urchin grazing is added. Mumby et al. (2007a) used the same model and found that if the total grazing level is ≥ 42 %, which required the presence of Diadema urchins, then the system always tends towards an equilibrium with high coral cover. However, if the grazing potential decreases to 5-40%, multiple stable states exist, such that if a reef starts off with high enough coral cover and low enough algal cover, it will tend towards a coraldominated state; otherwise, it tends towards an algal-dominated state. Thus, Mumby et al. were able to suggest a grazing level threshold for the existence of multiple states, using modelling. Overall, the model’s strength is that it has detailed benthic dynamics which are spatial and stochastic, and includes size and age structure. Also, it allows for the investigation of anthropogenic stressors such as fishing and increased nutrients. However, it still has weaknesses. First, there is no self-seeding (endogenous recruitment) of corals and hence no possibility of stock-recruitment dynamics. Second, fish and urchin dynamics are not included and piscivorous fish are not considered at all. This means that fishing cannot be modelled as a direct effect on fish biomass.
2.2.3 Regional Spatial Models
Apart from the local spatial models, there are also regional spatial models for reefs in a geographic region. Mumby and Dytham (2006) extended the spatial model in Mumby et al. (2006a) to a metapopulation context on a larger spatial scale. They parameterised this model for a metapopulation of four mid-depth Caribbean fore-reefs and investigated the effects of different hurricane frequencies, levels of fishing and connectivity regimes on benthic covers. One of their main conclusions was that fishing had a significant effect on final coral cover, whereas the type of connectivity did not. Mumby (2006) also investigated a metapopulation of four reefs using the same local spatial model as a 48
building block. Preece and Johnson (1993) constructed a cellular automaton model representing more than one reef and the connectivity between them. Preece and Johnson used their model to investigate the recovery of coral cover on reefs within the GBR after disturbances at different scales. Using the model results, their main conclusion was that recovery was dependent on connectivity and that there are strong interactions between factors acting at different scales. For example, if retention of larvae for one reef is high then connectivity is less important for recovery, whereas the threat of CoT starfish predation can exacerbate the effects of low connectivity. There are also regional models that include explicit hydrodynamics. For example, Wolanksi et al. (2004) combined the ecological model of McCook et al. (2001b) with a two-dimensional oceanographic model to create an eco-hydrological model. They applied it to a 400 km section of the GBR, and the reefs were subjected to the acute disturbances of river plumes and cyclones together with the chronic human disturbance of eutrophication. Historical data for plumes and cyclones were used for parameterisation. The authors found that the model was able to reproduce qualitatively the distribution of coral and algae on the modelled area. Wolanski and D’eath (2005) extended the regional model in Wolanski et al. (2004) by adding CoT dynamics and using this model, they predicted that global warming could lead to a total collapse of reefs in the GBR if global warming is not managed and if there is no biological adaptation.
2.3 Summary This review has shown the diverse ways in which existing coral reef models can be constructed and used to understand: (i) the key ecological processes underpinning reef degradation and recovery, (ii) the human activities that alter these processes and the socioeconomic context within which they occur, and (iii) how reefs can be better managed for sustainable use. The details vary between models, sometimes substantially. There are differences in the number and type of reef organisms modelled, the smallest taxonomic group considered, the number of size and age classes used, the types of disturbances modelled and the ecological interactions included, and this means that each model is appropriate for investigating only a subset of phenomena. This is to be expected because different types of models are required to answer different research and management questions. 49
However, there is a particular emphasis on the ecological processes determining the resilience of coral cover to natural and anthropogenic disturbances. The anthropogenic disturbances studied are typically fishing, increased nutrients and increased sediments, which are three key stressors affecting coral reefs (see Chapter 1, section 1.3).
50
Chapter 3. Thesis Contribution
3.1 Overview of Thesis In this thesis, dynamic differential equations models of coral reef ecosystems at a local, within-reef, scale are created and analyzed. A stepwise-refinement approach is used whereby the simplest models with minimum complexity are first constructed and then complexity is added to these models one step at a time. Complexity is added until a sufficient level is reached for the purposes of the users of the models (Bradbury et al. 2005). Here, the proposed users are reef managers who are interested in using models to aid in management and reef scientists who are interested in using models to study the dynamics underlying reef degradation. The stepwise-refinement approach maximises the ‘payoff’ of a model, which measures the utility of the model for specified aims, by ensuring that the model is not too simple and therefore unrepresentative of the real ecosystem (complexity is too low), and that the model does not become too difficult to analyze and obtain relevant conclusions from (complexity is too high). This helps to achieve an optimum balance between model complexity and analytical tractability – see Figure 3.1, which is based on the conceptual framework by Grimm et al. (2005) for agent-based models, which also recommends finding what they call a ‘Medawar zone’ of maximum payoff. Thus, this thesis begins by constructing a simple dynamic model of just the benthic component of coral reefs and a separate dynamic model of just the fish and urchin component of coral reefs. For each component, there are different model versions reflecting different levels of ecological complexity (Chapters 4-6). Subsequently, these components are linked in more complex, integrated models that couple benthic dynamics with fish and urchin dynamics (Chapter 7). The model dynamics investigated in this thesis are deterministic and non-spatial, and the models are generic in the sense that they include organisms and interactions which are generally expected in coral reefs around the world. Also, the models are generic in the sense that they are parameterised using data from reefs worldwide (Chapters 9-11). For the benthic models, the effects of fishing, nutrification (sensu Szmant 2002) and sedimentation are parameterised and the likelihood of continuous and discontinuous phase shifts from coral- to algal-dominance is investigated (Chapters 1314). A continuous phase shift involves a continuous (though relatively rapid) decrease 51
Figure 3.1. Conceptual graph of model complexity against model ‘payoff’, which measures the utility of a model for specified aims. Using a stepwise-refinement method means moving from left to right in the direction of increasing complexity. Graph is based on that by Grimm et al. (2005).
in coral cover and/or a continuous increase in algal cover; in contrast, a discontinuous phase shift involves a bifurcation that gives rise to discontinuous changes in benthic covers, with associated region of multiple equilibria and hysteresis. For all the benthic models and all the fish and urchin models constructed here, sensitivity analyses are applied and these determine how the benthic covers or biomasses of the modelled groups, the variables, change as the parameters change (Chapter 15). A sensitivity analysis is also applied to one of the integrated models, for some key parameters. Two techniques are used: the Elementary Effects method and Sobol’ Variance Decomposition (Morris 1991, Helton and Davis 2003). The Elementary Effects method allows the magnitude of the change in the response variables to be determined as a parameter is changed across its range, and also gives an indication of the degree of non-linearity in the variables. Sobol’ Variance Decomposition gives a quantitative measure of the degree of interaction between parameters, a component of non-linearity. This method is only applied to the simplest benthic model due to the need for independence of all the parameters. The implications of the results for understanding the ecological processes that drive reef degradation and for reef management are discussed in Chapters 8, 12 and 16. In the final part of this thesis, the results and discussions in all previous chapters are 52
synthesised and conclusions for the future of coral reefs in general are drawn, in the context of the need for sustainable reef management (Chapter 17).
3.2 Significance and Novelty of Thesis In a wider context, the work in this thesis forms part of the modelling work of the Modelling and Decision Support Working Group (MDSWG) of the Coral Reef Targeted Research and Capacity Building for Management Program (CRTR-CBMP; http://www.gefcoral.org/). The MDSWG is the integrative group of the CRTR-CBMP and aims to create a holistic scientific understanding of the interaction between coral reefs and the human societies that depend on them, through modelling. The MDSWG models are intended to aid decision-makers and reef-users manage the use of coral reefs in a more sustainable way (Bradbury et al. 2005). This is a necessarily interdisciplinary venture which focuses on both the biophysical and socioeconomic aspects of coral reefs. The MDSWG is developing biophysical models at local (within-reef), regional (reef system) and global levels. This thesis fits into the local biophysical modelling work of the MDSWG, and the local models constructed and analyzed in this thesis are embedded in a predominantly biophysical regional model currently in development by other members of the MDSWG (see Melbourne-Thomas et al. (2007) for more details on this model). These models will be linked to dynamic socioeconomic models, which are to be developed by other MDSWG members, to create dynamic socio-ecological models that operate at different scales but which are linked to each other. The generic nature of the base models constructed in this thesis allows the models to be refined and parameterised for different, specific, reef systems around the world. This is a design feature that meets a key objective of the MDSWG: model design should not restrict applicability of the models to reefs in one particular region of the world. This thesis is novel by addressing important questions relevant to reef science and management using techniques and analyses that have hitherto not been applied to coral reef models. These techniques and analyses are: mathematical equilibrium analyses using theorems such as the Poincaré-Hopf Theorem (Milnor 1965) and the Richeson-Wiseman Theorem (Richeson and Wiseman 2002); a novel form of genetic algorithm called the Immune-Inspired Algorithm (IIA; Kelsey and Timmis 2003) that is used to search parameter spaces for multiple equilibria; parameter sweeps that involve randomly sampling parameter spaces and calculating the proportion of parameter sets 53
sampled that give different types of phase shifts (according to different quantitative criteria); sensitivity analyses using the Elementary Effects method (Morris 1991) and Sobol’ Variance Decomposition (Helton and Davis 2003). The results generated significantly furthers the understanding of generic coral reef ecosystem dynamics and aids in identifying key management actions and future research opportunities that need to be pursued in order to build reef resilience and ensure the sustainability of coral reefs. Thus, this thesis advances the area of coral reef modelling, fills knowledge gaps in coral reef science and extends the capacity of reef managers and other stakeholders. The local models in this thesis can be used by local reef scientists and managers around the world to aid in reef management and also provides a basis for a novel regional model being developed by the MDSWG, as mentioned above. The significance and novelty of this thesis is now explained in more detail.
3.2.1 Systematic Investigation of Phase Shifts
The work in this thesis is a new contribution towards coral reef modelling in several ways. First, all versions of the benthic model are mathematically analyzed to investigate how many equilibria they could have and the stability of these equilibria. In addition to these mathematical analyses, a novel form of genetic algorithm, the IIA, is applied to each benthic model to search for the possibility and likelihood of multiple equilibria (multi-equilibria) within the parameter spaces that are derived using data from reefs worldwide. Since discontinuous phase shifts require multi-equilibria, the results are used to determine the likelihood of this type of shift with fishing, nutrification and sedimentation. The advantage of analyzing increasingly complex versions of the benthic model is that differences in behaviour between model versions can be clearly attributed to the complexity added at a particular stage. This is a powerful way of isolating the factors responsible for phenomena such as multi-equilibria. Although some existing models have greater complexity than the models in this thesis, such as the inclusion of spatial effects, these models do not address the issue of how increasing complexity affects the results. Thus, it is difficult in these studies to isolate the effects of added complexity on reef dynamics and resilience to human stressors. Parameter sweeps are performed for the two simplest benthic models and these 54
provide a quantitative measure of how likely different types of continuous and discontinuous phase shifts are with decreasing grazing pressure, which could arise due to increasing fishing pressure. Quantitative criteria for different types of continuous and discontinuous shifts are determined, with respect to the rate of decrease in coral cover and the rate of increase in algal cover. By randomly sampling parameter sets, the probability of each type of continuous and discontinuous phase shift is determined for a reef exposed to no nutrification or sedimentation, just nutrification or sedimentation, or both nutrification and sedimentation. Results from the sensitivity analyses (see section 3.1) give quantitative measures of which parameters have the most effect on coral and algal covers, and hence which ecological processes are most important in precipitating a phase shift. There are several software packages available that detect bifurcations, such as AUTO (http://cmvl.cs.concordia.ca/auto/) or CONTENT (http://www.enm.bris.ac.uk/staff/hinke/courses/Content/index.html). These could be used to determine whether a benthic model exhibits a bifurcation, required for a discontinuous phase shift, as one or more parameters are changed. For example, a software package such as AUTO can be used to determine whether a benthic model can exhibit a bifurcation for a specific set of parameters, as grazing pressure is decreased. However, for the purposes of this thesis, a significant limitation of these packages is that they work by changing chosen parameters by specified amounts, starting with a single parameter set. Thus, they cannot be used to efficiently find a parameter set in a large multi-dimensional parameter space which gives a bifurcation, when there is no prior knowledge as to whether any parameter set within the space can give a bifurcation. Therefore, the novel and powerful IIA methodology is used in this thesis for the purpose of determining whether a parameter set in a given parameter space can give a discontinuous phase shift. It does this by attempting to converge on parameter sets giving multi-equilibria by maximising a specified ‘fitness’ function; multi-equilibria are required for a discontinuous shift, so if no parameter sets are found to give multiequilibria, then discontinuous shifts are not possible within the parameter space. This method is fast and fairly easy to program in C++. In addition, packages such as AUTO cannot be used for randomly sampling a large number of parameter sets from a large multi-dimensional parameter space and determining whether each set can exhibit a bifurcation (and therefore a discontinuous phase shift), because parameters are not changed randomly. Such random sampling is required for the parameter sweeps. Furthermore, performing parameter sweeps requires 55
that each parameter set sampled be tested to see if it can give rise to a continuous or discontinuous phase shift as grazing pressure is decreased. To test whether a parameter set can give a discontinuous phase shift, it is necessary to calculate the maximum change in coral and algal covers at equilibrium for a given change in grazing pressure (see Chapter 14, section 14.1 for details). Similarly, the maximum changes in equilibrium covers are required to test if the parameter set can give a continuous phase shift (section 14.1). However, the maximum changes in equilibrium covers cannot be calculated using packages such as AUTO. Thus, the parameter sweeps were implemented using C++ programs, which allowed random sampling to be included and were straightforward to code; packages such as AUTO were not used. This part of the thesis, which investigates phase shifts, is important and new because although there is some theoretical support for discontinuous phase shifts with fishing (Mumby et al. 2007a), there is little theoretical or experimental support for such shifts with nutrification and/or sedimentation. Also, it is not known how likely continuous or discontinuous phase shifts are on coral reefs exposed to anthropogenic stress. Furthermore, it is unclear what processes are responsible for the existence of alternative stable states and the associated possibility of discontinuous phase shifts. The results in this thesis help to provide insight into these questions. This new insight is then translated into implications for reef management.
3.2.2 Determining the Potential for Non-linearity
This thesis investigates the potential for non-linearity, which includes synergy, in coral reefs in a quantitative way. Here, non-linearity means non-linear changes in a state variable (a type of benthic cover or biomass) when a parameter is changed, or nonadditive changes in a state variable when more than one parameter is changed or when more than one type of anthropogenic stressor is applied to a model reef system (with each type of stressor changing one parameter or a set of parameters). First, synergy between fishing, nutrification and sedimentation are investigated using the benthic models. Although other existing coral reef models have investigated synergy, such as McCook et al. (2001b) for acute and chronic disturbances and Mumby et al. (2006a) for decreases in grazing pressure, nutrification and hurricanes (see Chapter 2, sections 2.1.2 and 2.2.2 respectively), none have investigated the potential for synergy between fishing, nutrification and sedimentation. Simulations are performed with reef systems 56
exposed to one stressor or a combination of more than one of the stressors to determine whether synergy can occur, in terms of whether the change in benthic covers under more than one stressor is greater than the sum of the changes under each one of the stressors separately. Second, the parameter sweeps and sensitivity analyses give measures of the degree of non-linearity in the response variables in the models as parameters change. From this, the degree of non-linearity with increasing anthropogenic stress can be deduced. In particular, the Sobol’ Variance Decomposition sensitivity analysis gives a direct quantitative measure of the potential for interactions between parameters, which give non-linearity, for the simplest benthic model. Since the parameter sweeps and the two sensitivity analysis techniques that are used have not been applied to coral reef models before, these methods give novel types of results. Whether coral reefs exhibit non-linearity, and to what degree, are important research and management questions because the presence of non-linearity accelerates reef degradation as human stress increases. Also, non-linearity that leads to discontinuous phase shifts makes recovery much harder through the hysteresis effect, posing a much greater management problem. Thus, the theoretical results in this thesis have important management implications and build on previous coral reef studies which suggest that coral reefs are likely to exhibit non-linearity (e.g., Knowlton 1992, McClanahan et al. 2002b). The results also suggest important hypotheses on nonlinearity that should be tested experimentally.
3.3 Further Comparison with Existing Coral Reef Modelling Studies Although the work in this thesis is novel, it has similarities with existing coral reef modelling studies at a local scale, reviewed in Chapter 2. Most obviously, this thesis uses differential equations to represent the dynamics of functional groups, which is an approach used by all of the differential equations models reviewed. Also, the investigation of phase shifts under anthropogenic stress is a recurrent theme in existing models. However, there are also many differences in the scope and aims of the work in this thesis compared with those in other studies, and the precise methodology used. These similarities and differences are now examined in more detail. Unlike the ECOPATH studies reviewed in Chapter 2, this thesis does not focus on the flow properties between different groups within a coral reef ecosystem. The models built in this thesis are dynamic and not static (in the sense described in Chapter 57
2, section 2.1.1) like ECOPATH models, and are used to investigate the dynamics of reefs exposed to human stressors. The main focus of this thesis is on equilibrium dynamics, but some information on non-equilibrium dynamics is given by the sensitivity analyses, which examine benthic covers and fish and urchin biomasses after three different lengths of time, and a mathematical analysis of the possibility of periodic orbits for the simplest benthic model. Since this thesis investigates coral reef dynamics, it is important that fundamental coral reef ecosystem processes have been explicitly modelled. These include competition between corals and algae for space, biomassdependent negative effects of herbivorous fish and urchins on each other due to competition for algae, and coral recruitment, which are not explicitly included in ECOPATH models. Compared with existing ECOPATH models, the models in this thesis have fewer groups of organisms. The large number of groups used in ECOPATH models is not necessary for the research questions addressed in this thesis. The ECOSIM models of Gribble (2003) and Tsehaye and Nagelkerke (2008) are dynamic, unlike the ECOPATH models. However, they still have the disadvantage that they do not explicitly include ecological processes such as coral recruitment, unlike the models in this thesis. In common with both ECOSIM studies, this thesis examines the effects of fishing on reef dynamics, but unlike both these studies, the models in this thesis provide a dynamic account of how the covers of different benthic groups change. In particular, unlike Gribble (2003), the focus of this thesis is not on prawn trawling and the associated effects of discards. Rather, there is a focus on the removal of fish, which is a human stressor for coral reefs globally. The benthic models in this thesis are similar in spirit, though not in detail, to the differential equations models of McCook et al. (2001b) and Mumby et al. (2007a). However, fish and urchin dynamics are also investigated in the models of this thesis, which these two existing studies do not consider. The inclusion of fish and urchin dynamics allows the direct effects of fishing on the biomasses of different fish groups to be modelled. Explicit benthic, fish and urchin dynamics in the integrated models in this thesis allows investigation of dynamic feedbacks between the benthos and the grazers. The models in this thesis do not split corals into juveniles and adults as in McCook et al. (2001b). Instead, corals enter into the models at 1 yr old, such that the dynamics of juvenile corals less than 1 yr old are not considered. This is a simplification that helps to keep the models analytically tractable enough for the analyses described in section 3.2. In comparison with the model by Mumby et al. (2007a), the most complex benthic model in this thesis includes one more benthic group and more interactions between the 58
modelled groups, and is hence more complex in this sense. The differential equations model of McClanahan (1995) includes the dynamics of reef (CaCO3) accretion, whereas the models in this thesis do not because questions relating to reef accretion are not addressed here. The dynamics of invertivorous fish are included in McClanahan’s model, such that the predation rate on sea urchins can vary with time, whereas invertivorous fish are assumed to exert a constant predation rate on urchins in the models in this thesis and their dynamics are not modelled. This simplification is due partly to a need to keep the models simple enough for analysis, and partly because of time constraints. The most complex fish and urchin model in this thesis includes two different size classes for piscivorous fish, which allows the investigation of different fishing pressures on the two size classes – a feature not included in McClanahan’s model. Unlike in this thesis, McClanahan did not analyze his model mathematically for the number of equilibria and their stability. In addition, he did not investigate the effects of disturbances other than fishing, such as increased nutrients and increased sedimentation, and the interaction of these disturbances with each other and with fishing, as is done here. Furthermore, McClanahan included only one algal group in his model and thus he could not use his model to explore the effects of different algal groups with different ecological properties. In contrast, the most complex benthic model in this thesis includes two types of algae, turf algae and macroalgae, which exhibit different modes of growth and which have different effects on corals. Lastly, McClanahan’s model does not include exogenous coral, fish or urchin recruitment, which is modelled in this thesis. This thesis focuses on the biophysics of coral reefs and does not consider socioeconomics, which contrasts with the model of Kramer (2007). The biophysical part of Kramer’s model has the significant disadvantage that multiple equilibria were obtained by introducing an ad hoc non-linear competitive term of algae on coral (see Chapter 2, section 2.1.2). In contrast to this ‘hardwiring’, the models in this thesis are constructed from simple and natural representations of significant ecological interactions in coral reef ecosystems (see Chapters 4-6), and the existence of multiple equilibria, when they do exist, is an ‘emergent’ property of these interactions that is not exogenously forced by means of an ad hoc assumption. The local scale models in this thesis are not spatial, which is a fundamental difference to the local spatial models reviewed in Chapter 2, section 2.2. Spatial effects are not examined in this thesis due to time limitations. In spite of this, the models in this thesis have an advantage over the existing local spatial models reviewed in that they 59
explicitly include fish and urchin as well as benthic dynamics. In this thesis, models are constructed with the minimal complexity required to investigate certain research and management questions, in order to maximise the utility of the models and ensure analytical tractability for the application of techniques such as the IIA and the parameter sweeps. These models can be extended in several ways for the purposes of investigating more questions, such as the addition of size structure, spatial effects or stochasticity. This would make the models directly comparable to more of the studies considered in Chapter 2. However, in extending the models in this thesis, the step-wise refinement methodology should be pursued, because it makes transparent the effect of additional complexity on model behaviour. This aids in the interpretation of model results and hence the understanding of phenomena being modelled.
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Part II: Local Models of Coral Reef Ecosystems
Chapter 4. Benthic Model without Macroalgae
4.1 Modelling Methodology for Benthic Models In the benthic models (constructed in this chapter and Chapter 5), organisms are modelled as functional groups. A functional group consists of organisms that perform the same ecological function; organisms are not grouped according to taxa or morphology. For example, Steneck and Dethier (1994) identified seven functional groups for benthic marine algae and McCook et al. (2001a) used virtually the same groups to successfully classify the mechanisms that give rise to coral-algal competition. Thus, as Carpenter (1986) notes, a functional group approach can aid in understanding complex systems such as coral reefs. A functional group approach is appropriate for modelling coral reefs because modelling at, say, the species level would lead to computational overload, even if it were possible to obtain model parameters for all species. The approach is also justified by the fact that the ecological traits of reef organisms do not always correlate well with taxonomic groups above a species level (e.g. algae; Birrell et al. 2008). Furthermore, the use of functional groups retains the generic status of the models because the same functional groups are found on reefs worldwide, for benthic organisms, fish and urchins. For example, the Caribbean and the Great Barrier Reef (GBR) broadly share the same coral and fish functional groups (Bellwood et al. 2004). Because of its advantages, a functional group approach has often been used in existing coral reef models (see Chapter 2). In this thesis, the broad-scale indicator of ‘proportional cover’ is used to represent each of the benthic functional groups modelled. These benthic cover variables are incorporated into differential equations along with relevant parameters, which then allow their dynamics to be examined. This leads to an assessment, for example, of whether a reef area is coral-dominated or algal-dominated, and of whether a phase shift between these two states can occur. Since proportional covers are used, the reef area being modelled is not specified explicitly. Nevertheless, the absolute spatial scale 61
cannot be completely ignored. The non-spatial benthic models in this chapter average the benthic dynamics over space and hence, the greater the area modelled, the less likely it is that the averaged results will be representative. This is because of larger spatial variations in the dynamics arising from uneven interaction frequencies and strengths over space. Thus, the benthic models should be applied at a ‘local’ scale – on the order of tens of metres to a few kilometres – and in habitats that are relatively homogeneous in terms of substrate properties. The functional groups modelled are key groups the balance between which is thought to be important in maintaining the ‘health’ (or otherwise) of coral reefs, namely hard corals, turf algae and macroalgae. Fish and urchins are not modelled explicitly in the benthic models.
4.2 The Coral-Turf Model (CTm) The Coral-Turf model (CTm) is the simplest benthic model constructed. It has only two independent dynamic variables – the proportional covers for the two functional groups ‘hard corals’ and ‘turf algae’. Thus, it corresponds to a reef that always has insignificant amounts of macroalgae. Here, ‘macroalgae’ refers to algae which are structurally more complex and have larger thalli than turf algae, and is an important functional group that will be added later in more complex benthic models (Chapter 5). The CTm models hard corals and turf algae competing for space on a coral reef benthos, with space consisting of the hard substratum over which hard corals and turf algae can grow. Key ecological processes determining competition for space are modelled explicitly – these are growth and mortality for hard corals and turf algae, and recruitment for hard corals. Recruitment for turf algae is modelled implicitly, and contributes to space becoming turf algae when there is insufficient grazing. Mortality of hard corals in the CTm occurs due to a variety of background processes and human activities that enhance the sedimentation rate; in contrast, turf algae experience mortality due to grazing by herbivorous fish and sea urchins, such that they are limited by herbivores as well as space. The dynamics of grazers are not modelled, such that there is no feedback to the grazers from the benthos. This is a limitation that will be removed in the integrated models, which are more complex (see Chapter 7).
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4.2.1 Details of Functional Groups Modelled in the CTm
The hard corals functional group consists of reef-building corals. These corals have many different morphologies, for example: massive, branching, encrusting and tabular. Examples of reef-building corals include the massive coral species complex Montastraea annularis, an important framework builder on Caribbean reefs (Hughes 1994, Kramer 2003, Carpenter et al. 2008), the branching Acropora spp., which are common on Indo-Pacific reefs (Arceo et al. 2001, Arias-Gonzalez et al. 2006, Sandin et al. 2008), and the encrusting coral Porites astreoides, which is common in the western Atlantic (Aronson and Precht 2001, Kramer 2003, Idjadi et al. 2006). Turf algae are rapid colonisers of dead coral (e.g., McClanahan 1997, McCook 2001, Diaz-Pulido and McCook 2002) and are commonly found on coral reefs. Since turf algae have high rates of production, they are often the major contributor to reef primary production (Carpenter 1986, McCook 1999). Turf algae are characterised ecologically by a fast growth rate and a low canopy height (Klumpp and McKinnon 1989), and by having minor direct negative effects on hard corals (see section 4.2.2).This functional group consists mainly of filamentous algal species that form turfs approximately > 2-4 mm high on the hard substratum, but also includes simple unicells and crusts. Thus, turf algae correspond to the ‘Filamentous Algae’ and ‘Microalgae’ functional groups of Steneck and Dethier (1994). This definition of ‘turf algae’ is consistent with its usage in the literature, such as in Carpenter (1986) and Klumpp and McKinnon (1989). Examples of turf algal species are Hersiphonia secunda, which are common on Davies Reef in the GBR (Klumpp and McKinnon 1989), and Ceramium spp., which can be found on Glover’s Reef atoll in the Mesoamerican Barrier Reef System (MBRS) (McClanahan et al. 2007). Any part of the modelled hard reef substratum not covered by hard corals or turf algae is referred to as ‘space’, which forms the third ‘functional group’. This group is not just empty substrate but contains dead coral skeletons, non-geniculate crustose coralline algae (CCA) and turf algae that have been grazed back to a very low canopy height (approximately ≤ 2-4 mm). Turf algae often survive being grazed by fish because fish commonly do not remove the basal portions (Hay 1991), although this may not be the case for urchins, which are capable of greater erosion of the substrate while feeding (McClanahan and Shafir 1990). CCA are encrusting, commonly possessing a prostrate growth form, and secrete a calcareous skeleton (Steneck 1986, Spalding et al. 2001). CCA encrust hard substrata including dead coral skeletons, but are overgrown by 63
competitors such as fleshy algae, unless disturbance (often herbivory) keeps them clean (Steneck 1986). Examples of species of CCA are Lithothamnion spp. and Clathromorphum spp. (Steneck 1986). Live sessile organisms apart from hard corals, turf algae, macroalgae and CCA are not modelled and are assumed to be constant components of the benthos, with insignificant effects on coral-algal dynamics. These include: soft corals, sponges, bryozoans, tunicates, zoanthids, bivalves and clams. Soft corals lack the hard skeletons hard corals have, and are able to attain high abundance on many coral reefs (Fabricius 1997). They are rarely found on reefs outside the Indo-Pacific, but can be significant occupiers of the benthic substratum for Indo-Pacific reefs. Some justification for the assumption that soft corals are constant benthic components and do not affect coralalgal dynamics significantly is provided by a study of Fabricius (1997) on central GBR reefs. She compared the cover of 24 genera of soft corals on seven reefs which had been impacted by Crown-of-Thorns (CoT; i.e., Acanthaster planci, a starfish) outbreaks to the cover on seven reefs which had not. At the time the surveys were performed, for each impacted reef, the most recent CoT outbreak occurred 4-7 yrs previously. Fabricius found that even though the reefs impacted by CoT had much lower live hard coral cover, neither the number of soft coral colonies nor soft coral cover differed between impacted and non-impacted sites. This suggests that soft corals did not respond to the increase in space brought about by the CoT outbreaks, and hence that soft coral cover is not limited by space or competition with other organisms for space (Fabricius 1997). Sponges have the potential to be an important group with respect to benthic space dynamics because they can be successful competitors with corals for space (Aerts 1998, Hodgson 1999). However, sponges are not included in the CTm as a simplification, to achieve analytical tractability and because of a lack of data for parameterisation of the demographic rates for sponges. Also, a study by Hodgson (1999) found that sponge cover on most reefs around the world is usually low and this suggests that sponges are not usually an important component of the benthos with respect to spatial dynamics. Hodgson summarised survey results from over 300 reefs in 31 countries and territories worldwide and found that none of the sites had high sponge cover, with the majority of reefs having < 5 % cover of encrusting sponges. Less than 30% of sites in the Caribbean had sponge cover > 5 % and less than 10% of sites in the Indo-Pacific had a cover of > 5 %. This in part justifies leaving out sponges in the CTm. Studies of whether bryozoans, tunicates, zoanthids, bivalves and clams can be important competitors for space on coral reefs are rare and hence, as a simplification, 64
they are not modelled in the CTm. The total proportional cover in this model, which adds up to 1, is the sum of the hard coral, turf algal and ‘space’ covers. Thus, if a is the total area of the hard benthic substratum and b is the area taken up by non-modelled groups, then c = a − b is the area considered in the model, and the proportional covers considered in the model are proportions of c. From now on, ‘coral’ is used as a short-hand for ‘hard coral’.
4.2.2 Details of Interactions Modelled in the CTm
The main interactions between the three functional groups in the CTm are derived from the literature and modelled. Interactions initiated by corals are first considered. Corals are colonial organisms which exhibit indeterminate growth, sensu Sebens (1987), and which lack true senescence. However, there is partial and total mortality of coral colonies due to a variety of chronic background processes such as naturally occurring diseases and sediments (Bythell et al. 1993, Langmead and Sheppard 2004). Thus, corals experience background mortality in the CTm. This mortality rate may be enhanced by human activities such as coastal development, which can increase the sedimentation rate (see Chapter 1, section 1.3.3). Coral larvae, or planulae, can enter the modelled reef area, settle and survive to become part of the adult community. Corals reproduce via two main mechanisms. Some species reproduce by spawning eggs and sperm into the water, which is followed by external fertilisation; these corals are called spawners. In contrast, in other species eggs are fertilised within parent colonies; these are called brooders (Richmond 1997, Knowlton 2001). Richmond (1997) estimates that 85% of over 250 coral species studied are spawners. Whether a coral is a spawner or a brooder has an effect on its ecology. Brooder larvae are usually larger than spawner larvae, with zooxanthellae inherited from their parent colonies. These brooded larvae are able to settle and metamorphose (i.e., are competent) straight after release from parent colonies, and although the energy provided by the zooxanthellae gives these larvae the potential for long dispersal, evidence suggests that they usually settle near their parent colonies. For example, the brooding coral Favia fragum exhibits an average swimming time of only 4 min (Richmond 1997, Knowlton 2001). In contrast, the smaller spawner larvae become competent some time after they are produced. For many spawning Acropora spp., this time can be as long as 72 hrs, whereas for the spawning genera Montastraea and 65
Goniastrea, the time is approximately 18 hrs (Richmond 1997). This time before competency allows the larvae to disperse. Once competent larvae reach a suitable site, they can settle by attaching to the hard benthic substratum. Settled larvae then undergo morphological and biochemical changes, a process called metamorphosis (Richmond 1997, Birrell et al. 2008). Recruitment involves settlement of coral larvae, metamorphosis, and post-settlement survival and growth (Birrell et al. 2008). In the CTm, it is assumed that a coral larva becomes part of the extant community when it has settled and has survived for 1 yr, by which time it is referred to as a ‘recruit’. It is assumed that coral colonies less than 1 yr old are generally too small to contribute significantly to coral cover and hence to benthic dynamics. In the CTm, larvae originate from both reproduction by coral colonies external to the reef area modelled and reproduction by colonies on the reef area modelled. This gives rise to exogenous and endogenous recruitment respectively. Due to the evidence for short dispersal distances of brooders, it is assumed that all endogenous recruitment arises from reproduction by brooders. In contrast, exogenous recruitment arises due to reproduction by spawners, whose larvae are able to disperse long distances in the currents. Successful recruitment requires the availability of suitable substrata for settlement. CCA with cropped turf is a suitable surface for larvae to settle and grow, particularly since CCA can induce coral settlement (McCook et al. 2007). More than 20 species of CCA have been shown to have this ability to induce and at least 26 coral species have been observed to be induced to settle by CCA (Birrell et al. 2008). The presence of grazed turf has no significant effect on coral recruitment (Birrell et al. 2008). In addition, coral larvae can settle on ungrazed turf algae, albeit at a lower rate than on grazed turf (Birrell et al. 2005). Based on these considerations, coral recruitment on both space and turf algae are modelled. Coral colonies grow laterally by asexually forming new polyps (Richmond 1997). Colonies readily grow laterally over dead coral skeletons and CCA; in addition, the cropped turf algae that exist within ‘space’ are too short to impede the growth of corals (McCook et al. 2007). Thus, in the CTm, corals can grow laterally over available space. Corals can also overgrow turf algae by being competitively superior or, for branching corals, by reaching a size where their branches grow over turf algae (thus escaping direct competition but indirectly capturing space) (McCook et al. 2001a). Therefore, it is assumed in the CTm that corals can also laterally overgrow turf algae. 66
Interactions initiated by turf algae are now considered. The growth of turf algae is predominantly vertical, due to their short thalli, and hence there is no lateral growth of turf algae over benthic space in the CTm. In the absence of grazing, turf algae have the ability to regenerate themselves on suitable substrata such as dead coral. This may be because the environment is saturated with turf algal propagules, thus enabling turf algae to establish quickly onto new substrates, or because they are able to re-grow quickly from the intact basal portions left behind after grazing. Diaz-Pulido and McCook (2002) found that turf algae dominated algal recruitment onto dead corals and colonised them within months, which supports saturation of the environment with turf algal propagules. Thus, it is assumed that if ‘space’ is left ungrazed, then turf algae grow from it at a fixed rate, and hence there is no need to model turf algal recruitment explicitly. This approach to modelling turf algal growth is also used in Mumby et al’s (2006a) spatial simulation model. Generally, algal turfs do not overgrow and are poor competitors with live coral colonies (Jompa and McCook 2002a, Jompa and McCook 2003a, Birrell et al. 2008). Smith et al. (2006) used laboratory experiments to show that mixed turf algae can indirectly increase the mortality of coral species when they are in close contact. The mechanism is the release of dissolved organic compounds by the algae into the surrounding water. However, it is not known how prevalent this mechanism is in the field and how effective it is in causing coral mortality on an ecological scale. In addition, a study by Diaz-Pulido and McCook (2002) for an inshore central GBR reef found that turf algae cannot colonise live coral. Thus, there are no terms in the CTm representing coral mortality by turf algae due to overgrowth or other means. Algal turfs are grazed by both fish and urchins (Klumpp and McKinnon 1989). They are the preferred food of most herbivorous fish (Russ and St. John 1988, Choat 1991) and urchins can also be major consumers of algae (Sammarco 1982, Carpenter 1986). Thus, in the CTm, grazing on turf algae is included. Grazing on ‘space’ is also included because the cropped turfs (which are continually trying to grow vertically upwards) in space are grazed.
4.2.3 Equations for the CTm
The CTm is defined by two independent differential equations, 4.1 and 4.2 below, with one equation for the coral proportional cover (C) and one for the turf algal proportional cover (T). Short descriptions are given below the terms to aid in their ecological 67
interpretation.
dC = l Cs + l Cb C (S + ε C T ) + rC (S + α C T )C − 3 14 4244 3 dt 14442444
(
)
Coral recruitment
Coral growth
dCC {
Turf algal growth
Grazing on turf
(4.1)
.
(4.2)
Coral mortality
dT = ζ T (1 − θ )S − g T θT − ε C lCs + lCb C T − 14 4244 3 dt 14243 123
(
,
)
Coral recruitment on turf
rC αC TC 1 424 3 Coral growth over turf
Here, S denotes the proportional cover of ‘space’, which is a dependent variable satisfying C + T + S = 1 , and its dynamics are given by the differential equation
dS dC dT =− − . dt dt dt
(4.3)
In eqn. 4.1, d C is the background mortality rate of corals, which may be enhanced by human activities leading to increased sedimentation. l Cs and l Cb are the recruitment rates for spawning and brooding corals
respectively. For spawners, it is assumed that planulae contribute to a large external larval pool from which recruits are drawn, and which remains constant in size due to replenishment from exogenous sources. Thus, the external larval pool is assumed to be large enough for the contributions to and from the modelled reef area to be negligible, and is also assumed to be in equilibrium. The latter assumption is clearly a strong one and in reality, there will be significant seasonal and stochastic variation in larval pool size, and hence in recruitment. However, because only benthic dynamics averaged over seasons (see Chapter 9) is considered, and because recruitment from the larval pool occurs at a low rate (Chapter 9), these sources of variability are excluded. Thus, recruitment due to spawners occurs at a constant rate and explicit larval pool dynamics are not modelled. ε C is the coral recruitment rate onto turf algae relative to the rate onto space. From section 4.2.2, ε C < 1 . rC is the lateral growth rate of existing corals into space. There is also direct coral overgrowth of established turf algae, which leads to the additional growth term rC α C CT , where α C is the coral growth rate over turf algae relative to that over space. In eqn. 4.2, the achieved grazing pressure is represented by the parameter θ . An 68
explicit form of θ as a function of grazer biomass is detailed in Chapter 7. In the benthic models, θ is assumed to be constant, but θ is allowed to vary in the fish and urchin models as well as in the integrated models (see Chapters 6 and 7). θ lies in the range 0 ≤ θ < 1 and is an increasing function of the grazer biomass (Chapter 7). Thus,
g T is the maximum possible rate at which established turf algae can be grazed, with g T θ representing the achieved grazing rate. Space includes turf algae that have been grazed back to a low canopy height, referred to as ‘fine turf’ sensu Klumpp and McKinnon (1989). This fine turf grows to become turf algae if left ungrazed, and the rate at which this occurs is denoted by ζ T . This potential rate is reduced by herbivorous grazing effort, which hinders the growth of fine turf, and this is represented by the realized growth rate ζ T (1 − θ ) . Figure 4.1 shows a schematic representation of the functional groups in the CTm and the interactions modelled. Table 5.1 at the end of Chapter 5 summarises the parameters used in the CTm together with parameters for the other two benthic models.
G, R
Turf Algae
Hard Corals
M
F
G, R
G
‘Space’
Figure 4.1. Schematic diagram showing the functional groups and interactions in the CTm. Each arrow represents conversion of the proportional cover of one group to another due to the process(es) next to the arrow. F = feeding by grazers, G = growth, M = mortality, R = recruitment.
69
4.2.4 Mathematical Analysis of the CTm
Section A4.1 of Appendix A4 states and briefly explains all the mathematical theorems and conditions used in the mathematical analysis of the CTm. These are also used in the mathematical analysis of the two benthic models constructed in Chapter 5 (see Appendix A5). Section A4.1 also states and briefly explains the Richeson-Wiseman Theorem, which is used in the mathematical analysis of the fish and urchin models (see Appendix A6) and the integrated models (see Chapter 7, section 7.1.1). The first step in the analysis of the CTm is to ensure that the variables in the CTm stay within the biological domain, which is the simplex
{(C , T ) : 0 ≤ C , T ≤ C + T ≤ 1}. In section A4.2.1 of Appendix A4, it is shown that this is always true. Also, in section A4.2.2, it is shown that the CTm has either one equilibrium or three equilibria in the biological domain. To determine the stability of the equilibria, the Routh-Hurwitz conditions (Murray 2002) are used. In section A4.2.3, it is shown that if there is just one equilibrium, then it must be a locally asymptotically stable (LAS) sink node and if there are three equilibria, then in order of ascending T (the equilibrium value of T), the first and third equilibria are LAS sink nodes whereas the middle equilibrium is an unstable saddle node. Since an equilibrium is either a sink node or a saddle node, Hopf bifurcations, in which a LAS spiral sink becomes an unstable spiral source and a limit cycle is generated as a parameter is increased past a certain value (Murray 2002), are not possible. Section A4.2.3 also shows that if the empirically derived parameter ranges (see Tables 9.1 and 13.1 in Chapters 9 and 13 respectively) are used, then the CTm cannot exhibit periodic orbits and hence limit cycles.
70
Appendix A4
A4.1 Theorems and Conditions Used This section describes the mathematical techniques that are applied in the analyses of the dynamical coral reef ecosystem models constructed in the main text, in general terms. Let F : R n → R n be a continuously differentiable map. A point y ∈ R n is a regular value of F if the derivative dF ( x) : R n → R n is non-singular (i.e. Det ((∂Fi ∂x j )( x )) ≠ 0 , where Det is the determinant) for every x ∈ R n satisfying
F ( x ) = y (Milnor 1965). Note that any y for which F −1 ( y ) is empty is taken to be a regular value. Sard’s Theorem is a fundamental result which states that regular values are dense in R n . Sard’s Theorem. For a continuously differentiable map F : R n → R n , the set of regular values is dense in R n . (Weisstein 2009). x ∈ R n is a regular point if the derivative dF ( x) : R n → R n is non-singular (Milnor
1965). If x ∈ R n is a regular point, then the Inverse Function Theorem (Milnor 1965) implies that there is an open neighbourhood of x, U ⊂ R n , such that F maps U diffeomorphically onto the open set F (U ) . Thus, if y ∈ R n is a regular value, then each solution of F ( x ) = y is isolated as well as regular. Of particular interest are the equilibria of the vector field F, which are solutions of
F ( x ) = 0 . By Sard’s Theorem, it can be assumed that 0 is a regular value (otherwise, the system is not generic). Thus, every zero of F is regular and isolated. In this case, there is only a finite number of zeros of F in any compact subset X ⊂ R n . If there were infinitely many zeros in X, then this set of zeros would admit an accumulation point in X (by compactness). Such a point would also be a zero of F (by continuity), but would not be isolated, giving a contradiction. The coral reef ecosystem models studied in this thesis are dynamical systems of the form x& = F ( x, π ) for some continuously differentiable map F : R n × R m → R n , 71
where π ∈ R m is a vector of parameters. In this thesis, the interest is in the behaviour of such a model for values of π which lie in some specified parameter space Π ⊂ R m , defined by biological criteria that captures relevant knowledge about possible parameter values (for example, the growth rates, mortality rates and so forth must be non-negative). Also, the interest is in a convex set X ⊂ R n which, for each π ∈Π , is forward invariant under the flow of x& = F ( x, π ) . The state space X describes the set of biologically realistic values that the state variables can take; the set X is called the biological domain. In the case of the benthic models, X is a simplex and hence compact; in the case of the fish and urchin models, X is a positive orthant space R k+ . In the case of the integrated models, X is a product of a simplex (for the benthic variables) with a positive orthant space R k+ (for the fish and urchin variables). Consider the extended map Fˆ : R n × R m → R n × R m defined by Fˆ ( x, π ) = (F ( x, π ), π ) . By Sard’s Theorem, almost all points in the subspace {0}× R m
can be taken to be regular values of Fˆ . That is, for almost all parameter sets π , the zeros (in x) of F ( x, π ) are regular and isolated. This allows the Poincaré-Hopf Theorem, described below, to be applied. Of most interest are the zeros (equilibria) of F which lie in the biological domain X. A priori, there may be no equilibria in X. To investigate the existence of equilibria in X, it is convenient to use the Poincaré-Hopf Theorem. This theorem has two parts. The first is the index of a zero x of F. If the zero is regular, then the index is defined as
I F (x ) = sgn[Det (dF ( x ))] . (Hofbauer and Sigmund 1998). The second is the Euler characteristic of X, denoted by χ ( X ) . If X is homeomorphic to a finite simplicial complex K (for example, a simplex), then χ ( X ) is defined as
χ ( X ) = ∑ (− 1)k nk , k ≥0
where nk is the number of k-dimensional simplices in K (Hazewinkel 2002). It can be shown that χ ( X ) depends only on X and not on the particular choice of K (i.e. χ ( X ) is topologically invariant). For example, if K = ∆ ⊂ R n is a standard simplex, then
χ (X ) = 1. 72
Poincaré-Hopf Theorem. Let X ⊂ R n be a compact manifold, and let F : R n → R n be a continuously differentiable vector field on X for which there are isolated zeros. If X has a boundary ∂X , it is required that F is outward pointing on ∂X (this implies, in particular, that F has no zeros in ∂X ). Then
∑ I (x ) = χ ( X ) , F
x
where the summation is taken over the set of zeros x of F in X. (Milnor 1965). Note that if F is inward pointing on ∂X , then − F is outward pointing on ∂X and has the same zeros as F. Also, if the zeros are regular points, then I − F ( x ) = (− 1) I F ( x ) abd n
thus, the equation in the Poincaré-Hopf Theorem holds for − F with χ ( X ) replaced by
(− 1)n χ ( X ) . In the case in which X = ∆ , a simplex, then χ ( X ) = 1 , and if the requirements for the Poincaré-Hopf Theorem to be applied are satisfied, it follows from the equation in the Poincaré-Hopf Theorem that there must be an odd number of (isolated) equilibria x in the interior of X . Another technique for showing the existence of at least one equilibrium is the Richeson-Wiseman Theorem, which will be useful in the analyses of the fish and urchin models and the integrated models. Richeson-Wiseman Theorem. Let F : R n → R n be a continuously differentiable map. A compact subset W ⊂ R n is called a window for the dynamical system (discrete-time or continuous-time) on R n defined by F if the forward orbit of any x ∈ R n intersects W. If there exists a window W for the dynamical system defined by F, then there is a zero of F in W. (Richeson and Wiseman 2002).
The following mathematical theory is for two-dimensional dynamical systems. This is used when analyzing the two simplest benthic models in this thesis, which are both twodimensional. Let x = ( x1 , x 2 ) be a regular equilibrium of a two-dimensional dynamical system defined by a continuously differentiable map F : R 2 → R 2 . The well-known Routh73
Hurwitz conditions (Murray 2002) determine the local stability properties of x . First, consider the Jacobian matrix at x : ∂F1 (x ) ∂x1 J ( x ) = dF ( x ) = ∂F2 ∂x (x ) 1
∂F1 (x ) ∂x 2 . ∂F2 (x ) ∂x 2
Then x is locally asymptotically stable (LAS) if both the eigenvalues of J ( x ) have a negative real part, and x is unstable if J ( x ) has an eigenvalue with a positive real part (Robinson 2004). The Routh-Hurwitz conditions give sufficient conditions for these outcomes, which are usually expressed in terms of the trace (Tr) and determinant (Det) of J ( x ) : TrJ ( x ) = (∂F1 ∂x1 + ∂F2 ∂x 2 )( x ) and
DetJ ( x ) = (∂F1 ∂x1 )( x )(∂F2 ∂x 2 )( x ) − (∂F1 ∂x 2 )( x )(∂F2 ∂x1 )(x ) . These sufficient conditions give: 1. x is LAS if TrJ ( x ) < 0 and DetJ ( x ) > 0 . 2. x is unstable if either TrJ ( x ) > 0 or DetJ ( x ) < 0 . The eigenvalues of J ( x ) are
λ± =
{
1 TrJ ( x ) ± 2
}
[TrJ (x )]2 − 4 DetJ (x ) .
Thus, the eigenvalues of J ( x ) are both real if [TrJ ( x )] ≥ 4 DetJ ( x ) . This is always the 2
case if DetJ ( x ) < 0 , in which case, the (real) eigenvalues have opposite signs and the equilibrium x is an unstable saddle node. If [TrJ ( x )] ≥ 4 DetJ ( x ) > 0 , then both (real) 2
eigenvalues have the same sign. If the sign is positive ( TrJ ( x ) > 0 ), then x is an unstable source, and if it is negative ( TrJ ( x ) < 0 ), then x is a LAS sink or sink node. (Robinson 2004). The eigenvalues of J ( x ) are both complex if [TrJ ( x )] < 4 DetJ ( x ) . In this case, 2
Re(λ ± ) = 12 TrJ ( x ) , and x is called an unstable spiral source if TrJ ( x ) > 0 , and a LAS spiral sink if TrJ ( x ) < 0 (Robinson 2004). 74
It is possible that a dynamical system has a locally stable limit cycle, i.e. a closed, periodic trajectory that is locally attracting for nearby trajectories (Jordan and Smith 2005). However, for two-dimensional systems and in some circumstances, it is possible to rule out the existence of periodic trajectories using the following criterion: Bendixson’s Negative Criterion. If TrJ ( x ) = ∂F1 ∂x1 + ∂F2 ∂x 2 has a single sign for all x in a simply connected region D ⊂ R 2 , then there are no closed trajectories of
x& = F ( x ) in D. (Jordan and Smith 2005).
A4.2 The Coral-Turf Model (CTm) A4.2.1 Showing that 0 ≤ C (t ), T (t ), S (t ) ≤ 1 for the CTm The biological domain is a simplex, {(C , T ) : 0 ≤ C , T ≤ C + T ≤ 1} . Using eqns. 4.1-4.3, if C = 0 , then dC dt = lCs (1 − T + ε C T ) > 0 , since ε C > 0 and 0 ≤ T ≤ 1 . This shows that on the edge C = 0 of the simplex, the trajectories are pointing inwards. Also, if T = 0 and S ≠ 0 , then dT dt = ζ T (1 − θ )(1 − C ) > 0 , since 0 ≤ C , θ < 1 ; if T = S = 0 ,
then C = 1 and dT dt = 0 , with dC dt = −d C < 0 and dS dt = d C > 0 . This shows that on the edge T = 0 , the trajectories are pointing inwards except at the vertex
(C , T ) = (1, 0) . At this vertex, the trajectory points along the edge T = 0
towards the
origin (0, 0) ; after the trajectory leaves the vertex, it will then point inwards, since trajectories are inward pointing along the edge T = 0 except at (1, 0) . For θ ≠ 0 , if S = 0 , then dS dt = d C C + g T θT > 0 , since C + T = 1 . For θ = 0 , if S = 0 and C ≠ 0 ,
then dS dt = d C C > 0 ; if S = C = 0 , then T = 1 and dS dt = 0 , with dC dt = ε C l Cs > 0 and dT dt = −ε C l Cs < 0 . Since dS dt > 0 on the edge S = 0 ( C + T = 1 ) except at the vertex (0,1) when θ = 0 , this means that trajectories on the edge S = 0 ( C + T = 1 ) are pointing inwards except at the vertex (0,1) when θ = 0 . If θ = 0 , then at this vertex, the trajectory points along the edge S = 0 towards the vertex (1, 0) ; after the trajectory leaves the vertex (0,1) , it will then point inwards, since trajectories are inward pointing along this edge except at the vertex (0,1) . These calculations show that the dynamics stay within the simplex
75
{(C , T ) : 0 ≤ C , T ≤ C + T ≤ 1} and furthermore, the vector field is always inward pointing on the boundary of this simplex except at the two vertices (C , T ) = (1, 0) and (0,1) .
A4.2.2 Finding the Number of Equilibria for the CTm
From section A4.2.1, the dynamics for the CTm stay within the biological domain, which is a simplex, and there are no equilibria on the boundary of the simplex. Denote this simplex by Ω , such that
Ω = {(C , T ) : 0 ≤ C , T ≤ C + T ≤ 1} .
(A4.1)
Let ∂Ω be the boundary of this simplex and let the differential equations system for the CTm be denoted by x& = f ( x ) , where x = (C , T ) . From section A4.2.1, f is inward pointing on ∂Ω apart from the vertices (C , T ) = (1, 0) and (0,1) . The Poincaré-Hopf Theorem (section A4.1), applied below, requires that f is inward pointing on the entire boundary. Thus, instead of Ω , consider Ω′ , which is the closed area bounded by ∂Ω and the two lines C = 1 − ε and T = 1 − ε , where ε is small and positive. Thus, Ω′ excludes the two vertices (1, 0) and (0,1) . Ω′ has five edges. From section A4.2.1, the vector field is inward pointing along the edges C = 0 , T = 0 and S = 0 of Ω′ . Along the edge C = 1 − ε , (C , T ) can be written as (1 − ε , εξ ) , where 0 ≤ ξ ≤ 1 . Then eqns. 4.1 and 4.2 can be written as
[{
]
}
dC = ε l Cs + l Cb (1 − ε ) {1 − (1 − ε C )ξ } + rC (1 − ε ){1 − (1 − α C )ξ } + d C − d C , dt
[
{
}
]
dT = ε (1 − ξ )ζ T (1 − θ ) − εξ g T θ + ε C lCs + l Cb (1 − ε ) + rC α C (1 − ε ) . dt
Clearly, dT dt > 0 for ξ = 0 (when T = 0 ) and dS dt = − dC dt − dT dt = d C (1 − ε ) + g T θεξ > 0 for ξ = 1 (when S = 0 ), and for ε sufficiently small, dC dt < 0 for all 0 ≤ ξ ≤ 1 . This implies that the vector field is inward pointing along this edge for ε sufficiently small. The proof that the vector field 76
is inward pointing along the edge T = 1 − ε , for ε sufficiently small, is similar. Since there are no equilibria of the vector field on ∂Ω , ε > 0 can be chosen small enough such that the vector field is inward pointing along ∂Ω′ and also that there are no equilibria in the excised region Ω \ Ω′ . Thus, Ω′ is forward invariant under the flow of the dynamical system and contains all equilibria of the vector field in Ω . Ω and Ω′ are both homeomorphic to a ball and thus, they have the same Euler
characteristic (see section A4.1). Since Ω is a simplex, χ (Ω′) = χ (Ω ) = 1 . Let F : R 2 → R 2 denote the vector field defined by the CTm eqns. 4.1-4.3. Then F is
inward pointing on Ω′ , and it follows from the Poincaré-Hopf Theorem (section A4.1) that Ω′ , and hence Ω , contains an odd number of (interior) equilibria of F. It will now be shown that the maximum number of equilibria in the biological domain is three. First, eqns. 4.1 and 4.2 are written as:
(
)
C& = f (C , T ) = l Cs + lCb C {1 − C − (1 − ε C )T } + rC {1 − C − (1 − α C )T }C − d C C ,
(
)
T& = g (C , T ) = ζ T (1 − θ )(1 − C − T ) − g T θT − ε C l Cs + l Cb C T − rC α C TC .
(A4.2a)
(A4.2b)
The nullcline f (C , T ) = 0 can be written as C = C (1) (T ) , where C(1) (T ) is the positive root of the quadratic
(l
b C
[
)
]
+ rC C 2 − − l Cs + l Cb {1 − (1 − ε C )T } + rC {1 − (1 − α C )T } − d C C
− l Cs {1 − (1 − ε C )T } = 0 .
That is,
C (1) (T ) =
{
}
1 Y + Y 2 + 4 XZ , 2X
(A4.3)
where X = l Cb + rC ,
(A4.4a)
Y = −lCs + l Cb {1 − (1 − ε C )T } + rC {1 − (1 − α C )T } − d C ,
(A4.4b)
Z = l Cs {1 − (1 − ε C )T }.
(A4.4c) 77
X and Z are positive, but Y may be positive or negative. Furthermore, both Y and Z are linear decreasing functions of T, and X is independent of T. Thus,
C (′1) (T ) =
1 2X
YY ′ + 2 XZ ′ C (1) (T )Y ′ + Z ′ <0 Y ′ + = Y 2 + 4 XZ Y 2 + 4 XZ
for 0 ≤ T ≤ 1 . That is, C(1) (T ) is decreasing in T. Also, from eqns. A4.4a,b it is easily checked that 2 X − Y > 0 , and hence, from eqn. A4.3, C (1) (T ) < 1 if and only if Y + Z < X . From eqns. A4.4a,b,c, X − Y − Z = l Cb (1 − ε C )T + rC (1 − α C )T + d C + l Cs (1 − ε C )T , which is positive for 0 ≤ T ≤ 1 . Thus, it follows that 0 < C (1) (T ) < 1 for all 0 ≤ T ≤ 1 .
Since C (1) (T ) satisfies f (C (1) (T ), T ) = 0 , it follows that
C (′1) (T ) = −
∂f ∂T (C(1) (T ), T ) = − F , ∂f ∂C E
(A4.5)
where
E=−
∂f = C (l Cb + rC ) + l Cs + d C − l Cb (S + ε C T ) − rC (S + α C T ) , ∂C
(A4.6a)
F =−
∂f = (1 − ε C ) l Cs + l Cb C + rC (1 − α C )C , ∂T
(A4.6b)
(
)
with S = 1 − C − T and C = C (1) (T ) . Clearly, F is positive for 0 ≤ C ≤ 1 , and hence 0 ≤ T ≤ 1 , because 0 < C (1) (T ) < 1 for all 0 ≤ T ≤ 1 . Since C (′1) (T ) < 0 for 0 ≤ T ≤ 1 , E
must also be positive for all 0 ≤ T ≤ 1 . Similarly, the nullcline g (C , T ) = 0 can be written as C = C ( 2) (T ) , where
C ( 2) (T ) =
ζ T (1 − θ )(1 − T ) − (g T θ + ε C l Cs )T . ζ T (1 − θ ) + (ε C l Cb + α C rC )T
(A4.7)
78
C( 2) (T ) satisfies g (C ( 2) (T ), T ) = 0 , so it follows that
C(′2) (T ) = −
∂g ∂T H C( 2) (T ), T = − , ∂g ∂C G
(
)
(A4.8)
where
G=−
∂g = ζ T (1 − θ ) + ε C l Cb + α C rC T , ∂C
H =−
∂g = ζ T (1 − θ ) + g T θ + ε C l Cs + l Cb C + α C rC C , ∂T
(
)
(
(A4.9a)
)
(A4.9b)
with C = C ( 2) (T ) . Clearly, G and H are both positive, and hence C (′2) (T ) < 0 , for C,T ≥ 0 .
From eqn. A4.7, the denominator of C( 2) (T ) is positive for 0 ≤ T ≤ 1 . Thus,
C ( 2) (T ) ≥ 0 if and only if 0 ≤ T ≤ T0 , where
T0 =
ζ T (1 − θ ) . ζ T (1 − θ ) + g T θ + ε C lCs
(A4.10)
Clearly, 0 < T0 < 1 . Also, it follows from eqns A4.7, A4.8 and A4.9a,b that C( 2) (T ) is decreasing in T for 0 ≤ T ≤ T0 , with C ( 2) (0) = 1 and C ( 2) (1) < 0 . In addition, a direct calculation using eqn. A4.7 shows that 1 − T − C ( 2 ) (T ) > 0 , and hence C( 2) (T ) lies in the biological domain 0 ≤ C , T ≤ C + T ≤ 1 for all 0 ≤ T ≤ T0 . Furthermore, from eqns. A4.8 and A4.9a,b,
C (′2′ ) (T ) =
(
)
G ′H − GH ′ 2 ε C l Cb + α C rC H = > 0, G2 G2
and hence C( 2) (T ) is convex decreasing for 0 ≤ T ≤ T0 . Since 0 < C (1) (0) < 1 = C ( 2) (0) and C (1) (1) > 0 > C ( 2) (1) , it follows that the two
79
isoclines C (1) (T ) and C( 2) (T ) intersect at least once in the range 0 < T < T0 . Hence, there is at least one equilibrium of the CTm in the biological domain. However, since the convexity-concavity of the curve C = C (1) (T ) is indeterminate, it is not possible to conclude that there cannot be more than one such equilibrium. Similar results were found when the two nullclines were written as functions of C rather than T: one of the nullclines is convex decreasing in the biological domain (as C increases), but the other nullcline can be convex or concave (when each parameter takes any positive value or when parameters take values from their empirically derived ranges, as given by Tables 9.1 and 13.1 in Chapters 9 and 13 respectively), such that it is still not possible to rule out the case where there is more than one equilibrium in the biological domain. More generally, equilibrium values of T are solutions of the equation
C (1) (T ) = C ( 2) (T ) , and these lie in the biological domain 0 ≤ C , T ≤ C + T ≤ 1 provided 0 < T < T0 . By construction, C (1) (T ) satisfies XC 2 − YC − Z = 0 . Substituting
C = C ( 2) (T ) into this equation gives an equilibrium equation for T. From eqn. A4.7, C ( 2) (T ) has the form
C ( 2) (T ) =
a − bT , a + cT
(A4.11)
where a, b and c are positive constants (independent of T). Substituting C( 2) (T ) from eqn. A4.11 into XC 2 − YC − Z = 0 gives: X (a − bT ) − Y (a − bT )(a + cT ) − Z (a + cT ) = 0 . 2
2
(A4.12)
From eqns. A4.4a,b,c, X is a constant and Y and Z are linear in T, and hence eqn. A4.12 is a cubic in T. It follows that there can be at most three equilibrium points in the biological domain. In fact, it is possible to construct a numerical example in which three distinct equilibria exists in the biological domain. Using the Immune-Inspired Algorithm (see Appendix A13, sections A13.1 and A13.2) to search the parameter space given by the parameter ranges: d C = 0.001 − 0.01 yr-1; l Cb = 0.001 − 0.5 yr-1; l Cs = 0.001 − 0.01 yr-1;
80
rC = 0.001 − 0.2 yr-1; α C = 0.001 − 1 ; ε C = 0.001 − 0.15 ; g T = 0.001 − 15 yr-1;
ζ T = 0.001 − 80 yr-1; θ = 0 − 1 , positive parameter sets were found that give multiple equilibria in the biological domain. One such parameter set is: d C = 0.00813 yr-1, l Cb = 0.118 yr-1, l Cs = 0.00878 yr-1, rC = 0.174 yr-1, α C = 0.00762 , ε C = 0.0136 ,
g T = 0.00314 yr-1, ζ T = 0.0371 yr-1 and θ = 0.01 , which gives the three equilibria
(C , T ) = (0.0648, 0.927) , (0.210, 0.774) and (0.491, 0.488) . However, note that the above parameter ranges are not the empirically derived parameter ranges investigated in Chapters 13-15.
A4.2.3 Finding the Local Stability of Equilibria for the CTm Let (C , T ) be an equilibrium of the CTm (eqns. 4.1-4.3) in the biological domain. The Jacobian at this equilibrium is
− E J (C , T ) = − G
−F , − H
(A4.13)
where E , F , G , H are the expressions E, F, G, H given by eqns. A4.6a,b and A4.9a,b evaluated at equilibrium. As shown in section A4.2.2, these are all positive, except possibly E . However, the equilibrium equation
(
)
C& = f (C , T ) = l Cs + l Cb C {1 − C − (1 − ε C )T }+ rC {1 − C − (1 − αC )T }C − d C C = 0
gives l Cs {1 − (1 − ε C )T } s dC = − lC + l Cb {1 − C − (1 − ε C )T }+ rC {1 − C − (1 − α C )T }, C
and substituting into eqn. A4.6a gives
(
)
E = C l Cb + rC +
lCs {1 − (1 − ε C )T } , C
(A4.14)
81
which is positive. Using the Routh-Hurwitz conditions for a LAS equilibrium (section A4.1), the local stability properties of the equilibrium (C , T ) are determined by TrJ (C , T ) = −(E + H ) ,
(A4.15a)
DetJ (C , T ) = E H − F G .
(A4.15b)
From eqn. A4.15a, TrJ < 0 , and hence the equilibrium is LAS if E H > F G and is an unstable saddle node if E H < F G (section A4.1). Also, note that
[TrJ (C , T )]
2
− 4 DetJ (C , T ) = (E − H ) + 4 F G , 2
which is positive. Thus, both eigenvalues of J (C , T ) are real and if E H > F G , they are negative, which means that (C , T ) is a LAS sink node (section A4.1). From eqns. A4.5 and A4.8, C (′1) (T ) = − F E and C (′2) (T ) = − H G . The sufficient condition for an equilibrium to be LAS is therefore:
E H > FG
⇔
H F > G E
⇔ C (′1) (T ) > C (′2) (T ) .
(A4.16)
If this inequality is reversed, that is C (′1) (T ) < C (′2) (T ) , then the eigenvalues are real and have opposite signs, which means that the equilibrium is an unstable saddle node. Now suppose that the CTm has three equilibria, and that these are ordered in terms of increasing T . As shown in section A4.2.2, since 0 < C (1) (0) < 1 = C ( 2) (0) and both C(1) (T ) and C ( 2) (T ) are decreasing in T, it follows that C(′1) (T ) > C(′2) (T ) at the first and third equilibria, and C (′1) (T ) < C (′2) (T ) at the intermediate equilibrium. Thus, by the inequalities in A4.16, the first and third equilibria are LAS sink nodes, and the intermediate equilibrium is an unstable saddle node. Finally, it is shown that the CTm cannot exhibit periodic orbits in the biological domain, and hence limit cycles, when parameters take values from the empirically derived ranges. To do this, Bendixson’s Negative Criterion is used (see section A4.1). 82
Using eqns. A4.6a and A4.9b, ∂f ∂g = −(E + H ) , + ∂C ∂T
(A4.17)
where
(
)
E = C l Cb + rC + l Cs + d C − l Cb (S + ε C T ) − rC (S + α C T ) ,
(
)
H = ζ T (1 − θ ) + g T θ + ε C lCs + l Cb C + α C rC C .
(A4.18a)
(A4.18b)
From eqn. A4.18a, ∂E ∂C = 2(l Cb + rC ) and ∂E ∂T = l Cb (1 − ε C ) + rC (1 − α C ) , which are both positive. Hence, E takes its minimum value on the biological domain when C = T = 0 and S = 1 , giving E min = (l Cs + d C ) − (l Cb + rC ) . From eqn A4.18b,
H min ≥ min{ζ T , g T } (since 0 ≤ θ < 1 ). Thus, E + H ≥ E min + H min ≥ l Cs + d C − lCb − rC + min{ζ T , g T } . Hence, if min{ζ T , g T } + l Cs + d C > l Cb + rC ,
(A4.19)
then ∂f ∂C + ∂g ∂T is negative on the biological domain, and hence periodic orbits cannot exist within this domain. Using the empirically derived parameter ranges from Tables 9.1 and 13.1 in Chapters 9 and 13 respectively, it can easily be checked that condition A4.19 holds when parameters take values from the empirically derived ranges.
83
Chapter 5. Benthic Models with Macroalgae Macroalgae are common components of reefs that have undergone a phase shift. For example, during the phase shift seen in Jamaica between 1977-1993, macroalgal cover increased from 4 to 92% (Hughes 1994). Brown macroalgae such as Sargassum fissifolium and Lobophora variegata are common on reefs and have been implicated in reef degradation (Diaz-Pulido and McCook 2003). In this chapter, macroalgae is added to the CTm (constructed in Chapter 4) in two different ways, giving rise to two more benthic models, the Coral-Algae model (section 5.1) and the more complex Coral-TurfMacroalgae model (section 5.2).
5.1 The Coral-Algae Model (CAm) In the Coral-Algae model (CAm), macroalgae and turf algae are amalgamated into one group and their combined cover is represented by one dynamic variable. The rationale for doing this is to keep the differential equations system two-dimensional by restricting the number of independent variables to two. This model retains the analytical tractability of the CTm, making it easier to apply the Immune-Inspired Algorithm (IIA), a form of genetic algorithm, to search parameter space for multiple equilibria (see Chapter 13) and to apply parameter sweeps (Chapter 14). The key assumption in the CAm is that the ratio of turf to macroalgal cover remains constant through time. This is a fairly severe limitation which will be removed in the more complex, but less analytically tractable, Coral-Turf-Macroalgae model (section 5.2). Results from the two models can then be compared to examine how serious a limitation this is. The CAm models hard corals, turf algae and macroalgae competing for space on the hard benthic substratum of a coral reef, with the two algal groups considered as one group. As in the CTm, recruitment for hard corals is modelled explicitly whereas recruitment for turf algae is modelled implicitly; recruitment for macroalgae is not modelled due to a lack of data (see section 5.1.2). Mortality and growth are modelled explicitly for hard corals, turf algae and macroalgae. As in the CTm, hard corals undergo mortality due to background processes and human activities that enhance the sedimentation rate, whereas turf algae undergo mortality due to grazing. On top of this, hard corals can now undergo mortality due to overgrowth by macroalgae. Mortality of 84
macroalgae occurs due to grazing, which means that, like turf algae, macroalgae are limited by herbivores as well as space. In addition, coral growth is reduced by the presence of nearby macroalgae. Thus, the CAm models the key ecological processes determining competition for space between the three modelled types of organisms. In common with all three benthic models in this thesis, grazer dynamics are not modelled in the CAm, such that there is no feedback from the benthos to the grazers; however, this constraint is removed in the more complex integrated models (Chapter 7).
5.1.1 Details of the Macroalgae Functional Group
The macroalgae functional group consists of algae that are larger and more structurally complex than turf algae. Thus, macroalgae in this thesis corresponds to the Foliose Algae, Corticated Foliose Algae, Corticated Macrophytes, Leathery Macrophytes and Articulated Calcareous Algae groups of Steneck and Dethier (1994). All of these macroalgal groups are able to form tissues which are more than a cell wide, in contrast to turf algae. Also, the thallus size of macroalage is greater than that for turf algae and can be over a metre long (for leathery macrophytes) (Steneck and Dethier 1994, Hughes et al. 2007). Examples of macroalgae include Dictyota spp., which are corticated foliose algae common on Caribbean reefs (McCook et al. 2001a), Sargassum spp, which are leathery macrophytes abundant on inshore reefs (Steneck and Dethier 1994, McCook et al. 2001a, Jompa and McCook 2002a) and Halimeda spp., which are articulated calcareous algae that are major reef carbonate sediment producers (Delgado and Lapointe 1994).
5.1.2 Details of Extra Interactions Modelled in the CAm
The inclusion of macroalgae in the CAm gives rise to extra interactions, which will now be discussed. In terms of ecological interactions, a major difference between turf algae and macroalgae is that the latter can grow laterally into space, which is partly due to their more complex morphology. For example, Nugues and Bak (2006) found that L. variegata can grow laterally over dead corals. In addition, the CAm includes terms representing growth of macroalgae over live coral colonies. There is evidence that L. variegata is able to overgrow live colonies of the massive coral Porites cylindrica (Jompa and McCook 2002a) and the foliose coral Agaricia agaricites (Nugues and Bak 85
2006). De Ruyter van Steveninck et al. (1988) also observed that L. variegata can overgrow live coral tissue in Curaçao, for five coral species. Futhermore, McCook et al. (2001a) reviewed the literature on coral-algal interactions and proposed that overgrowth of corals by macroalgae is common or probable for encrusting and foliose corals, and possible for other coral life-forms (but less common). Although Nugues and Bak (2006) found that A. agaricites was the only coral species out of six tested that was overgrown by L. variegata, their experiments did not use cages to exclude herbivory. Also, surveys of natural interactions in the same study found that, for the five coral species not overgrown by L. variegata in the experiments, 10-40% of live coral colonies of each species were overgrown partially by the macroalga. Thus, the effect of macroalgal overgrowth was likely to be underestimated in the experiments (Mumby et al. 2007a). Overall, the studies considered show that overgrowth of live coral is possible and justifies the inclusion of this interaction in the CAm. However, there is the possibility that macroalgae are not always competitively superior to corals (Nugues and Bak 2006). Live coral colonies have mesenterial filaments (see Chapter 1, Figure 1.1) which Nugues et al. (2004) found were used as a short-range defence against nearby macroalgae, with contact causing discolouration of algal blades. Thus, the CAm includes a parameter representing overgrowth of live corals by macroalgae, but this can be taken to be low or zero. There is evidence that macroalgae can impose metabolic costs on corals they are in contact with, thus reducing their growth. For example, Jompa and McCook (2002a) found that the nearby presence of L. variegata significantly reduced the skeletal extension rate of P. cylindrica. Similarly, Lirman (2001) showed that Dictyota spp. significantly reduced growth of P. astreoides colonies. Thus, in the CAm, there is a term representing reduction of the coral growth rate due to the presence of nearby macroalgae affecting the growth of coral colonies over space or turf algae. There is also some evidence that corals can reduce the growth rate of macroalgae. Jompa and McCook (2002a) found that the growth of L. variegata over live P. cylindrica is significantly lower than over dead coral tissue, indicating growth inhibition of the macroalga by the coral. De Ruyter van Steveninck et al. (1988) found that five species of corals significantly lowered the growth rate of L. variegata close ( < 1 cm) to them. The proposed mechanisms were sweeper tentacles or mesenterial filaments damaging the algal blades and a chemically-mediated defensive process. However, away from the coral-algal interaction fringe (1-3 cm away), the growth of L. 86
variegata blades was unaffected. Thus, the effect is very localised and the studies do not show that corals reduce the growth of macroalgae into space away from the coral-algal margin. That corals can reduce the growth of macroalgae at the coral-algal margin is represented in the CAm simply by taking the growth rate of macroalgae over corals to be less than that over space. Macroalgal recruitment onto live corals is not modelled because evidence suggests that corals employ defensive mechanisms that keep algal recruits at bay. DiazPulido and McCook (2002) found that no algal recruits colonised live Porites corals, and similarly, Diaz-Pulido and McCook (2004) concluded that healthy corals prevented attachment and survival of macroalgal recruits (Sargassum spp. and L. variegata). Macroalgal recruitment onto space or turf algae is also not represented in the CAm due to a lack of knowledge of the supply-side dynamics of macroalgae, such as dispersal and recruitment (McCook 1999, Diaz-Pulido and McCook 2004). In the existing models reviewed in Chapter 2, macroalgal recruitment is also not modelled explicitly. Although coral larvae can settle onto macroalgae, they usually do not survive. For example, Nugues and Szmant (2006) found that larvae from the Caribbean coral F. fragum were able to settle on Halimeda opuntia; however, the experiments were only performed for 9 d and the long-term survivorship of the coral larvae on H. opuntia is unknown, but is likely to be low due to breakage of macroalgal segments (Nugues and Szmant 2006). Coral larvae have also been found to settle on the macroalgae Ulva fasciata, Sargassum polycystum and Laurencia papillosa, but again are unlikely to survive due to being dislodged (Birrell et al. 2008). Furthermore, macroalgae generally have negative effects on juvenile corals through shading, overgrowth, abrasion and chemical effects (McCook et al. 2001a, Birrell et al. 2008). Thus, overall, it is reasonable to assume that recruitment of coral larvae on macroalgae is insignificant. Hence, coral recruitment on macroalgae is not modelled in the CAm, consistent with all existing models reviewed in Chapter 2. Macroalgae are grazed by herbivores, and this is modelled in the CAm. These algae are sometimes chemically defended against grazing by secondary metabolites. For example, Dictyota spp. have the compound pachydictyol-A and Laurencia spp. have the compound elatol, both of which deter a range of reef fish. Pachydictyol-A is also effective against the grazing urchin Diadema antillarum (Hay 1991). Thus, the grazing rate on macroalgae can be lower than that on turf algae. Background mortality of macroalgae is not modelled due to a lack of quantified data for the significance of this effect. 87
5.1.3 Equations for the CAm
The CAm is defined by the two independent differential equations 5.1 and 5.2. C and S retain their meanings from the CTm, but there is a new variable A which represents algal proportional cover, that is, the combined covers of turf algae and macroalgae. Space, S, is a dependent variable equal to 1 − C − A , with dynamics given by eqn. 5.3. To aid with ecological interpretation, short descriptions are given below the terms
dC = lCs + lCb C {S + εC (1 − νM )A}+ rC (1 − βM νM A){S + αC (1 − νM )A}C − 42444444 3 dt 14444244443 144444
(
)
Coral recruitment
−
γ MC rM ν M AC 142 4 43 4
Coral growth
dCC { Coral mortality
,
(5.1)
Macroalgal growth over corals
dA = rMν M A(S + γ MC C ) + ζ T (1 − θ )S − {g Mν M + g T (1 − ν M )}θA − ε C (1 − ν M ) lCs + lCb C A 424444 3 144424443 42444 3 14243 1444 dt 144
(
Macroalgal growth
Turf algal growth
Grazing on macroalgae and turf
)
Coral recruitment on turf
− rC α C (1 − ν M )(1 − β Mν M A) AC , 144444244444 3
(5.2)
Coral growth over turf
dS dC dA =− − . dt dt dt
(5.3)
The coral parameters d C , l Cb , l Cs , rC , α C and ε C retain their meanings as in the CTm.
g T , ζ T and θ also retain the same meanings. However, there are now five new parameters. Four of these represent new interactions that arise due to the inclusion of macroalgae. One, ν M , is the fixed proportion of A which is macroalgal cover. Thus, ν M A is the macroalgal cover and
(1 − νM )A
is the turf algal cover. ν M takes values in the range 0-1. It is assumed that ν M
cannot take the extreme values 0 or 1, to ensure that the dynamics can have some turf algae or macroalgae; if ν M = 0 , then the CAm reduces to the CTm and if ν M = 1 , then there cannot be any turf algae at any point in time, which is unrealistic. rM is the rate at which established macroalgae grows laterally into available space whereas γ MC is the growth rate of macroalgae over corals relative to that over space. γ MC < 1 because of the 88
energetic costs associated with overgrowing live coral, as discussed in section 5.1.2.
β M is a parameter which measures the negative effect that macroalgae has on the growth of corals. Thus, for example, the term rC (1 − β M ν M A)CS in eqn. 5.1 represents the growth of corals into space taking into account the effect of (adjacent) macroalgae. Since the growth rate of corals cannot be negative, β M ≤ 1 . Finally, g M is the maximum grazing rate on macroalgae, with g M θ the achieved grazing rate when the grazing pressure is θ . The maximum grazing rate on (combined) algae is then simply the weighted average of g M and g T , namely g Mν M + gT (1 −ν M ) , as shown in eqn. 5.2. Figure 5.1 shows a schematic representation of the functional groups in the CAm and the interactions modelled. Table 5.1 at the end of this chapter summarises the parameters used for the CAm, together with those for the other two benthic models.
G, R
Algae
Hard Corals G
M
F
G, R
G
‘Space’
Figure 5.1. Schematic diagram showing the functional groups and interactions in the CAm. Each arrow represents conversion of the proportional cover of one group to another due to the process(es) next to the arrow. F = feeding by grazers, G = growth, M = mortality, R = recruitment.
89
5.1.4 Mathematical Analysis of the CAm
As for the CTm, the variables for the CAm always stay within the biological domain
{(C , A) : 0 ≤ C , A ≤ C + A ≤ 1} (see section A5.1.1 in Appendix A5). Section A5.1.2 in Appendix A5 shows that the CAm has an odd number of equilibria and that there is a maximum of five equilibria. In section A5.1.3, it is shown that, in order of ascending A (the equilibrium value of A), the first equilibrium is a locally asymptotically stable (LAS) sink node, and if there are more equilibria, they alternate between being an unstable saddle node and being a LAS sink node. Since an equilibrium is either a sink node or a saddle node, Hopf bifurcations are not possible, as for the CTm (Chapter 4, section 4.2.4).
5.2 The Coral-Turf-Macroalgae Model (CTMm) The Coral-Turf-Macroalgae model (CTMm) adds complexity to the CAm by separating the two algal components into independent dynamic variables, one for turf algal cover (as in the CTm) and one for macroalgal cover. In particular, it is no longer assumed that the ratio of turf to macroalgal cover (represented in the CAm by the parameter ν M ) is fixed. Another advantage of this decoupling is that interactions between turf algae and macroalgae can be explicitly modelled. Turf algae and macroalgae now compete for space and unlike in the CAm, the lateral growth of macroalgae over turf algae is modelled explicitly. The overgrowth of turf algae by macroalgae is also included in the models by Mumby et al. (2006a) and Mumby et al. (2007a). Thus, although there is little direct evidence which shows that macroalgae can directly overgrow algal turfs, the growth of macroalgae over algal turfs is included in the CTMm. The CTMm includes the organisms and interactions in the CAm (see section 5.1), and thus models hard corals, turf algae and macroalgae competing for space on the hard substratum of a coral reef benthos. However, because turf algae and macroalgae are modelled as separate groups in the CTMm, mortality of turf algae now also occurs due to macroalgal overgrowth.
90
5.2.1 Equations for the CTMm
The CTMm is defined by the three independent differential equations 5.4, 5.5 and 5.6 below. In these equations, C, T and S are the proportional covers of hard corals, turf algae and ‘space’ respectively, as in the CTm. A fourth variable M is introduced to represent macroalgal cover. S is not an independent variable but is given by S = 1 − C − T − M , and hence there are three independent variables. The terms in the
equations are the same as or similar to those in the CTm (Chapter 4, section 4.2.3) and the CAm (section 5.1.3).
dC = l Cs + l Cb C (S + ε C T ) + rC (1 − β M M )(S + α C T )C − d C C − γ MC rM MC , dt
(
)
dT = ζ T (1 − θ )S − g T θT − ε C l Cs + lCb C T − rC α C (1 − β M M )TC − γ MT rM MT , dt
(
)
(5.4)
(5.5)
dM = rM M (S + γ MC C + γ MT T ) − g M θM , dt
(5.6)
dS dC dT dM =− − − . dt dt dt dt
(5.7)
The coral parameters d C , l Cb , l Cs , rC , α C and ε C , the turf algal parameters g T and ζ T , and the grazing pressure parameter θ have the same meaning as in the CTm. The four macroalgal parameters g M , rM , β M and γ MC have the same meaning as in the CAm. In the CTMm, γ MT is an extra macroalgal parameter representing the lateral growth rate of macroalgae over turf algae relative to that over space. Since macroalgae expend more energy to overgrow turf algae than to overgrow space, due to greater competition from the developed turf algal strands, γ MT < 1 . Figure 5.2 shows a schematic representation of the functional groups modelled in the CTMm and the interactions between them. The definitions of the parameters for the CTMm are summarised in Table 5.1, which also shows the parameters for the CTm and the CAm.
91
G, R
Turf Algae
Hard Corals
G
G, R M
G
F
G G
Macroalgae
‘Space’ F
Figure 5.2. Schematic diagram showing the functional groups and interactions in the CTMm. Each arrow represents conversion of the proportional cover of one group to another due to the process(es) next to the arrow. F = feeding by grazers, G = growth, M = mortality, R = recruitment.
5.2.2 Mathematical Analysis of the CTMm
Section A5.2.1 in Appendix A5 shows that the variables in the CTMm always stay within the biological domain {(C , T , M ) : 0 ≤ C , T , M ≤ C + T + M ≤ 1} . Section A5.2.2 shows that there are two types of equilibria – those without macroalgae and those with macroalgae. It is also shown that the equilibria without macroalgae are the same as those for the CTm with the same coral and turf algae parameters, such that there is either one equilibrium or three equilibria without macroalgae. Furthermore, there are no equilibria with macroalgae if
g M θ ≥ rM .
(5.8)
That is, if the achieved grazing rate on macroalgae is greater than or equal to the macroalgal lateral growth rate over space, then there are no equilibria with macroalgae. 92
Section A5.2.3 examines the stability of equilibria without macroalgae for the CTMm. It is shown that if condition 5.8 holds, then the stability of these equilibria is exactly the same as for the CTm with the same coral and turf algae parameters. Thus, if 5.8 holds, the CTMm behaves like the CTm at equilibrium.
93
Parameter
Definition
dC
Coral mortality rate
l Cb
Rate at which coral larvae, produced by local established brooding corals,
Model Specificity CTm, CAm and CTMm
recruit onto space l Cs
Rate at which exogenous spawning coral larvae recruit onto space
rC
Lateral growth rate of corals over space
αC
Growth rate of corals over turf, relative to the rate over space
εC
Recruitment rate of corals onto turf, relative to the rate onto space
gT
Maximum rate at which turf algae is grazed
ζT
Growth rate of fine turf (occupying space)
θ
Grazing pressure
gM
Maximum rate at which macroalgae is grazed
rM
Lateral growth rate of macroalgae over space
βM
Coral growth is inhibited by the presence of nearby macroalgae and this is represented as depression of rC by the factor (1 − β M M ) , where M is the macroalgal cover
γ MC
Lateral growth rate of macroalgae over corals, relative to the rate over space 94
CAm and CTMm only
νM
The proportion of total algal cover that is macroalgal cover
γ MT
Lateral growth rate of macroalgae over turf, relative to the rate over space
CAm only CTMm only
Table 5.1. Parameters for the CTm, the CAm and the CTMm, with their ecological meanings
95
Appendix A5
A5.1 The Coral-Algae Model (CAm) A5.1.1 Showing that 0 ≤ C (t ), A(t ), S (t ) ≤ 1 for the CAm
Using eqns. 5.1-5.3 and similar calculations to those for the CTm (Appendix A4, section A4.2.1), if C = 0 , then dC dt > 0 . Also, if A = 0 and S ≠ 0 , then dA dt > 0 ; if A = S = 0 , then C = 1 and dA dt = 0 , with dC dt < 0 and dS dt > 0 . In addition, for
θ ≠ 0 , if S = 0 , then dS dt > 0 . For θ = 0 , if S = 0 and C ≠ 0 , then dS dt > 0 ; if S = C = 0 , then A = 1 and dS dt = 0 , with dC dt > 0 and dA dt < 0 . Analogous to
the CTm (section A4.2.1), this shows that the dynamics stay within the simplex
{(C , A) : 0 ≤ C , A ≤ C + A ≤ 1} and also that the vector field is always inward pointing on the boundary of this simplex except at the two vertices (C , A) = (1, 0) and (0,1) .
A5.1.2 Finding the Number of Equilibria for the CAm
From section A5.1.1, the dynamics for the CAm stay within the biological domain, defined by 0 ≤ C , A ≤ C + A ≤ 1 , and there are no equilibria on the boundary of the simplex. By using the same method as for the CTm, it can be shown that there is an odd number of equilibria within the interior of the biological domain (see Appendix A4, section A4.2.2 for details; calculations for the CAm are similar to those for the CTm). It will now be shown that the maximum number of equilibria in the biological domain is five. The nullcline equation C& = f (C , A) = 0 derived from eqn. 5.1 is a quadratic in C for given A: XC 2 − YC − Z = 0 ,
(A5.1)
where X = l Cb + rC (1 − β Mν M A) ,
(A5.2a) 96
Y = −lCs + lCb {1 − A + ε C (1 −ν M )A} + rC (1 − β Mν M A){1 − A + α C (1 −ν M ) A} − d C
− γ MC rMν M A ,
(A5.2b)
Z = l Cs {1 − A + ε C (1 − ν M ) A}.
(A5.2c)
X and Z are always positive for 0 ≤ A ≤ 1 , and are linear and decreasing functions of A. On the other hand, Y is a quadratic in A with coefficient of A 2 equal to rC β Mν M {1 − α C (1 − ν M )}, which is positive. The quadratic in eqn. A5.1 has only one positive root
C (1) ( A) =
{
}
1 Y + Y 2 + 4 XZ . 2X
(A5.3)
Similarly, the nullcline equation A& = g (C , A) = 0 derived from eqn. 5.2 gives
ζ (1 − θ ) + {g Mν M + g T (1 − ν M )}θ + ε C l Cs (1 − ν M ) ζ T (1 − θ ) − T A − rMν M (1 − A) C ( 2 ) ( A) = b ζ T (1 − θ ) + {rMν M (1 − γ MC ) + ε C lC (1 − ν M ) + α C rC (1 − ν M )(1 − β Mν M A)}A (A5.4)
This has the form
C ( 2 ) ( A) =
a − bA , a + cA
(A5.5)
where a is a positive constant (independent of A), and b and c are linear functions of A, with c positive for 0 ≤ A ≤ 1 . Since equilibrium values of A are solutions of
C (1) ( A) = C ( 2 ) ( A) , and C (1) ( A) satisfies eqn. A5.1, substituting C ( 2 ) ( A) from eqn. A5.5 into eqn. A5.1 gives an equilibrium equation X (a − bA) − Y (a − bA)(a + cA) − Z (a + cA) = 0 . 2
2
Since X, Z, b and c are linear in A, and Y is a quadratic in A, the left-hand side is a sextic polynomial in A. It follows that there can be at most six roots in the biological range 0 ≤ A ≤ 1 , giving at most six equilibria in the biological domain. However, it has
97
already been shown that there must be an odd number of equilibria in this domain, and hence there can be at most five equilibria in the biological domain.
A5.1.3 Finding the Local Stability of Equilibria for the CAm The Jacobian of the CAm at an equilibrium (C , A ) in the biological domain is
− E J (C , A ) = − G
−F , − H
(A5.6)
where
E =−
∂f (C , A ) = C lCb + rC (1 − β Mν M A ) + lCs + d C + γ MC rMν M A − lCb (S + ε C (1 −ν M )A ) ∂C
{
}
− rC (1 − β Mν M A )(S + α C (1 − ν M )A )
{
}
= C l Cb + rC (1 − β Mν M A ) + l Cs +
lCs {S + ε C (1 − ν M ) A} , C
(A5.7a)
(using the equilibrium equation C& = f (C , A ) = 0 to substitute for dC )
F =−
∂f (C , A ) = lCs + lCb C {1 − ε C (1 −ν M )} ∂A
(
)
β Mν M rC {S + α C (1 − ν M )A } +C , + rC (1 − β Mν M A ){1 − α C (1 − ν M )} + γ MC rMν M
G =−
∂g (C , A ) ∂C
= rMν M (1 − γ MC )A + ζ T (1 − θ ) + ε C lCb (1 −ν M ) A + α C rC (1 −ν M )(1 − β Mν M A )A ,
H =−
(A5.7b)
(A5.7c)
∂g (C , A ) = rMν M A + ζ T (1 − θ ) + {g Mν M + g T (1 −ν M )}θ ∂A + ε C (1 − ν M )(l Cs + lCb C ) + α C rC (1 − ν M )(1 − 2 β Mν M A )C − rMν M S − γ MC rMν M C .
(A5.7d) 98
Clearly, E , F and G are positive for 0 < C , A , S < 1 . Using the Routh-Hurwitz conditions (Appendix A4, section A4.1), the local stability properties of the equilibrium (C , A ) are determined by TrJ (C , A ) = −(E + H ) ,
(A5.8a)
DetJ (C , A ) = E H − F G .
(A5.8b)
From eqns. A5.8a,b, (C , A ) is an unstable saddle node if E H < F G (Appendix A4, section A4.1). On the other hand, if E H > F G , then H > 0 and hence TrJ (C , A ) < 0 , and so the equilibrium is LAS. Also, note that
[TrJ (C , A )]
2
− 4 DetJ (C , A ) = (E − H ) + 4 F G , 2
which is positive. It follows that for a LAS equilibrium, both eigenvalues of J (C , A ) are real and negative, and hence (C , A ) is a LAS sink node (Appendix A4, section A4.1). Differentiating the nullcline equations f (C , A) = 0 and g (C , A) = 0 with respect to A and evaluating at equilibrium gives
C (′1) (A ) = −
∂f ∂A (C , A ) = − F , ∂f ∂C E
(A5.9a)
C (′2) (A ) = −
∂g ∂A (C , A ) = − H . ∂g ∂C G
(A5.9b)
Since a sufficient condition for an equilibrium to be LAS is E H > F G , it follows from eqns. A5.9a,b that this condition is equivalent to the condition 0 > C (′1) (A ) > C (′2 ) (A ) . Also, it follows from eqns. A5.3 and A5.4 that 0 < C (1) (0) < 1 = C ( 2) (0) (cf. Appendix A4, section A4.2.2). Thus, if all equilibria in the biological domain are ordered in terms of increasing A , it follows that the first equilibrium is a LAS sink 99
node, with subsequent equilibria (if there are any) alternating between being an unstable saddle node and a LAS sink node.
A5.2 The Coral-Turf-Macroalgae Model (CTMm) A5.2.1 Showing that 0 ≤ C (t ), T (t ), M (t ), S (t ) ≤ 1 for the CTMm
Using eqns. 5.4-5.7 and similar calculations to those for the CTm (Appendix A4, section A4.2.1), if C = 0 , then dC dt > 0 . Also, if T = 0 and S ≠ 0 , then dT dt > 0 ; if T = S = 0 , then dT dt = 0 . In addition, if M = 0 , then dM dt = 0 . Furthermore, if S = 0 and C ≠ 0 , then dS dt > 0 ; if S = C = 0 , then dS dt = 0 . Analogous to the
CTm (section A4.2.1), this shows that the dynamics stay within the simplex
{(C , T , M ) : 0 ≤ C , T , M ≤ C + T + M ≤ 1} . Note that if θ ≠ 0 , then none of the vertices can be an equilibrium. However, if θ = 0 , then (C , T , M ) = (0, 0,1) is an equilibrium.
A5.2.2 Equilibrium Analysis for the CTMm
The equilibria are found by solving the three equations dC dt = 0 , dT dt = 0 and
dM dt = 0 , which are:
(l
s C
+ lCb C ){1 − C − M − (1 − ε C )T } + rC (1 − β M M ){1 − C − M − (1 − α C )T }C
− d C C − γ MC rM MC = 0 ,
(A5.10)
ζ T (1 − θ )(1 − C − M − T ) − g T θT − ε C (l Cs + lCb C )T − rC α C (1 − β M M )TC − γ MT rM MT = 0 ,
(A5.11)
M {rM (S + γ MC C + γ MT T ) − g M θ } = 0 ,
(A5.12)
From eqn. A5.12, two expressions for M are derived as: M = 0,
(A5.13a)
100
M =
rM (1 − C − T ) − g M θ + γ MC rM C + γ MT rM T . rM
(A5.13b)
When M = 0 (no macroalgae present), the CTMm eqns. 5.4 and 5.5 are the same as the equations for the CTm, eqns. 4.1 and 4.2 (Chapter 4, section 4.2.3). Thus, in this case, the analysis of the CTMm to determine the number of equilibria is the same as that for the CTm (see Appendix A4, section A4.2.2 for the CTm analysis). This means that the CTMm has either one equilibrium or three equilibria without macroalgae in the biological domain. In addition, these equilibria are the same as those for the CTm with the same parameters. To obtain solutions with non-zero macroalgal cover, substitute for M from eqn. A5.13b into eqns. A5.10 and A5.11 to obtain two equations in C and T. These equations cannot be solved analytically. They can potentially have no solutions, one solution or more than one solution for C and T in the biological range. In addition, the PoincaréHopf Theorem (Appendix A4, section A4.1) cannot be applied because the domain on which the system dynamics are defined, the simplex
{(C , T , M ) : 0 ≤ C , T , M ≤ C + T + M ≤ 1} , is not inward pointing at the face
M = 0 (see
section A5.2.1). However, from eqn. A5.13b, if
(1 − C − T ) + γ MC C + γ MT T − (g M
rM )θ ≤ 0 , then M is not positive, and so there cannot
be equilibria with macroalgae present in the biological domain. Since γ MC , γ MT < 1 , this is the case if
g M θ ≥ rM ,
(A5.14)
which is condition 5.8.
A5.2.3 Finding the Local Stability of Equilibria for the CTMm The Jacobian of the CTMm at an equilibrium (C , T , M ) in the biological domain is
− E J (C , T , M ) = − G − X
−F −H −Y
−V −W , − Z
(A5.15)
101
where: E = C {l Cb + rC (1 − β M M )}+ l Cs + d C + γ MC rM M − l Cb (S + ε C T ) − rC (1 − β M M )(S + α C T )
l Cs (S + ε C T ) = C l + rC (1 − β M M ) + l + , C
(A5.16a)
F = (1 − ε C ){lCs + l Cb C }+ rC (1 − α C )(1 − β M M )C ,
(A5.16b)
V = (lCs + lCb C ) + β M rC (S + α C T )C + rC (1 − β M M )C + γ MC rM C ,
(A5.16c)
{
b C
}
s C
G = ζ T (1 − θ ) + ε C l Cb T + rC α C (1 − β M M )C ,
(A5.16d)
H = ζ T (1 − θ ) + g T θ + ε C (l Cs + l Cb C ) + rC α C (1 − β M M )C + γ MT rM M ,
(A5.16e)
W = ζ T (1 − θ ) + γ MT rM T − β M rC α C C T ,
(A5.16f)
X = rM (1 − γ MC )M ,
(A5.16g)
Y = rM (1 − γ MT )M ,
(A5.16h)
Z = rM M + g M θ − rM S − γ MC rM C − γ MT rM T .
(A5.16i)
All of these are positive except possibly W , X , Y and Z . For an equilibrium without macroalgae, M = 0 and hence X = Y = 0 . Hence,
− E the eigenvalues of J are − Z together with the eigenvalues of K = − G
−F . Thus, − H
using the Routh-Hurwitz conditions (Appendix A4, section A4.1), for the eigenvalues to have negative real parts (i.e. for the equilibrium (C ,T ,0) to be LAS), necessary and sufficient conditions are: DetK > 0 ,
(A5.17a)
TrK < 0 ,
(A5.17b) 102
Z > 0.
(A5.17c)
Conditions A5.17a,b are the same as the stability conditions for the CTm, a twodimensional system (see Appendix A4, section A4.2.3). Since M = 0 , Z = g M θ − rM (1 − C − T ) − γ MC rM C − γ MT rM T and hence condition A5.17c is
equivalent to g M θ − rM (1 − C − T ) − γ MC rM C − γ MT rM T > 0 ,
(A5.18)
which is the negation of the numerator of A5.13b. Thus, if A5.14 holds, that is if 5.8 holds, then A5.18 always holds, such that stability of equilibria without macroalgae is determined by A5.17a,b. This means that if condition 5.8 holds (or, more generally, condition A5.18), then the CTMm has the same equilibria as the CTm with the same parameters (see section A5.2.2), with the same stability properties.
103
Chapter 6. Fish and Urchin Models
6.1 Modelling Methodology for Fish and Urchin Models The fish and urchin models use the same functional group approach as used for the benthic models (see Chapter 4, section 4.1). Due to the large number of fish species, a species-level approach is prohibitive because the data for parameterisation is not available and because of the computational costs involved. For example, there are approximately 27 species of herbivorous fish in the Caribbean and 91 in the GBR (Bellwood et al. 2004). Using functional groups is also appropriate because the ecological traits of fish species do not always correlate with taxonomic groupings. For example, the balistid (triggerfish) Melichthys niger is herbivorous whereas the majority of other fish in the family Balistidae are invertebrate feeders (Froese and Pauly 2007). Biomass is used as a broad-scale indicator for all functional groups in the fish and urchin models, because it is a common currency for all fish groups and for sea urchins. The non-spatiality of the models gives rise to the same disadvantages as mentioned for the benthic models (section 4.1). Thus, as with the benthic models, these fish and urchin models should be applied at the ‘local’ scale, on the order of tens of metres to a few kilometres. The functional groups modelled are those identified as potentially of significance in reef degradation, particularly in phase shifts between coral- and algaldominance. These groups are herbivorous fish and sea urchins, the main grazers on coral reefs, and piscivorous fish, which are predators on herbivorous fish. Two fish and urchin models of increasing complexity are constructed in this chapter. Both of these models assume that benthic dynamics are fixed.
6.2 The Herbivorous Fish-Piscivorous Fish-Urchins Model (HPUm) The Herbivorous Fish-Piscivorous Fish-Urchins model (HPUm) is the simplest fish and urchin model. It includes herbivorous fish, sea urchins and one piscivorous fish group. In the HPUm, ‘fish’ refer to reef-associated fish. For each of the three groups, the HPUm models the key ecological processes that determine the biomass dynamics – 104
feeding and subsequent growth, mortality and recruitment. In the model, herbivorous fish and sea urchins both feed on algae (turf and macroalgae) and compete with each other in doing so. Also, piscivorous fish predate upon herbivorous fish. For the two fish groups, mortality occurs due to fishing and natural processes excluding predation, such as senescence. Mortality of both fish groups also occur due to predation by piscivorous fish. For sea urchins, mortality occurs due to predation by invertivorous fish (not modelled explicitly and assumed to be of constant biomass) and other natural processes. Benthic dynamics are not modelled in the HPUm, such that the algal food resource is assumed to be of fixed size and there is no feedback from the herbivores to the benthos. This constraint is removed in the integrated models (Chapter 7).
6.2.1 Details of Functional Groups Modelled in the HPUm
There is much evidence that herbivory can be an important control of algal abundance. For example, at regional ( ≥ 50 km) and local ( ≈ 50 m) scales on the GBR, Sargassum abundance was found to depend strongly on the level of herbivory (McCook 1999). Miller et al. (1999) found that both turf and macroalgal covers significantly responded to decreases in herbivorous grazing, using caging experiments on an offshore reef in the Florida Keys. The increase in algal cover in the Caribbean following the 1983-4 mass mortality of the urchin D. antillarum, together with experimental studies in the 1970s and 1980s, provides evidence that grazing by this urchin is a key factor in controlling algal cover (Hughes 1994, Hughes et al. 1999). Herbivorous fish and sea urchins are the main herbivores on coral reefs. For example, Hay (1984) found that on fished reefs in Haiti and the U.S. Virgin Islands (USVI), urchins were the dominant grazers whereas on reefs with less fishing pressure, herbivorous fish were the dominant grazers. The major herbivorous coral reef fish families are the Acanthuridae (surgeonfishes), Pomacentridae (damselfishes), Scaridae (parrotfishes) and Siganidae (rabbitfishes) (Choat 1991). Sea urchins can be major grazers on reefs in both the western Atlantic and the Indo-Pacific. For example, D. antillarum can significantly affect the abundance, diversity and productivity of algae in the western Atlantic (Lessios 1988) and the boring sea urchin Echinometra mathaei is important on Kenyan reefs, where it lowers algal biomass and benthic diversity (McClanahan and Kurtis 1991, Carpenter 1997). Given the potential importance of herbivory in controlling algal abundance, herbivorous fish and herbivorous sea urchins 105
are modelled in the HPUm as two functional groups; these groups feed predominantly on algae. Piscivorous fish are common on coral reefs. For example, Goldman and Talbot (1976) found that piscivores comprised 54% of total fish biomass on One Tree Island reef in the GBR. Also, on Kingman and Palmyra atolls in the northern Line Islands in the Pacific, the biomass pyramid is inverted with the biomass of top predators dominating – at Kingman, top predators account for 85% of total fish biomass (Sandin et al. 2008). There is evidence that predation is an important process structuring reef fish communities. For example, the synodontid (lizardfish) Synodus englemani was found to be responsible for 65% of the mortality rate of planktivorous fish (Jones 1991). Hixon and Beets (1989) found a significant negative correlation between piscivorous fish density and their prey density on artificial reefs at St. Thomas in the USVI, over the course of 30 months. Similarly, at One Tree Island reef in the GBR, Thresher (1983) found a significant negative relationship between the serranid (grouper) Plectropomus leopardus and its planktivorous fish prey, the damselfish Acanthochromis polyacanthus. Thus, the piscivorous fish functional group is modelled because of their commonality in coral reef ecosystems and the importance of predation by these fish. The piscivorous fish group in the HPUm consists of those fish that predominantly predate fish groups included in this model, that is, herbivorous fish or other piscivorous fish. Invertivorous fish such as haemulids (grunts) and the majority of labrids (wrasses) are not modelled explicitly, but are assumed to have a constant biomass. This keeps the model complexity to a level that aids in the mathematical analyses and the parameterisation process. The assumption of a constant invertivorous fish biomass means that urchins suffer a constant predation mortality rate. There is evidence that the density of triggerfishes is dependent upon the density of their urchin prey, E. mathaei. McClanahan and Shafir (1990) examined Kenyan reef lagoons protected and unprotected from fishing, and found that the protected sites had a significantly higher density of triggerfishes, a more than 100 times lower sea urchin population density and a four times higher predation rate on E. mathaei. Furthermore, they found a significant negative correlation between triggerfish and sea urchin densities, and a significant positive correlation between triggerfish density and predation intensity on E. mathaei. Thus, for Kenyan reefs, a density-dependent mortality rate for urchins may be more appropriate. However, predation on D. antillarum may be low due to the spines on this urchin being able to reduce predation effectively and the rarity of D. antillarum predators in some locations (Carpenter 1997). In this case, the predation mortality rate 106
for urchins is largely independent of the density of urchins. This gives partial justification for a density-independent mortality rate for urchins.
6.2.2 Details of Interactions Modelled in the HPUm
The majority of reef fish, such as parrotfishes and groupers, reproduce by spawning, which involves releasing gametes into the water column (Leis 1991, Sale 1991). Once the larvae are formed from the gametes, they can be dispersed by currents in a pelagic stage. Cowen et al. (2006) used an individual-based model of larval dispersal with active larval behaviour and showed that for several reef fish species in the Caribbean, larval dispersal is typically on the order of 10-100 km. By splitting contiguous reef habitat into 9 km by 50 km sections, Cowen et al. also found that, on average, 21% of all recruits are due to self-recruitment. Even at smaller scales, self-recruitment can be significant. For example, Almany et al. (2007) found that at a 0.3 km2 coral reef in a small marine protected area (MPA) at Kimbe Island, Papua New Guinea, about 60% of settled juvenile fish originated from locally produced larvae. Thus, for the spatial scale of the HPUm, there is evidence that fish recruits can be of exogenous or endogenous origin. Therefore, for both herbivorous fish and piscivorous fish in the HPUm, both exogenous and endogenous recruitment are modelled. Only recruits that are 1 yr old or older are modelled, because younger fish are assumed to be too small on average to make a significant contribution to fish biomass. The process of growth is modelled differently for herbivorous fish and piscivorous fish. Herbivorous fish feed on turf and macroalgae and convert these into energy for growth. In the HPUm, the available algal cover grazed by herbivorous fish is determined by the realised grazing pressure exerted by herbivorous fish, θ H (1 − λU θ U ) ; the growth rate of herbivorous fish is then determined by factors that convert the amount of algal cover grazed into the energy used for growth. Here, θ H is the grazing pressure exerted by herbivorous fish, which is decreased by the factor 1 − λU θ U due to exploitative competition with sea urchins (Hay and Taylor 1985, Carpenter 1986, Carpenter 1988, Carpenter 1990). θU is the grazing pressure exerted by sea urchins and
λU , a fixed positive constant, measures the competitiveness of these urchins relative to herbivorous fish. θ H and θU are related to the grazer biomasses by: 107
θH =
H , iH + H
(6.1a)
θU =
U , iU + U
(6.1b)
where H and U are the biomasses per unit area of herbivorous fish and sea urchins respectively, and iH and iU are inversely related to the ‘accessibility’ of algae (turf and macroalgae) to herbivorous fish and sea urchins respectively. Note that θ H is an increasing function of fish biomass H and θ H → 1 as H → ∞ , and similarly for θU . The inaccessibility of the algal food resource to herbivorous fish and urchins, as measured by iH and iU respectively, depends on foraging abilities and the topographic complexity, or rugosity, of the reef. The higher the rugosity, the more individuals per unit planar area is needed to achieve a given grazing pressure, due to the greater benthic surface area per unit planar area. Thus, higher rugosity is reflected in higher inaccessibility parameters. Using values of iH from the empirically derived generic parameter range, which is 0.001-0.5 kg m-2 (see Chapter 10), Figure 6.1a shows how θ H changes as H increases from 0 to 0.282 kg m-2, which is the highest herbivorous fish biomass found from the literature (Williams and Hatcher 1983, Alino et al. 1993, McClanahan et al. 1996, Arias-Gonzalez et al. 1997, Arias-Gonzalez 1998, Letourner et al. 1998, Van Rooij et al. 1998, McClanahan et al. 1999, Williams et al. 2001, Friedlander and DeMartini 2002, Gribble 2003, Kramer 2003, MBRSP 2005, SAGIP Lingayen Gulf Project 2005, AriasGonzalez and Morand 2006, Mumby and Dytham 2006, Newman et al. 2006, Craig et al. 2008, Sandin et al. 2008, Tsehaye and Nagelkerke 2008; piscivorous fish biomasses, used below to derive the maximum piscivorous fish biomass, are also taken from these sources). Figure 6.1a shows that for small values of iH in the parameter range, θ H approaches its limit 1 quickly and in a non-linear fashion, and as iH increases, θ H increases at a slower and increasingly linear rate. Thus, depending on the value of iH ,
θ H can be very non-linear to nearly linear across the empirical range of H and its maximum value, found at the maximum H from the literature, varies from near the limit 1 to ≈ 0.4 . Figure 6.1b shows θU as U increases from zero to its maximum value found 108
from the literature, namely 0.390 kg m-2 at St. John in the USVI, before the mass mortality of D. antillarum in the western Atlantic (Levitan 1988). This corresponds to 14.4 urchins m-2, which is the highest density reported from the review by Lessios (1988) and the recent survey by Kramer (2003). Different values of iU from the empirically derived generic parameter range, 0.002-4 kg m-2 (Chapter 11), are used in Figure 6.1b.
(a)
(b)
θH
θU
H
U
Figure 6.1. Graphs showing how (a) θ H = H (i H + H ) changes with H for different values of iH in the empirically derived generic parameter range 0.001-0.5 kg m-2 and (b)
θU = U (iU + U ) changes with U for different values of iU in the empirically derived generic parameter range 0.002-4 kg m-2. For (a), H varies from zero to its highest value as found from the literature, 0.282 kg m-2. Also, the values for iH are, from top to bottom: 0.001 kg m-2, 0.005 kg m-2, 0.01 kg m-2, 0.05 kg m-2, 0.1 kg m-2, 0.25 kg m-2 and 0.5 kg m-2. For (b), U varies from zero to its highest value as found from the literature, 0.390 kg m-2. Also, the values for iU are, from top to bottom: 0.002 kg m-2, 0.01 kg m-2, 0.05 kg m-2, 0.1 kg m-2, 0.25 kg m-2, 0.5 kg m-2, 1 kg m-2 and 4 kg m-2.
Piscivorous fish grow by eating herbivorous fish or smaller piscivorous fish. In the HPUm, it is assumed that the growth rate of piscivorous fish is a function of the prey biomass and follows a Holling Type-III response with the exponent 2:
109
X2 Realised predator growth rate = rX g X 2 2 i PX + X
,
(6.2)
where rX is the proportion of the prey biomass eaten that is used for growth, g X is the maximum predation rate possible on the prey, X is the prey biomass per unit area and
i PX is a parameter which is inversely related to the ‘accessibility’ of prey to the predator. i PX is dependent upon the ability of the predator to catch prey and the ability of the prey to escape predation. These two abilities are in turn dependent on a number of factors. For example, several studies have shown that an optimal prey size for predatory fish is a length or width (depending on the orientation at which the prey is eaten) that is 60% of the width of the predator’s mouth (Gill 2003) – thus, the prey and predator sizes affect the ability of a predator to catch its prey. The Holling Type-III response shown in eqn. 6.2 means that at low and high prey biomasses, the rate of increase of the realized growth rate is slow. At low prey biomasses, this reflects the difficulty of prey capture due to low encounter rates of predators with prey, which offers a biomass refuge for prey to recover. At high prey biomasses, the slow rate of increase of the growth rate reflects prey saturation, such that predation, and therefore growth, is no longer limited by the availability of prey. Figure
(
)
2 6.2 shows the function X 2 i PX + X 2 as X increases from zero to 0.03 kg m-2, which is
well below the maximum biomass of either herbivorous or piscivorous fish found from the literature (maximum piscivorous fish biomass is 0.433 kg m-2), and for different values of i PX from the empirically derived generic parameter range 0.007-0.01 kg m-2 (see Chapter 10). As i PX increases, the sigmoidal shape becomes slightly more prominent, although overall, the variation is not large. Both fish groups in the HPUm are subject to natural mortality, which is mortality due to starvation, normal levels of disease or senescence; this corresponds to the ‘physiological’ mortality of Pauly (1980). In this thesis, natural mortality excludes predation mortality (modelled separately) and is thus used in a different sense from studies such as Pauly (1980) and studies using ECOPATH and/or ECOSIM. Since overfishing has been implicated in coral reef phase shifts (Chapter 1, section 1.3.1), fishing on both fish groups in the HPUm is included. Fishing decreases the fish biomass of each group through time and the amount of fish caught is dependent on fishing effort, the standing stock fish biomasses and the accessibility of each type of fish to the fishermen (Kramer 2007). Fishing effort is a function of the number of 110
X2 2 i PX +X2
X
(
)
2 Figure 6.2. Graph of the function X 2 i PX + X 2 in eqn. 6.2 as X increases from zero to
0.03 kg m-2. Values of i PX from the empirically derived generic parameter range 0.0070.01 kg m-2 are used. The values for i PX are, from top to bottom: 0.007 kg m-2, 0.008 kg m-2, 0.009 kg m-2 and 0.01 kg m-2.
fishing hours and the technology (type of fishing gear) used, whereas the accessibility is determined by the technology used and the structural complexity of the coral reef habitat. Herbivorous sea urchins, like most fish, reproduce by spawning and their larvae are capable of dispersing in the currents. For example, D. antillarum has a pelagic larval duration (PLD) of 28-60 d, which is very long (Lessios 1988, Hunte and Younglao 1988). Thus, urchin recruits in a particular reef area may have originated from other reef areas, i.e. exogenous sources. However, endogenous recruitment may also be important. For example, in Barbados, the density of D. antillarum recruits was found to be positively correlated with the density of the local adult spawning population, with urchin larvae being retained close to the same island by an eddy system (Hunte and Younglao 1988). Therefore, both exogenous and endogenous recruitment is modelled for the sea urchins functional group. Endogenous recruitment is assumed to be at a constant rate, with no density-dependence. In particular, possible Allee effects which may arise, for example, when the fitness of a population increases with increasing population size (Gascoigne 2004), are not included. Though an enhanced fertilisation effect with increasing population densities is possible for urchins, this may be offset by urchins growing to a larger size at lower densities and hence producing more gametes, which is the case for D. antillarum (Gascoigne 2004). As with fish, urchin recruits in 111
the HPUm are assumed to be 1 yr old or older, because it is assumed that younger urchins do not make a significant contribution to urchin biomass on average. As for hard corals in the benthic models (see Chapter 4, section 4.2.3), exogenous fish or urchin recruitment can be conceptualised as recruitment from exogenous larval pools (see Sale (1991) for reef fish), one for each functional group. Fish and urchins from other reef areas not modelled contribute larvae to and recruit from these exogenous pools. As for hard corals, each of these larval pools is assumed to be large enough for the contributions to and from the modelled reef area to be negligible, such that the exogenous recruitment rates for the modelled reef is independent of local fish and urchin biomasses. In addition, the sizes of the larval pools are assumed to be constant for each simulation, such that explicit larval pool dynamics are not modelled. This is primarily because of a lack of relevant data to model these dynamics and to facilitate model analysis. Sea urchins, like herbivorous fish, grow due to energy gained by feeding on turf and macroalgae. In the HPUm, the available algal cover grazed is determined by the realised grazing pressure exerted by sea urchins, θU (1 − λ H θ H ) , which includes exploitative competition from herbivorous fish through the fixed positive parameter λ H . From Chapter 7, λ H and λU must add up to 1. Mortality of urchins is represented as a constant rate, and includes predation mortality by invertivorous fish (see section 6.2.1). Fishing mortality for the urchins group is not modelled because there is generally no significant fishing on urchins living in coral reef environments (Andrew et al. 2002).
6.2.3 Equations for the HPUm
The HPUm is defined by three independent differential equations, one for the biomass of each functional group modelled. Below each term, there is a short description indicating its ecological interpretation. In these descriptions, “H” indicates “herbivorous” and “P” indicates “piscivorous”.
112
H U dH ( g M Mµ M + g T Tµ T + ζ T Sµ S ) 1 − λU = l Hex + l Hen H + 1424 3 i H + H dt iU + U H fish 14444444444 4244444444444 3 recruitment
−
H fish feeding and growth
dH H 1 2 3 H fish mortality by natural processes excluding predation
H2 H P − ρ H f , − g P 2 2 + i H i + H 2 FH PH 4 443 14 2443 144 H fish mortality by predation
(6.3)
H fish mortality by fishing
H2 dP ex en P − = l P + l P P + rP g P 2 dPP 2 { 1424 3 dt i + H PH P fish mortality 14442444 3 P fish recruitment
P fish predation of H fish and growth
by natural processes excluding predation
P2 P P − ρ P f , − ψ P g P (1 − rP ) 2 2 i FP + P i PP + P 3 14444244443 144244 P fish predation of P fish and growth
(6.4)
P fish mortality by fishing
U H dU 1 − λH ( g M Mµ M + g T TµT + ζ T Sµ S ) , = lUex + lUenU + κU 1424 3 dt i + U i + H U H 44444444444 44444444442 Urchin 14 3 recruitment
Urchin feeding and growth
− dU U {
(6.5)
Urchin mortality
where H, P, and U are the biomasses (per unit area) of herbivorous fish, piscivorous fish and sea urchins respectively. l Xex is the exogenous recruitment rate (per unit area) of 1 yr old juveniles of biomass type X ∈ {H , P,U }. l Xen is the endogenous recruitment rate, which arises from reproduction by the local population and subsequent recruitment of 1 yr old juveniles to the reef area being modelled.
g M , g T and ζ T have the same meaning as in the benthic models and represent the maximum grazing rates on macroalgae and turf algae, and the growth rate of turf algae respectively (see Chapter 5, Table 5.1). M, T and S are the proportional covers of macroalgae, turf algae and space respectively, as in the CTMm (see Chapter 5, section 5.2.1). µ M , µ T and µ S are the biomasses of herbivorous fish accumulated through growth by grazing on 100% cover of macroalgae, turf algae and space respectively.
θ H (1 − λU θ U )g M M , θ H (1 − λU θ U )g T T , and θ H (1 − λU θ U )ζ T S (where θ H and θU are 113
as specified by eqns. 6.1a,b) are the rates of grazing by herbivorous fish on proportional covers M of macroalgae, T of turf algae and S of space. Thus,
θ H (1 − λU θ U )( g M Mµ M + g T Tµ T + ζ T Sµ S ) is the total growth of herbivorous fish biomass due to grazing.
d H and d P are the natural mortality rates for herbivorous fish and piscivorous fish respectively, which exclude mortality by predation and fishing. g P is the maximum predation rate of piscivorous fish on herbivorous fish, whereas ψ P is the maximum predation rate of piscivorous fish on other piscivorous fish relative to that on herbivorous fish, such that ψ P g P is the maximum predation rate of piscivorous fish on other piscivorous fish. i PH and i PP measure the inaccessibility of herbivorous fish and piscivorous fish to predation by piscivorous fish respectively. rP is the proportion of prey consumed by piscivorous fish that is used for growth; rP ≤ 1 . f is the maximum total fish biomass catch rate, for a given fishing effort and using a given technology. i FH and i FP measure the inaccessibility of herbivorous fish and piscivorous fish to fishermen respectively. ρ H and ρ P are the proportions of f directed to herbivorous fish and piscivorous fish respectively, so that ρ H f and ρ P f are the maximum herbivorous fish biomass and maximum piscivorous fish biomass catch rate respectively. These satisfy
ρH + ρP = 1.
(6.6)
κ U is the biomass accumulated through growth by urchins grazing on 100% cover of macroalgae, turf or space, relative to that for herbivorous fish. Thus, it is assumed as a simplification that the growth accumulation parameters for urchins are proportional to those for herbivorous fish, namely κ U µ M , κ U µ T and κ U µ S . Finally, d U is the mortality rate for sea urchins, which includes any (constant) predation mortality by invertivores (see section 6.2.2). Figure 6.3 shows a schematic representation of the functional groups in the HPUm and the interactions modelled. Table 6.1 at the end of this chapter summarises the parameters used for this model.
114
H, M
F
Herbivorous Fish
Piscivorous Fish
F, R
H, M
R R
C
R
M
Sea Urchins F, R R
Figure 6.3. Schematic diagram showing the functional groups and interactions in the HPUm. Each solid arrow represents conversion of the biomass of one group to another due to the process(es) next to the arrow, or the biomass lost to or gained from sources not modelled explicitly (for arrows that do not point to or originate from a box). The dashed line represents competition for algae between herbivorous fish and sea urchins. C = competition for algae, F = feeding and subsequent growth, H = fishing, M = mortality due to processes other than predation and fishing for fish groups and due to all processes for urchins, R = recruitment.
6.2.4 Mathematical Analysis of the HPUm
Firstly, the dynamics need to be constrained to the biological domain H , P, U ≥ 0 . From eqns. 6.3-6.5, for X ∈ {H , P,U }, if X = 0 , then dX dt ≥ 0 with equality only if the exogenous recruitment rate l Xex = 0 . Thus, H , P, U ≥ 0 . Secondly, the biomasses should be finite, such that there is a need to ensure that they do not tend to infinity with time t. From eqn. 6.3, if l Hen < d H ,
(6.7)
115
then dH dt < 0 for H sufficiently large, such that H remains finite. From eqn. 6.4, if
l Pen + rP g P < d P ,
(6.8)
then P remains finite because dP dt < 0 for sufficiently large P, regardless of what H is. From eqn. 6.5, if lUen < d U ,
(6.9)
then dU dt < 0 for U sufficiently large, such that U remains finite. The ecological meaning of conditions 6.7-6.9 is that if the mortality rates of the three functional groups are large enough, then their biomasses cannot tend to infinity, because a positive feedback mechanism that leads to indefinitely increasing biomass is avoided. Conditions 6.7-6.9 are sufficient but not necessary, that is, they ensure that all the biomasses are finite, but if they do not hold, then the biomasses may still remain finite. For each X ∈ {H , P,U }, if there is no endogenous or exogenous recruitment, that is l Xen , l Xex = 0 , then X must tend to zero even if there is no fishing, because all organisms of biomass type X must eventually die due to senescence and other processes. From eqns. 6.3-6.5, this gives rise to three further conditions:
CB < dH , iH
(6.10)
rP g P < d P ,
(6.11)
κU CB iU
< dU ,
(6.12)
where C B = g M Mµ M + g T Tµ T + ζ T Sµ S (a constant). Condition 6.10 is derived from eqn. 6.3 by first substituting in l Hen , l Hex = 0 . The resulting expression for dH dt must be < 0 for all H > 0 . This expression is maximised when there is no predation or fishing
( P = f = 0 ) and when urchins have no negative effect on herbivorous fish grazing ( λU = 0 ), and the maximised expression is 116
dH dt = {H (i H + H )}C B − d H H = H [{C B (i H + H )} − d H ] . For this to be < 0 for all H > 0 , condition 6.10 must hold. Conditions 6.11 and 6.12 are derived in a similar way.
Note that if condition 6.8 holds, then so does condition 6.11. If conditions 6.7-6.9 hold, then dynamics for the HPUm remain bounded, and it is shown in section A6.1.1 in Appendix A6 that a compact set exists which is intersected by the forward trajectory of any initial point (H 0 , P0 ,U 0 ) ≥ (0, 0, 0) . Section A6.1.1 then shows how this implies that there is at least one equilibrium in the biological domain. For the HPUm, the stability of an equilibrium typically has to be determined numerically, because the Routh-Hurwitz conditions for the threedimensional HPUm (Murray 2002) are typically not analytically tractable.
6.3 The Herbivorous Fish-Small to Intermediate Piscivorous FishLarge Piscivorous Fish-Urchins Model (HSLUm) The Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous FishUrchins model (HSLUm) adds complexity to the HPUm by splitting the piscivorous fish group into two different size classes – small to intermediate piscivorous (SIP) fish and large piscivorous (LP) fish. The SIP fish include those species that do not grow to a large size, such as species from the family Scorpaenidae (scorpionfishes), and the smaller size classes from those species that do eventually grow large, such as large groupers (Froese and Pauly 2007). One reason for splitting piscivorous fish into two size classes is that these classes can have different life history characteristics. Larger piscivorous fish can be more fecund; for example, a female red lutjanid or snapper (Lutjanus campechanus) that is 61 cm long and has a biomass of 12.5 kg produces as many eggs as 212 females 42 cm long and with a total biomass of 233 kg (Birkeland 1997c). This may lead to a greater recruitment rate for LP fish, although other factors such as the fertilisation rate need to be factored in. In addition, piscivorous fish species that can attain a large size tend to have a predominantly K-selected life history strategy that includes a slow growth rate; such species include sharks and large groupers and snappers (Stevens et al. 2000, Scales et al. 2007). K-selection refers to the density-dependent part of natural selection which includes a focus on competitive ability for limiting resources (Pianka 1972). Furthermore, the (non-fishing) predation rate on LP fish, especially the apex predators, is lower than for the smaller piscivorous fish. However, fishing effort can be 117
directed disproportionately towards LP fish. For example, the live fish trade targets large reef fish (Bryant et al. 1998, Scales et al. 2007). From FishBase data (Froese and Pauly 2007), excluding sharks, piscivorous fish can have an asymptotic total length of up to 203 cm, and including sharks, up to 747 cm. However, from Friedlander and DeMartini (2002), Williamson et al. (2004), Rhodes and Tupper (2007), Frisch et al. (2008), Craig et al. (2008) and Sandin et al. (2008), the maximum total length of piscivorous fish (as identified from FishBase) found from different reefs is 200 cm, and excluding Kingman atoll, which is one of the most pristine reefs in the world, the maximum length is 90 cm. This justifies the assertion by Munro and Williams (1985) that a ‘large’ reef fish is 60-90 cm long and the classification used by Cheung and Sadovy (2004), whereby any reef-associated fish > 60 cm is considered ‘large’. Following these authors, the HSLUm assumes that any
fish with total length ≥ 60 cm is a LP fish and that any piscivorous fish with a lower total length is a SIP fish. This classification is used when finding parameter values for piscivorous fish (see Chapter 11). The herbivorous fish group is not split into two size classes because herbivorous fish tend to be small to intermediate in size. From FishBase data (Froese and Pauly 2007), only 22 species were identified that can grow to > 60 cm in total length, compared to over 90 species for piscivorous fish. Recruitment into the LP fish group occurs when fish in the SIP fish group grow large enough. There is still exogenous and endogenous recruitment into the SIP fish group, as in the HPUm. Both groups of piscivorous fish are able to contribute to endogenous recruitment, but possibly with different recruitment rates. In the HSLUm, it is assumed that the LP fish group does not suffer significant (non-fishing) predation mortality. It is also assumed that SIP fish only predate herbivorous fish, whereas LP fish predate both herbivorous fish and SIP fish. Thus, the HSLUm models the dynamics of the biomasses of herbivorous fish, SIP fish, LP fish and sea urchins by modelling the key processes of feeding and subsequent growth, mortality and recruitment for each group. The processes for herbivorous fish and sea urchins are as for the HPUm, except that herbivorous fish are now predated upon by two piscivorous fish groups. SIP fish experience mortality due to predation by LP fish whereas LP fish do not experience mortality due to predation. Both piscivorous fish groups experience mortality due to fishing and natural processes apart from predation.
118
6.3.1 Equations for the HSLUm
The HSLUm has four independent variables: H and U are as in the HPUm, Ps is the biomass (per unit area) of SIP fish, and Pl is the biomass (per unit area) of LP fish. The dynamics of these variables are determined by the system of four differential equations shown below. The terms in the equations are the same as or similar to those seen in the HPUm (section 6.2.3).
H U dH 1 − λU ( g M Mµ M + g T Tµ T + ζ T Sµ S ) − d H H = l Hex + l Hen H + dt i H + H iU + U
H2 − ( g Ps Ps + g Pl Pl ) 2 2 i PH + H
− ρ H
H , f i FH + H
H2 dPs ex en en = l Ps + l Ps Ps + l Pl Pl + (1 − φ Ps )rPs g Ps 2 2 dt i PH + H Ps − ρ Ps f i FPs + Ps
P2 Ps − d Ps Ps − ψ Pl g Pl 2 s 2 Pl i PlPs + Ps
,
H2 dPl = φ Ps rPs g Ps 2 2 dt i PH + H
Pl − ρ Pl f i FPl + Pl
(6.13)
(6.14)
H 2 Ps + rPl g Pl 2 2 i PH + H
P2 + ψ Pl 2 s 2 i + P s PlPs
Pl − d Pl Pl
,
(6.15)
U H dU 1 − λ H ( g M Mµ M + g T Tµ T + ζ T Sµ S ) = lUex + lUenU + κ U dt i + U i + H H U
− dU U ,
(6.16)
The parameters for herbivorous fish and sea urchins have the same meaning as in the HPUm (section 6.2.3). In particular, the sea urchin equation is unchanged. However, in the HSLUm, the HPUm piscivorous fish parameters are replaced by separate parameters for SIP fish and LP fish. Thus, d Ps and d Pl are the natural mortality rates for SIP fish ex en and LP fish respectively, l Ps is the exogenous recruitment rate for SIP fish, and l Ps and
119
l Plen are the endogenous recruitment rates to the small to intermediate size class of
piscivorous fish due to reproduction by each size class. g Ps and g Pl are the maximum predation rates of SIP fish and LP fish on herbivorous fish respectively. i PH measures the inaccessibility of herbivorous fish prey to predators, for simplicity assumed the same for each of the two predator groups. ψ Pl is the maximum predation rate on SIP fish by LP fish, relative to that on herbivorous fish. Thus, ψ Pl g Pl is the maximum predation rate of LP fish on SIP fish. i PlPs measures the inaccessibility of SIP fish prey to LP fish. rPs and rPl are the proportions of the prey biomass consumed that are used for growth, for SIP fish and LP fish respectively. Due to this growth, some SIP fish make the transition to LP fish, and φ Ps is a constant that determines how quickly these SIP fish make the transition; φ Ps is non-negative and can take values greater than 1 (see Appendix A6, section A6.2.1 for details on how the term with φ Ps is derived by considering predation and growth of SIP fish, and why φ Ps can be greater than 1). As in the HPUm, f represents the maximum rate of extraction of fish biomass by fishing, and ρ H , ρ Ps and ρ Pl are the proportions of f directed to herbivorous fish, SIP fish and LP fish respectively. The ρ parameters must satisfy
ρ H + ρ Ps + ρ Pl = 1 .
(6.17)
i FPs and i FPl measure the inaccessibility of SIP fish and LP fish to fishermen respectively. Figure 6.4 shows a schematic representation of the functional groups in the HSLUm and the interactions modelled. Table 6.1 summarises the parameters for the HSLUm, together with those for the HPUm.
6.3.2 Mathematical Analysis of the HSLUm
As for the HPUm, the biomasses in the HSLUm must be non-negative, that is H , Ps , Pl ,U ≥ 0 , and must remain finite. From eqns. 6.13-6.16, for X ∈ {H , Ps , Pl ,U } , if X = 0 , then dX dt ≥ 0 . Thus, H , Ps , Pl ,U ≥ 0 . From eqn. 6.13, if
120
H, M
Small to Intermediate Piscivorous Fish
F
Herbivorous Fish F, R
H, M
R
C R
R
R
F F, R H, M
M
Large Piscivorous Fish
Sea Urchins F, R R
Figure 6.4. Schematic diagram showing the functional groups and interactions in the HSLUm. Each solid arrow represents conversion of the biomass of one group to another due to the process(es) next to the arrow, or the biomass lost to or gained from sources not modelled explicitly (for arrows that do not point to or originate from a box). The dashed line represents competition for algae between herbivorous fish and sea urchins. C = competition for algae, F = feeding and subsequent growth, H = fishing, M = mortality due to processes other than predation and fishing for fish groups and due to all processes for urchins, R = recruitment.
l Hen < d H ,
(6.18)
then dH dt < 0 for H sufficiently large, such that H remains finite. In addition, from
eqns. 6.14-6.15, if en l Ps + rPs g Ps < d Ps
(6.19)
and l Plen + rPl g Pl < d Pl ,
(6.20) 121
then d (Ps + Pl ) dt < 0 for Ps + Pl sufficiently large, such that Ps and Pl remain finite. Furthermore, from eqn. 6.16, if lUen < d U ,
(6.21)
then dU dt < 0 for U sufficiently large, such that U remains finite. Conditions 6.186.21 are all sufficient but not necessary, and conditions 6.18 and 6.21 are the same as conditions 6.7 and 6.9 for the HPUm. Conditions 6.18-6.21 mean that if the mortality rate for each group is large enough, then a runaway increase in biomass(es) due to recruitment and growth is prevented. For each group, if there is no endogenous or exogenous recruitment, then the biomass must tend to zero, since the population cannot be sustained by feeding and subsequent growth alone. This is because in the absence of recruitment, all organisms in a population must eventually die due to senescence and other processes, even if food is always available and there is no fishing or predation. Thus, for each X ∈ {H , Ps ,U }, if l Xen , l Xex = 0 , then X should tend to zero and if φ Ps = 0 , then Pl should tend to zero. From eqns. 6.13-6.16, this gives rise to four further conditions (calculations are similar to those used to derive conditions 6.10-6.12 for the HPUm – see section 6.2.4):
CB < dH , iH
(1 − φ Ps )rPs g Ps
(6.22)
< d Ps ,
(6.23)
rPl g Pl (1 + ψ Pl ) < d Pl ,
(6.24)
κU CB iU
< dU ,
(6.25)
where the benthic parameter, C B = g M Mµ M + g T Tµ T + ζ T Sµ S , is assumed constant. Conditions 6.22 and 6.25 are the same as conditions 6.10 and 6.12 for the HPUm. Also, if condition 6.19 holds, then so does 6.23. 122
If conditions 6.18-6.21 hold, then dynamics for the HSLUm remain bounded and in Appendix A6, section A6.2.2, it is shown that a compact set exists which is intersected by the forward trajectory of every point (H 0 , Ps 0 , Pl 0 ,U 0 ) ≥ (0, 0, 0, 0) , and how this implies that there is at least one equilibrium in the biological domain. The stability of an equilibrium typically has to be determined numerically, since the RouthHurwitz conditions for the four-dimensional HSLUm (Murray 2002) typically do not lead to analytically tractable conditions.
123
Parameter(s)
Definition
Model Specificity
µ M , µT , µ S
The herbivorous fish biomass accumulated through growth from grazing
HPUm and HSLUm
on 100% cover of macroalgae, turf or space (the turf within space is what is consumed) respectively
dH
The mortality rate of herbivorous fish from all factors other than predation and fishing
iH
A parameter which measures the inaccessibility of algae (turf and macroalgae) to herbivorous fish grazing
i FH
A parameter which measures the inaccessibility of herbivorous fish to fishermen
i PH
A parameter which measures the inaccessibility of herbivorous fish to predation by piscivorous fish
l Hen
The endogenous recruitment rate of herbivorous fish
l Hex
The exogenous recruitment rate of herbivorous fish
λH
Competitiveness of herbivorous fish relative to sea urchins
ρH
The proportion of the total fishing pressure f which acts on herbivorous fish
f dU
The maximum catch rate due to fishing The mortality rate of sea urchins 124
iU
A parameter which measures the inaccessibility of algae (turf and macroalgae) to sea urchin grazing
lUen
The endogenous recruitment rate of sea urchins
lUex
The exogenous recruitment rate of sea urchins
κU
A parameter which measures the biomass accumulated by urchin grazing that contributes to growth, relative to that for herbivorous fish grazing
λU
Competitiveness of sea urchins relative to herbivorous fish
dP
The mortality rate of piscivorous fish from all factors other than predation and fishing
gP
The maximum predation rate of piscivorous fish on herbivorous fish
i FP
A parameter which measures the inaccessibility of piscivorous fish to fishermen
i PP
A parameter which measures the inaccessibility of piscivorous fish to predation by other piscivorous fish
l Pen
The endogenous recruitment rate of piscivorous fish
l Pex
The exogenous recruitment rate of piscivorous fish
rP
The proportion of biomass consumed by piscivorous fish used for growth
ψP
The predation rate on piscivorous fish by other piscivorous fish, relative
125
HPUm only
to that on herbivorous fish
ρP
The proportion of the total fishing pressure f which acts on piscivorous fish
d Ps
The mortality rate of SIP fish from all factors other than predation and fishing
g Ps
The maximum predation rate of SIP fish on herbivorous fish
i FPs
A parameter which measures the inaccessibility of SIP fish to fishermen
i PlPs
A parameter which measures the inaccessibility of SIP fish to predation by LP fish
en l Ps
The endogenous recruitment rate of SIP fish due to reproduction by SIP fish
ex l Ps
The exogenous recruitment rate of SIP fish
rPs
The proportion of biomass consumed by SIP fish used for growth
φ Ps
A constant parameter that determines how quickly SIP fish biomass becomes LP fish biomass due to predation and subsequent growth
ρ Ps
The proportion of the total fishing pressure f which acts on SIP fish
d Pl
The mortality rate of LP fish from all factors other than predation and fishing
126
HSLUm only
g Pl
The maximum predation rate of LP fish on herbivorous fish
i FPl
A parameter which measures the inaccessibility of LP fish to fishermen
l Plen
The endogenous recruitment rate of SIP fish due to reproduction by LP fish
rPl
ψ Pl
The proportion of biomass consumed by LP fish used for growth The predation rate on SIP fish by LP fish, relative to that on herbivorous fish
ρ Pl
The proportion of the total fishing pressure f which acts on LP fish
Table 6.1. Parameters for the HPUm and the HSLUm, with their ecological meanings
127
Appendix A6 A6.1 The Herbivorous Fish-Piscivorous Fish-Urchins Model (HPUm) A6.1.1 Proving the Existence of at Least One Equilibrium for the HPUm
From eqn. 6.3,
(
) (
)
dH ≤ l Hex + C B − d H − l Hen H , dt
(A6.1)
where C B = g M Mµ M + g T Tµ T + ζ T Sµ S is a constant. Integrating A6.1 from 0 to t gives:
en l ex + C B H (t ) ≤ H 0 e −(d H −lH )t + H en d H − lH
{
}
en 1 − e −(d H −lH )t ,
(A6.2)
where H 0 = H (0) . Thus, if condition 6.7 holds, then d H − l Hen > 0 and for any δ > 0 ,
l ex + C B H (t ) ≤ H en d H − lH
+ δ
(A6.3)
for a sufficiently large finite value of t. From eqn. 6.4,
(
)
dP ≤ l Pex − d P − l Pen − rP g P P . dt
(A6.4)
Analogous to the derivation of A6.3, if condition 6.8 holds, then d P − l Pen − rP g P > 0 and for any δ > 0 ,
l Pex en d P − l P − rP g P
P(t ) ≤
+ δ
(A6.5)
for a sufficiently large finite value of t. 128
Lastly, from eqn. 6.5,
(
) (
)
dU ≤ lUex + κ U C B − d U − lUen U . dt
(A6.6)
Analogous to the derivation of A6.3, if condition 6.9 holds, then d U − lUen > 0 and for any δ > 0 ,
lUex + κ U C B U (t ) ≤ en dU − lU
+ δ
(A6.7)
for a sufficiently large finite value of t. Thus, if conditions 6.7-6.9 hold, then for any initial condition
(H 0 , P0 ,U 0 ) ≥ (0, 0, 0) , the forward trajectory will eventually intersect a compact set of the form W = {(H , P,U ) : (0, 0, 0) ≤ (H , P,U ) ≤ (H max , Pmax ,U max )}, where the elements of (H max , Pmax ,U max ) are positive and finite. Furthermore, trajectories that start at
(H 0 , P0 ,U 0 ) < (0, 0, 0) can easily be defined to link smoothly with dynamics across the boundary of the positive quadrant – that is, any trajectory which starts at
(H 0 , P0 ,U 0 ) < (0, 0, 0) can be defined such that it moves continuously across to the positive quadrant and then joins up continuously with a trajectory in the positive quadrant. If this is done, then for any initial condition in R 3 , the forward trajectory will eventually enter W. Using the Richeson-Wiseman Theorem (Appendix A4, section A4.1), this implies that there is an equilibrium in W and hence the biological domain.
A6.2 The Herbivorous Fish-Small to Intermediate Piscivorous FishLarge Piscivorous Fish-Urchins Model (HSLUm) A6.2.1 Derivation of the φ Ps Term
Firstly, the maximum predation rate of small to intermediate piscivorous (SIP) fish on herbivorous fish, g Ps , can be written as λg m , where λ is the expected herbivorous fish biomass consumed every time a SIP fish feeds, g is the rate at which a SIP fish feeds and m is the average weight of a SIP fish. Now, if there are N SIP fish, such that 129
Ps = mN , then in a small time interval δt , the expected number of SIP fish that feed is:
H2 N . gδt 2 2 + i H PH
(A6.8)
Each SIP fish that feeds is expected to consume herbivorous fish biomass λ , a proportion rPs of which is used for somatic growth. Thus, the increase in Ps due to predation and subsequent growth is:
H2 H 2 Ps H2 P . rPs λgδt 2 N = r λ g δ t = r g δ t Ps Ps Ps 2 2 2 s i2 + H 2 m i PH + H PH i PH + H
(A6.9)
The biomass of the expected number of SIP fish that feed is:
H2 H2 gδt 2 N × m = gδt 2 2 2 + i H PH i PH + H
H2 m Ps = g Ps δt 2 P . 2 s λ + i H PH
(A6.10)
The summation of the expressions in A6.9 and A6.10 gives the biomass of SIP fish that feed in the time interval δt after feeding, and is equal to:
m 1 + λrPs
H2 rPs g Ps δt 2 2 i PH + H
Ps .
(A6.11)
Let τ Ps be the rate at which SIP biomass becomes large piscivorous (LP) fish biomass due to predation and subsequent growth. Then:
m τ Ps δtPs = φˆPs 1 + λrPs
H2 rPs g Ps δt 2 2 i PH + H
Ps ,
(A6.12)
where φˆPs is the proportion of feeding SIP fish biomass that becomes LP fish biomass. From A6.12,
130
H2 , 2 2 i PH + H
τ Ps = φ Ps rPs g Ps
(A6.13)
with
φ Ps = φˆPs 1 +
m λrPs
.
(A6.14)
Using A6.13, τ Ps Ps is the φ Ps term in eqns. 6.14 and 6.15 for the HSLUm. It is clearly seen from A6.12 that φ Ps determines the rate at which feeding SIP fish become LP fish. In addition, from A6.14, φ Ps can be larger than 1.
A6.2.2 Proving the Existence of at Least One Equilibrium for the HSLUm
From eqn. 6.13,
(
) (
)
dH ≤ l Hex + C B − d H − l Hen H , dt
(A6.15)
and as for the HPUm, if condition 6.18 holds, then d H − l Hen > 0 and for any δ > 0 ,
l ex + C B H (t ) ≤ H en d H − lH
+ δ
(A6.16)
for a sufficiently large finite value of t. From eqns. 6.14 and 6.15: d (Ps + Pl ) ex en ≤ l Ps − d Ps − l Ps − rPs g Ps Ps − d Pl − l Plen − rPl g Pl Pl . dt
(
)
(
)
(A6.17)
For Ps + Pl ≠ 0 , by letting P = Ps + Pl and X = Pl (Ps + Pl ) , A6.17 can be written as
131
{(
)
(
) }
dP ex en ≤ l Ps − d Ps − l Ps − rPs g Ps (1 − X ) + d Pl − l Plen − rPl g Pl X P . dt
(A6.18)
en Since 0 ≤ X ≤ 1 and (d Ps − l Ps − rPs g Ps )(1 − X ) + (d Pl − l Plen − rPl g Pl )X is linear in X,
(d
Ps
)
(
)
en − l Ps − rPs g Ps (1 − X ) + d Pl − l Plen − rPl g Pl X is minimised at either 0 or 1, i.e. the
en minimum value is min{d Ps − l Ps − rPs g Ps , d Pl − l Plen − rPl g Pl }. Thus,
{
}
dP ex en ≤ l Ps − min d Ps − l Ps − rPs g Ps , d Pl − l Plen − rPl g Pl P . dt
(A6.19)
Analogous to the derivation of A6.3, if conditions 6.19 and 6.20 hold, then
{
}
en min d Ps − l Ps − rPs g Ps , d Pl − l Plen − rPl g Pl > 0 and for any δ > 0 ,
ex l Ps + δ P(t ) ≤ en en min{d Ps − l Ps − rPs g Ps , d Pl − l Pl − rPl g Pl }
(A6.20)
for a sufficiently large finite value of t. For Ps + Pl = P = 0 , eqns. 6.14 and 6.15 show that dP dt ≥ 0 , such that P either stays at zero or becomes positive. If P stays at zero, then the trajectory is obviously bounded and if P becomes positive, then A6.20 shows that the trajectory must be bounded. Lastly, from eqn. 6.16,
(
) (
)
dU ≤ lUex + κ U C B − d U − lUen U , dt
(A6.21)
and as for the HPUm, if condition 6.21 holds, then d U − lUen > 0 and for any δ > 0 ,
l ex + κ U C B U (t ) ≤ U en dU − lU
+ δ
(A6.22)
for a sufficiently large finite value of t. Thus, if conditions 6.18-6.21 hold, then for any initial condition
(H 0 , Ps 0 , Pl 0 ,U 0 ) ≥ (0, 0, 0, 0) , the forward trajectory will eventually intersect a compact 132
set of the form W = {(H , Ps , Pl ,U ) : (0, 0, 0, 0) ≤ (H , Ps , Pl ,U ) ≤ (H max , Ps max , Pl max ,U max )} , where the elements of (H max , Ps max , Pl max ,U max ) are positive and finite. Furthermore, analogous to the analysis for the HPUm (see section A6.1.1), trajectories that start at
(H 0 , Ps 0 , Pl 0 ,U 0 ) < (0, 0, 0, 0) can easily be defined to link smoothly with dynamics across the boundary of the positive quadrant. If this is done, then for any initial condition in R 4 , the forward trajectory will eventually enter W. Using the RichesonWiseman Theorem (Appendix A4, section A4.1), this implies that there is an equilibrium in W and hence the biological domain.
133
Chapter 7. Integrated Models Chapters 4 and 5 detailed the three benthic models which included benthic dynamics but assumed fixed fish and urchin dynamics. In contrast, Chapter 6 detailed the two fish and urchin models which included fish and urchin dynamics but assumed fixed benthic dynamics. To construct models where both the benthic component and the fish and urchin component vary dynamically, it is necessary to link a benthic model with a fish and urchin model. This is done by combining the differential equations for both models and writing total grazing pressure θ explicitly as a function of herbivorous fish and sea urchin biomasses. θ is the sum of the realised grazing pressures exerted by herbivorous fish and sea urchins (see Chapter 6, section 6.2.2.):
θ = θ H (1 − λU θU ) + θ U (1 − λ H θ H ) ,
(7.1)
where θ H and θU are functions of the grazer biomasses as given by eqns. 6.1a,b in Chapter 6. Thus eqn. 7.1 gives θ as an explicit function of grazer biomasses. The constraint
λ H + λU = 1
(7.2)
is imposed such that θ lies in the range 0 ≤ θ < 1 , θ is an increasing function of both
θ H and θU (increasing biomass of grazers always leads to increased total grazing pressure), and θ → 1 as either θ H → 1 or θ U → 1 . Thus, as either the herbivorous fish biomass or sea urchin biomass increases to infinity, the total grazing pressure tends towards its maximum limit of 1. Clearly, eqns. 7.1 and 7.2 imply that
θ = θ H + θU − θ H θ U , which is independent of λ H and λU . However, these competition parameters serve to specify the explicit decomposition of θ in eqn. 7.1 into separate components due to herbivorous fish and urchins. It is clearly seen that if both types of grazer are present, θ is less than the sum of the grazing pressures of fish and urchins grazing on their own, reflecting competition for algal food. Models formed by the combination of a benthic model with a fish and urchin model are referred to as integrated models. In these integrated models, the benthic covers and the fish and urchin biomasses are treated as variables. Since there are three 134
different benthic models and two different fish and urchin models, there are a total of six different integrated models (Table 7.1). This chapter first mathematically analyzes these integrated models generally and then analyzes a specific integrated model in greater detail (section 7.1).
Integrated Model
Benthic Model
Fish and Urchin Model
CT-HPUm
CTm
HPUm
CT-HSLUm
CTm
HSLUm
CA-HPUm
CAm
HPUm
CA-HSLUm
CAm
HSLUm
CTM-HPUm
CTMm
HPUm
CTM-HSLUm
CTMm
HSLUm
Table 7.1. Table showing the six integrated models. For each integrated model, the benthic model and fish and urchin model from which it was created are given in the second and third columns. C, T, A, M, H, P, U, S, L and m stand for Coral, Turf, Algae (Turf and Macroalgae), Macroalgae, Herbivorous fish, Piscivorous fish, Urchins, Small to intermediate piscivorous fish, Large piscivorous fish and model respectively. See Chapters 4, 5 and 6 for more details of these constituent benthic models and fish and urchin models.
7.1 Mathematical Analysis of Integrated Models 7.1.1 General Analysis for All Integrated Models
For any of the six integrated models in Table 7.1, it is required that if there is no exogenous or endogenous recruitment, then the fish and urchin biomasses must tend to zero (see Chapter 6, sections 6.2.4 and 6.3.2). The conditions that ensure this is the case are as for the fish and urchin models, except that C B , which is a function of the benthic proportional covers, is replaced by its theoretical maximum value C B max . This is necessary because the proportional covers, and hence C B , are now variables . The equilibrium analysis for an integrated model proceeds by considering the equations for the benthic functional groups and those for the fish and urchin groups 135
separately. Using eqn. 7.1 together with eqns. 6.1a,b, θ is:
H
U
U
H
1 − λU + 1 − λ H . θ = i H + H i H + H iU + U iU + U
(7.3)
Also, the explicit form of C B , as given in Chapter 6, is: C B = g M Mµ M + g T Tµ T + ζ T Sµ S .
(7.4)
If the CTm is used, then M = 0 and if the CAm is used, then T and M are defined in terms of the total algal cover A. Since the proportion of A that is macroalgal cover is fixed at ν M , where 0 < ν M < 1 (see Chapter 5, section 5.1.3), then for the CAm,
M = ν M A . Also, since A = T + M , T = (1 − ν M ) A . First, consider just the equations for the benthic groups at equilibrium (i.e., with derivatives set to zero). For any fixed θ in 0 ≤ θ < 1 , the same analysis as for the corresponding benthic model (sections 4.2.4, 5.1.4 and 5.2.2 in Chapters 4 and 5) shows that there is at least one solution (C , T , M , S ) in the biological domain and hence at least one value of C B . This gives at least one curve C B (θ ) in the (θ , C B ) plane. Similarly, for the fish and urchin equations at equilibrium and any fixed C B in 0 ≤ C B ≤ C B max , the same analysis as for the corresponding fish and urchin model (sections 6.2.4 and 6.3.2) shows that there is at least one solution in the biological domain and hence at least one value of θ (with 0 ≤ θ < 1 ). This gives at least one curve
θ (C B ) in the (θ , C B ) plane. Equilibria for the integrated model are then determined by intersections of the curve(s) C B (θ ) and the curve(s) θ (C B ) . Using the fact that the benthic dynamics always remain bounded within a simplex (see Appendices A4 and A5, sections A4.2.1, A5.1.1 and A5.2.1) and the same calculations as in sections A6.1.1 and A6.2.2 in Appendix A6, except with C B max as specified in eqn. 7.15 below instead of C B , a forward trajectory for an integrated model which starts at any initial point in the biological domain (all variables ≥ 0 and benthic covers adding up to 1) must intersect a compact set with no negative values. Thus, analogous to the fish and urchin models (see sections A6.1.1 and A6.2.2), this means that the Richeson-Wiseman Theorem (Appendix A4, section A4.1) can be applied, which proves that there is at least one equilibrium in the biological domain for an 136
integrated model. Therefore, there is at least one intersection of the curve(s) C B (θ ) and the curve(s) θ (C B ) . The stability of any equilibrium for an integrated model is determined by the eigenvalues of the Jacobian J, which can be computed numerically for a given set of parameters.
7.1.2 Example Analysis for the Most Complex Integrated Model
As an illustration, the analysis described in section 7.1.1 is applied to the most complex integrated model, the CTM-HSLUm (see Table 7.1). From sections 5.2.1 and 6.3.1 in Chapters 5 and 6, the eight differential equations defining this system are:
(
)
dC = l Cs + l Cb C (S + ε C T ) + rC (1 − β M M )(S + α C T )C − d C C − γ MC rM MC , dt
(
(7.5)
)
dT = ζ T (1 − θ )S − g T θT − ε C l Cs + lCb C T − rC α C (1 − β M M )TC − γ MT rM MT , dt
(7.6)
dM = rM M (S + γ MC C + γ MT T ) − g M θM , dt
(7.7)
dS dC dT dM =− − − , dt dt dt dt
(7.8)
H2 U H dH 1 − λU − d H H − ( g Ps Ps + g Pl Pl ) 2 = l Hex + l Hen H + C B 2 dt i H + H iU + U i PH + H
H , − ρ H f i FH + H
(7.9)
H2 dPs ex en = l Ps + l Ps Ps + l Plen Pl + (1 − φ Ps )rPs g Ps 2 2 dt i PH + H Ps − ρ Ps f i FPs + Ps
,
P2 Ps − d Ps Ps − ψ Pl g Pl 2 s 2 Pl i PlPs + Ps (7.10)
137
H2 dPl = φ Ps rPs g Ps 2 2 dt i PH + H
Pl − ρ Pl f iFPl + Pl
H 2 Ps + rPl g Pl 2 2 i PH + H
P2 + ψ Pl 2 s 2 i + P s PlPs
Pl − d Pl Pl
,
(7.11)
U H dU 1 − λ H − d U U , = lUex + lUenU + κ U C B dt i + U i + H H U
(7.12)
where θ and C B are as given by eqns. 7.3 and 7.4 respectively. Since S = 1 − C − T − M , there are seven independent variables. Using the analyses of
sections 5.2.2 and 6.3.2 in Chapters 5 and 6, the four conditions which ensure that the dynamics stay within the biological domain are given by conditions 6.18-6.21. Following section 7.1.1, the conditions which ensure that the fish and urchin biomasses tend to zero if there is no recruitment are given by conditions 6.23-6.24 and
C B max < dH , iH
κ U C B max iU
(7.13)
< dU ,
(7.14)
where C B max = max{g M µ M , g T µ T , ζ T µ S } .
(7.15)
First, consider the benthic equations at equilibrium. From the mathematical analysis of the CTMm (section 5.2.2), for any fixed θ in 0 ≤ θ < 1 , there is either one solution or three solutions without macroalgae and zero or more solutions with macroalgae. Thus, there is at least one curve C B (θ ) in the range 0 ≤ θ < 1 for solutions without macroalgae and perhaps more curves for solutions with macroalgae. Next, consider the fish and urchin equations at equilibrium. From the mathematical analysis of the HSLUm (section 6.3.2), if conditions 6.18-6.21 hold, then there is at least one solution in the biological domain and hence at least one θ (C B ) curve.
138
Figure 7.1a illustrates C B (θ ) for solutions without macroalgae, for an arbitrary set of parameters that satisfies conditions 6.18-6.21, 6.23-6.24 and 7.13-7.14. It also illustrates the one and only θ (C B ) curve for the same set of parameters. The two curves intersect once, which shows that there is exactly one equilibrium without macroalgae in this case. It can be checked numerically that this equilibrium is LAS. In this case, there are no equilibria with macroalgae in the biological domain. Similarly, Figure 7.1b illustrates C B (θ ) for solutions with macroalgae and θ (C B ) , using another arbitrary set of parameters that satisfies all the conditions for the CTM-HSLUm. This parameter set is the same as that used for Figure 7.1a except that the growth rate of macroalgae is increased. In this case, there is exactly one equilibrium with macroalgae, which can be shown numerically to be LAS, and no equilibria without macroalgae that are LAS. Figure 7.1c illustrates two C B (θ ) curves, one for solutions with macroalgae and one for solutions without, for another arbitrary set of parameters for the CTM-HSLUm that satisfies all the conditions. It also illustrates the one and only θ (C B ) curve. Unlike the previous two examples, there are three equilibria in this case, as shown by θ (C B ) intersecting the two C B (θ ) curves three times. It can be shown numerically that the equilibrium without macroalgae is LAS and the equilibrium with macroalgae that has a higher C B value (equilibrium value of C B ) is also LAS, with the remaining equilibrium being unstable. Thus, perturbations from an equilibrium may result in convergence to a new equilibrium, with the loss or appearance of macroalgae. In Figure 7.1, the slope of θ (C B ) is a measure of how responsive grazing pressure is to the benthic state represented by the modelled benthic covers. The less this slope is, the closer θ (C B ) is to a straight line in the (θ , C B ) plane and the less responsive is the grazing pressure. Thus, if grazing pressure is very unresponsive, as in a system where grazer biomasses at equilibrium are limited by food to a very small extent, θ (C B ) approximates a straight line. In this case, the number of possible equilibria for the CTM-HSLUm can be taken to be the same as for the CTMm, because the analysis for the CTMm assumes that grazing pressure is fixed across all benthic covers. Note that for an integrated model with the benthic part based on the CTMm, it is easy to see that condition 5.8 in Chapter 5 becomes: g M θ ≥ rM ,
(7.16) 139
where θ is the equilibrium value of θ . If condition 7.16 holds for an equilibrium with
θ , then the equilibrium cannot have any macroalgae. (a)
CB
θ
(b)
CB
θ
(c)
CB
θ Figure 7.1. Graphs of C B (θ ) for solutions without macroalgae (black curves), C B (θ ) for solutions with macroalgae (blue curves) and θ (C B ) (red curves) for the most complex integrated model, the CTM-HSLUm. The intersections of C B (θ ) and θ (C B ) denote equilibria. Each graph is drawn using an arbitrary set of parameters that are all positive except for the total fishing pressure f, which is zero. (a) System with one LAS equilibrium that has no macroalgae, (b) system with one LAS equilibrium that has macroalgae and (c) system with two LAS equilibria (one with and one without macroalgae) and one unstable equilibrium. 140
Chapter 8. Discussion for Part II In Part II, three benthic models of increasing complexity and two fish and urchin models of increasing complexity were constructed using differential equations. From a review of the literature, functional groups and interactions important in reef degradation were identified and assigned variables and parameters in the models. The models all operate at a ‘local’ scale – on the order of tens of metres to a few kilometres – and are deterministic and non-spatial. Mathematical analyses of each of these models found equilibrium properties such as the number of equilibria and their stability. In addition, the analyses derived constraints that ensure the dynamic variables stay within their biologically meaningful ranges. Integrated models were then constructed that dynamically combine a benthic model with a fish and urchin model. The integrated models were analyzed with respect to their equilibria. The simplest benthic model, the Coral-Turf model (CTm; Chapter 4), has two independent groups competing for space, hard corals and fast growing turf-algae, and does not include macroalgae. This models, for example, some offshore Indo-Pacific reefs, which have either no or insignificant amounts of macroalgal species. The CoralAlgae model (CAm) and Coral-Turf-Macroalgae model (CTMm) add complexity to the CTm by adding macroalgae in two different ways (Chapter 5). In the CAm, macroalgae are amalgamated with turf algae into one group whose dynamics are represented as one variable. In the CTMm, macroalgae and turf algae are represented by two independent dynamic variables. For all of the benthic models, the mathematical analyses show that the dynamics always stay in the biological domain for any set of parameter values that make biological sense. The analyses also suggest that all the benthic models have the potential to exhibit multiple stable equilibria. The CTm can have up to two stable equilibria (and one unstable equilibrium), the CAm can have up to three stable (and two unstable) equilibria and the CTMm can have up to two stable equilibria (and one unstable equilibrium) without macroalgae and an unknown number of stable equilibria with macroalgae. In addition, section A4.2.2 in Appendix A4 shows numerically that it is possible for the CTm to exhibit three equilibria for positive parameter sets, two of which must be stable according to the mathematical analysis of the CTm. Furthermore, sections A4.2.3 in Appendix A4 and A5.1.3 in Appendix A5 show that for the CTm and the CAm, any stable equilibrium is a sink node and any unstable equilibrium is a saddle 141
node, such that Hopf bifurcations are not possible. However, the mathematically analyses are incomplete and more analyses could be done; for example, although it was shown using Bendixson’s Negative Criterion that the CTm cannot exhibit periodic orbits and hence limit cycles when the empirically derived parameter ranges are used (section A4.2.2), the CAm could be analyzed further to see if there are any conditions for the parameters under which periodic orbits can occur. Furthermore, the CTMm could be further analyzed to find out more about the possibility of periodic orbits. In addition, the equations for each of the benthic models can be further analyzed to try to find, and examine the biological relevance of, conditions for the parameters under which multiple stable equilibria are not possible, using structures in the equations. The possibility of multiple stable equilibria is important because if this is the case, then there can be shifts from a stable state of high coral cover and low algal cover to an alternative stable state with lower coral cover and higher algal cover. Such a phase shift may be brought about, for example, by acute disturbances that affect benthic covers, such as tropical storms. Thus, the presence of multiple stable equilibria means that recovery after an acute disturbance such as a hurricane is not guaranteed if the perturbed system moves into the domain of attraction of a new equilibrium (Holling 1973). In contrast, if there is just one stable equilibrium, the system will eventually recover in the absence of further disturbances. However, although multiple stable equilibria are possible in principle, it remains to determine whether they are possible within narrower, empirically derived parameter ranges. For the CTMm, the mathematical analysis shows that if the achieved grazing rate on macroalgae is greater than or equal to the lateral growth rate of macroalgae (Chapter 5, condition 5.8), then at equilibrium, there are no macroalgae present and the number and stability of equilibria are exactly the same as for the CTm with the same set of benthic parameters for hard corals and turf algae. This demonstrates the importance of managing nutrient loads that can potentially increase macroalgal growth rates (e.g., Delgado and Lapointe 1994) and of maintaining grazing pressure on macroalgae. Thus, anthropogenically-derived nutrients such as those from agriculture, aquaculture and urban run-off need to be managed. Also, herbivorous fish species that feed significantly on macroalage need to be identified and their stocks managed, particularly in light of the paucity of information on fish taxa that do feed significantly on macroalgae (Bellwood et al. 2006a). Ideally, these macroalgal grazers would be non-specialists which can subsist on other food sources (such as turf algae), such that they can survive at an equilibrium with no macroalgae and help to sustain it. This result shows that both 142
bottom-up and top-down processes must be managed to minimise macroalgal cover. The simplest fish and urchin model, the Herbivorous Fish-Piscivorous FishUrchins model (HPUm) represents piscivorous fish as one group, whereas the more complex model, the Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous Fish-Urchins model (HSLUm) adds complexity by representing piscivorous fish as two size classes (Chapter 6). The advantage of this extra complexity is that it allows different life history traits for the two size classes of piscivorous fish to be represented, as well as different fishing rates on the two fish groups. Conditions which ensure that dynamics stay within the biological domain and that biomasses remain finite were derived for both fish and urchin models. In addition, conditions were derived which ensure that for each group, a positive biomass is unsustainable in the absence of (endogenous and exogenous) recruitment. Under these biologically plausible conditions, it was found that for both the HPUm and the HSLUm, there is at least one equilibrium in the biological domain. For both models, the stability of an equilibrium must be determined numerically. The integrated models combine a fish and urchin model with a benthic model by expressing the grazing pressure parameter as a function of herbivorous fish and urchin biomasses, and by expressing food availability for herbivorous fish and urchins as a function of the benthic state variables (Chapter 7). Combinations of different benthic models with different fish and urchin models give six different integrated models, each of which inherits some of the properties of the component models from which they are constructed. For the integrated models, a method was found to determine graphically how many equilibria there are. In this method, equilibria are given by the intersection points of two types of planar curves, one type derived from the equilibrium equations for the benthic groups and the other type from the equilibrium equations for the fish and urchin groups (Chapter 7). In addition, the mathematical analysis shows that for an integrated model, conditions analogous to those for the fish and urchin models ensure that there is at least one equilibrium. The stability of any equilibrium of an integrated model must be determined numerically. It was noted that if herbivore biomasses are not strongly influenced by changes in algal food supply, then grazing pressure is not very sensitive to such changes. This represents a situation where there is little dynamic feedback from the benthos to the fish and urchins. In this case, the level of grazing pressure at equilibrium is determined by factors other than food availability, such as fish recruitment. The mathematical analysis 143
shows that, in this case, it is likely that an integrated model can support only the same number of equilibria as the corresponding benthic model. Thus, this proves that each of the integrated models can theoretically exhibit multiple stable equilibria. Several studies show that fish populations can be recruitment limited. For example, Russ (1991) cites a study (Williams 1986) where herbivorous fish numbers did not respond positively three years after a large increase in turf algal cover. Recruitment limitation is commonly assumed in demographic models of coral reef fish populations (Ault and Johnson 1998). However, there are also several studies showing a positive response of herbivorous fish to an increase in the algal food resource. For example, studies showing how fish responded positively to the decline of urchin competitors (Hay and Taylor 1985, Carpenter 1988, Carpenter 1990) are consistent with a food-limitation hypothesis. It was also noted that for an integrated model whose benthic part is based on the CTMm, if the achieved grazing rate on macroalgae at an equilibrium is greater than or equal to the lateral growth rate of macroalgae (Chapter 7, condition 7.16), then there is no macroalgae present at that equilibrium. Analogous to the implications of condition 5.8 for the CTMm (see above), condition 7.16 highlights the importance of managing both bottom-up and top-down processes in order to minimise macroalgal cover. In constructing the models detailed in Part II, several assumptions were made which limit the realism of the models. The most problematic assumptions concern recruitment. For the benthic part of the models, it is assumed that exogenous coral recruitment arises from an exogenous larval pool which contributes to several reef areas, including the area modelled, and to which these reef areas may contribute. The larval pool size is assumed to be constant through time and hence, so is the exogenous recruitment rate to the modelled reef area. In reality, exogenous recruitment is likely to be both seasonal and highly stochastic. To facilitate model analysis, seasonal effects are averaged and stochastic effects are not modelled. The constant larval pool assumption is also made for exogenous fish and urchin recruitment. Neither soft corals nor sponges are modelled. Soft corals are not modelled because there is evidence that they do not affect coral-algal dynamics significantly, and hence can be treated as a fixed component of the benthos (Chapter 4, section 4.2.1). Sponges are not modelled because there is evidence that they are generally uncommon (section 4.2.1). Nonetheless, sponges can be superior competitors to corals for space (Aerts 1998) and the models will therefore likely overestimate the amount of coral cover on reefs with significant sponge cover. However, none of the existing non-spatial differential equations models reviewed in Chapter 2, section 2.1.2, explicitly model soft 144
corals or sponges, so the models in this thesis are not at a disadvantage in this respect. The analyses in Part II found conditions which ensure that all models make biological sense, determined that there is at least one equilibrium for each model if the models make biological sense, and determined the number of possible equilibria and their stability in most cases for the benthic models. These analyses are unique in relation to comparable existing studies that use non-spatial differential equations models (see section 2.1.2). Another unique aspect of the work in Part II is that several models of different complexity have been constructed, a procedure that provides a framework for investigating the implications of differing levels of complexity for reef degradation and management – this will be explored further in Parts III and IV. Modelling two different types of algae, turf algae and macroalgae, explicitly competing for space and modelling two different size classes for piscivorous fish are aspects of complexity not considered in comparable studies.
145
Part III: Quantifying the Parameters In Part III, model parameters are quantified using a variety of methods and data obtained from journal papers, surveys and the online database FishBase (Froese and Pauly 2007) (Chapters 9-11). The data used are from reefs worldwide and different reef habitats such as the reef flat, reef slope and lagoon. Thus, the parameter ranges derived are for a ‘generic’ reef. Part III concludes with a discussion of the process of quantifying the parameters.
Chapter 9. Parameter Ranges for the Benthic Groups The parameters for hard corals (section 9.1) and algae (section 9.2) are considered in turn. For each parameter, the parameterisation methodology is described and the data sources given. At a particular geographic location, data for different species in a functional group and data for similar reef habitats are pooled where possible and appropriate. Also, data from the western Atlantic are not used in conjunction with data from the Indo-Pacific where possible, because these two biogeographic regions can have very distinct assemblages of hard corals and fish (see Chapter 1, section 1.1). The parameter ranges derived are summarised in Table 9.1 at the end of this chapter together with the parameter definitions. All parameter values are rounded to the nearest significant figure unless otherwise stated. For all of the rates in Chapters 9-11, the average yearly rate is first derived to average out seasonal effects such as different algal growth rates, due to differing seasonal light levels for photosynthesis, and different seasonal recruitment rates for hard corals or fish. To numerically solve the differential equations for each model, required for the sensitivity analyses in Chapter 15, these average yearly rates are scaled to the smaller updating time steps used. Also, some of the yearly rates are derived by first considering discrete time intervals. By assuming that, within a discrete time interval ∆t , there are n small time steps of size δt and that the change in proportional cover or biomass within each of these n time steps is simply a proportion δt ∆t of the total change within that interval (i.e. the change within an interval is evenly spread out), discrete changes are converted into instantaneous rates. 146
The advantage of using continuous differential equations rather than discrete difference equations, in terms of parameterisation, is that individual processes can be considered in isolation. This is because small time intervals of size δt are considered, such that the probability of two processes occurring simultaneously is of the order (δt ) , 2
which is negligible. Thus, parameter values for each process can be determined as if the process was independent of other processes.
9.1 Hard Coral Parameters
The coral mortality rate d C is parameterised using studies that measure the percentage decrease in coral tissue cover. Bythell et al. (1993) measured this percentage decrease, due to unknown causes and naturally occurring black band disease and tissue necrosis, over 26 months for a coral population at St. Croix in the U.S. Virgin Islands (USVI). Mortality due to bleaching and sedimentation was negligible. Lirman (2003) reported the percentage decrease in the cover of Acropora palmata colonies in Florida, in the absence of storms. Similarly, Nugues and Roberts (2003) measured the percentage decrease, due to unknown causes and naturally occurring diseases, in cover of Colpophyllia natans and Siderastrea siderea in St. Lucia over 15 months. These studies give three estimates of coral mortality due to background processes. The range of d C used is the set of values between the lowest and highest estimates, which is 0.02-0.1 yr-1 (Table 9.1). The recruitment rates l Cb and l Cs are parameterised using a study of coral recruitment onto settlement tiles (Dunstan and Johnson 1998), studies that report coral recruitment rates (Hughes 1984, Lirman 2003, Langmead and Sheppard 2004, Mumby 2006) and studies reporting coral cover (see below for studies). First, within a settlement tile, space is not a limiting factor for coral recruitment in the sense that recruits can settle on any of the tile area. Thus, the space proportional cover S can be taken to be 1. Similarly, for studies reporting recruitment rates, it is assumed that space is not limiting either. For a particular study, let a be the average area of a 1 yr coral recruit in cm2, b be the total area onto which corals can recruit in cm2 and N be the total number of coral recruits into the 1 yr age class every year. If brooding corals are being studied, then
147
l Cb C =
Na , b
(9.1)
and if spawning corals are being studied, then
l Cs =
Na , b
(9.2)
where C is the coral cover of the study location. Coral recruits are assumed to have a circular planimetric area, which allows their cover area to be calculated in terms of their radii. Sets of a, b and N values are taken from different studies where possible. For some studies, a cannot be calculated, in which case a is calculated using the mean diameter growth rates for juvenile corals of different families, for western Atlantic and Indo-Pacific reefs (Edmunds 2007). Values of C, used in eqn. 9.1, are taken from the settlement studies themselves or from other studies. For the Dunstan and Johnson (1998) settlement tile study on the Great Barrier Reef (GBR) in Australia, C is taken to be 0.250, which is the average of values at 1991 and 1995 (Bellwood et al. 2004), 1991 and 1995 being the years between which the Dunstan and Johnson (1998) study was undertaken. For the Langmead and Sheppard (2004) and Mumby (2006) studies for the Caribbean, C is taken to be 0.100, which is the mean coral cover for the Caribbean in 2001 (Gardner et al. 2003). To determine whether corals are brooding or spawning, the classification by Richmond (1997) is used. The range for each of l Cb and l Cs is taken to be the set of values between the minimum and maximum values found, giving ranges of 0.009-0.5 yr-1 and 0.00006-0.01 yr-1 for l Cb and l Cs respectively (Table 9.1). The lateral coral growth rate rC is parameterised by first assuming that coral occurs as n hemispherical colonies. These colonies grow laterally over space and it is assumed that this space is not limiting. Denote the initial proportional cover of coral colony i by C i and its radius by ri , where 1 ≤ i ≤ n . After one year, colony i grows to have a proportional cover of C i′ and a radius of ri′ . Let d be the increase in the radius, such that ri′ = ri + d , and A be the planer reef area containing the colonies. Then:
Ci =
πri 2 A
,
(9.3)
148
C i′ =
π (ri′)2 A
,
(9.4)
n
∑ (C ′ − C ) i
rC =
i
i =1
n
∑C
yr-1.
(9.5)
i
i =1
From eqns. 9.3 and 9.4,
{
}
(
)
C i′ − C i = (π / A) (ri′) − ri 2 = (π / A) ri 2 + 2dri + d 2 − ri 2 = d (π / A)(2ri + d ) . 2
(9.6)
Using eqn. 9.6, eqn. 9.5 can be written as:
n
rC =
d ∑ (2ri + d ) i =1
n
∑r
yr-1.
(9.7)
2
i
i =1
Assume that each ri is randomly distributed such that ri = X i + rmin , where X i are independent, identically distributed random variables that have non-negative values and
rmin is the minimum radius of a coral colony in the model. Using ri = X i + rmin in eqn. 9.7 gives:
n
rC =
d ∑ {2( X i + rmin ) + d } i =1
n
∑ (X
i
+ rmin )
n
yr-1 =
2d ∑ X i + nd (2rmin + d )
2
i =1
2d =
i =1
n
∑ (X
i
+ rmin )
yr-1
2
i =1
1 n ∑ X i + d (2rmin + d ) -1 n i =1 yr . 1 n 2 ∑ ( X i + rmin ) n i =1
(9.8)
The denominator of the expression on the right-hand side of eqn. 9.8 is:
1 n ( X i + rmin )2 ∑ n i =1
149
=
(
1 n 2 X i2 + 2 X i rmin + rmin ∑ n i =1
)
1 n 2 = ∑ X i2 − X 2 + X 2 + 2rmin X + rmin . n i =1
(9.9)
Thus, for large n,
rC ≈
2dX + d (2rmin + d )
σ + ( X + rmin ) 2 X
2
yr-1,
(9.10)
where X and σ X2 are the mean and variance of the X i respectively. For the modelled reef area, which is on the scale of tens of metres to kilometres, a large number n of noninterfering colonies can reasonably be assumed. Thus, eqn. 9.10 can be used to find values of rC , using values of rmin , d, X and σ X2 . From Edmunds (2007), 0.442 cm and 1.14 cm are the average radii for 1 yr old coral recruits in the western Atlantic and Indo-Pacific respectively. Thus, rmin is 0.442 cm and 1.14 cm for western Atlantic reefs and Indo-Pacific reefs respectively. d is the linear skeletal extension rate for corals and 118 values are taken from reefs in the western Atlantic and Indo-Pacific (Huston 1985, Chornesky and Peters 1987, Babcock 1991, Koop et al. 2001, Loch et al. 2002, Langmead and Sheppard 2004). The median values of d are used, which are 0.973 cm and 0.875 cm for western Atlantic corals and Indo-Pacific corals respectively. In the calculation of the medians, the mid-point of any ranges reported is used. The median is used rather than the mean because it is not distorted by a few large figures (such as those for Acropora spp.). The mean and variance of the radii of different coral populations in the western Atlantic and Indo-Pacific are determined using data from the literature and surveys (MBRSP 2005, Sandin et al. 2008). The mean and variance of the radii minus rmin can then be calculated for each population; i.e. X and σ X2 . Using the values of rmin , d, X and σ X2 for western Atlantic reefs, 25 estimates of rC were derived from eqn. 9.10 and using the values of rmin , d, X and σ X2 for Indo-Pacific reefs, a further 4 estimates of rC were derived. The range of values of rC was taken to lie between the minimum and maximum values found. This range is 0.04-0.2 yr-1 (Table 9.1). Some coral species, particularly branching species, may grow up and out and 150
this allows them to grow into the space above turf algae, thus overtopping them. Therefore, corals may be unimpeded by turf when they are growing up and over them. This corresponds to the case where corals can grow as quickly over turf algae as they can over space, that is α C = 1 . In contrast, some species may not be able to overgrow turf algae at all due to a prostrate morphology, and this corresponds to the case where
α C = 0 . Thus, the range of α C explored is 0-1 (Table 9.1). Birrell et al. (2005) studied recruitment of Acropora millepora coral recruits onto two turf assemblages of different canopy heights. One assemblage had a height of < 3 mm, which corresponds to fine turf within the space group modelled, and the other
had a height of 5-8 mm, which corresponds to the turf algae group modelled. It was found that recruitment onto the turf assemblage of greater height was lower by a factor of 0.25 / 1.88 = 0.133 . This suggests that the recruitment rate onto turf algae is lower than the rate onto space by a factor of 0.133, which gives ε C = 0.1 to the nearest significant figure. To account for uncertainty, this figure for ε C is extended by 50% in each direction, giving 0.05-0.15 (Table 9.1).
9.2 Algae Parameters
To parameterise the maximum grazing rates on turf algae and macroalgae, g T and g M respectively, the study by Hughes et al. (2007) was first used. In this study, 5 m by 5 m cages were used to cultivate large stands of Sargassum sp., a fleshy macroalgae. After the cages were removed, herbivorous fish reduced macroalgal cover from 53% to ≈ 0% after 30 d. 1-3 adults of the batfish Platax pinnatus, the herbivorous fish species mainly responsible for the reduction, were regularly captured on video in the plots after cage removal, which equates to ≈ 0.1-0.3 kg m-2 (Bellwood et al. 2006a). Values in this range are high considering that 0.282 kg m-2 is the maximum herbivorous fish biomass found from the literature on a larger scale (see Chapter 6, section 6.2.2). Thus, it is assumed that the herbivorous fish in the Hughes et al. study exert a grazing pressure that is near to the maximum limit of 1, that is θ ≈ 1 . The change in macroalgal cover due to grazing is M ′ − M = −( g M δt )M , where M and M ′ are the initial and final macraolgal covers for a small time period δt . Using this for the Hughes et al. study gives g M = 1 (30 d)-1. Assuming that this grazing rate is maintained throughout the year, g M over a year can be found by scaling up, giving 151
g M = 12.2 yr-1. For parrotfish, Mumby et al. (2006a) recorded the ratio of bites on macroalgae to bites on turf algae as 1: 1.05, which suggests that herbivorous fish can graze on both types of algae at an equal rate. Thus, it is assumed that g T is also 12.2 yr1
. Rounding to the nearest significant gives g T = 10 yr-1 and this is extended to a +/–
50% range to account for uncertainty, giving 5-15 yr-1 (Table 9.1). To determine a range of g M , the fact that some macroalgae have chemicals that deter grazers needs to be considered. In addition, macroalgae have greater structural toughness in relation to turf algae and this also makes them potentially less palatable than turf algae (Hay 1991). Russ and St. John (1988) found that for the gut contents of 91 herbivorous fish species from the Caribbean and Indo-Pacific, the contents of only 7% of these species were dominated by macroalgae, whereas 51% of species had contents dominated by filamentous algal turfs. This lends further support for the greater palatability of turf algae over macroalgae. Furthermore, in the Hughes et al. (2007) study, only ≈ 7% of herbivorous fish species (3 out of 44 species) had a significant effect in the reduction of macroalgae after cage removal. Cvitanovic and Bellwood (2009) studied reefs along Orpheus Island in the GBR and also found that ≈ 7% of herbivorous fish species (3 out of 45 species) fed significantly on macroalgae. Given this evidence that turf algae is a more important part of the diet of herbivorous reef fish, it is assumed that g M ≤ g T . The Hughes et al. (2007) and Cvitanovic and Bellwood (2009) studies found that ≈ 7% of herbivorous fish species fed predominantly on macroalgae, which suggests that
g M can be 7% of g T . However, these studies were conducted on Indo-Pacific reefs and the number of species that feed predominantly on macroalgae may be lower in the western Atlantic due to lower species diversity and hence lower functional redundancy (see Chapter 1, section 1.1). Theoretically, an area may have no fish species that feed predominantly on macroalgae. However, this is not guaranteed to give g M = 0 yr-1 because fish that feed predominantly on turf may have macroalgae as part of their diet. The precise ratio of g M to g T would also depend on the distribution of total fish biomass among fish species that feed predominantly on turf or macroalgae. Thus, in light of these considerations, and in the absence of data to derive an exact figure for how low g M can be relative to g T , it is assumed that g M can be up to 99% less than
g T , that is, it ranges from (0.01)g T - gT (Table 9.1). Turf algae are capable of growing very quickly. McClanahan (2002) states that
152
turf algae have a regeneration time (sensu Grubb (1977)) on the order of 20 d. This means that it takes on the order of 20 d for sexually mature adult turf algae to be replaced by its offspring. Mumby et al. (2006a) assumed in their simulation model of a coral reef that algal turfs can completely cover dead corals within 6 months. Thus, from these two studies, the upper limit on the regeneration time is assumed to be 6 months, such that in the absence of grazing, it takes 20 d to 6 months for a reef area with 100% space cover to become 100% turf algal cover. For a benthic model, the change in space cover in the absence of grazing and due to turf algal growth is S ′ − S = −(ζ T δt )S , where S and S ′ are the initial and final space covers for a short time period δt , and ζ T is the turf algal growth rate. This gives ζ T = 1 (20 d)-1 to 1 (6 months)-1. Assume that this growth rate is maintained throughout the year, such that ζ T over a year can be found by scaling up. This gives the range 2-20 yr-1 (Table 9.1). The lateral growth rate of macroalgae, rM is obtained by first considering macroalgae as n patches on a reef area with proportional space cover S. Let M i and M i′ be the macroalgal covers of patch i (in cm2) at the beginning and end of a year respectively. Then rM is given by
n
∑ (M ′ − M ) i
rM =
i =1
n
S∑ M i
i
yr-1.
(9.11)
i =1
Mumby et al. (2005) studied the mean lateral extension rates of two types of macroalgal patches on an exposed fore-reef at Glover’s Reef in Belize. Specifically, the two sites Long Cay and Middle Cay were studied. They found that the extension rate of Lobophora variegata and Dictyota pulchella patches was 7.3 cm2 yr-1 and 40.15 cm2 yr-1 respectively. In the study, these macroalgae grew over dead parts of Montastraea annularis ramets that were covered with encrusting coralline algae or cropped turf algae, corresponding to space in the benthic models, or filamentous algae, corresponding to turf algae in the benthic models. In addition, they may have grown over live ramet parts (Nugues and Bak 2006), although the extent to which this occurred was not recorded. Since macroalgae grow more slowly over turf algae and live coral than space (see Chapter 5, sections 5.1.2 and 5.2.1), it is assumed that in the study, half of the increase in macroalgal cover was due to growth over encrusting coralline algae or cropped turf 153
n
algae (space), such that
1 n
∑ (M ′ − M ) = 3.65-20.1 cm2. This assumption guarantees i
i
i =1
that the macroalgal growth rate over space is greater than that over turf algae or live coral in the study, because the turf algal cover is assumed to be the same as space cover (see below) and from Mumby et al. (2005), coral cover was greater than space cover on average. For Long Cay and Middle Cay, Mumby et al. (2005) also give the percentage of D. pulchella patches in different size categories. Using 2,130 cm2 for the > 2,130 cm2 size category and the mid-points of all other categories, together with the reported total reef area considered, the mean values of M i for Long Cay and Middle Cay are derived as 507 cm2 and 356 cm2 respectively. Assume these figures apply to macroalgal patches n
for L. variegata as well. Thus,
1 n
∑ M i = 507 cm2 for Long Cay and i =1
n
1 n
∑M
i
= 356
i =1
cm2 for Middle Cay. From Mumby et al. (2005), the average proportional cover of dead parts of ramets covered with encrusting coralline algae, cropped turf algae or filamentous algae for Long Cay is 18.8% and that for Middle Cay is 38.4%. Assume that half of this is dead parts of ramets with encrusting coralline algae or cropped turf algae (space), such that S = 0.094 for Long Cay and S = 0.192 for Middle Cay. This assumption is necessary because the cover of dead parts of ramets covered with just encrusting coralline algae or cropped turf algae was not reported. Using the values of 1 n
n
n
i =1
i =1
∑ (M i′ − M i ) together with the values of 1n ∑ M i
and S for Long Cay and Middle
Cay in eqn. 9.11 gives the range of rM as 0.05-0.4 yr-1 (Table 9.1). The inhibition parameter β M (Table 9.1) is parameterised using studies measuring the reduction in coral growth due to macroalgae. Lirman (2001) described an experiment where macroalgae of Dictyota spp. were added bi-weekly to plots with the coral Porites astreoides. The experiment was conducted within a cage that excluded herbivores and the added macroalgae led to a macroalgal biomass that is equal to the maximum biomass found on reefs in the Northern Florida Reef Tract, where the study took place. In the control plots without macroalgae, the percentage increase in surface area per month of P. astreoides was 5.42% mo-1, which is equivalent to
{(1.0542)
12
}
− 1 100% yr-1 = 88.4% yr-1 = 0.884 yr-1. Since there is herbivorous grazing in
these control plots, assume that the inhibition of coral growth by macroalgae is 154
negligible. In the caged treatments, the percentage increase of P. astreoides was 0.39%
{
}
mo-1, which is equivalent to (1.0039 ) − 1 (100 )% yr-1 = 4.78% yr-1 = 0.0478 yr-1. On 12
reef areas with the maximum macroalgal biomass, macroalgae completely covered corals, including P. astreoides. Thus, in the caged treatments, it is assumed that the macroalgal biomass added corresponds to a macroalgal proportional cover of 1, that is M = 1 . Thus, 1 − β M M = 1 − β M × 1 = 0.0478 0.884 = 0.0541 , which gives β M = 0.946 .
Jompa and McCook (2003a) studied the reduction in growth of Porites cylindrica branches, this time by the macroalga Hypnea pannosa, which was the dominant macroalga at the study location, an inshore GBR reef at 3-5 m depth. In the experimental treatment with H. pannosa, there were extensive mats of the macroalga growing around the coral branches. Thus, it is assumed that horizontal macroalgal cover is not limiting, such that M = 1 . In the treatment without H. pannosa, the macroalga was cleared from the experimental plot. By comparing results from the two treatments, it was found that the skeletal extension rate of P. cylindrica was decreased by 0.11 mm per month by H. pannosa, that is, the rate was decreased by 6.04% in comparison to the case without H. pannosa. A similar growth study was performed by Jompa and McCook (2002a) on a 7-8 m deep reef slope at Great Palm Island in the GBR. In this study, P. cylindrica was used again, but this time, the macroalga L. variegata was studied. The skeletal extension rate of P. cylindrica was measured in caged treatments with and without L. variegata. In the treatments with L. variegata, macroalgae completely surrounded the coral branches and hence, horizontal macroalgal cover is again assumed not to be limiting, such that M = 1 . In the caged treatment without L. variegata, macroalgae was removed manually. The experiment was conducted during two time periods and it was found that the skeletal extension rate of P. cylindrica was reduced by 16.2% and 20.1% for the two periods. Thus, from the two studies, the skeletal extension rate of P. cylindrica is reduced by 14.1% on average due to macroalgae. In the absence of macroalgae, the coral growth rate in the benthic models, rC , can be approximated by eqn. 9.10. In the presence of macroalgae, this growth rate becomes rC (1 − β M ) (when M = 1), and using the same method used to derive eqn. 9.10, rC (1 − β M ) can be approximated by:
rC (1 − β M ) ≈
2d 2 X + d 2 (2rmin + d 2 )
σ X2 + ( X + rmin )
2
,
(9.12)
155
where d 2 is the reduced skeletal extension rate due to macroalgae. Thus: rC (1 − β M ) 2d X + d 2 (2rmin + d 2 ) = 1− βM ≈ 2 . rC 2dX + d (2rmin + d )
(9.13)
The studies for P. cylindrica were conducted in the Indo-Pacific, so the values d = 0.875 cm and rmin = 1.14 cm, obtained using Indo-Pacific data, are used (see
parameterisation of rC in section 9.1). Also, d 2 = (0.875)(1 − 0.141) cm = 0.752 cm. From eqn. 9.13, 1 − β M is independent of σ X2 . The values of X used are the same as those used in the derivation of rC , for Indo-Pacific reefs. Note that 1 − β M is an increasing function of X since d > d 2 . Hence, only the minimum and maximum values of X are needed. Using these values in eqn. 9.13 gives β M as ≈ 0.2 . The last growth study used to parameterise β M is Box and Mumby (2007). In this study, the effects of L. variegata and D. pulchella on juvenile corals of Agaricia spp. on shallow fore-reefs in Roatán, Honduras, were investigated. The corals tested had a mean area of 1.25 cm2, which is greater than the area of a 1 yr old Agaricia recruit in the western Atlantic (0.882 cm2; Edmunds 2007). Thus, it is assumed that the corals in the Box and Mumby (2007) study are more than 1 yr old, and hence the results are relevant for the benthic models, which only consider corals 1 yr old or older. The macroalgae in the caged experiments completely surrounded the corals, so assume M = 1. It was found that the skeletal extension rate of the coral was reduced by 40% and 69% by L. variegata and D. pulchella respectively. Since the study was performed in the western Atlantic, d = 0.973 cm, the median d value for western Atlantic reefs (see section 9.1). This means that d 2 = (0.973)(1 − 0.4) cm = 0.584 cm and
d 2 = (0.973)(1 − 0.69) cm = 0.302 cm are used for the two types of macroalgae studied. The X values used are the same as those used in the derivation of rC , for western Atlantic reefs. Eqn. 9.13 then gives the lowest value of β M as 0.4 and the highest value as 0.7. Thus, overall, the range of β M is taken to be in the range 0.2-0.9 (Table 9.1). As in the parameterisation of rC , to parameterise γ MC and γ MT (Table 9.1), first consider a reef area where corals occur as n hemispherical colonies and consider macroalgae growing over these colonies. Nugues and Bak (2006) investigated the growth of L. variegata over the perimeters of live corals of six different species. They 156
found that across 5.95 cm of the perimeter of a coral colony, the growth rate of L. variegata was 0-6.86 cm2 yr-1. Let this growth rate be denoted by M C . Also, let m be the rate of increase in macroalgal cover due to overgrowth of corals in cm2 yr-1, c be the proportion of the perimeter of coral colonies where there is contact with macroalgae and A be the planar reef area modelled in cm2. Then:
n MC , m = c 2π ∑ ri i =1 5.95 cm
(9.14)
and
n n M C c 2π ∑ ri 2c ∑ ri (m / A) = i =1 5.95 cm = i =1 γ MC rM = n MC n M ∑ ri 2 M π ∑ ri 2 i =1 i =1
MC . 5 . 95 cm
(9.15)
Using the same notation as in the derivation of rC in section 9.1, let ri = X i + rmin . Then eqn. 9.15 can be written as 1 n X i + rmin ∑ 2c n M C , γ MC rM = ni =1 M 1 2 5.95 cm ( X i + rmin ) n ∑ i =1
(9.16)
and for large n,
γ MC rM ≈
M C X + rmin 2c . 2 2 M σ X + (X + rmin ) 5.95 cm
(9.17)
c and M are derived using data from Lirman (2001), who surveyed c for Dictyota spp. on reefs in Biscayne National Park, Florida and total algal cover for the same reefs. In this study, it was found that c = 0.197 and by assuming that the proportion of total algal cover that is due to Dictyota spp. is the same as the proportion of the total length where there is contact between algae and corals that is due to Dictyota spp., M = 0.166 .
157
Assume that M C for Dictyota spp. is the same as for L. variegata, such that it takes values in the range 0-6.86 cm2 yr-1. Since the Nugues and Bak (2006) and Lirman (2001) studies are for western Atlantic reefs, rmin = 0.442 cm is used, which is the value for western Atlantic coral recruits (see section 9.1), as are the pairs of values of X and σ X2 for western Atlantic reefs used in the derivation of rC (section 9.1). Using eqn. 9.17 then gives the range of γ MC rM as 0-0.275. From the parameterisation of rM above, the range of rM is 0.294-0.422 yr-1 for D. pulchella. It is assumed that this range applies for Dictyota spp. as well, giving a range for γ MC of 0-0.9 (Table 9.1). In the absence of relevant data to parameterise the growth rate of macroalgae over turf algae, it is assumed that macroalgae can grow over turf algae at the same rate as over corals. This means that γ MT also lies in the range 0-0.9 (Table 9.1).
ν M is used only in the Coral-Algae model (CAm) and represents the proportion of total algal cover that is macroalgae. Thus, ν M can take any value between 0 and 1 (Table 9.1). However, for reasons given in section 5.1.3 in Chapter 5, ν M cannot take the extreme values of 0 and 1. The grazing pressure θ is used in all the benthic models. From the literature, the lowest herbivorous fish biomass found in the western Atlantic is 0.00184 kg m-2 in Sapodilla Cayes in Belize (MBRSP 2005; see Chapter 6, section 6.2.2 for all references examined). Using the lowest sea urchin biomass in Belize, which is zero (Kramer 2003), with the highest value of i H for western Atlantic reefs, which is 0.5 kg m-2 (see Chapter 10, section 10.2), and the explicit form of θ in Chapter 7 gives θ = 0.00367 , which is close to the theoretical minimum of 0 (when there are no herbivorous fish or urchins). From section 6.2.2 in Chapter 6, the highest herbivorous fish biomass found from the literature is 0.282 kg m-2 for a northern reef lagoon in New Caledonia (Letourneur et al. 1998). Assuming there are no urchins, then this biomass together with the lowest value of i H for Indo-Pacific reefs, which is 0.001 kg m-2 (section 10.2), gives θ = 0.996 , which is close to 1. This figure will be higher if there are also urchins, since θ is an increasing function of urchin biomass. Thus, θ can be ≈ 1. Furthermore, Carpenter (1986) found that an average of 97% of turf algal production was taken up by herbivorous grazing on a back-reef/reef-crest site in St. Croix, which again suggests that
θ can be ≈ 1. Because of these considerations, the range of θ explored for the benthic models is 0 ≤ θ < 1 (Table 9.1), which is the full theoretical range.
158
Parameter
Definition
Empirically Derived
Model Specificity
Ranges dC
Coral mortality rate
l Cb
Rate at which coral larvae, produced by local established brooding corals,
0.02 - 0.1 yr-1
CTm, CAm and
0.0009 - 0.5 yr-1
CTMm
recruit onto space 0.00006 - 0.01 yr-1
l Cs
Rate at which exogenous spawning coral larvae recruit onto space
rC
Lateral growth rate of corals over space
αC
Growth rate of corals over turf, relative to the rate over space
εC
Recruitment rate of corals onto turf, relative to the rate onto space
0.05 - 0.15
gT
Maximum rate at which turf algae is grazed
5 - 15 yr-1
ζT
Growth rate of fine turf (occupying space)
2 - 20 yr-1
θ
Grazing pressure
gM
Maximum rate at which macroalgae is grazed
(0.01)g T -
rM
Lateral growth rate of macroalgae over space
0.05 - 0.4 yr-1
βM
Coral growth is inhibited by the presence of nearby macroalgae and this
0.04 - 0.2 yr-1 0-1
0-1
0.2 - 0.9
is represented as depression of rC by the factor (1 − β M M ) , where M is the macroalgal cover
γ MC
Lateral growth rate of macroalgae over corals, relative to the rate over 159
gT
0 - 0.9
CAm and CTMm only
space
νM
The proportion of total algal cover that is macroalgal cover
γ MT
Lateral growth rate of macroalgae over turf, relative to the rate over space
0-1
CAm only
0 - 0.9
CTMm only
Table 9.1. Parameters for the CTm, the CAm and the CTMm, with their ecological meanings and empirically derived ranges
160
Chapter 10. Parameter Ranges for the Fish Groups The recruitment parameters for all fish groups in the fish and urchin models are considered first (section 10.1). This is followed by consideration of the feeding (grazing for herbivorous fish and predation for piscivorous fish) and growth parameters (section 10.2), and then the mortality parameters (section 10.3). As for the benthic models, data for different species in a functional group and for similar reef habitats are pooled where possible and appropriate, and data from the western Atlantic is not used with data from the Indo-Pacific where possible. The parameter ranges derived are summarised in Table 10.1 at the end of this chapter, together with the parameter definitions. All parameter values are rounded to the nearest significant figure unless otherwise stated. In the derivation of the fish and urchin parameters in this chapter and Chapter 11, data from several ECOPATH (Christensen et al. 2005) models are used. The groups in these ECOPATH models do not always correspond to the groups in the models in this thesis. For example, in this thesis, herbivorous and piscivorous fish feed predominantly on algae and (herbivorous and/or piscivorous) fish respectively (see Chapter 6, section 6.2.1). However, in the ECOPATH models, fish that predominantly feed on algae are sometimes amalgamated with fish that feed on detritus (e.g., Arias-Gonzalez 1998). In addition, though piscivorous fish groups in ECOPATH may feed predominantly on fish, predation on non-modelled fish groups such as invertivorous fish is also included. In these cases, the ECOPATH groups closest to the herbivorous and piscivorous fish groups in the models in this thesis are used as the best (and only) available option. In addition, for the ECOPATH models used, it is assumed that there is no biomass accumulated during the time interval considered and no migration, because it is not stated in any of the corresponding studies that these processes are included and reef fish usually make localised movements (Williams 1991). In other words, it is assumed that for all the ECOPATH models, Ai = Ei = 0 in eqn. 2.1a (Chapter 2) for all groups i. It is also assumed that there is no recruitment into the 1 yr age class in the ECOPATH models used. These models are geared towards fisheries management (Christensen et al. 2005), where recruitment is usually into an older age class that represents fish large enough for fisheries to catch, such that earlier recruitment is not explicitly mentioned in any of the ECOPATH studies. Thus, it is reasonable to assume that there is no recruitment into the 1 yr age class in these ECOPATH models. Furthermore, for the ECOPATH models, it is sometimes necessary to split 161
biomass production P for a particular fish group to find the production that is used for somatic growth, somatic production PS , and production that is used to form gametes, gamete production PG . To do this, formulae that relate P to PS and PG were derived. First, the proportion of P that is used to produce mucus (and therefore unavailable for somatic growth or formation of gametes) was found and subtracted from P. Many species of labrids (wrasses) and scarids (parrotfishes) produce mucus bubbles that they live in at night (Videler et al. 1999). These bubbles once penetrated may warn these fish of the presence of predators (Videler et al. 1999). However, using diet information for all reef-associated fish with ranges that include part of the area between latitudes 30oN and 30oS (the approximate geographic distribution of coral reefs, see section 1.1 in Chapter 1), obtained from FishBase (Froese and Pauly 2007), only 3 wrasse species were identified that are in the modelled groups in this thesis (cf. 52 parrotfish species). Thus, only parrotfish are assumed to produce significant amounts of mucus in the models in this thesis. From Johnson et al. (1995), herbivorous fish use 5% of their production to produce mucus, and are the only fish group taken to produce significant amounts of mucus. It is therefore assumed that for the models in this thesis, for herbivorous fish groups with parrotfish, only (1 − 0.05) P = (0.95) P is available for somatic growth or gamete formation. Next, formulae relating P and PG were derived. From Johnson et al. (1995), herbivorous fish use an equivalent of 13.7% of their biomass for reproduction per year. Thus, from this study, PG = (0.137 yr-1)B, where B is the biomass of the fish group being considered. From Van Rooij et al. (1998), PG for the parrotfish Sparisoma viride was 6.21-63.4% of P. Thus, on average, PG = {(0.0621 + 0.634) 2}P = (0.348) P . For each fish group in an ECOPATH model, PG is taken to be the average of the two PG estimates found using the formulae derived from these two studies. Lastly, formulae relating P and PS were derived. Since PS is P minus any production used to produce mucus and gametes, using Johnson et al. (1995), PS = P – (0.05)P – (0.137 yr-1)B = (0.95)P – (0.137yr-1)B for herbivorous fish groups with parrotfish, and PS = P – (0.137yr-1)B for other fish groups. Similarly, using Van Rooij et al. (1998) with Johnson et al. (1995), PS = P – (0.05)P – (0.348)P = (0.602)P for herbivorous fish groups with parrotfish, and PS = P – (0.348)P = (0.652)P for other fish groups. For each fish group in an ECOPATH model, PS is taken to be the average of 162
the two PS estimates found using the formulae derived from these two studies. For a particular group in an ECOPATH model, the consumption due to a particular food item is xQ , where x is the proportion of the total diet which is due to that food item and Q is the total consumption. The production due to a particular food item is assumed to be xP , where P is the total production. Thus, the production to consumption ratio due to a particular food item is equal to xP xQ = P Q . The somatic production due to a particular food item is xPS and hence, the somatic production to consumption ratio is PS Q .
10.1 Fish Recruitment Parameters The endogenous and exogenous recruitment rates for herbivorous fish, l Hen and l Hex respectively, were first parameterised. Estimates for these two parameters were derived by first calculating the recruitment rates for three acanthurids (surgeonfishes) in East Panama and for one pomacentrid (damselfish) in the GBR, using data primarily from Robertson (1988), Doherty and Fowler (1994), and FishBase (Froese and Pauly 2007). The upper bound of the recruitment rates found were than extended by scaling the results to derive recruitment rates for the cases when there are more herbivorous fish species. To do this, data on the number of herbivorous fish species on coral reefs from Bellwood et al. (2004) was used. Robertson (1988) measured the density of recently settled juveniles for three surgeonfish species, for patch reefs at Punta de San Blas, East Panama. Censuses were taken over 8 yrs in the period 1978–1986. The three species studied were Acanthurus bahianus, A. chirurgus and A. coeruleus. Six patch reefs were studied, with areas in the range 0.14-0.67 ha. Assuming that each patch reef takes the average value of 0.41 ha, the total area is 2.46 ha (24,600 m2). This assumption is necessary because the area for each patch reef is not given. The mean of the total number of adults over the eight year period is assumed to be the mean of the total number of adults in 1978 and 1986, which is 276, 67 and 99 for the three fish species respectively. This assumption is necessary because adult abundances for all six patch reefs are only given in 1978 and 1986. Over 8 yrs in 1978-1986, the average number of recruits per year was 1,630, 1,020 and 183 respectively. Russ (1991), by reviewing recruitment studies of different reef fish species in the Caribbean and the Indo-Pacific, reported that the mean mortality rate of newly 163
settled fish in their first year was around 50%. By assuming this rate for the three surgeonfish species studied in Robertson (1988), the yearly number of recruits into the 1 yr age class was 815, 510 and 92 respectively. From FishBase (Froese and Pauly 2007), the biomass of a 1 yr recruit is 17.0 g, 54.7 g and 22.5 g for A. bahianus, A. chirurgus and A. coeruleus respectively. Using the total study area as derived earlier, this gives the total biomass of 1 yr recruits in 1 yr as 0.563 g m-2, 1.13 g m-2 and 0.0842 g m-2 respectively. Assuming that the length of an average adult fish is halfway between the length at 1 yr old and the asymptotic length, the latter two lengths derived from FishBase data (Froese and Pauly 2007), the average biomass of an adult fish is 64.3 g, 424 g and 348 g for A. bahianus, A. chirurgus, and A. coeruleus respectively. This gives the total biomass of adults as 0.721 g m-2, 1.15 g m-2 and 1.40 g m-2 respectively. Thus, over a year, and considering all three species of surgeonfish, l Hex + l Hen H = l Hex + l Hen (0.721 g m-2 + 1.15 g m-2 + 1.40 g m-2) = 0.563 g m-2 + 1.13 g m-2 + 0.0842 g m-2, i.e. l Hex + l Hen (3.27 g m-2) = 1.78 g m-2. From Swearer et al. (1999), otolith signatures showed that at least 44.5% of Thalassoma bifasciatum recruits were of local origin on leeward reef sites in St. Croix, during the summer recruitment period. At Lizard Island in the GBR, Jones et al. (1999) found that 15-60% of Pomacentrus amboinensis recruits were of local origin. For a reef area at Kimbe Island in Papua New Guinea, Almany et al. (2007) found that ≈ 60% of settled juveniles were of local origin, for Amphiprion percula (pelagic larval duration of ≈ 11 d) and Chaetodon vagabundus (longer pelagic larval duration of 38 d). Thus, assume that up to 60% of fish recruitment can be of local origin. Then solving l Hen (3.27 g m-2) = 0.6 × 1.78 g m-2 gives the maximum value of l Hen as 0.327 over a year, giving a range for l Hen of 0-0.327 yr-1. Also, the minimum value of l Hex = 0.4 × 1.78 g m-2 = 0.712 g m-2 over a year, giving a range for l Hex of 0.712-1.78 g m-2 yr-1. This estimate of l Hex is also assumed to hold for an equivalent number of surgeonfishes (three) on Indo-Pacific reefs, in the absence of surgeonfish recruitment data in the Indo-Pacific. Another estimate of l Hex can be obtained from Doherty and Fowler (1994). From this study, for patch reefs in seven lagoons in the GBR, the recruit density over a year for the damselfish P. moluccensis was found to be 0.028-0.404 m-2 yr-1. Most recruits colonised the reefs between November and February each year, and the surveys were taken in April. Thus, assume that on average, the recruits are 3.5 months old. The mean mortality rate of newly settled fish in their first year is 50% (Russ 1991); thus, assume 164
that the recruits in the study have suffered 50 × 3.5 12 = 14.6% mortality already. Thus, the 1 yr old recruit density over a year can be found by multiplying the recruit density over a year by 0.5 (1 − 0.146 ) . This gives a range of 0.0164-0.237 m-2 yr-1 for the 1 yr old recruit density over a year. From FishBase, the biomass of a 1 yr old recruit of this species is 1.19 g (Froese and Pauly 2007). Thus, the biomass of 1 yr old recruits is 0.0195-0.282 g m-2. As before, assume that up to 60% of recruitment can be of local origin. This gives an estimate of 0.00780-0.282 g m-2 yr-1 for the range of l Hex . The two estimates for l Hex are probably underestimates because it is likely that only a subset of all herbivorous fish species were considered (three acanthurids or one pomacentrid). From Bellwood et al. (2004), the total number of species of herbivorous fish in the Caribbean and the GBR is estimated to be 27 and 91 respectively. Thus, the upper bound for the first estimate of l Hex is increased by a factor of 91/3 to 54.0 g m-2 yr1
, and that for the second is increased by a factor of 91 to 25.7 g m-2 yr-1. This gives a
range for l Hex of 0.00000780-0.0540 kg m-2 yr-1. l Hen is not changed because it is assumed that all additional herbivorous fish species recruit endogenously at the same rate as for the three acanthurid species considered in Robertson (1988), due to a lack of endogenous recruitment data for these additional fish species. Rounding to the nearest significant figure, l Hen is in the range 0-0.3 yr-1 and l Hex is in 0.000008-0.05 kg m-2 yr-1 (Table 10.1). The endogenous and exogenous recruitment rates for piscivorous fish, l Pen and l Pex respectively, were parameterised next. First, to derive estimates for l Pen , it is assumed that the endogenous recruitment rates are proportional to the rate at which gametes are produced locally. Thus, l Pen can be derived by multiplying l Hen by the ratio of PG B for piscivorous fish to PG B for herbivorous fish. For a fringing back-reef flat in Bolinao, Alino et al. (1993) found a PG B ratio of 0.975 yr-1 for herbivorous fish and 0.405 yr-1 for piscivorous fish. Thus, piscivorous fish produce gametes at a rate which is 0.405 0.975 = 0.415 times that for herbivorous fish. Using the same method of dividing PG B for piscivorous fish by PG B for herbivorous fish, and using data from Opitz (1996), Arias-Gonzalez et al. (1997), AriasGonzalez (1998), Gribble (2003), Arias-Gonzalez and Morand (2006), and Tsehaye and Nagelkerke (2008) for reefs in the Caribbean, French Polynesia, the Mexican Caribbean, the GBR, French Polynesia and the Red Sea respectively, piscivorous fish produce 165
gametes at a rate which is up to 1.26 times that for herbivorous fish. The range for l Pen is then estimated to be 0- (1.26)(0.327 ) yr-1 = 0-0.412 yr-1 (recall that 0.327 yr-1 is the maximum value of l Hen , the endogenous recruitment rate for herbivorous fish). Estimates for l Pex could not be derived directly due to a lack of data and thus, was derived from l Hex . To do this, it was assumed that the exogenous recruitment rates derived from Robertson (1988), for three surgeonfish species, and from Doherty and Fowler (1994), for one damselfish species, also apply for piscivorous fish species. Similar to the parameterisation of l Hex , these rates then need to be scaled up according to the number of piscivorous fish species that could be at a particular reef area. Bellwood et al. (2004) does not give data on the number of piscivorous fish species in the Caribbean or the GBR. However, 166 species of piscivorous fish were identified from FishBase (Froese and Pauly 2007; only fish that are reef-associated and which are found within the area between latitudes 30oN and 30oS, the approximate geographic distribution of coral reefs – see section 1.1. in Chapter 1 – were considered), which suggests that the number of species of piscivorous fish at a particular reef can be as many as the number of species of herbivorous fish. Thus, it is assumed that the upper bound of the range for l Pex is the same as that for the range of l Hex , such that the range of l Pex is the same as the range of l Hex . Rounding to the nearest significant figure, l Pen is in the range 0-0.4 yr-1 and l Pex is in 0.000008-0.05 kg m-2 yr-1 (Table 10.1). The endogenous recruitment rates for small to intermediate piscivorous (SIP) en fish due to reproduction by SIP fish and large piscivorous (LP) fish, l Ps and l Plen
respectively, were derived from l Hen using the same method used to derive l Pen . Thus, it is assumed that endogenous recruitment rates are proportional to the rate at which en gametes are produced locally, such that l Ps is calculated from l Hen by multiplying by the en ratio of PG B for SIP fish to PG B for herbivorous fish. Similarly, l Ps is calculated
from l Hen by multiplying by the ratio of PG B for LP fish to PG B for herbivorous fish. From Opitz (1996), for a Caribbean reef, the PG B ratio for herbivorous fish, SIP fish and LP fish are 0.230 yr-1, 0.207 yr-1 and 0.133 yr-1 respectively. Thus, SIP fish produce gametes at a rate that is 0.207 0.230 = 0.900 times that for herbivorous fish, whereas for LP fish, it is 0.133 0.230 = 0.578 times. From Arias Gonzalez et al. (1997), for a French Polynesian fringing reef, SIP fish produce gametes at a rate which is 1.21 166
times that for herbivorous fish, whereas for LP fish, it is 0.350 times. In the same study, for a French Polynesian barrier reef, SIP fish produce gametes at a rate which is 1.20 times that for herbivorous fish, whereas for LP fish, it is 0.369 times. Thus, the range en for l Ps is estimated to be 0- (1.21)(0.327 ) yr-1 = 0-0.396 yr-1 and that for l Plen is estimated
to be 0- (0.578)(0.327 ) yr-1 = 0-0.189 yr-1. ex could not be Estimates for the exogenous recruitment rate for SIP fish, l Ps
derived directly due to a lack of data and thus, was derived from l Hex in the same way as ex l Pex . This gives a range for l Ps which is the same as the range for l Hex . en Rounding to the nearest significant figure, l Ps is in the range 0-0.4 yr-1, l Plen is in ex 0-0.2 yr-1 and l Ps is in 0.000008-0.05 kg m-2 yr-1 (Table 10.1).
10.2 Fish Feeding and Growth Parameters
The parameters for herbivorous fish were obtained first. Values for i H , which measures the ‘inaccessibility’ of algae to herbivorous fish, were found by initially calculating how much grazing pressure, θ , is exerted by a given biomass of herbivorous fish H on reefs with a range of rugosities. A range for i H was then calculated by equating each θ value to θ expressed as a function of H (Chapters 6 and 7, eqns. 6.1a,b and 7.1) and rearranging. To calculate the θ values, consider the Coral-Turf model (CTm; Chapter 4) with no corals (i.e. coral cover C is fixed at 0) and only herbivorous fish as grazers. For this system, at equilibrium:
ζ T (1 − θ )S − g T θT = ζ T S − (ζ T S + g T T )θ = 0 ⇒ θ=
ζTS , gT T + ζ T S
(10.1)
where S + T = 1 . The study by Williams et al. (2001) showed that on a 12 m fore-reef with a rugosity of 1.2, at Ambergris Caye in Belize, a herbivorous fish population with a biomass of 0.0156 kg m-2 can keep 50% of the reef substratum in a cropped state. Thus, T = S = 0.5 and eqn. 10.1 gives:
167
θ1 =
ζT gT + ζ T
,
(10.2)
where θ1 is θ for a rugosity of 1.2 and a fish population of biomass 0.0156 kg m-2. If the rugosity R changes by a factor p, then between two points on a reef area, the benthic length changes by the factor p. Therefore, it is assumed that the reef area changes by the factor p 2 . This is assumed to lead to θ1 changing by the factor 1 p 2 , because θ1 measures the amount of algae that can be grazed per unit planar reef area and this is dependent on the surface area per unit planar reef area. Using the explicit form of the grazing pressure (eqns. 6.1a,b and 7.1),
θ1 p
2
=
H , iH + H
(10.3)
where p = R / 1.2 and H is the herbivorous fish biomass per m2, which is 0.0156 kg m-2. Rearranging eqn. 10.3 gives:
iH =
(
H p 2 − θ1
θ1
).
(10.4)
From reefs worldwide, R takes values in the range 1.12-2.43 (Klumpp and McKinnon 1989, McClanahan and Shafir 1990, Ostrander et al. 2000, Edmunds 2002, Miller and Gerstner 2002, Beltran-Torres et al. 2003, Mumby et al. 2005, Irizarry-Soto 2006, Mangi and Roberts 2007). From section 9.2 in Chapter 9, g T and ζ T take values in the ranges 5-15 yr-1 and 2-20 yr-1 respectively. Thus, using eqn. 10.4, i H is in the range 0.001-0.5 kg m-2 (Table 10.1). Next, parameter values for µ M , µ T , µ S , which measure the biomass accumulated by herbivorous fish due to grazing on macroalgae, turf algae and ‘space’ respectively, were found. For simplicity, it is assumed that the amount of biomass that is consumed, and that contributes to herbivorous fish growth, is the same for each food source: macroalgae, turf or space. That is, µ M = µ T = µ S = µ . Somatic production per year due to algal consumption, PSA , was first calculated for herbivorous fish groups from different ECOPATH models (Alino et al. 1993, Opitz 1996, Arias-Gonzalez et al. 1997, Arias-Gonzalez 1998, Gribble 2003, Arias-Gonzalez and Morand 2006, Tsehaye and 168
Nagelkerke 2008). PSA is PS multiplied by the fraction of the diet which is algae. Then, assuming as a simplification that urchins have no effect on herbivorous fish grazing (i.e.
λU = 0 ; see Chapter 11, section 11.2), PSA is equivalent to
µ ( g M M + g T T + ζ T S ){H (i H + H )} (the grazing term for herbivorous fish, see eqns 6.3 and 6.13 in Chapter 6), where M, T and S are the proportional covers of macroalgae, turf algae and space respectively and H is the herbivorous fish biomass. Also, it is assumed that M = T = S = (1 3)C . A range of values of µ was then found for each PSA value (and hence each reef in the ECOPATH models) by equating PSA to
µ ( g M M + gT T + ζ T S ){H (iH + H )} and rearranging for µ , using estimates of proportional covers and H from the literature for the reefs in the ECOPATH models, parameter ranges derived in section 9.2 for the benthic parameters ( g M , gT and ζ T ) and values of i H in the range derived above. For the reef in Alino et al. (1993), C = 0.1 , as given in the same study; for the reef in Opitz (1996), C = 0.1 (Gardner et al. 2003), which is the mean coral cover of reefs in the Caribbean basin in 2001; for the reefs in Arias-Gonzalez (1998), C = 0.128 (Gardner et al. 2003), which is the mean coral cover of reefs in the Caribbean basin in 1998; for the French Polynesian fringing and barrier reefs in Arias-Gonzalez et al. (1997), C = 0.2 and 0.35 respectively, as given in the same study; for the reef in Gribble (2003), C = 0.200 (Bellwood et al. 2004), which is the mean coral cover of reefs in the GBR in 2003; for the reef in Tsehaye and Nagelkerke (2008), C = 0.5 , as given in the same study. The herbivorous fish biomasses in the ECOPATH models are used as the H values. From section 9.2, g M takes values in the range (0.01)g T - gT , g T = 5-15 yr-1 and ζ T = 2-20 yr-1. Also, using the same method used above to derive the range of i H for reefs worldwide, but only using rugosity values from western Atlantic reefs, i H for western Atlantic reefs lies in the range 0.003-0.5 kg m-2. This range of values was used for reefs in the ECOPATH models that are from the western Atlantic, such as the Caribbean reef in Opitz (1996). Similarly, using the same method and rugosity values from Indo-Pacific reefs only, i H for Indo-Pacific reefs lies in the range 0.001-0.4 kg m-2, and this range was used for reefs in the ECOPATH models that are from the IndoPacific.
µ was then calculated using these values, for each of the reefs in the ECOPATH models. The total range for µ derived is 0.00006-0.5 kg m-2 (Table 10.1). 169
The upper limit of this range is modified by eqn. 11.7 in Chapter 11, due to the need to satisfy constraints that ensure the fish and urchin models and integrated models make biological sense (specifically, constraints 10.6c and 11.4). Note that for the fish and urchin models, the amount of herbivorous fish biomass gained due to grazing on algae, C B = µ ( g M M + g T T + ζ T S ) (section 6.2.4), is taken to range from its theoretical minimum zero (when M + T + S = 0 ) to its theoretical maximum 10 kg m-2 yr-1 (when S = 1 , ζ T = 20 yr-1 and µ = 0.5 kg m-2); this maximum value is then modified by an equation similar to eqn. 11.7. The parameters for piscivorous fish were then parameterised. First, the parameters that measure inaccessibility of prey groups to predator groups, i PH , i PP and i PlPs (see eqns. 6.4 and 6.14 in Chapter 6), were quantified. To parameterise i PH , it was assumed that the predation rate of piscivorous fish on herbivorous fish saturates (i.e.
(
)
2 H 2 i PH + H 2 ≈ 1 – see eqn. 6.4 in Chapter 6) between a quarter of the maximum
herbivorous fish biomass (H) reported in the literature and half of this maximum. This is not much more than a guess, but this assumption was necessary because of a lack of data. From section 6.2.2 in Chapter 6, the highest herbivorous fish biomass is 0.282 kg m-2 (Letourner et al. 1998). This gives:
H2 = 0.99 ≈ 1 , 2 i PH +H2
(10.5)
where H varies between 0.0705-0.141 kg m-2. Using eqn. 10.5, i PH = 0.007-0.01 kg m-2 (Table 10.1). It is assumed that piscivorous fish can be as accessible to other piscivorous fish predators as herbivorous fish, such that i PP = i PH = 0.007-0.01 kg m-2 (Table 10.1).
(
)
2 Figure 6.2 in Chapter 6 shows the graph of X 2 i PX + X 2 against X for X ∈ {H , P}
and different values of i PX in the empirically derived range. Similarly, it is assumed that SIP fish can be as accessible to LP fish predators as herbivorous fish are to piscivorous fish, such that i PlPs = i PH = 0.007-0.01 kg m-2 (Table 10.1). Next, the maximum predation rates were parameterised. For g P , the maximum predation rate of piscivorous fish on herbivorous fish, this was done by first obtaining the QH B ratios of piscivorous fish groups from ECOPATH models. Here. QH is the consumption of herbivorous fish, which is Q multiplied by the fraction of the 170
piscivorous fish diet that is herbivorous fish. This QH B ratio is equivalent to
{
(
)} (see eqn. 6.3 in Chapter 6), where H is the herbivorous fish biomass, and thus Q B was equated to g {H (i + H )}. Using values of H from 2 g P H 2 i PH +H2
2
H
P
2 PH
2
the same ECOPATH models and the range of i PH derived above then allows a range for g P to be calculated for each QH B ratio, and hence each ECOPATH model. For example, from the ECOPATH model of Alino et al. (1993), for a fringing back-reef in Bolinao and for piscivorous fish, QH B is 0.658 yr-1. From above, i PH is in the range 0.007-0.01 kg m-2 and from the model, H = 0.00275 kg m-2. Solving
{
(
)}
2 g P H 2 i PH + H 2 = QH B for g P gives a range 5-9 yr-1. Using data from the
ECOPATH models in Opitz (1996), Arias-Gonzalez et al. (1997), Arias-Gonzalez (1998), Gribble (2003), Arias-Gonzalez and Morand (2006), and Tsehaye and Nagelkerke (2008), and using the same method, gives 14 more ranges for g P . The total range of g P is then 0.9-17 yr-1 (Table 10.1). The upper limit of this range is modified by constraint 10.11 (see below), due to the need to satisfy (sufficient, but not necessary) constraints that ensure the piscivorous fish biomass remains finite. The maximum predation rate of piscivorous fish on other piscivorous fish is
ψ P g P , where ψ P is the maximum predation rate of piscivorous fish on other piscivorous fish relative to that on herbivorous fish. ψ P was parameterised by first obtaining the QH B and QP B ratios of piscivorous fish groups from ECOPATH models. Here, QP is the consumption of piscivorous fish, which is Q multiplied by the fraction of the piscivorous fish diet that is other piscivorous fish, and is equivalent to 2 ψ P g P {P 2 (i PP + P 2 )} (see eqn. 6.4 in Chapter 6). Here, P is the piscivorous fish
biomass, which can be obtained from the ECOPATH models. QP B divided by QH B
{ (
2 is equivalent to ψ P g P P 2 i PP + P2
)} divided by g {H (i 2
P
2 PH
+H2
)} and equating the
two expressions gives an equation independent of g P . Using values of H from the same ECOPATH models and the ranges of i PH and i PP derived above then allows a range for ψ P to be derived for each ECOPATH model. For example, from the ECOPATH model of Alino et al. (1993), QH B is 0.658
{ (
2 yr-1 and QP B is 0.651 yr-1. Thus, 0.651 0.658 is equal to ψ P g P P 2 i PP + P2
{
(
)}
)}
2 divided by g P H 2 i PH + H 2 . As determined above, i PH and i PP are in the range
0.007-0.01 kg m-2 and from the model, H = 0.00275 kg m-2 and P = 0.00143 kg m-2. 171
Using the fact that ψ P is a decreasing function of i PH and an increasing function of i PP , solving for ψ P gives a range 2-7. Using data from the ECOPATH models in Opitz (1996), Arias-Gonzalez et al. (1997), Arias-Gonzalez (1998), Gribble (2003), AriasGonzalez and Morand (2006), and Tsehaye and Nagelkerke (2008), and using the same method, gives 14 more ranges for ψ P . The total range of ψ P is 0.01-7 (Table 10.1). To parameterise the maximum predation rates of SIP fish and LP fish on herbivorous fish (in the HSLUm), g Ps and g Pl respectively, a method analogous to that for g P was employed, using data from Opitz (1996) for an ECOPATH model of a Caribbean reef. For this reef and for SIP fish, the QH B ratio on herbivorous fish is 2 + H 2 )}. Using H = 0.0443 kg m-2 (from the 1.44 yr-1 and is equivalent to g Ps {H 2 (i PH
model) and the range of i PH derived above, solving for g Ps gives the range 1-2 yr-1. For the same study, for LP fish, the QH B ratio on herbivorous fish is 0.477 yr-1 and is 2 equivalent to g Pl {H 2 (i PH + H 2 )}. Solving for g Pl gives 0.5 yr-1, and this is extended to
a +/– 50% range to account for uncertainty, which gives the range 0.25-0.75 yr-1 (Table 10.1). To parameterise the maximum predation rate of LP fish on SIP fish relative to that on herbivorous fish, ψ Pl , a method analogous to that used to parameterise ψ P is used, again using data from the ECOPATH model of Opitz (1996). From this model, for LP fish, the QPs B ratio, where QPs is consumption of SIP fish, is 0.447 yr-1 and is 2 equivalent to ψ Pl g Pl {Ps2 (i PlPs + Ps2 )} (see eqn. 6.14 in Chapter 6). Similarly, the QH B 2 + H 2 )}. Thus, ratio is 0.477 yr-1 and is equivalent to g Pl {H 2 (i PH 2 2 ψ Pl g Pl {Ps2 (i PlPs + Ps2 )} divided by g Pl {H 2 (i PH + H 2 )} is equal to 0.447 0.477 , and
the resulting equation is independent of g Pl . From earlier in this section, i PlPs was found to be 0.007-0.01 kg m-2 and from the model, Ps = 0.00565 kg m-2. Using the fact that ψ Pl is a decreasing function of i PH and an increasing function of i PlPs , solving for
ψ Pl gives a range 2-4 (Table 10.1). Note that self-predation for the SIP fish group is not modelled. This can be justified by Opitz (1996), who showed that the proportion of the diet of small SIP fish due to self-predation is 0.002, a very low number. Also, selfpredation for the LP fish group is not modelled. From Opitz (1996), the proportion of their diet due to self-predation is 0.052, which is quite low. The parameters rP , rPs and rPl measure the proportion of consumed biomass 172
that is used for somatic growth for the model piscivorous fish groups, and were parameterised by calculating PS Q ratios for piscivorous fish groups from ECOPATH models. The PS Q ratio for a piscivorous fish group with biomass X represents the proportion of biomass consumed, of any prey type, which is used for somatic growth, and is thus equal to rX . Using data from the ECOPATH models of Alino et al. (1993), Opitz (1996), Arias-Gonzalez et al. (1997), Arias-Gonzalez (1998), Gribble (2003), Arias-Gonzalez and Morand (2006), and Tsehaye and Nagelkerke (2008), PS Q for piscivorous fish groups is in the range 0.03-0.2, which is used as the range for rP (Table 10.1). From Opitz (1996), for an ECOPATH model of a Caribbean reef and for SIP fish, the ratio PS Q is 0.0691, which is equal to rPs . Arias-Gonzalez et al. (1997) created an ECOPATH model for a fringing reef and a barrier reef in French Polynesia. Using data for the two reefs from this study and the same method, two further values of PS Q for SIP fish were derived. Rounding to the nearest significant figure gives the range 0.070.1, which is used as the range of rPs (Table 10.1). Using data from Opitz (1996) and Arias-Gonzalez et al. (1997) again, three values for the PS Q ratio for LP fish were derived. Rounding to the nearest significant figure gives the range 0.03-0.06, which is used as the range for rPl (Table 10.1). The final predation and growth parameter for fish is φ Ps , which determines how quickly SIP fish biomass becomes LP fish biomass due to predation and subsequent somatic growth. φ Ps has a minimum value of zero, which corresponds to the case where there are no piscivorous fish species that can become large. However, the upper limit of the range of φ Ps cannot be derived using empirical data, due to a lack of relevant data. This upper limit is assumed to be 10 – this is a reasonably large value which means that SIP fish biomass can become LP fish biomass at a rate that is 10 times the rate at which SIP fish accumulate biomass through predation (see eqn. A6.13 in Appendix A6). Thus, the range for φ Ps is taken to be 0-10 (Table 10.1).
10.3 Fish Mortality Parameters The natural mortality rates for fish, which do not include mortality due to predation or fishing, were parameterised first. Values for these natural mortality rates were estimated 173
using total (instantaneous) mortality rates for fish species from FishBase (Froese and Pauly 2007). Only fish species that correspond to the herbivorous and piscivorous fish in this thesis (see Chapter 6, section 6.2.1) were considered. In addition, only reefassociated fish with ranges that include part of the area between latitudes 30oN and 30oS (the approximate geographic distribution of coral reefs, see section 1.1 in Chapter 1) were considered – whether a fish species is reef-associated or not is given in FishBase, as is the range (Froese and Pauly 2007). All species in 78 families were checked; the 78 chosen families have at least one fish species that have been recorded in coral reef fish surveys by Loreto et al. (2003), the MBRSP (2005) and the SAGIP Lingayen Gulf Project (2005). For 39 of these families, no species were found that were in the model groups in this thesis. FishBase mortality rates include predation mortality and mortality due to other causes apart from fishing, and are calculated from the empirical equations of Pauly (1980). They refer to the late juvenile and adult phases of a population (Froese and Pauly 2007). These FishBase mortality rates are used as upper bound estimates for the natural mortality rates for the models in this thesis, but are likely to be overestimates because they include mortality from predation. A SIP fish in this thesis has a total length of < 60 cm, and a LP fish has a total length of ≥ 60 cm (Chapter 6, section 6.3). Thus, to estimate the natural mortality rate for the SIP fish group, the total mortality rates of piscivorous fish species with an asymptotic length < 60 cm from FishBase were used. Similarly, to estimate the natural mortality rate for the LP fish group, the total mortality rates for piscivorous reef fish species that have an asymptotic length of ≥ 60 cm from FishBase were used. Estimates for the LP fish group may be an overestimate not only because predation mortality is included, but because the FishBase rates may include the life cycle of the fish when they are < 60 cm and hence more susceptible to mortality. Using this method, the natural mortality rates for herbivorous fish, piscivorous fish, SIP fish and LP fish – d H , d P , d Ps and d Pl respectively – lie in the ranges 0.2-5 yr-1, 0.07-3 yr-1, 0.2-3 yr-1 and 0.07-1 yr-1 respectively (Table 10.1). The upper bounds of these ranges were modified such that the mortality rates always satisfy the constraints which ensure the models make biological sense (Chapters 6 and 7, sections 6.2.4, 6.3.2 and 7.1). If necessary, the upper bounds of other parameters are modified as well. From the mathematical analysis of the integrated models (section 7.1), the following constraints ensure that the herbivorous fish biomass remains finite and tends to zero with no recruitment: 174
µ max{g M , g T , ζ T } iH
< dH ,
(10.6a)
l Hen < d H ,
(10.6b)
where µ M = µ T = µ S = µ . Eqn. 10.6b is also required for either of the fish and urchin models to make biological sense (sections 6.2.4 and 6.3.2). From sections 10.1 and 10.2,
i H = 0.001-0.5 kg m-2 and l Hen = 0-0.3 yr-1. Since g M ≤ g T (Chapter 9, section 9.2), eqn. 10.6a is equivalent to:
µ max{g T , ζ T } iH
< dH .
(10.6c)
Eqn. 10.6c sets a lower limit to d H . This lower limit varies from 0.0006-10,000 yr-1, so to prevent the situation where no value of d H in 0.2-5 yr-1 will satisfy condition 10.6c, an upper limit is placed on µ , µ max 1 kg m-2, given by
µ max 1 =
i H (5 − 0.000001) . max{g T , ζ T }
The straight brackets
(10.7)
means that the numerical value of the term inside the brackets
is taken – for example, 0.5 kg = 0.5 . Also, the 0.000001 is required to ensure that the strict inequality in condition 10.6c holds; if 000001 is not included, then equality between the left- and right-hand sides of 10.6c may occur. From eqn. 10.7,
µ max 1 ≥ 0.00006 is guaranteed, 0.00006 being the lowest empirically derived magnitude of µ . Note that constraint 11.4 from Chapter 11 sets another upper limit to µ (see eqn. 11.7). Together with eqn. 10.6b, which sets another lower limit to d H , d H takes values in the range d H min -5 yr-1, where d H min = max{0.2, d H min 1 } and
175
µ max{g T , ζ T } d H min 1 = max + 0.000001, i H
l Hen + 0.000001 .
(10.8)
If the fish and urchin models are being considered, rather than the integrated models, then µ max{g M , gT , ζ T } in eqns. 10.6a and 11.4 are replaced by C B (see sections 6.2.4 and 6.3.2) and analogous to µ , an upper bound is placed on C B which has an explicit form similar to µ max in section 11.3. In addition, µ max{g T , ζ T } in eqns. 10.8 and 11.6 is replaced by C B , and since C B can take values down to zero, there is no need to set a lower limit to iU , such that iU min 1 specified by eqn. 11.8 (see section 11.3) is redundant. From the mathematical analysis of the HPUm (section 6.2.4), the following constraint ensures that the piscivorous fish biomass remains finite. l Pen + rP g P < d P .
(10.9)
From sections 10.1 and 10.2, l Pen takes values in the range 0-0.4 yr-1, rP is in the range 0.03-0.2 and g P is in the range 0.9-17 yr-1. Condition 10.9 sets a lower limit for d P . This lower limit varies from 0.02703.80 yr-1, so to prevent the situation where no value of d P in 0.07-3 yr-1 will satisfy 10.9, an upper limit is placed on g P , g P max yr-1, where g P max = min{17, g P max 1 } and
g P max 1 =
3 − l Pen − 0.000001 . rP
(10.10)
The range for d P is then d P min -3 yr-1, where d P min = max{0.07, d P min 1 } and
d P min 1 = l Pen + rP g P + 0.000001 .
(10.11)
For the two size-class piscivorous fish model (HSLUm), the mathematical analysis (section 6.3.2) shows that the following constraints ensure that the SIP fish biomass and LP fish biomass remain finite and tend to zero with no recruitment: en l Ps + rPs g Ps < d Ps ,
(10.12a) 176
l Plen + rPl g Pl < d Pl ,
(10.12b)
rPl g Pl (1 + ψ Pl ) < d Pl .
(10.12c)
en From sections 10.1 and 10.2, l Ps takes values in the range 0-0.4 yr-1, rPs is in the range
0.07-0.1, g Ps is in the range 1-2 yr-1, rPl is in the range 0.03-0.06, g Pl is in the range 0.25-0.75 yr-1, ψ Pl is in the range 2-4 and l Plen takes values in the range 0-0.2 yr-1. Equation 10.12a sets a lower limit to d Ps and eqns. 10.12b,c set a lower limit to d Pl . The lower limit to d Ps varies between 0.0700-0.600 yr-1 and that for d Pl varies between 0.0075-0.245 yr-1. Thus, let d Ps take values in the range d Ps min -3 yr-1, where d Ps min = max{0.2, d Ps min 1 } and
en d Ps min 1 = l Ps + rPs g Ps + 0.000001 .
(10.13)
Similarly, let d Pl take values in the range d Pl min -1 yr-1, where d Pl min = max{0.07, d Pl min 1 } and
{
}
d Pl min 1 = max l Plen + rPl g Pl + 0.000001, rPl g Pl (1 + ψ lc ) + 0.000001 .
(10.14)
Next, mortality due to fishing was considered for all fish groups. First, the inaccessibility of the fish groups to fishermen, i FH , i FP , i FPs and i FPl , were parameterised. From section 6.2.2 in Chapter 6, the highest herbivorous fish biomass is 0.282 kg m-2 (Letourner et al. 1998). To parameterise i FH , it is assumed that the availability of herbivorous fish biomass per unit of fishing pressure saturates (i.e. H (i FH + H ) ≈ 1 – see eqns. 6.3 and 6.13 in Chapter 6) between a quarter of the maximum herbivorous fish biomass (H) reported and half of this maximum. As in the derivation of i PH , the parameter measuring the inaccessibility of herbivorous fish to piscivorous fish (see section 10.2), this is not much more than a guess, but this assumption is necessary because of a lack of data. Then: 177
H = 0.99 ≈ 1 , i FH + H
(10.15)
where H varies between 0.0705-0.141 kg m-2. Using eqn. 10.15, i FH = 0.0007-0.001 kg m-2 (Table 10.1). It is assumed that piscivorous fish can be as accessible to fishermen as herbivorous fish, and hence i FP = i FH = 0.0007-0.001 kg m-2 (Table 10.1). For the most complex fish and urchin model, it is assumed that SIP fish and LP fish can be as accessible to fishermen as herbivorous fish, such that i FPs = i FPl = i FH = 0.0007-0.001 kg m-2 (Table 10.1). X (i FX + X ) , X ∈ {H , P, Ps , Pl } , is the fraction of the maximum possible catch rate achievable with fish biomass X. Figure 10.1 shows how X (i FX + X ) changes with X ∈ {H , P, Ps , Pl } for X from 0 to 0.005 kg m-2, which is well within the range of biomass of herbivorous, piscivorous, SIP or LP fish found from the literature (see section 6.2.2; SIP and LP fish biomasses are derived from the same sources used to derive herbivorous and piscivorous fish biomasses). The graph shows that for all values of iFX in the empirically derived range, X (i FX + X ) reaches 0.8 at relatively low biomasses of < 0.005 kg m-2, so a large proportion of the maximum catch rate possible is achieved at relatively low biomasses. However, for biomasses < 0.001 kg m-2, there is a steep decrease in X (i FX + X ) , which allows recovery of fish biomass. X i FX + X
X
Figure 10.1. Graph showing how X (i FX + X ) , X ∈ {H , P, Ps , Pl } , changes as X increases from zero to 0.005 kg m-2. Values of i FX from the empirically derived parameter range 0.0007-0.001 kg m-2 are used. The values for i FX are, from top to bottom: 0.0007 kg m-2, 0.0008 kg m-2, 0.0009 kg m-2, and 0.001 kg m-2.
178
Values of f, the maximum catch rate, were estimated using fish catches from different reef fisheries around the world (Stevenson and Marshall 1974, Luchavez et al. 1984, Russ 1991, McManus et al. 1992, Alino et al. 1993, Koslow et al. 1994, Amar et al. 1996, Dalzell and Adams 1997, Laroche et al. 1997, Russ and Alcala 1998, PetSoede et al. 1999, Licuanan and Gomez 2000, White et al. 2000, Halls et al. 2002, Rodwell et al. 2003, Cheung and Sadovy 2004, Arias-Gonzalez et al. 2004, Kuster et al. 2005, Alcala and Russ 2006, Zeller et al. 2006, Rhodes and Tupper 2007, Tsehaye and Nagelkerke 2008, Craig et al. 2008, Rhodes et al. 2008). When possible, only herbivorous and piscivorous fish are considered from a particular study. From FishBase data (Froese and Pauly 2007), the families Balistidae, Haemulidae, Labridae, Lethrinidae, Mullidae, Priacanthidae and Sparidae are considered to consist mainly of invertivores and therefore fishing figures for these groups are excluded if possible. For Indo-Pacific reefs, the family Mugilidae is excluded if possible, because no herbivorous fish species were found for this family for the Indo-Pacific, using FishBase data (Froese and Pauly 2007). In the studies of fishing, it is not always clear whether the area fished comprises only coral habitats or whether it includes other habitats such as seagrass beds and mangroves. If other habitats are included, it is assumed that the catch per unit area is the same for all habitats, such that the reported figures can be used for coral habitats only. Also, it is not always clear that the fish caught are only herbivorous and piscivorous fish. If there are other types, it is unclear whether the fishermen can switch to catching the same amount of herbivorous and piscivorous fish if they stopped catching these other types. Thus, in these cases, the figures reported are used as estimates of upper limits for the amount of herbivorous and piscivorous fish that can be caught. Finally, the standing stock biomasses of the fish populations are not usually given together with the fish biomass that is caught. Thus, it is assumed that the catch values reported are achieved with a large enough standing stock biomass such that accessibility to fish biomass for fishers is not the main limiting factor for the fish biomass that can be caught; rather, the main limiting factors are the number of fishing hours and the capacity of the type of gears used. This is a reasonable assumption because from earlier in this section, the proportion of the maximum catch rate achieved is high ( > 0.8 ) at low fish biomasses of < 0.005 kg m-2. The range of catch values can then be used as direct estimates of f.
Using data from the sources listed above, the range of catch values is 0.0000790.064 kg m-2 yr-1. The lower limit is close to zero, which is the theoretical situation where there is no fishing. Thus, the lower limit is extended to zero to explore this no179
fishing situation as well. Then the range for f is 0-0.06 kg yr-1 to the nearest significant figure (Table 10.1). Lastly, the proportions of f that are herbivorous fish biomass, piscivorous fish biomass, SIP fish biomass and LP fish biomass – ρ H , ρ P , ρ Ps and ρ Pl respectively – were parameterised. Using the composition of landed fish catches from Alino et al. (1993), Amar et al. (1996), Laroche et al. (1997), Russ and Alcala (1998), Kuster et al. (2005), Rhodes and Tupper (2007), Craig et al. (2008) and Rhodes et al. (2008), ρ H was estimated to vary from 0.0190 to 0.888 and ρ P was estimated to vary from 0.112 to 0.981. This suggests that the entire theoretical range of ρ H and ρ P can be approximately attained in reality. Thus, ρ H and ρ P are taken from the entire theoretical range 0-1, with the necessary constraint ρ H + ρ P = 1 (Table 10.1). Since
ρ P = ρ Ps + ρ Pl and takes values in 0-1, it was assumed that ρ Ps and ρ Pl can both take values in 0-1 as well, subject to the necessary constraint ρ H + ρ Ps + ρ Pl = 1 (Table 10.1). These ranges for ρ allow all the theoretically possible fishing strategies for the modelled fish groups to be investigated.
180
Parameter
µ M , µT , µ S
Definition The herbivorous fish biomass accumulated
Empirically Derived Ranges
Model Specificity
0.00006 - µ max kg m-2
HPUm and
through growth from grazing on 100% cover of
HSLUm
macroalgae, turf or space (the turf within space is what is consumed) respectively
dH
The mortality rate of herbivorous fish from all
d H min - 5 yr-1
factors other than predation and fishing
iH
A parameter which measures the inaccessibility
0.001 - 0.5 kg m-2
of algae (turf and macroalgae) to herbivorous fish grazing
i FH
A parameter which measures the inaccessibility
0.0007 - 0.001 kg m-2
of herbivorous fish to fishermen
i PH
A parameter which measures the inaccessibility
0.007 - 0.01 kg m-2
of herbivorous fish to predation by piscivorous fish l Hen
0 - 0.3 yr-1
The endogenous recruitment rate of herbivorous fish
l Hex
The exogenous recruitment rate of herbivorous
0.000008 - 0.05 kg m-2 yr-1
fish
ρH
The proportion of the total fishing pressure f 181
0 - 1, with ρ H + ρ P = 1
(HPUm) or ρ H + ρ Ps + ρ Pl = 1
which acts on herbivorous fish
(HSLUm) f
dP
0 - 0.06 kg m-2 yr-1
The maximum catch rate due to fishing The mortality rate of piscivorous fish from all
d P min - 3 yr-1
factors other than predation and fishing
gP
The maximum predation rate of piscivorous
0.9 - g P max yr-1
fish on herbivorous fish
i FP
A parameter which measures the inaccessibility
0.0007 - 0.001 kg m-2
of piscivorous fish to fishermen
i PP
A parameter which measures the inaccessibility
0.007 - 0.01 kg m-2
of piscivorous fish to predation by other piscivorous fish l Pen
The endogenous recruitment rate of piscivorous
0 - 0.4 yr-1
fish l Pex
The exogenous recruitment rate of piscivorous
0.000008 - 0.05 kg m-2 yr-1
fish
rP
The proportion of biomass consumed by
0.03 - 0.2
piscivorous fish used for growth
ψP
The predation rate on piscivorous fish by other
182
0.01 - 7
HPUm only
piscivorous fish, relative to that on herbivorous fish
ρP
The proportion of the total fishing pressure f
0 - 1, with ρ h + ρ P = 1
which acts on piscivorous fish d Ps
The mortality rate of SIP fish from all factors
d Ps min - 3 yr-1
other than predation and fishing g Ps
The maximum predation rate of SIP fish on
1 - 2 yr-1
herbivorous fish i FPs
A parameter which measures the inaccessibility
0.0007 - 0.001 kg m-2
of SIP fish to fishermen i PlPs
A parameter which measures the inaccessibility
0.007 - 0.01 kg m-2
of SIP fish to predation by LP fish en l Ps
The endogenous recruitment rate of SIP fish
0 - 0.4 yr-1
due to reproduction by SIP fish ex l Ps
The exogenous recruitment rate of SIP fish
rPs
The proportion of biomass consumed by SIP
0.000008 - 0.05 kg m-2 yr-1 0.07 - 0.1
fish used for growth
φ Ps
A constant parameter that determines how quickly SIP fish biomass becomes LP fish biomass due to predation and subsequent 183
0 - 10
HSLUm only
growth
ρ Ps
The proportion of the total fishing pressure f
0 - 1, with ρ H + ρ Ps + ρ Pl = 1
which acts on SIP fish d Pl
The mortality rate of LP fish from all factors
d Pl min - 1 yr-1
other than predation and fishing g Pl
The maximum predation rate of LP fish on
0.25 - 0.75 yr-1
herbivorous fish i FPl
A parameter which measures the inaccessibility
0.0007 - 0.001 kg m-2
of LP fish to fishermen l Plen
The endogenous recruitment rate of SIP fish
0 - 0.2 yr-1
due to reproduction by LP fish rPl
The proportion of biomass consumed by LP
0.03 - 0.06
fish used for growth
ψ Pl
The predation rate on SIP fish by LP fish,
2-4
relative to that on herbivorous fish
ρ Pl
The proportion of the total fishing pressure f
0 - 1, with ρ H + ρ Ps + ρ Pl = 1
which acts on LP fish
Table 10.1. Fish parameters for the HPUm and the HSLUm, with their ecological meanings and empirically derived ranges. Explicit expressions for
d H min , d P min , g P min , d Ps min and d Pl min are given in the main text of section 10.3. An explicit expression for µ max is given in Chapter 11, section 11.3. 184
Chapter 11. Parameter Ranges for the Sea Urchin Group Sea urchin parameters for recruitment, feeding and growth, and mortality are considered in turn (sections 11.1, 11.2 and 11.3 respectively). As for the fish parameters, data from ECOPATH models are used, with the same assumptions (see Chapter 10). The parameter ranges derived are summarised in Table 11.1 at the end of this chapter, together with the parameter definitions.
11.1 Urchin Recruitment Parameters To derive the endogenous and exogenous recruitment rates of sea urchins, lUen and lUex respectively, the studies by Hunte and Younglao (1988) and Karlson and Levitan (1990) were used. Hunte and Younglao (1988) studied recruitment of Diadema antillarum onto reefs in Barbados, in the Caribbean. Barbados is up-current from areas known to have D. antillarum in the western Atlantic. In addition, it is unlikely that larvae can travel from the coast of Africa and reach Barbados (Lessios 1988). Thus, it is assumed that urchin recruits in Barbados originate from the reproduction of the endogenous adult population. It is possible that urchin larvae are trapped in an eddy in the island’s lee and then subsequently recruit back to the natal reef (Lessios 1988). Hunte and Younglao (1988) performed their study from Oct 1984–Dec 1985. A recruit was assumed to be between 10-15 mm in test (shell) diameter in the study. Since newly metamorphosed D. antillarum have an initial growth rate of 1 mm wk-1 (Hunte and Younglao 1988), assume that the recruits are 12.5 wk (i.e. (10 + 15) 2 wk) old. From the study, the total recruit density over a year is 0.048 m-2. Also from the study, the initial adult density is 3.3 m-2; thus, it was assumed that this is the density of the adult population which produced all the recruits in the study. By measuring from graphs of the frequency distributions of adult urchin test diameters presented in the study, the mean test diameter of the initial adult population was derived as 45.2 mm. Ebert (1982) measured the probability of post-settlement urchins surviving to a certain age. For urchins of the family Diadematidae, he found that the probability of surviving to an age x was e − Zx , with Z = 0.315 for Echinothrix diadema (at Kapapa Island, Hawaii), Z = 0.288 for Diadema setosum (in Zanzibar) and Z = 0.654 for D. setosum (in Eilat, Israel). The average Z is 0.419. Thus, for D. antillarum, the 185
probability of surviving to age x is assumed to be e − (0.419 )x . Since the urchin recruits in Hunte and Younglao (1988) are assumed to be 12.5 wk old, a proportion e − (0.419 ){(52−12.5 ) / 52} = 0.727 survive to become 1 yr old. Thus, the total 1 yr old recruit
density is (0.048)(0.727 ) m-2 = 0.0349 m-2. From Carpenter (1997), D. antillarum recruits reach a diameter of 25-30 mm in a year. Thus, assume 1 yr recruits are of diameter 27.5 mm (i.e., (25 + 30) 2 mm). Using the allometric equation from Levitan (1988) for D. antillarum, which is log10 live weight (g) = 2.99 × log10 size (mm test diameter) – 3.20,
(11.1)
a 1 yr recruit weighs 12.7 g. Also from eqn. 11.1, the weight of a 45.2 mm diameter average adult is 56.1 g. Thus, for Barbados, the adult population has a biomass of 185 g m-2 ( 56.1 × 3.3 g m-2) and the biomass of 1 yr recruits in a year is 0.443 g m-2 ( 12.7 × 0.0349 g m-2). Then lUen = 0.443 185 yr-1 = 0.00239 yr-1. Thus, the range of lUen used is 0-0.002 yr-1 (Table
11.1), which includes the extreme case of no endogenous recruitment. This case might arise because D. antillarum have a long pelagic larval duration of 28-60 d (Lessios 1988, Hunte and Younglao 1988), and hence, provided the current regime is appropriate, all larvae might disperse from the natal reef. Furthermore, since recruitment for temperate sea urchins is very unpredictable (Lessios 1988), assuming this is also true for tropical urchins, it can be argued that there may be no relationship between recruitment and the endogenous population for tropical urchins. After the mass mortality of D. antillarum in the Caribbean, Karlson and Levitan (1990) found 0.017-0.534 recruits m-2 yr-1 at different sites in Lameshur Bay, St. John, USVI. An urchin recruit in this study was < 50 mm in test diameter. Assume that the recruits in this study had an average diameter of 25 mm (i.e., (0 + 50) 2 mm), which is within the diameter range for 1 yr recruits (Carpenter 1997). Then it follows that the recruitment rate of 1 yr old D. antillarum recruits in Lameshur Bay is 0.017-0.534 individuals m-2 yr-1. Using eqn. 11.1, a 1 yr recruit which has a test diameter of 25 mm has a weight of 9.55 g. This gives a recruitment rate of 0.162-5.10 g m-2 yr-1. Then lUex has an upper limit of 5.10 g m-2 yr-1 = 0.00510 kg m-2 yr-1. However, from earlier considerations, in Barbados, in which all recruitment is endogenous, lUex can be zero. Thus, the range of lUex is 0-0.005 kg m-2 yr-1. This range was derived using data after the 186
mass mortality. Before the mass mortality, lUex is likely to be greater. Thus, the upper range is increased by an order of magnitude, giving the range 0-0.05 kg m-2 yr-1 (Table 11.1).
11.2 Urchin Feeding and Growth Parameters
The inaccessibility parameter iU , which measures the inaccessibility of algae to urchins, was derived in a similar way to i H , which measures inaccessibility of algae to herbivorous fish (see Chapter 10, section 10.2). A system with only turf algae and space was considered, that is a system specified by the CTm (Chapter 4) with coral cover C fixed at 0 (so that S + T = 1 ). For this system at equilibrium, eqn. 10.1 in Chapter 10 can be derived. Carpenter (1986) found that over Feb–Dec 1983 on a back-reef/reef-crest site at Tague Bay Reef, St. Croix, USVI, the ratio of the daily algal biomass consumption to the daily algal biomass production varied between 0.75-1.43 each month, with a mean of 1.05 (a ratio > 1 means that more algal biomass is grazed than produced, such that standing algal biomass decreases). This was for a treatment with 4.9 D. antillarum urchins per m2 and microherbivores (such as amphipods), with herbivorous fish excluded using cages. Since Carpenter concluded that microherbivores have minimal effect on the algal community, the results suggest that urchins at that density are able to keep ≈ 100 % of the substratum in a cropped state (i.e. S ≈ 1 and T ≈ 0 ). Furthermore, the test diameters of the urchins in this study were similar to those in the surrounding area, which had a mean of 68.4 mm. Using eqn. 11.1 then gives a total urchin biomass of 948 g m-2. Thus, using eqn. 10.1:
θ1 =
ζTS ζ T (0.99) = , g T T + ζ T S g T (0.01) + ζ T (0.99)
(11.2)
where θ1 is θ for an urchin population with a biomass of 0.948 kg m-2. Here, S = 0.99 ≈ 1 is used rather than 1, because using S = 1 gives iU = 0 from eqn. 11.3
below, and iU = 0 implies that the maximum grazing pressure is achieved no matter what the urchin biomass is, which is biologically unrealistic. It is assumed that θ1 is the
θ that corresponds to the lowest rugosity, 1.12 (from section 10.2), which ensures that 187
θ remains less than 1 (and hence retains biological realism) when the entire range of rugosities is considered. Analogous to the parameterisation of i H , as rugosity increases from 1.12 by a factor p, assume that the turf algal and space cover that can be grazed decreases by the factor p 2 . Then p = R / 1.12 , where R is the rugosity, and:
iU =
(
U p 2 − θ1
θ1
),
(11.3)
where U is the urchin biomass per m2 (cf. Chapter 10, eqn. 10.4), which is 0.948 kg m-2. Using eqn. 11.3 with the R values from section 10.2, which are 1.12-2.43, and the empirically derived values of g T and ζ T , which are 5-15 yr-1 and 2-20 yr-1 (see Chapter 9, section 9.2), gives iU in the range 0.002-4 kg m-2 (Table 11.1). The lower limit of this range is modified using eqn. 11.8 below. The parameters λ H and λU determine the competitiveness of herbivorous fish and sea urchins with respect to each other for algae, and must sum to 1 (Chapter 7, eqn. 7.2). Hay (1984) showed that on several (lightly fished) reefs, fish usually grazed experimentally attached Thalassoma testudinum (a seagrass) at greater rates than urchins (Diadema) did. From this and other studies, Hay concluded that fish appeared to be the primary grazers on tropical reefs worldwide. However, in the same study, it was found that for heavily fished reefs, urchins grazed at a greater rate. Since the biomasses for fish and urchins were not reported, λ H and λU cannot be parameterised using this study, and the study is consistent with λ H ≥ λU or λ H < λU . Also, McClanahan (1997) found that, on Kenyan reefs, sea urchins were more effective than herbivorous fish at preventing successional development of algae and were able to keep algal biomass to lower levels. Based on this and results from other studies, the McClanahan concluded that it has been consistently shown that urchins are better competitors than fish for algae. Thus, in the models, the entire theoretical range of λ H , 0-1, is used. λU is then given by
1 − λ H (Table 11.1). The parameter κ U measures the somatic growth of urchins relative to that of herbivorous fish, and is parameterised by considering PS Q for these two types of grazer. From Opitz (1996), for an ECOPATH model of a Caribbean reef, the ratio PS Q for herbivorous fish, which represents the proportion of algal biomass consumed 188
that is used for somatic growth, is 0.0311. From the same model, for echinoids, the
P B ratio is 1.10 yr-1 and the Q B ratio is 3.70 yr-1. Levitan (1989) describes a field experiment where D. antillarum are caged and then the production of gametes measured at the end of 26 wk, using forced KCl injections that induced spawning of all gametes for urchins ready to spawn. For three treatments with different initial urchin sizes and densities, the average test diameters were 33.75 mm, 28.75 mm and 31.75 mm. Using eqn. 11.1, this corresponds to 23.4 g, 14.5 g and 19.5 g respectively. For those urchins that spawned at the end of the experiment when injected with KCl, the mean gamete volumes, per urchin per year, were 0.02 mL, 0.06 mL and 0.06 mL respectively (Levitan 1989). No data could be found to convert these volumes to masses, so it is assumed that for the gametes, 1 mL = 1 g, as for water. The mean gamete biomasses produced (per urchin per year) are then 0.02 g, 0.06 g and 0.06 g. The production of gametes expressed as a proportion of the total urchin biomass (B) is therefore estimated as (0.000855 yr-1)B, (0.00414 yr-1)B and (0.00308 yr-1)B respectively, which gives an average of (0.00269 yr-1)B. Assume this is true for other reef-associated sea urchins. Then production which goes to somatic growth for urchins is PS = P – (0.00269 yr-1)B . Therefore, for echinoids in the Opitz (1996) study, the ratio PS B = P B − 0.00296 yr-1 = 1.10 yr-1. Then the ratio PS Q = (PS B ) (Q B ) = 1.10 3.7 0 = 0.297 . Thus, given the consumption of the same amount of algae, sea urchins produce 0.297 0.0311 = 9.55 times more somatic biomass than herbivorous fish per unit of algal biomass consumed, which is κ U . From Arias-Gonzalez et al. (1997), for an ECOPATH model of a fringing reef in French Polynesia, herbivorous fish have a PS Q ratio of 0.0360 and echinoids, using PS = P – (0.00269 yr-1)B as before, have a PS Q ratio of 0.114. This gives an estimate of 3.17 for κ U . For a French Polynesian barrier reef ECOPATH model, Arias-Gonzalez et al. (1997) found that herbivorous fish have a PS Q ratio of 0.0358 and echinoids, using PS = P – (0.00269 yr-1)B, have a PS Q ratio of 0.0606. This gives an estimate of 1.69 for κ U . Finally, from Johnson et al. (1995), for a tropho-dynamic network model of a shallow front slope at Davies Reef in the GBR, herbivorous fish have a PS Q ratio of 0.555 and sea urchins (assumed to be equivalent to the ‘invertebrate grazers’ group in the study), using PS = P – (0.00269 yr-1)B, have a ratio of 0.380. This gives an estimate of 0.685 for κ U . Overall, the range of κ U is 0.7-10 to the nearest significant figure 189
(Table 11.1).
11.3 Urchin Mortality Parameter
The only mortality parameter for sea urchins is d U , which includes predation mortality. For D. antillarum, Karlson and Levitan (1990) found a constant mortality rate of 0.270.47 yr-1 for five fringing reef sites and five years in St. John, USVI. Levitan (1989) found a mean mortality rate of 0.46 yr-1 on a fringing reef at 4 m depth, also in St. John. Ebert (1975) reported instantaneous mortality rates for D. antillarum of 1.30 yr-1 for the Virgin Islands and Florida Keys, and 1.39 yr-1 for Barbados. Ebert also reported rates of 0.64 yr-1 for E. diadema, 0.05 yr-1 and 0.2 yr-1 for Echinometra mathaei, and 0.07 yr-1 and 0.18 yr-1 for Echinometra oblonga, all for Kapapa Island in Hawaii. Thus, to the nearest significant figure, d U is in the range 0.05-1 yr-1 (Table 11.1). However, d U also has to satisfy two constraints to ensure that the models in this thesis make biological sense. First, the condition which ensures that the urchin biomass remains finite is lUen < d U (Chapter 6, sections 6.2.4 and 6.3.2). Using the empirically derived values for lUen and d U (see section 11.1 for the parameterisation of lUen ), this condition always holds. Second, from the mathematical analysis of the integrated models (Chapter 7, section 7.1), the following condition is required to ensure that the dynamics make biological sense:
κ U µ max{g M , g T , ζ T } iU
< dU ,
(11.4)
where µ M = µ T = µ S = µ . From sections 9.2, 10.2 and 11.2, g M takes values in the range (0.01)g T - gT yr-1, g T = 5-15 yr-1, ζ T = 2-20 yr-1, µ = 0.00006-0.5 kg m-2 and iU = 0.002-4 kg m-2. Condition 11.4 sets a lower limit to d U , which varies between 0.000052550,000 yr-1. To prevent the situation where no value of d U in 0.05-1 yr-1 will satisfy 11.4, an upper limit is placed on µ , µ max kg m-2, given by µ max = min{0.5, µ max 1 }. Here,
190
µ max 1 =
iU (1 − 0.000001) . κ U max{g T , ζ T }
(11.5)
d U takes values in the range d U min -1 yr-1, where d U min = max{0.05, d U min 1 } and
d U min 1 =
κ U µ max{g T , ζ T } iU
+ 0.000001 .
(11.6)
Note that eqn. 10.7 in Chapter 10 sets another upper limit to µ , such that
µ max = min{0.5, µ max 2 } is used, where i H (5 − 0.000001) iU (1 − 0.000001) , . κ U max{g T , ζ T } max{g T , ζ T }
µ max 2 = min
(11.7)
Also, from eqn. 11.5, it is not guaranteed that µ max 1 ≥ 0.00006 , as required. Thus, a lower limit is set for iU , iU min kg m-2, where iU min = max{0.002, iU min 1 } . Here,
iU min 1 =
κ U max{g T , ζ T }(0.00006) 1 − 0.000001
.
(11.8)
191
Parameter
λH
Definition Competitiveness of herbivorous fish relative
Empirically Derived Ranges
Model Specificity
0-1
HPUm and HSLUm
to sea urchins dU
The mortality rate of sea urchins
d U min - 1 yr-1
iU
A parameter which measures the
iU min - 4 kg m-2
inaccessibility of algae (turf and macroalgae) to sea urchin grazing lUen
0 - 0.002 yr-1
The endogenous recruitment rate of sea urchins
lUex
The exogenous recruitment rate of sea urchins
κU
A parameter which measures the biomass
0 - 0.05 kg m-2 yr-1 0.7 - 10
accumulated through growth by urchin grazing, relative to that for herbivorous fish grazing
λU
Equal to 1 − λ H
Competitiveness of sea urchins relative to herbivorous fish
Table 11.1. Urchin parameters for the HPUm and the HSLUm, with their ecological meanings and empirically derived ranges. Explicit expressions for d U min and iU min are given in section 11.3.
192
Chapter 12. Discussion for Part III In Part III, the parameters for the models in this thesis were quantified. In deriving the parameter ranges, a number of assumptions had to be made and new methods had to be developed that make use of available data. For example, in the derivation of the coral growth rate rC , it was assumed that all coral colonies can be approximated by hemispheres. This assumption was also made by Mumby (2006) for his spatial model. Although the shape of the colonies of some species can be approximated by a hemisphere, such as Colpophyllia natans and Siderastrea siderea colonies, there are likely to be colonies at a particular area that cannot be, particularly colonies of branching species such as Acropora spp. Thus, this simplifying assumption may lead to either an underestimate or overestimate of rC . The assumption of hemispherical colonies was also used to derive the growth rate of macroalgae over corals relative to over space, γ MC , thus leading to a possible underestimate or overestimate of this parameter as well. It is possible to derive more accurate ranges for rC and γ MC by measuring the lateral growth of corals over space and of macroalgae over corals and space experimentally, perhaps using photographs taken at several time intervals and computer programs (such as CPCe, http://www.nova.edu/ocean/cpce/) to track the change in covers. The same can be done to obtain more accurate estimates for the growth rate of corals over turf algae relative to that over space, α C , for which the full theoretical range (0 to 1) is explored in this thesis, and the growth rate of macroalgae over turf relative to over space, γ MT , for which the range was assumed to be the same as for γ MC due to a lack of data for direct parameterisation. Some estimates of the reduction of coral growth by macroalgae, β M , were derived in this thesis using a similar method to that used in the derivation of rC , which again assumes that coral colonies are hemispherical. This may lead to inaccuracies as mentioned in relation to rC , and a more accurate range could be obtained by conducting more experiments that directly measure the change in coral cover growth due to the presence of macroalgae, such as those in Lirman (2001). The derivation of the fish parameters relies on the use of ECOPATH models, in particular estimates of the production (P), consumption (Q) and biomass (B) for different fish groups. Some of the groups in the ECOPATH models do not correspond 193
directly to the herbivorous fish or piscivorous fish groups in the models in this thesis because their diets are not predominantly algae or fish (herbivorous and/or piscivorous). In these cases, the estimates for ECOPATH groups closest to the groups in the thesis models were used, and this may have given rise to inaccuracies in the derived parameter ranges. For example, if estimates for an ECOPATH group were used for the herbivorous fish group in this thesis, but not all fish in the ECOPATH group are herbivorous, then B for the ECOPATH group is an overestimate of the herbivorous fish biomass. This would affect the accuracy of parameter values that were derived using B, such as the maximum predation rates of the different piscivorous fish groups. Similar considerations apply for piscivorous fish parameter values that were derived using ECOPATH groups that do not feed predominantly on modelled fish groups (e.g., groups which include fish that eat invertivorous fish). Recruitment rates for the fish groups were found by scaling up rates for three surgeonfish and one damselfish species, according to the number of herbivorous fish or piscivorous fish species that could be present at a particular local reef area. Different fish species are likely to recruit at different rates because the 1 yr old juveniles for each species will have different biomasses and because of different reproductive rates (Sale 1991). Thus, to obtain more accurate ranges for the recruitment rates, experiments need to be done, similar to those in Robertson (1988) and Doherty and Fowler (1994), that measure the recruitment of all herbivorous and/or piscivorous fish species at a local scale. The natural mortality rates for the fish groups in this thesis exclude predation mortality, but their ranges were estimated using figures from FishBase that included predation. However, the natural mortality rates for the fish groups in this thesis take into account the mortality that arises due to starvation and it is unclear that this is captured in the FishBase figures, which are based on the empirical equations by Pauly (1980). Thus, although it is likely that the upper limits of the parameter ranges for the fish mortality rates are overestimates, it is nevertheless possible that they are actually underestimates. To obtain better estimates, experiments need to be done that track the mortality of a herbivorous or piscivorous fish population in the absence of recruitment, food, predation and fishing. Perhaps the most problematic parameters are those that measure ‘inaccessibility’ of fish prey to a predator and the inaccessibility of fish to fishermen, the ranges of which had to be estimated using just the maximum herbivorous fish biomass found from the literature. The derived ranges, especially for the parameters measuring 194
inaccessibility of fish to fishermen, do not give very different responses for the achieved predation and fishing rates as fish biomass increases (see Figures 6.2 and 10.1 in Chapters 6 and 10 respectively). These estimates were based on the rather arbitrary assumption (essentially a guess) that the maximum rates approximately saturate between a quarter and a half of the maximum herbivorous fish biomass. Different assumptions would give different ranges that could result in response functions that vary more significantly from each other, as the inaccessibility parameters are changed across their ranges, than those assumed in this chapter. Despite the sizeable number of studies that investigate reef fish predation and fishing of reef fish, there is a lack of suitable data to parameterise the inaccessibility parameters directly because it is unclear from these studies how the predation and fishing rates change with the biomass of the fish group being predated upon or fished. Thus, in order to obtain more accurate parameters, future studies would need to experimentally determine these changes quantitatively. Another problematic parameter was φ Ps , which measures the rate at which small to intermediate piscivorous (SIP) fish become large piscivorous (LP) fish through predation and subsequent growth, and the upper bound of the range used is essentially a guess. In order to obtain a more realistic estimate for the upper bound of this parameter range, experiments could be done that track SIP fish and measure how fast they become LP fish due to predation and subsequent growth. Overall, the parameter ranges were derived for highly aggregated functional groups that incorporate many species and which may be composed of different species at different localities. This means that there is scope for wide variation in the estimated parameter values and is probably a significant reason why wide variation was found for many of the parameter ranges, some of which vary by an order of magnitude or more. Thus, it may be the case that resolving inaccuracies in a parameter, such as in rC (see above), does not actually make much of a difference to the parameter range derived in this thesis.
195
Part IV: Exploring the Parameter Space In Part IV, the effects of three anthropogenic stressors commonly implicated in phase shifts in coral reefs – fishing, nutrification (sensu Szmant 2002), and sedimentation – are quantified and an Immune-Inspired Algorithm (IIA: a kind of genetic search algorithm; Kelsey and Timmis 2003) is used to determine sub-regions of parameter space where discontinuous shifts can and cannot occur under these stressors. The potential for continuous shifts and synergy between the three stressors is also examined (Chapter 13). Sweeps of parameter space are then undertaken for two of the benthic models to determine the likelihood of different types of coral-algal phase shifts, on reefs with and without macroalgae (Chapter 14). Part IV concludes with the application of two types of sensitivity analyses and a discussion of the results in Part IV (Chapters 15 and 16).
Chapter 13. Parameterising and Investigating Fishing, Nutrification and Sedimentation In this chapter, a general discussion of what is meant by a continuous and a discontinuous phase shift, the stressors investigated and how phase shifts using the benthic models were found is first presented (section 13.1). The effects of fishing, nutrification (sensu Szmant 2002) and sedimentation are then parameterised (section 13.2). Next, the search methodologies used to find phase shifts are detailed (section 13.3). The phase shift results found using the parameterised effects and search methodologies are then presented (section 13.4). In the last part of this chapter, the potential for synergy between the three anthropogenic stressors studied is investigated (section 13.5).
196
13.1 Continuous and Discontinuous Phase Shifts under Anthropogenic Stress 13.1.1 Defining Phase Shifts
A change from coral-dominance to algal-dominance is often characteristic of reef degradation, and is associated with a loss in the quality and/or quantity of ecosystem goods and services (Chapter 1, section 1.3). Thus, the ‘phase shift’ of this chapter refers to a coral-dominated reef changing to an algal-dominated one, because this is relevant to reef degradation and hence management. Coral and algal covers at equilibrium are considered, reflecting a focus on the total effect of anthropogenic stress on benthic covers, rather than transient changes. Specifically:
•
A coral-dominated reef is defined to be a reef with a low algal (proportional) cover at equilibrium that is < 0.25 and a coral cover at equilibrium that is at least 0.25 greater than this algal cover.
•
An algal-dominated reef is defined to be a reef with a low coral (proportional) cover at equilibrium that is < 0.25 and an algal cover at equilibrium that is at least 0.25 greater than this coral cover.
•
A phase shift in this chapter is defined to be a reef changing from a coral-dominated reef to an algal-dominated one due to stress.
Here, algal cover refers to turf algal cover T for the Coral-Turf model (CTm; Chapter 4), algal cover A for the Coral-Algae model (CAm; Chapter 5) and the sum of turf algal and macroalgal covers T + M for the Coral-Turf-Macroalgae model (CTMm; Chapter 5). Note that algal-dominance here does not necessarily mean that macroalgal cover has to be > 50% . Bruno et al. (2009) found that only 4% of 1,851 reefs surveyed worldwide had > 50% macroalgal cover, which suggests that shifts resulting in > 50% macroalgal cover are relatively rare. However, 20% of reefs surveyed had > 25% macroalgal cover, such that if turf algal cover is added, algal-dominance as defined in this chapter is a much more common occurrence. The definition of ‘phase shift’ used in this chapter is refined in Chapter 14, in order to investigate different types of shifts that, although reflecting reef degradation, do not necessarily mean a change from coral- to algal-dominance as defined in this chapter. 197
Two types of phase shifts are investigated in this chapter: continuous and discontinuous. A continuous phase shift is a phase shift where the benthic covers at equilibrium change smoothly (see Figure 13.1a), whereas a discontinuous phase shift involves a discontinuous change in benthic covers at equilibrium (Figure 13.1b). For a discontinuous phase shift, there is a range of stress levels over which multiple stable equilibria occurs and the path from coral- to algal-dominance is different to that from algal- to coral-dominance (see Figure 13.1b). Thus, identifying the possibility of discontinuous phase shifts is important for management because after such a shift, stress levels have to be decreased beyond the threshold at which the shift occurred for recovery of coral cover (Beisner et al. 2003). The definitions of a continuous phase shift and a discontinuous one are related to bifurcation analysis. From Figure 13.1b, a phase shift is discontinuous if there is a saddle-node bifurcation as the stress level is increased, which involves a locally asymptotically stable (LAS) equilibrium and an unstable equilibrium colliding and ceasing to exist (Jordan and Smith 2005) – this occurs as the equilibrium covers change discontinuously. In contrast, a phase shift is continuous if there is no bifurcation as the stress level is increased, as shown in Figure 13.1a.
13.1.2 Stressors Investigated
Three chronic anthropogenic stressors are investigated in this chapter: fishing, nutrification and sedimentation. These were chosen because they are often implicated in coral-algal phase shifts.
13.1.2.1 Fishing
Fishing may be an important contributor to phase shifts by decreasing the grazing pressure. For example, on many Caribbean reefs, fishing is thought to have depleted stocks of herbivorous fish, such that the main grazers on these reefs were sea urchins. When the 1983-84 mass mortality of Diadema antillarum urchins occurred, the lack of herbivorous fish would have then contributed to the phase shifts to macroalgal dominance that were observed (Bellwood et al. 2004). In addition, since there is evidence that algal abundance on coral reefs is controlled by herbivory (Chapter 6, section 6.2.1), it follows that fishing of herbivorous fish could lead to greater algal 198
(a) Proportional cover at equilibrium
Stress level (b) Proportional cover at equilibrium
Stress level
Figure 13.1. Conceptual diagrams of (a) a continuous phase shift and (b) a discontinuous phase shift with increasing levels of stress, as defined in section 13.1.1. The red and green curves represent the proportional covers for hard corals and algae respectively; solid and dashed curves represent stable and unstable equilibria respectively. For the discontinuous phase shift in (b), there is a range of stress levels for which there are three equilibria (two stable and one unstable).
abundance, favouring a phase shift (Mumby and Steneck 2008).
13.1.2.2 Nutrification
Nutrification is the increase in the flux of nutrients into coral reef waters because of human activities (Szmant 2002; see section 1.3.2 in Chapter 1 for more details on these activities). Where nutrients are limiting, nutrification can increase algal growth rates (Done 1992), thus favouring a phase shift. Lapointe et al. (1997) used experimental 199
results (e.g., from experiments that determined the nitrogen content of macroalgae) and implicated nutrification as a casual factor in the coral-macroalgal phase shift seen in Discovery Bay, Jamaica, in the 1980s. Using a similar experimental approach, Lapointe (1997) and Lapointe et al. (2005a,b) implicated nutrification as a cause of algal blooms on coral reefs in southeast Florida (but see Hughes et al. (1999) for criticism of the results of Lapointe (1997)).
13.1.2.3 Sedimentation
Sedimentation refers to the increase in the rate at which sediments are delivered to coral reef waters due to human activities, with coastal development being an important source of anthropogenic sediments (see section 1.3.3 in Chapter 1). This is relevant to reef degradation because increased sediments can smother corals and reduce light levels for photosynthesis, leading to an increased coral mortality rate, decreased coral recruitment rates and a decreased coral growth rate (see section 1.3.3). Thus, sedimentation can favour algae over corals, and hence a phase shift. In the review of sedimentation on coral reefs by Rogers (1990), sedimentation is implicated as potentially one of the most important factors in reef degradation.
13.1.2.4 Caveats
Although the three anthropogenic stressors studied are important in reef degradation, it is important to acknowledge that they are by no means the only important stressors. There are other anthropogenic or anthropogenically-enhanced stressors that can also affect reefs significantly, but which are not considered due to time constraints and/or because of a lack of data to translate the stressors into quantitative changes in model parameters. For example, as mentioned in section 1.3.4 in Chapter 4, an increased frequency of coral bleaching in the future and an increased effect of ocean acidification, driven by increased atmospheric carbon dioxide concentrations, is projected to have widespread impacts on reefs. Oil spills can result in reduced growth, reproduction and/or recruitment of corals, because of exposure of corals to the oil, and hence may also be important in reef degradation (Haapkylä et al. 2007). An example is a major oil spill near the mouth of the Panama Canal in 1986, which resulted in declines in coral cover in heavily affected reef areas (Bryant et al. 1998). Also, for a reef site at Eilat in the Red Sea, low coral recolonisation was probably due to the chronic effects of a series 200
of oil spills from a nearby loading terminal (Connell 1997). For reef areas where the mining of live corals is prevalent, such as the Maldives (Moberg and Folke 1999, Sluka and Miller 1998), the direct loss of live corals due to mining may have significant effects on coral cover. In addition, tropical cyclones affect coral reefs worldwide and can be a major structuring force for impacted reefs (Wilkinson 1999). Examples are Hurricane Mitch in 1998, which together with coral bleaching reduced coral cover on Belizean reefs by an average of 50% (McField et al. 2008), and Cyclone Rona in 1999, which reduced coral cover at Low Isles in the Great Barrier Reef (GBR) by 48-67% (Cheal et al. 2002). There is evidence for increases in cyclone intensities around the world in the future, due to increased SST (Elsner et al. 2008, Wilkinson and Souter 2008). For reefs in Southeast Asia, destructive fishing methods using sodium cyanide and dynamite are a common threat, and their use can cause an increase in coral mortality (McManus et al. 1997, Burke et al. 2002). In this thesis, the effects of fishing, nutrification and sedimentation are assumed to act independently. However, in reality, the effects of the three stressors are interdependent. For example, fish may act as a source of nutrients for benthic organisms by excretion of nutrients through their gills and production of faeces (Meyer and Schultz 1985). Also, fish may produce significant amounts of sediment whilst feeding on the benthos, in particular parrotfish that remove carbonate from the benthos (Bellwood et al. 2003). Thus, the increase in the nutrient or sediment flux due to nutrification or sedimentation respectively may be dampened by fishing. In addition, sediments can adsorb nutrients and form localised nutrient sources for macroalgae (Larned and Stimson 1996, Larned 1998), such that sedimentation may enhance the increased algal growth effect of nutrification through this mechanism. For the models in this thesis, such interdependent effects are assumed to be insignificant. This is to simplify the model analyses and because of a lack of data for quantification of the dependence of each stressor on the other two stressors. Furthermore, nutrients originating from non-modelled groups are assumed to be insignificant. However, in reality, some of these groups can be significant nutrient sources; for example, sponges can fix nitrogen through their bacterial symbionts and this may be an important nutrient source (Mohamed et al. 2008), and non-modelled fish groups may also be important nutrient sources due to their excretory products (Meyer and Schultz 1985). Hence, the modelled effects of nutrification may be significantly different from the actual effects, although the model results would still serve as a useful baseline for comparison with empirical data. 201
13.1.3 Searching for Phase Shifts under Anthropogenic Stress
In this chapter, the three benthic models are used to investigate whether continuous or discontinuous phase shifts can occur on reefs with or without macroalgae, under the effects of the three stressors discussed in section 13.1.2. First, the effects of fishing, nutrification and sedimentation on ‘pristine’ model reef areas are parameterised (see section 13.2). These three stressors are modelled indirectly through their effects on model parameters and the ‘pristine’ model reef area is one which is unaffected by the three stressors, and which is exposed to low amounts of nutrient and sediment fluxes. Although a pristine reef does not exist in reality (Hughes et al. 2003), the three studied stressors are applied to pristine model reef areas to investigate the potential of the stressors to cause phase shifts starting from the time when reefs were unaffected by humans. This allows an assessment of how important these stressors have been in reef degradation, and the potential future risk to reefs from these stressors. Parameter values for a ‘pristine’ model reef area are given in section 13.2. For each benthic model, the possibility of continuous phase shifts with each of the three stressors is determined using a manual search of the parameter space, and the possibility of discontinuous phase shifts with each of the stressors is determined using the Immune-Inspired Algorithm (IIA; Kelsey and Timmis 2003). Section 13.3 gives details on these search methodologies. If discontinuous shifts are possible for a benthic model, then sub-regions of parameter space are searched using the IIA to determine if these sub-regions can give discontinuous shifts (section 13.3). This is done to see if there are any particular combinations of parameters that favour discontinuous shifts.
13.2 Parameterising Fishing, Nutrification and Sedimentation 13.2.1 Fishing
The direct effect of fishing is to decrease fish biomass. In the benthic models, fishing is modelled indirectly through changes in the grazing pressure θ , which is a function of the herbivorous fish biomass that can be affected by fishing. From section 6.2.2 in Chapter 6, the highest herbivorous fish biomass (H) found from the literature is 0.282 kg m-2 (Letourner et al. 1998). This high biomass was for reefs in the northern lagoon of New Caledonia, where human density is low and only 202
artisanal fisheries exist (Letourner et al. 1998). Thus, this biomass reflects a reef under minimal fishing pressure and it is assumed that this biomass is typical of that found on a pristine reef. From Table 10.1 in Chapter 10, the parameter measuring ‘inaccessibility’ of algae to herbivorous fish, i H , can take values in the range 0.001-0.5 kg m-2. Considering only herbivorous fish, θ = H (i H + H ) (using eqns. 6.1a,b and 7.1 in Chapters 6 and 7) and thus, θ for a pristine reef lies between 0.4 and 1. The presence of urchins can only increase θ , since θ is an increasing function of sea urchin biomass (Chapter 7), and hence θ will still lie between 0.4 and 1. Therefore, 0.4 ≤ θ < 1 is taken as the range of grazing pressures for a pristine reef. For the benthic models, it is assumed that fishing can decrease herbivorous fish biomass down to the theoretical minimum of zero. Because urchin biomass can be zero at a local reef area (e.g., Kramer (2003) for western Atlantic reefs), investigating the extreme case of fishing decreasing fish biomass to zero corresponds to θ being decreased to zero. Thus, fishing can decrease θ from a pristine value to any value down to zero.
13.2.2 Nutrification
Studies by Larned and Stimson (1996), Larned (1998) and Schaffelke and Klumpp (1998a,b) suggest that reef macroalgae can be nutrient limited. Using laboratory experiments, Larned and Stimson (1996) showed that Dictyosphaeria cavernosa was nitrogen limited, although this may not be the case in the field because there is evidence that this algal species can derive nitrogen from bottom sediments on the reef substratum. Larned (1998) showed that eight out of nine macroalgal species examined were nitrogen limited, using laboratory experiments that raised the nitrogen concentration to much above the field concentrations. By combining these results with those from six other studies, Larned found that 22 out of 36 reef macroalgal species studied were nitrogen limited and 17 out of the same 36 species were phosphorus limited. Schaffelke and Klumpp (1998a,b) studied Sargassum baccularia on inshore reefs in the GBR and found that its growth was limited by both nitrogen and phosphorus. The macroalgal species from these studies come from both the Atlantic and Pacific regions, suggesting that nutrient limitation of reef algal growth occurs globally. From Larned (1998), the proportional change in the mass-specific growth rates of nine reef macroalgal species under nutrient enrichment (ammonium and phosphate) 203
were calculated. The proportional change was not calculated for species with a negative mass-specific growth rate when there was no nutrient enhancement. This is because the proportional change with nutrient enrichment would be negative, which is incorrect and biologically meaningless. Species with negative growth rates in the laboratory are able to maintain positive growth rates in the field possibly because these species utilise nutrients from sources other than the water column. The range for the proportional change shows that the mass-specific growth rate of reef macroalgae can increase by a factor of 1.35-3.98, disregarding data for Kappaphycus alvarezii. Data for K. alvarezii is not used because it gives an unusually large change, over six times greater than the next largest figure, and was thus assumed to be an extreme case that is not encountered in the field. Also, Schaffelke and Klumpp (1998a) showed that nutrient enhancement of S. baccularia can increase its mass-specific growth rate by a maximum factor of
9.50 3.40 = 2.79 . Thus, it is assumed that the lateral growth rate of macroalgal cover, rM (Chapter 5, Table 5.1), can increase by a factor of up to 4 for a pristine reef under nutrification. The growth rate of turf algal cover, ζ T (Chapter 5, Table 5.1), is also assumed to increase by a factor of up to 4 due to nutrification, in the absence of direct data. If turf algae and macroalgae coexist, then the relative response of these two algal groups to increased nutrients depends on local variables such as how efficiently nutrients are delivered to different microhabitats. The relative response could also depend on the species composition of each group, which could affect nutrient uptake kinetics. Given the uncertainty in the values of the local variables and the precise uptake kinetics of each algal group, as a simplification, it is assumed that if ζ T and rM can increase by up to ζ T max and rM max respectively, then as nutrients are increased, the proportion of ζ T max by which ζ T increases is the same as the proportion of rM max by which rM increases. There is some evidence from laboratory experiments to suggest that nutrients can affect coral growth. However, the effects in the field are uncertain and furthermore, it is unclear how nutrients can affect coral growth. Some studies have shown that increased nutrients depresses coral growth (e.g., Marubini and Davies 1996, FerrierPages et al. 2000), whereas other studies have shown increased coral growth with increased nutrients (see review by Szmant (2002)). Hence, this effect is not included in the nutrification regime for the local models. In addition, the effect of nutrients on coral reproductive rates is not considered. This is because of the equivocal evidence for a decrease in reproductive rates due to increased nutrients (Szmant 2002). 204
The parameter ranges for ζ T and rM for a pristine reef area are assumed to be those derived in section 10.2 in Chapter 10. Thus, for ζ T , the figure reported by McClanahan (2002) and that used in the model of Mumby et al. (2006a) are assumed to hold for a reef under pristine conditions. Values of rM were derived using data from Glover’s Reef, which is one of the most remote atolls in the Caribbean, being about 30 km offshore (McClanahan and Muthiga 1998), and which has been protected since 1993 (Spalding et al. 2001). Thus, it is reasonable to use figures from this reef as approximating those from a pristine reef without nutrification.
13.2.3 Sedimentation
Nugues and Roberts (2003) studied coral mortality of populations of Colpophyllia natans and Siderastrea siderea at two locations in St Lucia, Soufrière and Anse La Raye. They measured the loss of coral tissue as a percentage of the initial tissue area, for reef sites near to (approx. 0.5 km away), at mid-distance from (approx. > 0.5 km and ≤ 1 km away) and far from (approx. > 1 km away) a river mouth. These sites encompass a range of sedimentation rates and the mortality due to sediments is taken for each site, as is the mortality due to background processes (naturally occurring diseases and unknown causes). By comparing mortality due to sediments with the background mortality, it was found that chronic sedimentation can result in an increase in the coral mortality rate, d C (Chapter 5, Table 5.1), by up to a factor of 3. Since the original range of d C was derived using data where the effects of sedimentation were either not known to be significant or were excluded (see Chapter 9, section 9.1), this range is taken to be the range for a pristine reef. Cortes and Risk (1985) showed that the linear extension rate of the coral Montastraea annularis halves as sedimentation rates increase from < 1 mg cm-2 d-1 to > 50 mg cm-2 d-1, i.e. from a low to a high rate sensu Brown (1997). In the derivation of
the growth rate of coral cover, rC (Chapter 5, Table 5.1), a linear extension rate of 0.973 cm yr-1 was used for western Atlantic reefs and a rate of 0.875 cm yr-1 was used for Indo-Pacific reefs (section 9.1). The method used gave a range of 0.04-0.2 yr-1 for rC , which is assumed to be the range for pristine reefs. Using the results from Cortes and Risk (1985), it is assumed that the linear extension rates for western Atlantic and IndoPacific reefs can decrease by up to half due to sedimentation. Using the methodology of 205
Chapter 9, section 9.1, it was found that rC also can decrease by up to 0.5 due to sedimentation. Babcock and Smith (2000) studied recruitment of the coral Acropora millepora. They found that 8 months after settlement, the number of A. millepora recruits at sites with elevated sediment rates was 39% of the number at control sites. It is assumed that 1 yr after settlement, the number of recruits also decreases by 39% with increased sediments. Thus, it is assumed that for a pristine reef, the coral recruitment rates, l Cb and l Cs (Chapter 5, Table 5.1), can each decrease by a factor of up to 0.6 with increased
sedimentation. The effects of sedimentation on brooding and spawning corals are assumed to be independent, such that the values of l Cb and l Cs can decrease by different factors. This is justified by the possibility that brooding and spawning species may respond differently to sedimentation (Rogers 1990). As for rC , the empirically derived ranges for l Cb and l Cs in section 9.1 are assumed to be the pristine ranges. The increase in d C and the decreases in l Cb and l Cs derived are likely to be conservative estimates. This is because the studies from which they were derived, Nugues and Roberts (2003) and Babcock and Smith (2000), considered sedimentation rates of < 8 and < 12 mg cm-2 d-1 respectively, whereas sedimentation rates of > 200 mg cm-2 d-1 have been recorded in the field (e.g., Cortes and Risk 1985). As a simplification, it is assumed that if d C can increase by up to d C max , l Cb and l Cs can decrease by up to l Cb max and l Cs max respectively, and rC can decrease by up to rC max , then as sediments are
increased, the proportions of d C max , l Cb max , l Cs max and rC max by which d C increases, l Cb decreases, l Cs decreases and rC decreases respectively, are the same. For the parameters affected by the three stressors, the pristine ranges are given in Table 13.1 together with the extended ranges due to application of the stressors.
13.3 Detecting Phase Shifts 13.3.1 Finding Continuous Phase Shifts
To find continuous phase shifts for each of the benthic models, the parameter spaces for each model are manually searched as follows. For each model, the starting point is a pristine reef area with all parameters taking their mid-range pristine values. To this reef 206
Parameter
Pristine Range
Stressor
Extended Range
θ
0.4 - 1
Fishing
0-1
ζT
2 - 20 yr-1
Nutrification
2 - 80 yr-1
rM
0.05 - 0.4 yr-1
dC
0.02 - 0.1 yr-1
l Cb
0.0009 - 0.5 yr-1
0.00036 - 0.5 yr-1
l Cs
0.00006 - 0.01 yr-1
0.000024 - 0.01 yr-1
rC
0.04 - 0.2 yr-1
0.02 - 0.2 yr-1
0.05 - 1.6 yr-1 0.02 - 0.3 yr-1
Sedimentation
Table 13.1. Parameters for the benthic models that are affected by fishing, nutrification or sedimentation, together with the pristine ranges and the ranges extended by the stressors
area, increasing levels of fishing are applied (as described in section 13.2.1) to see if a continuous phase shift can occur. If a continuous phase shift does not occur even with the maximum effects of fishing, then the model parameters are changed until a continuous phase shift is found. For example, if using the mid-range pristine values gives a coral cover at equilibrium that is too high for there to be a continuous phase shift with fishing, then the coral mortality rate, d C , was increased to see if that makes a difference, and then other parameters are changed if necessary. This exercise is repeated with increasing levels of nutrification and then increasing levels of sedimentation. The aim of these searches is first to see if each model is able to give continuous phase shifts under each one of the three stressors investigated, and second to see how close parameters have to be to their mid-range pristine values for continuous shifts to occur, if such shifts are possible. The latter gives a measure of how likely continuous phase shifts are if the mid-range parameter values are taken to be typical of a pristine reef system.
13.3.2 The Immune-Inspired Algorithm
To detect discontinuous phase shifts, an Immune-Inspired Algorithm (IIA), a type of 207
‘genetic algorithm’, is used and this is now described. For a specified function with parameters that take values within specified ranges, the IIA converges on the parameter set that maximises (or minimises) the function. The IIA is an adapted form of the B-Cell Algorithm (BCA) of Kelsey and Timmis (2003) and Kelsey et al. (2003), which is based on the immunological response of (selective) clonal expansion of B-cells in response to a pathogen challenge. The BCA algorithm has been found to be effective at converging on the minimum of many equations, including the 20-dimensional Griewangk function (Kelsey and Timmis 2003, Kelsey et al. 2003). It is novel because it uses a unique, immunologically inspired mutation step that increases the likelihood of finding the global maximum (or minimum) of a function rather than just a local maximum. Details on how the modified form of the BCA used here, called the IIA, works is given in section A13.1 of Appendix A13. For each of the benthic models, the IIA is used to converge to the maximum of a ‘fitness’ function which is defined in terms of the parameters in the model. If this function can be positive for parameter values within a given parameter space V, then for at least one set of parameters within V, there are multiple stable equilibria (for the CTm and CAm) or the possibility of multiple stable equilibria (for the CTMm). If this function can take only negative values, then multiple stable equilibria cannot occur within V. This also means that discontinuous phase shifts cannot occur using parameter values within V, since a discontinuous phase shift requires multiple stable equilibria. The fitness functions for the CTm, CAm and CTMm, together with the technical details behind the derivation of each function, are found in sections A13.2, A13.3 and A13.4 in Appendix A13 respectively.
13.3.3 Finding Discontinuous Phase Shifts
To determine whether a benthic model can exhibit a discontinuous phase shift, the IIA is applied to different parameter spaces to determine if multiple stable equilibria are possible within these spaces. If multiple stable equilibria are not possible within a space, then a discontinuous phase shift cannot occur in that space. The use of the IIA to search for parameter sets, within a parameter space, that give multiple equilibria (required for a discontinuous phase shift) is a novel method. First, for each benthic model, the parameter space defined by the pristine parameter ranges is searched, denoted by V p . This is followed by the parameter spaces 208
defined by the pristine parameter ranges extended by just fishing ( V f ), just nutrification ( Vn ), just sedimentation ( Vs ), fishing and nutrification ( V f + n ), fishing and sedimentation ( V f + s ), nutrification and sedimentation ( Vn + s ) and all three stressors ( V f + n + s ). Then, for each benthic model, if a parameter space can exhibit multiple stable equilibria, sub-regions of that space are searched to determine how far the parameters have to be from their mid-range values for there to be the possibility of multiple stable equilibria, and hence the possibility of discontinuous shifts. Specifically, the subregions defined by the middle x% of the pristine ranges, as extended by the stressor(s) corresponding to the parameter space, if there are any stressors associated with the space, are searched, where x takes values from 5 to 100 in increments of 5. It has been suggested that the presence of macroalgae may be particularly important in facilitating discontinuous phase shifts. For example, Mumby et al. (2007a) investigated models parameterised using data for Caribbean reefs and found that alternative stable states (ASS) were possible at certain levels of grazing. This was attributed to ecological feedbacks that drive the benthic dynamics towards either a coral-dominated or an algal-dominated state. They suggested that macroalgae are important in creating positive feedbacks that drive the system to an algal-dominated state because they are able to reduce coral recruitment by decreasing the reef area suitable for coral settlement. This in turn stifles the increase in coral and hence helps macroalgae compete more effectively with corals for space (Mumby and Steneck 2008). In light of the possible importance of macroalgae for ASS, for the two models with macroalgae (the CAm and the CTMm), sub-regions of parameter space are explored to determine how large the effect of macroalgae on the other benthic groups has to be for there to be the possibility of multiple stable equilibria, and hence ASS. Specifically, for the CAm, the sub-regions with the key macroalgal parameters rM , β M and γ MC (see Table 5.1 in Chapter 5 for definitions of these parameters) taking values from the lowest y% of their pristine ranges and g M (see Table 5.1 for definition) taking values from the highest y% of its pristine range are explored, where y takes values from 5 to 100 in increments of 5. In addition, since ν M (Table 5.1) measures the strength of macroalgal effects per unit of algal cover, the sub-regions with 0 < ν M < z are explored, where z takes values from 0.05 to 1 in increments of 0.05. All other parameters take values from their pristine ranges as extended by fishing, nutrification and sedimentation (Table 13.1). Further, to determine whether ASS can occur with just macroalgal growth 209
over live corals or reduction of coral growth by macroalgae, or when none of these processes is present, the full parameter space ( V f + n + s ) is searched but with β M = 0 ,
γ MC = 0 , or β M = γ MC = 0 . Similarly, for the CTMm, the sub-regions with the macroalgal parameters rM ,
β M , γ MC and γ MT (Table 5.1) taking values from the lowest y% of their pristine ranges and g M taking values from the highest y% of its pristine range are explored, where y takes values from 5 to 100 in increments of 5. All the other parameters take values from their pristine ranges as extended by the effects of the three stressors. Also, to see whether ASS can occur with just macroalgal growth over live corals, macroalgal growth over turf algae or reduction of coral growth by macroalgae, or when none of these processes is present, the full parameter space ( V f + n + s ) is searched but with
β M = γ MT = 0 , β M = γ MC = 0 , γ MC = γ MT = 0 or β M = γ MC = γ MT = 0 . Table 13.2 summarises the sub-regions of the parameter spaces that are searched.
13.4 Phase Shift Results 13.4.1 Continuous Phase Shifts
Continuous phase shifts were found for each of the benthic models, for pristine reef areas that are exposed to just fishing, just nutrification or just sedimentation. For the CTm, continuous phase shifts were found for pristine reef areas with all the parameters taking their mid-range pristine values, except for d C in the fishing case and θ , d C , l Cb and α C (see Chapter 5, Table 5.1 for definitions) for the nutrification case. In these cases it was necessary to increase d C above its mid-range value and decrease θ , l Cb and
α C to below their mid-range values in order to decrease coral cover sufficiently for a shift to algal-dominance to occur. Nonetheless, θ , d C , l Cb and α C remain within the middle 50% of their pristine ranges. For these shifts, fishing decreases θ down to 0.05, and nutrification and sedimentation change the affected parameters up to their maximum limits (see sections 13.2.2 and 13.2.3). For the CTMm, with the mid-range pristine values for θ , g M and rM (Chapter 5, Table 5.1), condition 5.8 in Chapter 5 ( g M θ ≥ rM ) holds and thus, the CTMm has the same equilibria as for the CTm if shared parameters have the same values. Furthermore, 210
Parameter
Models
Sub-regions explored
Space(s) V p , V f , Vn ,
CTm, CAm, CTMm
Sub-regions defined by middle x% of the pristine ranges, as extended by
Vs , V f + n ,
the stressor(s) corresponding to the V f + s , Vn + s ,
parameter space (if there are any
V f +n+s
corresponding stressors), where x takes values from 5 to 100 in increments of 5.
V f +n+s
CAm, CTMm
Sub-regions with rM , β M , γ MC and
γ MT taking values from the lowest y% of their pristine ranges and g M taking values from the highest y% of its pristine range, where y takes values from 5 to 100 in increments of 5 ( γ MT only applies for CTMm) V f +n+s
CAm
Three sub-regions: first with β M = 0 , second with γ MC = 0 and third with
β M = γ MC = 0 V f +n+s
CTMm
Four sub-regions: first with β M = γ MT = 0 , second with
β M = γ MC = 0 , third with γ MC = γ MT = 0 and fourth with β M = γ MC = γ MT = 0
Table 13.2. Summary of sub-regions of parameter spaces searched using the IIA, and which models they apply for. See Table 5.1 in Chapter 5 for definitions of the parameters and the text in section 13.3.3 for definitions of the parameter spaces.
condition 5.8 always holds if θ is decreased as described above in the search for continuous phase shifts for the CTm. Thus, the same continuous phase shifts with just 211
fishing, just nutrification and just sedimentation as found for the CTm (with the stressors having the same effects) were produced using the CTMm, using the same values for parameters that are shared between the two models and the mid-range pristine values for all CTMm-specific parameters (which include g M and rM ). These phase shifts have no macroalgae at equilibrium. However, if g M is decreased to 0.5 yr-1, with all other parameter values remaining the same, then there are continuous phase shifts that have macroalgae at equilibrium for the CTMm, with just fishing and with just nutrification. Similarly, if
g M is decreased to 0.2 yr-1, holding all other parameter values constant, then there is a continuous phase shift with just sedimentation that has macroalgae at equilibrium. For the CAm, continuous shifts were found for pristine reef areas with midrange parameter values, under the effects of fishing, nutrification or sedimentation. For these shifts, the stressors have the same effects as for the shifts for the CTm and CTMm. Figure 13.2 graphically shows the continuous phase shifts found for the CTm, as described above. The graphs of shifts for the CAm are similar, except that benthic covers change more non-linearly. For the CTMm, the graphs are the same as for the CTm when g M takes its mid-range value (because the shifts are the same, as discussed above). When g M is decreased to 0.5 yr-1 or 0.2 yr-1 to give shifts with macroalgae (see above), the benthic covers change more non-linearly, with macroalgal cover increasing from zero and either matching turf algal cover or exceeding it as fishing, nutrification or sedimentation increases.
13.4.2 Discontinuous Phase Shifts
The IIA results show that the CTm cannot exhibit multiple stable equilibria, and hence ASS, for any parameter set within the full parameter space V f + n + s . Thus, from the mathematical analysis (Chapter 4, section 4.2.4), there can only be one stable equilibrium and hence the CTm cannot exhibit discontinuous phase shifts. This result also means that for the CTMm, there is only one equilibrium without macroalgae (which may or may not be stable). In particular, the CTMm cannot exhibit ASS unless macroalgae is present. In contrast, for each of the eight parameter spaces searched for the CAm and the CTMm, parameter sets giving ASS were found (for the CTMm, stability of equilibria was checked numerically). 212
(a) Proportional cover at equilibrium
θ (b) Proportional cover at equilibrium
ζ T / yr-1 (c) Proportional cover at equilibrium
d C / yr-1 Figure 13.2. Graphs showing continuous phase shifts for the CTm, for a pristine reef area exposed to (a) fishing of herbivores, (b) nutrification and (c) sedimentation. Reduction of grazing pressure θ is used as a proxy for fishing, an increase in the turfalgal growth rate ζ T is used as a proxy for nutrification and an increase in the coral death rate d C is used as a proxy for sedimentation. The red and green curves represent the proportional covers for hard corals and turf algae respectively; solid curves represent 213
stable equilibria. In (a), the parameters for the pristine system take their mid-range values (see Chapter 9, Table 9.1) except for d C = 0.08 yr-1, and fishing decreases θ to 0.05. In (b), the pristine system parameters take their mid-range values except for d C = 0.08 yr-1, l Cb = 0.15 yr-1, α C = 0.25 and θ = 0.55 ; nutrification increases ζ T by up to a factor of 4. In (c), the pristine system parameters take their mid-range values and sedimentation increases d C by up to a factor of 3, decreases lCb and l Cs by up to a factor of 0.6 and decreases rC by up to a factor of 0.5. These parameters all change simultaneously as sedimentation increases (see section 13.2.3), such that in the graph, when d C increases along the x-axis, all three of the other parameters decrease simultaneously, reflecting increasing sedimentation.
For the values of x tested (see Table 13.2), the maximum x value for which there is no possibility of ASS are given in Table 13.3, for each of the eight parameter spaces. For the CAm, it was also found that the maximum value of y tested for which there cannot be ASS is 45. However, for the CTMm, all values of y tested (Table 13.2) gave the possibility of ASS. Thus, extra values of y lower than 5 were tested and it was found that if y ≤ 1 , then there cannot be ASS. For the CAm, the IIA searches show that ASS can occur for a pristine reef area under the effects of all three stressors together if just one of macroalgal growth over live corals or macroalgal reduction of coral growth occurs. If neither of these processes is operating however, there can be no ASS. For the CTMm, the IIA searches show that there is the possibility of ASS for a pristine reef area under the effects of all three stressors together if just one of macroalgal growth over live corals, macroalgal growth over turf algae or macroalgal reduction of coral growth is operating. However, if all three processes are absent, then there can be no ASS. Finally, for the CAm, the maximum value of z tested (Table 13.2) for which there are no ASS is 0.3, that is there are no ASS for ν M < 0.3 – a proportion of macroalgae in the total algal cover of less than 0.3. Figure 13.3 shows examples of discontinuous phase shifts for the CAm under each one of the three stressors. These examples were found by using three parameter sets that give multiple stable equilibria, as found from the IIA searches for V f , Vn and Vs , and then manually changing them to derive sets for pristine reef areas that give discontinuous phase shifts (where the reef system passes through a region of multiple 214
Model CAm
CTMm
V
Vp
Vf
Vn
Vs
V f +n
V f +s
Vn + s
V f +n+s
x*
85
80
75
85
75
75
75
75
V
Vp
Vf
Vn
Vs
V f +n
V f +s
Vn + s
V f +n+s
x*
80
25
55
80
10
20
55
10
Table 13.3. Part of the IIA results for the CAm and CTMm. V is the parameter space searched and for each V, sub-regions are explored as described in the first row of Table 13.2. x * is the maximum value of x tested for which there is no possibility of ASS. The definition of x is given in the first row of Table 13.2.
stable equilibria) under each of the three stressors. Figure 13.4 shows examples of discontinuous phase shifts for the CTMm under each one of the three stressors. For these examples, the parameters for the pristine reef areas are based on the parameter sets found for the CAm. Thus, starting from an example for the CAm, say for the case with fishing, for the 13 parameters shared by the two models, the same parameter values are used in the CTMm. The one parameter unique to the CTMm, γ MT , is set to its midrange value. Then, the parameters are changed manually to try and find sets giving discontinuous phase shifts. This was achieved by altering all the parameters only slightly, except for d C , g M , γ MT and θ . Note that examples of discontinuous phase shifts for the CTMm can be derived directly using parameter sets found from the IIA searches of parameter spaces for this model, but the value of finding examples based on the CAm parameter sets is that this procedure shows that there is some consistency between results for the two models.
13.5 Investigating Synergy between Fishing, Nutrification and Sedimentation 13.5.1 Finding Synergy
For each of the benthic models, a pristine reef area was subjected to each of the three 215
(a) Proportional cover at equilibrium
θ (b) Proportional cover at equilibrium
rM / yr-1 (c) Proportional cover at equilibrium
d C / yr-1 Figure 13.3. Graphs showing discontinuous phase shifts for the CAm, for a pristine reef area exposed to (a) fishing of herbivores, (b) nutrification and (c) sedimentation. A decrease in grazing pressure θ is used as a proxy for fishing, an increase in the macroalgal lateral growth rate rM is used as a proxy for nutrification and an increase in the coral death rate d C is used as a proxy for sedimentation. The red and green curves represent the proportional covers for hard corals and algae respectively; solid and 216
dashed curves represent stable and unstable equilibria respectively. In (a), the pristine system has the parameters d C = 0.02 yr-1, l Cb = 0.0009 yr-1, l Cs = 0.00006 yr-1, rC = 0.2 yr-1, α C = 0.25 , ε C = 0.1 , g T = 10 yr-1, ζ T = 5 yr-1, g M = 3.5 yr-1, rM = 0.35 yr-1,
β M = 0.9 , γ MC = 0.1 , ν M = 0.9 and θ = 0.7 , and fishing decreases θ to 0.2. In (b), d C = 0.02 yr-1, l Cb = 0.0009 yr-1, l Cs = 0.00006 yr-1, rC = 0.2 yr-1, α C = 0.2 , ε C = 0.05 ,
g T = 9 yr-1, ζ T = 9 yr-1, g M = 1.6 yr-1, rM = 0.35 yr-1, β M = 0.9 , γ MC = 0.1 ,
ν M = 0.95 and θ = 0.85 , and nutrification increases rM and ζ T by a factor of up to 2. Both algal growth parameters change simultaneously as nutrification increases (see section 13.2.2), such that in the graph, when rM increases along the x-axis, ζ T increases simultaneously, reflecting increasing nutrification. In (c), d C = 0.03 yr-1, l Cb = 0.01 yr-1, l Cs = 0.00006 yr-1, rC = 0.2 yr-1, α C = 0.25 , ε C = 0.05 , g T = 5 yr-1,
ζ T = 10 yr-1, g M = 0.05 yr-1, rM = 0.3 yr-1, β M = 0.82 , γ MC = 0.01 , ν M = 0.98 and θ = 0.99 , and sedimentation increases d C by up to a factor of 1.4, decreases lCb and l Cs by up to a factor of 0.13 and decreases rC by up to a factor of 0.1. All four of these coral parameters change simultaneously as sedimentation increases (see section 13.2.3), such that in the graph, when d C increases along the x-axis, all three of the other parameters decrease simultaneously, reflecting increasing sedimentation.
stressors (fishing, nutrification and sedimentation) in turn, and then all combinations of these three stressors in turn. Each time, the changes in benthic covers at equilibrium are recorded. The changes in coral covers and total algal covers (T for the CTm, A for the CAm and T + M for the CTMm) at equilibrium are compared between the different cases. When more than one stressor is applied, if the changes are greater than the sums of the changes when each stressor is applied on its own, then there is synergy. The parameters for the pristine reef area tested for each model all take their mid-range values (see Chapter 9, Table 9.1); this is taken to represent a typical pristine reef area. For the CTMm, g M is also decreased to below its mid-range value in order to obtain macroalgae at equilibrium, to see if this affects the results. For the CAm and CTMm, which can both exhibit discontinuous phase shifts with fishing, for a pristine reef area that undergoes a discontinuous shift with fishing, the effect of applying nutrification is examined. The discontinuous phase shifts used for 217
(a) Proportional cover at equilibrium
θ (b) Proportional cover at equilibrium
rM / yr-1 (c) Proportional cover at equilibrium
d C / yr-1 Figure 13.4. Graphs showing discontinuous phase shifts for the CTMm, for a pristine reef area exposed to (a) fishing of herbivores, (b) nutrification and (c) sedimentation. A decrease in grazing pressure θ is used as a proxy for fishing, an increase in the macroalgal lateral growth rate rM is used as a proxy for nutrification and an increase in the coral death rate d C is used as a proxy for sedimentation. The red, green and brown curves represent the proportional covers for hard corals, turf algae and macroalgae 218
respectively; solid and dashed curves represent stable and unstable equilibria respectively. In (a), the pristine system has the parameters d C = 0.05 yr-1, l Cb = 0.0009 yr-1, l Cs = 0.00006 yr-1, rC = 0.2 yr-1, α C = 0.25 , ε C = 0.1 , g T = 10 yr-1, ζ T = 5 yr-1,
g M = 0.5 yr-1, rM = 0.35 yr-1, β M = 0.9 , γ MC = 0.1 , γ MT = 0.9 and θ = 0.7 , and fishing decreases θ to 0.2. In (b), d C = 0.05 yr-1, l Cb = 0.0009 yr-1, l Cs = 0.00006 yr-1, rC = 0.2 yr-1, α C = 0.2 , ε C = 0.05 , g T = 9 yr-1, ζ T = 11 yr-1, g M = 0.5 yr-1,
rM = 0.35 yr-1, β M = 0.9 , γ MC = 0.1 , γ MT = 0.9 and θ = 0.6 , and nutrification increases rM and ζ T by a factor of up to 2. Both algal growth parameters change simultaneously as nutrification increases (see section 13.2.2), such that in the graph, when rM increases along the x-axis, ζ T increases simultaneously, reflecting increasing nutrification. In (c), d C = 0.05 yr-1, l Cb = 0.01 yr-1, l Cs = 0.00006 yr-1, rC = 0.2 yr-1,
α C = 0.25 , ε C = 0.05 , g T = 5 yr-1, ζ T = 10 yr-1, g M = 0.2 yr-1, rM = 0.3 yr-1,
β M = 0.82 , γ MC = 0.01 , γ MT = 0.9 and θ = 0.9 , and sedimentation increases d C by up to a factor of 2, decreases lCb and l Cs by up to a factor of 0.3 and decreases rC by up to a factor of 0.25. All four of these coral parameters change simultaneously as sedimentation increases (see section 13.2.3), such that in the graph, when d C increases along the x-axis, all three of the other parameters decrease simultaneously, reflecting increasing sedimentation.
the CAm and the CTMm are those graphed in Figures 13.3a and 13.4a respectively. This is done to see whether the grazing pressure at which a discontinuous phase shift occurs as fishing is applied increases when nutrification is added to the system. The analytical model of Mumby et al. (2007a) was able to show this behaviour, which is a type of synergy in that the presence of both stressors can give a much greater change in coral and algal covers for a given decrease in grazing pressure than the presence of just fishing or just nutrification.
13.5.2 Synergy Results
For the CTm, the changes in coral and turf algal covers at equilibrium when combinations of more than one stressor are applied are always greater than the sums of 219
the changes seen when each stressor is applied on its own (Figure 13.5a). The same result was found for the CAm (Figure 13.5b), considering algal cover instead of turf algal cover. For the CTMm, the same result was found with g M taking either its midrange pristine value or a value (0.5 yr-1) much below its mid-range, and considering the sum of turf algal and macroalgal covers instead of turf algal cover. For the example with the mid-range value of g M , there is never any macroalgae at equilibrium since condition 5.8 in Chapter 5 never holds. However, for the example with the lower g M value, macroalgae does appear at equilibrium when nutrification is applied in combination with one or more of the other stressors (Figure 13.5c). Using the examples of discontinuous phase shifts with fishing found for the CAm and CTMm (see Figures 13.3a and 13.4a), nutrification was applied and it was found that the discontinuous phase shift was retained, but occurred at a higher value of θ (Figure 13.6). For the CTMm, the size of the hysteresis effect also increases with nutrification.
220
(a) Proportional cover at equilibrium
(b) Proportional cover at equilibrium
(c) Proportional cover at equilibrium
Figure 13.5. Graphs showing synergy between fishing of herbivores, nutrification and sedimentation on coral and algal covers at equilibrium, for the (a) CTm, (b) CAm and (c) CTMm. P = pristine reef system, F = pristine system with fishing, N = pristine 221
system with nutrification and S = pristine system with sedimentation, and a ‘+’ sign indicates that the specified stressors occur simultaneously. The red portion of the bars represents coral cover, the green portion represents turf algal cover in (a) and (c) and algal cover in (b), the brown portion represents macroalgal cover and the blue portion represents space cover. For (a)-(c), the pristine system has all parameters taking their mid-range values, except for g M = 0.5 yr-1 in (c). In (a), fishing decreases θ to 0.5, nutrification increases ζ T by a factor of 4 and sedimentation increases d C by a factor of 1.7, decreases l Cb and l Cs by a factor of 0.21 and decreases rC by a factor of 0.175. In (b), fishing decreases θ to 0.6, nutrification doubles rM and ζ T , and sedimentation affects the coral parameters as in (a). In (c), fishing decreases θ to 0.4, nutrification triples rM and ζ T , and sedimentation affects the coral parameters as in (a).
222
(a) Coral proportional cover at equilibrium
θ (b) Coral proportional cover at equilibrium
θ Figure 13.6. Graphs showing how nutrification increases the value of θ at which a discontinuous phase shift occurs with fishing, for the (a) CAm and (b) CTMm. For both graphs, the red curves represent coral cover at equilibrium in the case without nutrification and the brown curves represent coral cover at equilibrium in the case with nutrification; solid curves and dashed curves represent stable and unstable equilibria respectively. For (a) and (b), the cases without nutrification are the same as the discontinuous phase shift examples given in Figures 13.3a and 13.4a. The cases with nutrification are derived from the cases without nutrification by doubling ζ T and rM .
223
Appendix A13
A13.1 Immune-Inspired Algorithm Details Let the (bounded) parameter space being considered be denoted by V ⊂ R k and the (fitness) function be denoted by W, with W : V → R . Then the Immune-Inspired Algorithm (IIA) is used to find the parameter vector in V which maximises W. For each of the benthic models, for v ∈ V , v = (v1 ,K, v k ) with vi , 0 ≤ vi ≤ vi ,1 for 1 ≤ i ≤ k . The IIA searches V by mutating the initial parameter set v . For this mutation process, each vi is represented as a bit string of length l i . For different parameters, l i may be different to fine-tune the granularity required for each parameter, but for the benthic models, l i is fixed at 64. For vi , the bits are represented as a circular chromosome, such that successive bits are ordered sequentially 0,1,K, l i − 1 with the 0 coming after li − 1 . Between bit r0 and r1 = r0 + l mod(l i ) , where 0 ≤ r0 , l ≤ l i − 1 , let the continuous region of the circular bit string, including the end-points, be vi [r0 , l ] . Then a mutation of vi follows a procedure with three steps:
•
M1. Choose integers r0 , l in 0 ≤ r0 , l ≤ l i − 1 randomly, in a uniform and independent way
•
M2. With probability 1/2, each bit in vi [r0 , l ] is independently flipped ( 0 → 1 or 1 → 0 ) to obtain a new parameter vi′
•
M3. If vi′ is such that vi , 0 ≤ vi′ ≤ vi ,1 , then retain vi′ ; otherwise repeat M1 and M2
This process applied to each vi gives a new mutated vector v ′ = (v1′ ,K, v ′k ) ∈ V . The steps in the IIA are then as follows: •
Step 0. Initialisation. From V, choose a population P of N vectors randomly, in a uniform and independent way
•
Step 1. For each v ∈ P , do Steps 2-4 and then go to Step 5
•
Step 2. Clonal expansion. For v, create n clones 224
•
Step 3. Mutation. For the n clones of v, perform mutation (M1-3) to obtain v (1) ,K, v ( n ) ∈ V
{
}
•
Step 4. If W ( v * ) = max j W ( v ( j ) ) ≥ W ( v) , replace v by v * ; otherwise, retain v
•
Step 5. Variation. Replace the least fit member of P with a new member randomly chosen from V
•
Step 6. Evolution. Repeat Steps 1-5 for g – 1 more generations
For the benthic models, N = 3 and n = 3 are used to give efficient performance of the IIA (Kelsey et al. 2003). Kelsey and Timmis (2003) found that for several functions, including the 20-dimensional Griewangk function, g = 50 is sufficient to converge to the minimum. Thus, for the benthic models, g = 100 is used to ensure efficiency. Furthermore, for each search, 1,000 runs are performed and the maximum value of W out of all runs taken, to make it even more likely that the maximum W will be found.
A13.2 Applying the Immune-Inspired Algorithm to the Coral-Turf Model (CTm) From eqns. 4.1 and 4.2 for the CTm (Chapter 4, section 4.2.3), possible equilibria in the biological domain, (C , T ), satisfy the two nullcline equations
(l
)
+ l Cb C (S + ε C T ) + rC (S + α C T )C − d C C = 0 ,
(A13.1a)
ζ T (1 − θ )S − g T θT − ε C (l Cs + l Cb C )T − rC α C CT = 0 ,
(A13.1b)
s C
where S = 1 − C − T . As shown in Appendix A4, section A4.2.2, eqn. A13.1a defines a nullcline C = C (1) (T ) , with C(1) (T ) a decreasing function of T satisfying 0 < C (1) (T ) < 1 for all 0 ≤ T ≤ 1 . Similarly, it is shown in section A4.2.2. that eqn. A13.1b defines a nullcline C = C ( 2) (T ) , with C( 2) (T ) a decreasing function of T satisfying C ( 2) (0) = 1 and
C ( 2) (T ) + T < 1 for all 0 ≤ T ≤ T0 ( T0 is defined by eqn. A4.10 in section A4.2.2). C( 2) (T ) is negative for T0 < T ≤ 1 , and the threshold value T0 = T0 (θ ) is a decreasing function of θ with T0 (1) = 0 and 0 < T0 (0) < 1 (see eqn. A4.10). 225
Substituting C = C (1) (T ) into eqn. A13.1b and solving for θ as a function of T gives
ζ T S (1) (T ) − ε C (lCs + l Cb C (1) (T ))T − α C rC C (1) (T )T , θ (T ) = ζ T S (1) (T ) + g T T
(A13.2a)
where S (1) (T ) = 1 − C (1) (T ) − T . Also, for given θ , eqn. A13.1b is satisfied at
C = C ( 2) (T ) by definition. That is,
ζ T S ( 2) (T ) − ε C (l Cs + l Cb C ( 2) (T ))T − α C rC C ( 2) (T )T . θ= ζ T S ( 2) (T ) + g T T
(A13.2b)
Subtracting eqn. A13.2b multiplied by ζ T S ( 2) (T ) + gT T from eqn. A13.2a multiplied by
ζ T S (1) (T ) + gT T gives: θ (T )(ζ T S (1) (T ) + g T T ) − θ (ζ T S ( 2) (T ) + g T T )
{
(
)}
= −(C(1) (T ) − C( 2 ) (T )) ζ T + ε C lCb + α C rC T .
(A13.2c)
An equilibrium of the CTm with grazing pressure θ is a solution of C (1) (T ) = C ( 2) (T ) . If T = T is such a solution, then S (1) (T ) = S ( 2) (T ) = S (T ) and eqn. A13.2c gives
(θ (T ) − θ )(ζ S (T ) + g T ) = 0 , and hence θ (T ) = θ . Conversely, if T is a solution of θ (T ) = θ , eqn. A13.2c gives (C (T ) − C (T )){ζ (1 − θ ) + (ε l + α r )T } = 0 , and hence C (T ) = C (T ) . That is, T is an equilibrium of the CTm. Thus, T
T
(1)
(1)
( 2)
T
b C C
C C
( 2)
T = T is an equilibrium of the CTm with grazing pressure θ if and only if it is a solution of θ (T ) = θ . Note that, to give an equilibrium in the biological domain, a solution T = T of θ (T ) = θ must satisfy C (1) (T ) = C ( 2) (T ) and 0 < T < T0 (θ ) (see section A4.2.2). Since there are no solutions T = T of C (1) (T ) = C ( 2) (T ) with T0 (θ ) ≤ T ≤ 1 or T = 0 (section A4.2.2), 226
any solution of θ (T ) = θ found in the range 0 ≤ T ≤ 1 must give equilibria in the biological domain. Now differentiate eqn. A13.2a multiplied by ζ T S (1) (T ) + gT T with respect to T to give
(ζ
T
S (1) (T ) + g T T )θ ′(T ) + {− ζ T (1 + C (′1) (T )) + g T }θ (T )
(
)
= −ζ T (1 + C (′1) (T )) − ε C l Cb C (′1) (T )T − α C rC C (′1) (T )T − ε C l Cs + l Cb C (1) (T ) − α C rC C (1) (T ) . (A13.3a) Differentiating eqn. A13.2b multiplied by ζ T S ( 2) (T ) + gT T with respect to T gives:
{− ζ (1 + C ′ (T )) + g }θ = −ζ (1 + C ′ (T )) − ε l C ′ (T )T − α T
( 2)
T
( 2)
T
b C C
( 2)
(
)
r C (′2) (T )T − ε C lCs + l Cb C (2 ) (T ) − α C rC C (2 ) (T ) .
C C
(A13.3b) Let (C , T ) be an equilibrium in the biological domain for which θ (T ) = θ and
C = C (1) (T ) = C ( 2) (T ) . Then, evaluating eqns. A13.3a,b at T = T and subtracting gives
(ζ
T
{
(
)}
S + g T T )θ ′(T ) = −(C (′1) (T ) − C (′2) (T )) ζ T (1 − θ ) + ε C l Cb + α C rC T ,
where S = 1 − C − T . Since T , S > 0 (Appendix A4, section A4.2.2), it follows that
θ ′(T ) < 0 ⇔ C (′1) (T ) > C (′2 ) (T ) ,
(A13.4a)
θ ′(T ) > 0 ⇔ C(′1) (T ) < C (′2 ) (T ) .
(A13.4b)
However, from section A4.2.3 in Appendix A4, a generic equilibrium (C , T ) is LAS if and only if C(′1) (T ) > C(′2) (T ) , and is unstable if and only if C(′1) (T ) < C(′2 ) (T ) . It therefore follows from A13.4a,b that A generic equilibrium (C , T ) in the biological domain for which θ (T ) = θ is LAS if and only if θ ′(T ) < 0 .
(A13.5a) 227
A generic equilibrium (C , T ) in the biological domain for which θ (T ) = θ is unstable if and only if θ ′(T ) > 0 .
(A13.5b)
In section A4.2.2, it was shown that for a given θ , there is either one (generic) equilibrium or three (generic) equilibria of the CTm in the biological domain. If there is one equilibrium, then it must be LAS (section A4.2.3), and hence θ ′(T ) < 0 . On the other hand, if there are three equilibria, then two of them are LAS and the intermediate equilibrium is unstable (section A4.2.3), such that there are alternative stable states (ASS). At the unstable equilibrium, θ ′(T ) > 0 . It follows that an unstable equilibrium in the biological domain exists, and hence multiple equilibria in the biological domain exist, for a grazing pressure θ if and only if there is a T = T in the range 0 ≤ T ≤ 1 for which θ (T ) = θ and θ ′(T ) > 0 . As discussed above, such a T will then necessarily lie in the range 0 < T < T0 (θ ) . Figure A13.1 illustrates the two cases when there are no values of θ or T (in the biologically relevant ranges) for which multiple equilibria exist, and when there is a range of values of θ and a range of values of T for which multiple equilibria do exist.
A13.2.1 Deriving and Computing the Fitness Function for the CTm
Let V denote a parameter space for the CTm, with parameter vectors
(
)
v = d C , lCb , l Cs , rC , α C , ε C , g T , ζ T , where each parameter is specified by a given range to
be searched; these vectors exclude θ . Let θ (T ; v ) denote the function A13.2a for the parameter vector v ∈ V . Also, let Θ ⊆ [0,1) be a range of values of θ to be searched. To apply the IIA (as described in section A13.1), the fitness function is taken as:
W ( v) = max{θ ′(T ; v ) : 0 ≤ T ≤ 1 and θ (T ; v ) ∈ Θ}.
(A13.6)
In W ( v ) , the explicit form of θ ′(T ; v ) is used, which is derived by directly differentiating θ (T ; v ) (as given by eqn. A13.2a) with respect to T and using the explicit forms of C(1) (T ) and C (′1) (T ) given in Appendix A4, section A4.2.2. Let
228
(a)
θ
T
(b)
θ
T
Figure A13.1. For the CTm, for a parameter vector v ∈V , conceptual diagrams showing the cases where (a) θ ′(T ; v ) < 0 for 0 ≤ T ≤ 1 and (b) θ ′(T ; v ) > 0 for some T in 0 ≤ T ≤ 1 with θ (T ; v ) ∈ [0,1) . For both graphs, the solid curves represent stable equilibria. For the graph in (b), the dashed curve represents unstable equilibria and the range of θ for which there are alternative stable states (ASS) is approximately 0 .2 ≤ θ ≤ 0 .3 .
W * = max{W ( v ) : v ∈ V } . If W * > 0 , then there is a v ∈ V and a T ∈ [0,1] such that
θ = θ (T ; v ) ∈ Θ and θ ′(T ; v ) > 0 . Thus, from A13.5b, there are multiple equilibria in the biological domain for at least one parameter vector (v,θ ) ∈V × Θ . On the other hand, if W * < 0 , then A13.5a implies that there is a unique (LAS) equilibrium in the biological domain for all parameter vectors (v, θ ) in V × Θ . To compute W ( v ) , m + 1 equally spaced points Ti = i m , 0 ≤ i ≤ m , in the 229
interval 0 ≤ T ≤ 1 are chosen; θ (Ti ; v ) and θ ′(Ti ; v ) are then computed, and the maximum of these values taken, as in eqn. A13.6. The value of m used is 200.
A13.3 Applying the Immune-Inspired Algorithm to the Coral-Algae Model (CAm) The nullcline equations for the CAm eqns. 5.1 and 5.2 (Chapter 5, section 5.1.3) are
(l
s C
)
+ l Cb C {S + ε C (1 − ν M )A} + rC (1 − β M ν M A){S + α C (1 − ν M )A}C − d C C
− γ MC rM ν M AC = 0 ,
(A13.7a)
(
)
rMν M A(S + γ MC C ) + ζ T (1 − θ )S − {g M ν M + g T (1 − ν M )}θA − ε C (1 − ν M ) l Cs + lCb C A
− rC α C (1 − ν M )(1 − β Mν M A) AC = 0 ,
(A13.7b)
where S = 1 − C − A . The (positive) solution of eqn. A13.7a gives an explicit nullcline
C = C (1) ( A) (see Appendix A5, section A5.1.2). Substituting this into eqn. A13.7b and solving for θ as a function of A gives
(
)
rMν M S (1) ( A) + γ MC rMν M C (1) ( A) − ε C (1 − ν M ) l Cs + l Cb C (1) ( A) ζ T S (1) ( A) + A − α C rC (1 − ν M )(1 − β Mν M A)C (1) ( A) , θ ( A) = ζ T S (1) ( A) + {g Mν M + g T (1 − ν M )}A (A13.8) where S (1) ( A) = 1 − C (1) ( A) − A . Using the analysis of Appendix A5, section A5.1.3, the development of the theoretical underpinning of the IIA as applied to the CAm proceeds analogously to that for the CTm in section A13.2, with θ ( A) playing the analogous role to θ (T ) for the CTm. In particular, the analogues of A13.5a,b in section A13.2 hold: A generic equilibrium (C , A ) in the biological domain for which θ (A ) = θ is LAS if and only if θ ′(A ) < 0 .
(A13.9a)
230
A generic equilibrium (C , A ) in the biological domain for which θ (A ) = θ is unstable if and only if θ ′(A ) > 0 .
(A13.9b)
The IIA is applied in a way analogous to that described for the CTm in section A13.2.1. The parameter space V consists of the parameter vectors
(
)
v = d C , lCb , l Cs , rC , α C , ε C , g T , ζ T , g M , rM , β M , γ MC ,ν M , where each parameter is
specified by a given range to be searched, and to compute the fitness function, 201 equally spaced points in 0 ≤ A ≤ 1 are used.
A13.4 Applying the Immune-Inspired Algorithm to the Coral-TurfMacroalgae Model (CTMm) The nullcline equations for the CTMm are
(l
s C
)
+ lCb C (S + ε C T ) + rC (1 − β M M )(S + α C T )C − d C C − γ MC rM MC = 0 ,
(A13.10a)
ζ T (1 − θ )S − g T θT − ε C (l Cs + lCb C )T − rC α C (1 − β M M )TC − γ MT rM MT = 0 ,
(A13.10b)
M {rM (S + γ MC C + γ MT T ) − g M θ } = 0 ,
(A13.10c)
where S = 1 − C − T − M . Clearly, M = 1 (and C = T = S = 0 ) is a solution of eqns. A13.10a,b, but is a solution of eqn. A13.10c only if the grazing pressure θ = 0 ( g M , the maximum grazing rate on macroalgae, is positive in the empirically derived range – see Table 9.1 in Chapter 9). This equilibrium, which arises when θ = 0 , may or may not be stable. The only other solution of eqn. A13.10c with θ = 0 is M = 0 . Recall that a necessary condition for an equilibrium without macroalgae ( M = 0 ) of the CTMm to be LAS is Z = g M θ − rM (S + γ MC C + γ MT T ) > 0 , where S = 1 − C − T (see Appendix A5, section A5.2.3, conditions A5.17c and A5.18). But, for an equilibrium in the biological domain without macroalgae, it is impossible for this condition to hold when θ = 0 . Thus, when θ = 0 , there are no multiple stable equilibria. 231
Now, consider the case θ > 0 . Recall that equilibria of the CTMm without macroalgae are equilibria (C , T ) of the CTm with the same parameters (Appendix A5, section A5.2.2). However, it has been shown in section 13.4.2 that the CTm does not have multiple equilibria for the empirically derived parameter ranges. This implies that the CTMm does not have multiple equilibria without macroalgae for the empirically derived parameter ranges. (See Tables 9.1 and 13.1 in Chapters 9 and 13 respectively for the empirically derived parameter ranges for the CTm and CTMm.) Note that in the condition for an equilibrium without macroalgae to be LAS ( Z = g M θ − rM (S + γ MC C + γ MT T ) > 0 ), C , T and S are independent of gM . Thus, such an equilibrium can be LAS only if g M is larger than some threshold g M* > 0 which depends only on the CTm parameters together with rM , γ MC and γ MT . To find equilibria with macroalgae present, only solutions of eqn. A13.10c with 0 < M ≤ 1 are considered, and these must satisfy
gM =
rM
θ
{1 − (1 − γ MC )C − (1 − γ MT )T − M } .
(A13.11)
Next, eqns. A13.10a and A13.10b are solved for T as functions of C and M. From eqn. A13.10a,
T=
{l
s C
}
+ l Cb C + rC (1 − β M M )C (1 − C − M ) − d C C − γ MC rM MC , (1 − ε C ) lCs + lCb C + rC (1 − α C )(1 − β M M )C
(
)
(A13.12a)
and from eqn. A13.10b,
T=
ζ T (1 − θ )(1 − C − M ) . ζ T (1 − θ ) + g T θ + ε C (lCs + lCb C ) + α C rC (1 − β M M )C + γ MT rM M
(A13.12b)
Equating these two expressions for T leads to an equation F (C , M ) = 0 , where
F (C , M ) = K (C , M ) − L(C , M ) ,
(A13.13)
with
232
K (C , M )
{ ( + {g θ + ε (l
)
}
= ζ T (1 − θ ) ε C lCs + lCb C + α C rC (1 − β M M )C (1 − C − M ) T
C
s C
)
}{
+ l Cb C + α C rC (1 − β M M )C + γ MT rM M l Cs + l Cb C + rC (1 − β M M )C
(1 − C − M )
}
(A13.14a)
and
L(C , M )
{
(
)
}
= ζ T (1 − θ ) + g T θ + ε C l Cs + lCb C + α C rC (1 − β M M )C + γ MT rM M (d C + γ MC rM M )C .
(A13.14b) First, consider the case with fixed 0 < M < 1 . In this case, F (C , M ) is a cubic in C with negative leading coefficient − {ε C l Cb + α C rC (1 − β M M )}{lCb + rC (1 − β M M )}. Clearly,
L(C , M ) > 0 for all C > 0 and L(0, M ) = 0 . Since K (0, M ) > 0 , it follows that F (0, M ) > 0 . Also, K (C , M ) ≤ 0 for all C ≥ 1 − M . Hence, F (C , M ) < 0 for all C ≥ 1 − M . It follows that there is at least one solution of F (C , M ) = 0 in the range 0 < C < 1 − M , and there are no solutions with C in the range C ≥ 1 − M .
In fact, for the empirically derived parameter ranges, it can be shown that there is exactly one solution of F (C , M ) = 0 in the biological range 0 < C < 1 − M . Denote this solution by C = C (1) (M ) . Then 0 < C (1) (M ) < 1 − M . Note that the derivative C (′1) (M ) can be calculated by differentiating the identity
F (C (1) (M ), M ) = 0 to obtain
C (′1) (M ) = −
∂F ∂M (C(1) (M ), M ). ∂F ∂C
(A13.15)
The partial derivatives can be calculated explicitly from eqn. A13.13 and eqns. A13.14a,b. Let T(1) (M ) denote the value of T obtained by substituting C = C (1) (M ) into either of the two (equal) expressions A13.12a,b for T as a function of C and M. Consider
S (1) (M ) = 1 − M − C (1) (M ) − T(1) (M ) . Then, using the form of T(1) (M ) given by eqn. 233
A13.12b:
1 − S (1) (M ) = M + C (1) (M ) + T(1) (M ) = M + C (1) (M ) +
ζ T (1 − θ )(1 − C(1) (M ) − M ) ζ T (1 − θ ) + gT θ + ε C (l + lCb C(1) (M )) + α C rC (1 − β M M )C(1) (M ) + γ MT rM M s C
(
)
g T θ + ε C l Cs + l Cb C (1) (M ) + α C rC (1 − β M M )C (1) (M ) ζ T (1 − θ ) + (C (1) (M ) + M ) + γ MT rM M = ζ T (1 − θ ) + g T θ + ε C l Cs + l Cb C (1) (M ) + α C rC (1 − β M M )C (1) (M ) + γ MT rM M
(
)
< 1,
since C (1) (M ) + M < 1 . Thus, S (1) (M ) > 0 . Also, using eqn. A13.12b, T(1) (M ) > 0 , since
C (1) (M ) + M < 1 . Hence, the equilibrium (C (1) (M ), T(1) (M ), M ) lies in the biological domain for all 0 < M < 1 . In particular, substituting C = C (1) (M ) and T = T(1) (M ) into eqn. A13.11 defines a function
g M (M ) =
rM
=
rM
θ θ
{1 − (1 − γ )C (M ) − (1 − γ )T (M ) − M } MC
{S (M ) + γ (1)
(1)
MC
MT
(1)
C (1) (M ) + γ MT T(1) (M )}.
(A13.16)
The above discussion implies that g M (M ) > 0 for all 0 < M < 1 . Since C (′1) (M ) can be calculated from eqn. A13.15, it follows that T(1′) (M ) can be calculated using either eqn. A13.12a or eqn. A13.12b, and
T(1′) (M ) =
∂T(1) ∂C
(C (M ), M )C ′ (M ) + ∂T (C (M ), M ) . ∂M (1)
(1)
(1)
(1)
(A13.17)
Hence, g ′M (M ) can be calculated. Second, consider the case with M = 1 . Using similar calculations to the case with
234
0 < M < 1 , it can be shown that there is exactly one solution of F (C , M ) = 0 in the
biological range, which is C (1) (M ) = 0 . This gives T(1) (M ) = 0 and S (1) (M ) = 0 , and hence, the equilibrium (C (1) (1), T(1) (1),1) lies in the biological domain. Also, using eqn. A13.16, g M (1) = 0 . g ′M (M ) is calculated in the same way as for the case with 0 < M < 1.
Note that, using similar calculations to the case with 0 < M < 1 , it can also be shown that g M (M ) is defined at M = 0 with g M (0) > 0 , and that the equilibrium
(C (0), T (0),0) lies in the biological domain. From eqn. A13.10c, this equilibrium satisfies Z = g θ − r (S + γ C + γ T ) = 0 and furthermore, satisfied eqns. (1)
(1)
M
M
MC
MT
A13.10a,b with M = 0 . Thus, g M (0) = g M* .
A13.4.1 Deriving and Computing the Fitness Function for the CTMm Let V denote a parameter space for the CTMm, with parameter vectors
{
}
v = d C , lCb , l Cs , rC , α C , ε C , g T , ζ T , rM , β M , γ MC , γ MT , θ , where each parameter is specified
by a given range to be searched. Denote by g M (M ; v ) the function A13.16 with parameters vector v ∈V . Let G = [g M min , g M max ] be a range of values of g M to be searched. For θ = 0 , multiple stable equilibria are not possible, as shown above. For 0 < θ < 1 , clearly, if g ′M (M ; v ) < 0 for all 0 ≤ M ≤ 1 , then g M (M ; v ) is decreasing, and
hence there can be at most one stable equilibrium in the biological domain (see Figure A13.2a). On the other hand, if there are values of M in 0 ≤ M ≤ 1 for which
g ′M (M ; v ) > 0 , then there is the possibility of multiple stable equilibria in the biological domain (see Figure A13.2b). Note that M = 0 is considered because g M (0; v ) > 0 , such that if g ′M (0; v ) > 0 , then there is the possibility of multiple stable equilibria in the biological domain, since g M (M ; v ) initially increases above g M (0; v ) and g M (1) = 0 . The implementation of the IIA for the CTMm requires finding the equilibrium solution C = C (1) (M ) for different values of M, for a given parameter set v. This is done by solving the cubic F (C , M ) = 0 using the formulae from Press et al. (2002), to obtain three explicit expressions for C in terms of M, which are the three roots, and then numerically identifying the unique root which lies in the biological range 0 ≤ C ≤ 1 − M . 235
The fitness function
W ( v) = max{g ′M (M ; v ) : 0 ≤ M ≤ 1 and g M (M ; v ) ∈ G}
(A13.18)
is used for all θ in the empirically derived range except for θ = 0 , when W ( v ) is defined to be negative. W ( v ) is computed by taking 201 equally spaced values of M covering the interval 0 ≤ M ≤ 1 . If W * = max{W ( v) : v ∈ V } < 0 , then there are no parameter vectors (v, g M ) ∈ V × G for which multiple stable equilibria exists in the biological domain. On the other hand, if W * > 0 , then there is the possibility of multiple stable equilibria in the biological domain for at least one parameter vector
( v, g M ) ∈ V × G .
236
(a)
g M max
gM
g M min M
(b)
g M max
gM
g M min M
Figure A13.2. For the CTMm, for a parameter vector v ∈V , conceptual diagrams showing the cases where (a) g ′M (M ; v ) < 0 for 0 ≤ M ≤ 1 and (b) g ′M (M ; v ) > 0 for some M in 0 ≤ M ≤ 1 with g M (M ; v ) ∈ G = [g M min , g M max ] . In (a), g M min = 0.05 and g M max = 1.1 , whereas in (b), g M min = 0.1 and g M max = 0.7 . In both graphs, g M min and g M max are represented as dotted lines. Also, the blue line represents stable equilibria without macroalgae, as calculated using M = 0 from eqn. A13.10c, whereas the red line represents equilibria with macroalgae (which may or may not be stable), as calculated using rM (S + γ MC C + γ MT T ) − g M θ = 0 from eqn. A13.10c. All equilibria plotted are in the biological domain.
237
Chapter 14. Parameter Sweeps A parameter sweep involves randomly sampling parameter sets from the parameter space of a benthic model and testing each parameter set to see if it gives a specified type of phase shift. In this chapter, the mathematics behind a parameter sweep is first detailed for the two simplest benthic models, the CTm and the CAm, (section 14.1) and the results from the sweeps are then presented (section 14.2). The most complex benthic model, the CTMm, is far less analytically tractable than the CTm or the CAm (see Chapters 4 and 5). This means that sweeps for the CTMm are much more computationally expensive and hence time-consuming, such that sweeps for this model are not performed due to time constraints.
14.1 Mathematics of Parameter Sweeps The parameter sweeps in this chapter aim to find phase shifts when the grazing pressure
θ changes, which reflects changing grazer (herbivorous fish and sea urchin) biomass (Chapter 7). This may result from fishing or from other factors such as changes in fish or urchin recruitment, and is therefore of interest to reef management. For the parameter sweeps in this chapter, each parameter is sampled from a Uniform distribution with the minimum and maximum values for the parameter range used as the distribution limits. To apply a parameter sweep, it is necessary to define quantitative criteria for different types of shifts for a coral reef. For the CTm, the numerical results from section 13.4.2 in Chapter 13 show that only continuous shifts with changing θ are possible, because parameter sets giving multiple equilibria were not found using the empirically derived parameter ranges. However, for the CAm, numerical results from section 13.4.2 show that discontinuous shifts are possible in addition to continuous ones, because parameter sets giving multiple equilibria were found using the empirically derived parameter ranges. Thus, for the CAm, each sampled parameter set is tested to see if discontinuous phase shifts are possible with changing θ (see section 14.1.3 for more details).
238
14.1.1 Defining Different Types of Continuous Phase Shifts
Phase shifts with changing grazing pressure θ can be characterised by changes in coral cover or algal cover, or both. First, phase shifts characterised by changes in coral cover at equilibrium are defined quantitatively. Equilibrium cover is examined in order to determine the final effects of a change in θ , rather than just transient effects. The curve
θ = θ (C ) , where C is the coral cover at equilibrium (overbar denotes equilibrium value in the biological range), is used because it describes how coral cover responds to changing θ . For the CTm and the CAm, θ (C ) can be derived from the equilibrium equations, and explicit forms of θ (C ) for both models are given in section A14.1 in Appendix A14. A continuous phase shift is defined according to the maximum change in C that can occur for a given change in θ , ∆θ . Let C 0 be the smallest C in the biological range 0 ≤ C ≤ 1 for which θ (C ) = 0 and C1 be C in the biological range for which
θ (C ) = 1 . C1 is the maximum C in the biological range because it corresponds to T = 0 or A = 0 and for all equilibria in the biological domain, C is a decreasing
function of T or A (Appendices A4 and A5, sections A4.2.3 and A5.1.3). C 0 must then be the minimum C because otherwise, there will be an even number of equilibria for some θ in 0 ≤ θ < 1 , which is not allowed (Appendices A4 and A5, sections A4.2.2 and A5.1.2). To determine the maximum change in C as θ decreases from 1, n + 1 equally spaced C points in the interval C 0 ≤ C ≤ C1 are taken. In the applications, n = 200 . For a given change in θ , ∆θ , and for each of these n + 1 C points, the equation
θ (C ) − θ (C − ∆C ) = ∆θ
(14.1)
is solved for ∆C and the maximum ∆C value recorded. The maximum possible value of ∆C is C1 − C 0 and hence, ∆C is expressed as a proportion of C1 − C 0 :
E (C ) =
∆C . C1 − C 0
(14.2) 239
E (C ) ∈ [0, 1] and finding the maximum ∆C is equivalent to finding E * = max E (C ) .
(14.3)
C0 ≤ C ≤ C1
For E * , let the corresponding C value be denoted by C * . E * is computed for several different values of ∆θ = tθ , to obtain a set of values of E * = E * (tθ ) . The ratio
t * Rmin = min * θ tθ E (tθ )
(14.4)
is then computed. The smaller this ratio is, the larger is the maximum change in C for a given change of θ , and this is used as a measure of the strength of a phase shift. If * Rmin ≤ RC* ,
(14.5)
for some threshold RC* that has to be chosen, then an RC* -continuous phase shift is said to have occurred. Figure 14.1a,b shows graphs of θ (C ) and E (C ) for an arbitrary set of parameters for the CTm, and ∆θ , C 0 , C1 , C * and E * are shown for tθ = ∆θ = 0.1 . Instead of considering ∆θ with θ decreasing from 1, ∆θ can be considered with θ increasing from 0, and the maximum ∆C calculated using the equation
θ (C + ∆C ) − θ (C ) = ∆θ .
(14.6)
As before, ∆C can be expressed as a proportion of C1 − C 0 to give
F (C ) =
∆C , C1 − C 0
(14.7)
with F (C ) ∈ [0, 1] . The maximum ∆C is equivalent to F * = max F (C ) .
(14.8)
C0 ≤ C ≤ C1
240
Denote by C* the value of C corresponding to F * . Analogous to eqn. 14.4, the ratio
t Rmin* = min * θ tθ F (tθ )
(14.9)
can be computed and if Rmin* ≤ RC* ,
(14.10)
then an RC* -continuous phase shift is said to have occurred. If there is no region of multiple equilibria as θ changes between 0 and 1 (which is always the case for the CTm, based on the numerical results in Chapter 13, section 13.4.2), then the maximum ∆C as θ decreases from 1 is the same as the maximum ∆C as θ increases from 0.
Thus, if there is a continuous phase shift in this case, then E * = F * and * hence Rmin = Rmin* . Therefore, there is only a need to consider ∆θ in the direction of
decreasing θ in this case. Instead of using coral cover as a defining variable, turf algal cover, T , for the CTm or total algal cover, A , for the CAm can be used. The change in turf algal cover, ∆T , or the change in algal cover, ∆A , can be considered, using the functions θ (T ) and
θ (A ) given explicitly by the formulae A13.2a and A13.8 in Appendix A13, except with overbars for the arguments to denote equilibrium values of T and A in the biological ranges. If ∆θ is considered in the direction of decreasing θ , then analogous to eqns. 14.2 and 14.3, for the CTm, G and G * can be defined as
G (T ) =
∆T , T0 − T1
(14.11)
G * = max G (T ) ,
(14.12)
T1 ≤T ≤T0
where T0 and T1 are the maximum and minimum values of T in the biological range. The maximum in eqn. 14.12 is computed by taking 201 equally spaced T values in the range T1 ≤ T ≤ T0 . If ∆θ is considered in the direction of increasing θ , then for the 241
CTm, analogous to eqns. 14.7 and 14.8, H and H * can be defined as
H (T ) =
∆T , T0 − T1
(14.13)
H * = max H (T ) .
(14.14)
T1 ≤T ≤T0
For G * and H * , let the corresponding values of T be denoted by T * and T* respectively. G, G * , H and H * are the same for the CAm except that T is replaced by * A . Analogous to eqns. 14.4 and 14.9, ratios S min and S min* can be defined:
t * = min * θ , S min tθ G (tθ )
(14.15)
t S min* = min *θ . tθ H (tθ )
(14.16)
Analogous to conditions 14.5 and 14.10, if * S min ≤ S A* ,
(14.17)
then there is a S *A -continuous phase shift, and if
S min* ≤ S A* ,
(14.18)
then there is a S A* -continuous phase shift. If there is no region of multiple equilibria as
θ changes between 0 and 1, then the maximum ∆T (or ∆A ) as θ decreases from 1 is the same as the maximum ∆T (or ∆A ) as θ increases from 0. Thus, if there is a * continuous phase shift in this case, then G * = H * and hence S min = S min* . Therefore,
there is only a need to consider ∆θ in the direction of decreasing θ in this case. Figure 14.1c,d shows graphs of θ (T ) and G (T ) for an arbitrary set of parameters for the CTm, and ∆θ , T0 , T1 , T * and G * are shown for tθ = ∆θ = 0.1 . 242
(a)
θ
∆θ
C*
C0 C * − ∆C
(b)
C1
C
C1
C
E
E*
C0
C*
243
(c)
θ
∆θ
T1
(d)
T*
T * + ∆T
T0
T
G G*
T1
T*
T0
T
Figure 14.1. Graphs of (a) θ (C ) , (b) E (C ) , (c) θ (T ) , and (d) G (T ) for an arbitrary set of parameters for the CTm. ∆θ , C 0 , C1 , T0 , T1 , C * , E * , T * and G * are shown on the graphs for tθ = ∆θ = 0.1 . Definitions are given in the main text.
244
14.1.2 Defining Different Types of Discontinuous Phase Shifts
For the CAm, if there is a region of θ for which there are multiple equilibria, then there could be multiple solutions of ∆C for eqn. 14.1 or 14.6, for given C and ∆θ . Thus, E (C ) and F (C ) need to be redefined as
∆C E (C ) = min : θ (C ) − θ (C − ∆C ) = ∆θ , C1 − C 0
(14.19)
∆C F (C ) = min : θ (C + ∆C ) − θ (C ) = ∆θ . C1 − C 0
(14.20)
Similarly,
∆A G (A ) = min : θ (A ) − θ (A + ∆A) = ∆θ , A0 − A1
(14.21)
∆A H (A ) = min : θ (A − ∆A ) − θ (A ) = ∆θ . A0 − A1
(14.22)
Also, when multiple equilibria exist, because of hysteresis, E * need not be equal to F * and G * need not be equal to H * . This means that there could be different types of phase shifts when θ decreases from 1 and when θ increases from 0. With the new definitions 14.19-22, if condition 14.5 holds, then there is an RC* phase shift. If ∆C corresponding to RC* occurs across a discontinuity, that is, this ∆C includes a discontinuous change in C as θ changes by ∆θ , then the phase shift is defined to be a discontinuous one. Otherwise, the phase shift is continuous. Figure 14.2a shows the graph of θ (C ) for an arbitrary set of CAm parameters for which the maximum ∆C occurs across a discontinuity as θ decreases from 1, using tθ = ∆θ = 0.1 . If condition 14.10 holds, then there is an RC* -phase shift and if the ∆C corresponding to RC* occurs across a discontinuity, the shift is discontinuous. Similarly, if condition 14.17 holds, then there is a S *A -phase shift and if 245
condition 14.18 holds, then there is a S A* -phase shift. If the ∆A corresponding to S *A or
S A* occurs across a discontinuity, then there is a discontinuous S *A - or S A* -phase shift respectively. Figure 14.2b shows the graph of θ (A ) for an arbitrary set of CAm parameters for which the maximum ∆A occurs across a discontinuity as θ decreases from 1, using tθ = ∆θ = 0.1 .
14.1.3 Sweeps Performed for the Benthic Models
For the CTm, parameter sets were randomly sampled from the parameter space (a hypercube) defined by the parameter ranges in Chapter 9, Table 9.1, which are for a reef unaffected by nutrification or sedimentation (see Chapter 13, section 13.2). Denote this parameter space by V f . Each parameter set sampled was tested to see if it gave a continuous phase shift, using five tθ values of increasing magnitude (see Table 14.1). To determine if there was a phase shift, and of what type, three different sets of criteria are used. The first set considers only changes in coral cover and uses six different threshold values of RC* of decreasing magnitude (Table 14.1), representing phase shifts of increasing severity. The lowest value of RC* used is 0.1, which means that relative to ∆θ , the maximum value of the scaled change in coral cover, E, must be at least 10
times greater. This represents a very sharp change in coral cover. The second set of criteria considers only changes in turf algal cover and uses six different threshold values of S *A of decreasing magnitude. Values of S *A used are the same as those for RC* . The final set of criteria considers changes in both coral and turf algal covers. The same six values of RC* and S *A are used, but this time, a phase shift is recorded as occurring only * * if both Rmin ≤ RC* and S min ≤ S A* , that is, the change in both coral and turf covers must
be sufficiently high in response to a given change of θ . This is referred to as an
(R
* C
)
, S A* -continuous phase shift.
The percentages of parameter sets that give each type of phase shift, as defined by the three sets of criteria, are the outputs. For each set of criteria, N = 20,000 parameter sets are sampled, which gives one parameter sweep. To determine whether an output has converged, the maximum absolute percentage difference between an output derived using N − 1000 parameter sets and outputs derived using N − 1000 + i 246
(a)
θ
∆θ
C0 C − ∆C
(b)
C
C1
*
C
*
θ
∆θ
A1
A*
A * + ∆A
A0
A
Figure 14.2. Graphs of (a) θ (C ) and (b) θ (A ) for an arbitrary set of parameters for the CAm. ∆θ , C 0 , C1 , A0 , A1 , C * and A * are shown on the graphs for tθ = ∆θ = 0.1 . Definitions are given in the main text. For (a) and (b), the maximum ∆C and the maximum ∆A both occur across a discontinuity.
parameter sets, with 1 ≤ i ≤ 1000 , was calculated, and if this does not exceed 5%, then the output result was taken to have converged. The value derived using N input sets was then recorded. This criterion is chosen as a balance between accuracy and 247
computational time, and to be consistent with that used for the sensitivity analyses (see Chapter 15, section 15.1.3). For a parameter sweep, if there was no convergence for any output using N = 20,000 , then 5,000 more parameter sets are sampled iteratively until convergence was achieved for all outputs. When convergence has been achieved, a Principal Components Analysis (PCA) is performed for all the sampled parameter sets that give phase shifts, using the correlation matrix for the parameters. A PCA finds uncorrelated variables, the principal components (PCs), that are a function of potentially correlated variables (here, the parameters), with the first PC accounting for as much variability in the data set (here, the sampled parameter values that give phase shifts) as possible and subsequent PCs accounting for less and less variability. Thus, a PCA can be used to obtain variables that capture more variability than the original variables in the same number of variables. The aim of applying PCA in this chapter is to try and obtain two PCs that can capture a substantial amount of variation in the data; the data can then be plotted in twodimensions to reveal any trends. This exercise was repeated using three more parameter spaces: V f extended by nutrification alone, by sedimentation alone, and by both nutrification and sedimentation (see Table 13.1 in Chapter 13 for how these two stressors extend the parameter ranges). These three spaces are denoted by V f + n , V f + s and V f + n + s respectively For the CAm, the same parameter sweeps were performed as for the CTm, except that each parameter set sampled was also tested to see if there is a discontinuous phase shift. To determine whether a parameter set can give a discontinuous phase shift, it was first determined whether the set can give multiple equlibria. The same procedure as that used in the application of the IIA to the CAm was used to determine if there are multiple equilibria, i.e. compute θ ′(A ) using 201 equally spaced values of A in 0 ≤ A ≤ 1 and if max{θ ′(A )} > 0 for the 201 values of A , then multiple equilibria are
possible (see Appendix A13, section A13.3). If multiple equilibria are possible, then the region of θ where multiple equilibria occurs is noted, and whether a discontinuous phase shift occurs is determined as described in section 14.1.2. When testing for discontinuous phase shifts, the same six values of RC* and S *A were used as for continuous shifts, with RC* set equal to RC* and S A* set equal to S *A . A set of parameters is classified as giving a discontinuous shift if it gives a discontinuous shift when θ decreases from 1, a discontinuous shift when θ increases from 0 or both.
248
This gives six different types of discontinuous phase shifts using just coral criteria, six using just algal criteria and six using both types of criteria. In practice, N > 20,000 (30,000) was only required for one sweep and most outputs would have converged with 1% instead of 5%, with the N values used. The parameter sweeps performed are summarised in Table 14.1.
14.2 Parameter Sweep Results 14.2.1 CTm Results
For the CTm, using just coral cover criteria, for each parameter space tested, the percentage of parameter sets giving any type of continuous phase shift is quite high, above 40% (Figure 14.3a). However, this figure is highest for V f at over 70%, closely followed by the figure for V f + n + s , at a little under 70%. The next highest figure is for V f + s , which is still high at over 60%. There is a large drop to about 45% for V f + n . This
parameter space also has the lowest percentages of sets giving phase shifts with 0.2 ≤ RC* ≤ 0.5 . V f + s has the highest percentage of sets giving phase shifts with RC* = 0.5 and V f + n + s has the highest percentage of sets giving phase shifts with RC* ≤ 0.15 .
When just the turf algal cover criteria are used to classify shifts, the percentage of parameter sets giving any type of continuous shift is less compared to when just coral cover criteria are used, and this difference is much greater for V f + s and V f + n + s compared to the other two spaces (Figure 14.3b). The percentage is between 40-45% for V f + n and V f + s , and about 55% for V f + n + s . For V f , the percentage is high at over 70%.
The percentage of sets giving shifts with S *A = 0.5 is similar for all four spaces and the percentage of sets giving shifts with S *A = 0.05 is the greatest for V f , although this percentage is low at below 3%. Compared to V f , the percentages of sets giving shifts with S A* ≤ 0.3 for the other three spaces are lower, with the differences being greatest for V f + n and V f + s . The result that the percentage of parameter sets giving shifts is less when nutrification and/or sedimentation is added to fishing, compared to when there is just 249
Values of RC* Used (for the
Values of S *A Used (for the
CAm, RC * = RC* is also used)
CAm, S A* = S A* is also used)
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
n/a
2
n/a
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
3
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
n/a
5
n/a
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
6
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
n/a
8
n/a
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
9
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
n/a
11
n/a
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
12
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
0.1, 0.15, 0.2, 0.3, 0.4, 0.5
Sweep
Parameter
Number
Space Sampled
1
Vf
4
7
10
V f +n
V f +s
V f +n+ s
Values of tθ Used
0.05, 0.1, 0.15, 0.2, 0.25
Table 14.1. Summary of the 12 parameter sweeps performed for each of the CTm and the CAm. For each sweep, the parameter space sampled is given, together with the values of tθ , RC* and S *A used to test for phase shifts. If only coral criteria are used to test for phase shifts, then S *A is irrelevant and this is denoted by n/a; if only algal criteria are used, then RC* is irrelevant and this is denoted by n/a. See text in section 14.1.3 for more details on these sweeps. 250
fishing, appears odd. However, this can be explained by nutrification and/or sedimentation decreasing coral cover at equilibrium, C , and increasing turf algal cover at equilibrium, T , in such a way that the profiles of the θ = θ (C ) and θ = θ (T ) curves are flatter. This would mean that the maximum ∆C and ∆T is actually less as θ changes. Thus, even though C decreases and T increases across θ , the frequency of phase shifts decrease because the rate of change of C and T as θ changes decreases. Using the criteria that uses both coral and turf algal covers, the percentage of sets giving phase shifts of all types is lower than when using just coral cover or turf algal cover, for all four parameter spaces. This is necessarily the case. The percentages of sets giving the different types of shifts is very similar to that seen when just turf algal cover criteria are used, with the main difference being that V f + s shows a lower percentage of sets giving phase shifts of any type than V f + n (Figure 14.3c). This similarity suggests that a sharp decline in coral cover is not necessarily accompanied by a sharp increase in turf algal cover, but that a sharp increase in turf algal cover tends to be accompanied by a sharp decrease in coral cover. A possible reason is that corals tend to have less of an effect on turf algae than turf algae has on corals, due to the far slower growth rates of coral that make them less effective in competition for available space. For each of the 12 parameter sweeps, a PCA on the correlation matrix of parameters using only parameter sets giving phase shifts shows that the first two PCs only account for 0.287-0.339 of the total variance; i.e. up to about a third of the total variance. Because these proportions are so low, plotting the parameter sets giving shifts using the first two PCs as axes does not give readily interpretable trends. Therefore, these graphs are not presented. The low proportion of total variance accounted for by the first two PCs means that for parameter sets that give phase shifts, the parameters are mainly uncorrelated, such that there is little relationship between them. This suggests that parameter sets with diverse combinations of parameter values are able to give phase shifts. Figures 14.4 and 14.5 show the frequency distributions of C * and T * for continuous phase shifts found using the coral cover criteria and turf algal cover criteria respectively, for each of the four parameter spaces. For V f , the distribution of C * peaks at C * = 0.2 -0.3, with an otherwise even spread between C * = 0.15 and 0.75 and a tailing off at both ends (Figure 14.4a). With V f + n , there is a slight shift in the distribution towards higher C * categories (Figure 14.4b) and with V f + s , there is a large 251
(a) % of sampled parameter sets giving an RC* continuous phase shift
Vf
V f +n
V f +s PS
V f +n+ s
Vf
V f +n
V f +s PS
V f +n+ s PNS
Vf P
V f +n
V f +s PS
V f +n+ s PNS
(b) % of sampled parameter sets giving an S *A continuous phase shift
(c) % of sampled parameter sets giving an (RC* , S A* )continuous phase shift
Figure 14.3. For the CTm, graphs showing the % of sampled parameter sets giving (a) an RC* -continuous phase shift, (b) an S *A -continuous phase shift and (c) an (RC* , S A* )252
continuous phase shift. V f , V f + n , V f + s and V f + n + s are the parameter spaces as defined in section 14.1.3. The colours red, orange, light green, green, light blue and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of RC* and S *A in Table 14.1.
shift towards lower C * categories (Figure 14.4c). For V f + n + s , there is again a large shift towards lower C * categories, particularly those with C * ≤ 0.15 , and the more severe shifts tend to occur at lower C * values (Figure 14.4d). For V f , the frequency distribution of T * has a similar shape to that for C * except mirrored, with the distribution peaking at 0.6-0.7 (Figure 14.5a). With V f + n , there is a strong shift towards the 0-0.05 category, which accounts for over 40% of shifts (Figure 14.5b), whereas with V f + s , there is generally a shift towards higher T * categories, although the 0-0.05 category also increases in frequency (Figure 14.5c). With V f + n + s , there is an even stronger shift towards the 0-0.05 category than with V f + n – this category now accounts for some 70% of shifts (Figure 14.5d). These results show that nutrification tends to change the equilibrum curves
θ (C ) and θ (T ) in such a way that the changes in C and T are most non-linear at higher C values and lower T values respectively. This could be because nutrification promotes algal growth, leading to a sharper increase in algal cover at a lower T threshold that corresponds to a higher C . In contrast, sedimentation tends to change the curves in such a way that the changes in C and T are most non-linear at lower C values and higher T values respectively. This could be because sedimentation, by increasing coral mortality and decreasing coral recruitment and growth, decreases the coral cover by the greatest amount when it takes medium-high values. This could mean that the sharpest change in coral cover now occurs at a lower C value that corresponds to a higher T value. When both nutrification and sedimentation are added, most of the shifts tend to occur at lower C and T values. Thus, the trend for C is qualitatively the same as that seen when just sedimentation is added whereas the trend for T is qualitatively the same as that seen when just nutrification is added. This suggests that for C , the effects of sedimentation predominates whereas for T , the effects of nutrification predominates, 253
(a) Frequency
C*
(b) Frequency
C*
(c) Frequency
C*
254
(d) Frequency
C*
Figure 14.4. Graphs showing the frequency distributions of C * for continuous phase shifts found for the CTm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s . Bins of size 0.05 are used. The colours red, orange, light green, green, light blue
and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of RC* in Table 14.1.
which would make sense since sedimentation and nutrification directly affect the coral and algal parameters respectively.
14.2.2 CAm Results
For the CAm, Figure 14.6 shows the percentage of parameter sets sampled giving the six different types of continuous phase shifts, for the four parameter spaces described in section 14.1.3. Using just coral criteria, the percentage of parameter sets giving any type of continuous phase shift is very high at over 85% for each of the four parameter spaces, with V f + n + s exhibiting the highest percentage (Figure 14.6a). For V f + n and V f + n + s , the percentage of sets giving the most severe phase shifts, with RC* = 0.1 , is over 25% greater than for the other two spaces. Using just algal cover criteria gives a lower percentage of parameter sets giving any type of phase shift, for all four parameter spaces. However, this percentage remains 255
(a) Frequency
T
*
(b) Frequency
T* (c) Frequency
T* 256
(d) Frequency
T* Figure 14.5. Graphs showing the frequency distributions of T * for continuous phase shifts found for the CTm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s . Bins of size 0.05 are used. The colours red, orange, light green, green, light blue
and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of S *A in Table 14.1.
very high, over 80%, except for V f + s , for which the percentage drops sharply to about 50% (Figure 14.6b). Excluding V f + s , the percentages of sets giving shifts of different types are generally similar, although the percentages of sets giving shifts with S *A ≤ 0.2 are noticeably lower for V f . When criteria incorporating both types of cover are used, the percentages of the parameter sets giving the different types of shifts are very similar to that seen when just algal cover criteria are used, for all four parameter spaces (Figure 14.6c). These results show that when coral criteria are used, the frequency of continuous shifts of any type does not change much with nutrification and/or sedimentation, although the frequency of the most severe shifts increases with nutrification. This means that nutrification increases the non-linearity of the coral equilibrium curve θ (C ) . When algal cover criteria are considered, the frequency of shifts remains similar except for when just sedimentation is considered, when it is much less than the frequency obtained using coral cover criteria. This means that sedimentation tends to shift the algal 257
(a) % of sampled parameter sets giving an RC* continuous phase shift
Vf
V f +n
V f +s PS
V f +n+ s
Vf
V f +n
V f +s PS
V f +n+ s PNS
Vf P
V f +n
V f +s PS
V f +n+ s PNS
(b) % of sampled parameter sets giving an S *A continuous phase shift
(c) % of sampled parameter sets giving an (RC* , S A* )continuous phase shift
Figure 14.6. For the CAm, graphs showing the % of sampled parameter sets giving (a) an RC* -continuous phase shift, (b) an S *A -continuous phase shift and (c) an (RC* , S A* )258
continuous phase shift. V f , V f + n , V f + s and V f + n + s are the parameter spaces as defined in section 14.1.3. The colours red, orange, light green, green, light blue and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of RC* and S *A in Table 14.1.
equilibrium curve θ (A ) upwards in a way that decreases non-linearity. As for the CTm (section 14.2.1), the similarity in results when using algal cover criteria and when using coral cover criteria together with algal cover criteria suggests that corals tend to have less of an effect on algae than algae have on corals, such that a large change in algal cover is usually accompanied by a large change in coral cover, but not vice versa. For the CAm, macroalgae is included, which could potentially increase the effect of algae on coral, through direct overgrowth of corals and reduction of coral growth, and decrease the effect of coral on algae, because corals cannot overgrow macroalgae. Figure 14.7 shows the percentage of parameter sets sampled giving the six different types of discontinuous phase shifts, for the four parameter spaces. The percentage of parameter sets giving any type of discontinuous shift is very low for all four parameter spaces, less than 0.2%, no matter if just coral cover criteria is used, just algal cover criteria is used or criteria incorporating both types of cover is used. For V f and V f + s , the percentage is just 0.005%, under any one of the three sets of criteria. For V f + n + s , the percentage increases to about 0.05% under any one of the three sets of
criteria. V f + n gives the highest percentage, between about 0.1 and 0.15% for the three sets of criteria. Discontinuous phase shifts found are usually of the most severe type. In addition, the results show for discontinuous phase shifts, the region of θ with multiple equilibria is very small. For discontinuous shifts found by sampling V f and V f + s , the region of θ has a length of about 0.001. Shifts found by sampling V f + n have a
region of θ with a maximum length of about 0.03, whereas shifts found from V f + n + s have a region of θ with a maximum length of about 0.003. This shows that any hysteresis effect is small in relation to changing θ . By computing the number of turning points for θ (A ) , it was found that for parameter sets exhibiting multiple equilibria, there are always three equilibria, such that the theoretical case of five equilibria (see Chapter 5, section 5.1.4) never occurs. 259
(a) % of sampled parameter sets giving an RC* discontinuous phase shift
Vf
V f +n
V f +s PS
V f +n+ s
Vf
V f +n
V f +s PS
V f +n+ s PNS
Vf P
V f +n
V f +s PS
V f +n+ s PNS
(b) % of sampled parameter sets giving an S *A discontinuous phase shift
(c) % of sampled parameter sets giving an (RC* , S A* )discontinuous phase shift
Figure 14.7. For the CAm, graphs showing the % of sampled parameter sets giving (a) an RC* -discontinuous phase shift, (b) an S *A - discontinuous phase shift and (c) an 260
(R
* C
, S A* )-discontinuous phase shift. V f , V f + n , V f + s and V f + n + s are the parameter spaces
as defined in section 14.1.3. The colours red, orange, light green, green, light blue and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of RC* = RC * and S *A = S A* in Table 14.1.
From the PCA, it was found that the first two PCs only account for 0.193-0.221 of the total variance for the 12 parameter sweeps, which is less than for the first two PCs for the CTm. Therefore, as for the CTm results (section 14.2.1), the PCA results for the CAm are not presented, because any trends identified would not signify trends for the parameter sets giving shifts as a whole. Also, as for the CTm, this result suggests that parameter sets with diverse combinations of parameter values are able to give phase shifts. Figures 14.8 and 14.9 show the frequency distributions of C * and A * for continuous phase shifts found using the coral cover criteria and algal cover criteria respectively, for each of the four parameter spaces. The distribution of C * for V f peaks at 0.25-0.35, with a fairly steep decrease either side of this peak (Figure 14.8a). More severe shifts tend to occur at higher values of C * . For the V f + n , the distribution shifts to higher categories of C * , with severe shifts dominating these categories (Figure 14.8b). With V f + s , the distribution shifts to lower categories of C * and the shape of the distribution remains broadly similar, peaking at 0.2-0.3 (Figure 14.8c). For V f + n + s , there is a noticeable increase in the frequency of lower categories of C * , and the two most severe types of shift now occur at relatively high frequencies (compared to less severe shifts) across most categories of C * (Figure 14.8d). For V f , the A * distribution peaks at 0.45-0.6 and tails off at either end fairly steeply (Figure 14.9a). With V f + n , there is a large shift towards lower A * categories, such that a peak at 0-0.05 appears (Figure 14.9b). Between 0.05-0.5, the categories have roughly the same frequency and after 0.5, the distribution tails off. Severe shifts tend to occur at lower A * values. With V f + s , the distribution flattens, with peaks appearing at 0-0.05 and 0.85-0.9 (Figure 14.9c). For V f + n + s , there is a very large shift towards lower A * categories, such that the peak at 0-0.05 accounts for some 45% of all shifts (Figure 14.9d). 261
(a) Frequency
C*
(b) Frequency
C*
(c) Frequency
C*
262
(d) Frequency
C*
Figure 14.8. Graphs showing the frequency distributions of C * for continuous phase shifts found for the CAm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s . Bins of size 0.05 are used. The colours red, orange, light green, green, light blue
and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of RC* in Table 14.1.
These results, and their interpretation, parallel that for the CTm for C * and T * (section 14.2.1); they suggest that for C , the effects of sedimentation predominates whereas for A , the effects of nutrification predominates, which would make sense since sedimentation and nutrification directly affect the coral and algal parameters respectively.
263
(a) Frequency
A*
(b) Frequency
A* (c) Frequency
A* 264
(d) Frequency
A* Figure 14.9. Graphs showing the frequency distributions of A * for continuous phase shifts found for the CAm from randomly sampling (a) V f , (b) V f + n , (c) V f + s and (d) V f + n + s . Bins of size 0.05 are used. The colours red, orange, light green, green, light blue
and blue represent six phase shift types of increasing severity, as represented by the six decreasing values of S *A in Table 14.1.
265
Appendix A14
A14.1 Derivation of the Explicit Forms of θ (C ) In this section, explicit forms of θ (C ) are derived for the CTm and the CAm, where the overbar for C denotes an equilibrium value in the biological range. These are required in eqns. 14.1, 14.6, 14.19 and 14.20 in sections 14.1.1 and 14.1.2, in order to solve these for ∆C . In turn, ∆C values are required in the parameter sweeps to determine the maximum change in coral cover at equilibrium for a given change in grazing pressure θ (see sections 14.1.1 and 14.1.2). For the CTm, the explicit form of θ (C ) is derived directly from the two equilibrium equations for the CTm (eqns. A13.1a and A13.1b in Appendix A13), and is given by:
θ (C ) =
ζ T (1 − C − T(1) ) − ε C (lCs + lCb C )T(1) − rC α C C T(1) , g T T(1) + ζ T (1 − C − T(1) )
(A14.1)
where
T(1) = T(1)
(C ) = (l + l C + r C )(1 − C ) − d C . (l + l C )(1 − ε ) + r (1 − α )C s C s C
b C b C
C
C
C
C
(A14.2)
C
For the CAm, there are two equilibrium equations as specified by eqns. A13.7a and A13.7b in Appendix A13. A direct calculation from eqn. A13.7b gives rMν M (1 − C − A(1) (C )) + γ MC rMν M C s b − ε C (1 − ν M ) l C + l C C A(1) (C ) + ζ T (1 − C − A(1) (C )) − rC α C (1 − ν M )(1 − β Mν M A(1) (C ))C . θ (C ) = {g Mν M + g T (1 − ν M )}A(1) (C ) + ζ T (1 − C − A(1) (C ))
(
)
(A14.3)
In eqn. A14.3, A(1) (C ) is derived from eqn. A13.7a of Appendix A13 as follows. From eqn. A13.7a, a quadratic equation in A can be derived as: 266
A 2 [rC β Mν M {1 − α C (1 − ν M )}C ]
(
)
l s + l b C {1 − ε C (1 − ν M )} + rC {1 − α C (1 − ν M )}C + rC β Mν M (1 − C )C − A C C + γ MC rMν M C + {(l Cs + l Cb C )(1 − C ) + rC (1 − C )C − d C C } = 0 .
(A14.4)
The expression on the left-hand side of eqn. A14.4, for fixed 0 ≤ C ≤ 1 , has a positive coefficient of A 2 and a negative coefficient of A. There are two solutions for A in terms of C from this equation, A(1) (C ) and A( 2) (C ) . Assume that at a particular C in the biological range 0 ≤ C ≤ 1 , both solutions of A give equilibria in the biological range. Let these two values of A be A1 and A2 . From the mathematical analysis of the CAm in section A5.1.2 in Appendix A5, the equilibrium value C in the biological range is equal to C (1) (A ) , as given by eqn. A5.3 in Appendix A5 except with A instead of A (overbar used because only equilibrium values of A in the biological range are considered). Thus, eqn. A14.4 can be inverted to give C = C (1) (A1 ) = C (1) (A2 ) . But, since C (′1) (A ) < 0 (see Appendix A5, section A5.1.3), then C (1) (A1 ) ≠ C (1) (A2 ) , which gives a contradiction. Thus, there can be at most one solution of A giving an equilibrium in the biological range for each 0 ≤ C ≤ 1 . Also, there is a unique C * , with 0 < C * < 1 , such that the constant coefficient in eqn. A14.4 is 0. At C * , the smaller solution of A is 0, and using A = 0 in eqn. A14.3 gives θ = 1 . Since any equilibrium value of C is equal to C(1) (A ) and C (′1) (A ) < 0 , C * is the maximum possible value of C in the biological range. Thus, for the quadratic equation A14.4, there is no need to consider C > C * , i.e. the case where the constant coefficient is < 0 . For the case when the constant coefficient of eqn. A14.4 is > 0 , there are two positive roots for A for a given C with 0 ≤ C ≤ 1 . If the larger root for A gives an equilibrium in the biological range, then so must the smaller root for A – but this cannot be the case from the above argument. Thus, only the smaller root can give an equilibrium in the biological range. This is given by:
A(1) (C ) =
{
}
1 Y + Y 2 + 4 XZ , 2X
(A14.5) 267
where X = rC β Mν M {1 − α C (1 − ν M )}C ,
(
(A14.6a)
)
Y = l Cs + lCb C {1 − ε C (1 − ν M )} + rC {1 − α C (1 − ν M )}C + rC β Mν M (1 − C )C + γ MC rMν M C ,
(A14.6b)
(
)
Z = lCs + lCb C (1 − C ) + rC (1 − C )C − d C C .
(A14.6c)
A(1) (C ) in eqn. A14.3 is simply A(1) (C ) given by eqn. A14.5 with C replaced by C .
268
Chapter 15. Sensitivity Analyses The mathematics behind the two types of sensitivity analysis used, the Elementary Effects method and Sobol’ Variance Decomposition, are first described (section 15.1). The Elementary Effects method is applied to each of the three benthic models, each of the fish and urchin models, and the most complex integrated model (section 15.2). Sobol’ Variance Decomposition is then applied to the simplest benthic model (section 15.3).
15.1 Mathematics of Sensitivity Analysis Techniques Each model in this thesis is defined by a set of differential equations. In a sensitivity analysis for a particular model, the interest is in how the values of the variables change with the parameters and the initial conditions. Thus, define each model as (Helton and Davis 2003):
y = f (x, t ) ,
(15.1)
where y = ( y1 , y 2 ,..., y m ) is the output vector for the variables at time t, x = ( x1 , x 2 ,K , x n ) is the input vector of parameters and initial conditions and f denotes the application of the model to x for time t. Also, x ∈ X and y ∈ Y , where X and Y are the sets of all possible input vectors and all possible output vectors respectively. For each model run, the 4th-order Runge-Kutta method with a daily time step ( t = 1 365 yr) was used to solve the differential equations. One-at-a-time or local sensitivity analysis techniques change one input at a time whilst keeping all others fixed, and measure the resulting change in y. This approach is inappropriate for the models in this thesis because it does not take into account possible interactions between inputs when more than one input is changed at the same time. Instead, two global analysis techniques are used, which vary all the inputs simultaneously (Cariboni et al. 2007). These are the Elementary Effects method and Sobol’ Variance Decomposition. They are chosen because they give useful results that have a clear meaning. 269
To generate input sets for use in the two chosen sensitivity analysis techniques, he simplest way to sample across the input probability space is to do random sampling (RS) across each input range. Since there is no prior knowledge of the type of distribution each input has, each input is taken from a Uniform distribution over its range. For the parameters, the ranges used are those from the empirically derived ones from Chapters 9-11,13 and for the initial conditions, the ranges are derived using data from papers and surveys. The parameter ranges as extended by fishing, nutrification and sedimentation are explored to investigate how the variables can change by varying each parameter according to the values it can take under natural variation and human stress.
15.1.1 Elementary Effects Method
The Elementary Effects method computes elementary effects that measure the change in y with changes in the inputs across the input space (Morris 1991). For an example of its application to an ecological model, see Cariboni et al. (2007). To calculate the elementary effect of the jth input for y i (x ) (dropping the t for notational convenience), 1 ≤ i ≤ m , 1 ≤ j ≤ n , x is standardised such that each input is between 0 and 1. This is
necessary to enable direct comparison of the effects of different inputs (possibly measured in different units) on the outputs. Let z = ( z1 , z 2 ,K, z n ) denote the vector of standardised inputs, which is an element of Z, the set of all possible standardised input vectors; y i can then be written as a function of z. The elementary effect for y i for the jth input is defined as:
d ij (z ) =
y i (z1 , z 2 ,K, z j −1 , z j + ∆, z j +1 ,K, z n ) − y i (z ) ∆
,
(15.2)
where ∆ is the increment in z j and z = ( z1 , z 2 ,K, z n ) ∈ Z except that z j ≤ 1 − ∆ . For the jth input, if there is another input whose range is dependent upon it, then changing the standardised value of the jth input, z j , will mean that the (nonstandardised) range of this other input will change. Thus, if the standardised value of this other input remains the same when z j changes, then it will represent a different non-standardised value. This means that, from eqn. 15.2, d ij would not measure the
270
change in yi due to a change in just the jth input, but the change in yi due to changes in the jth input and the input whose range is dependent upon the jth input. For example, for the Coral-Algae model (CAm), g T has a range of 5-15 yr-1 and the range of g M is
(0.01)g T - g T
(see Table 9.1 in Chapter 9), which is dependent on g T . If the
standardised value of g T changes from 0 to 0.5, then this means that g T changes from 5 yr-1 to 10 yr-1, such that the range of g M changes from 0.05-5 yr-1 to 0.1-10 yr-1. If the standardised value of g M stays constant at 0, then this represents the non-standardised value 0.05 yr-1 before g T changes and 0.1 yr-1 after g T changes. Thus, changing the standardised value of g T changes the non-standardised value of g M . If the ith output is coral cover and the jth input is g T , then d ij would measure the change in coral cover due to changes in both g T and g M , confounding interpretation of the results. Therefore, if an input is such that the range of at least one other input is dependent upon it, then the elementary effects for this input are not calculated because the ecological meaning of the elementary effects is unclear. When the Elementary Effects method is applied in this chapter to a model, the elementary effects d ij are computed for all 1 ≤ i ≤ m and for all 1 ≤ j ≤ n corresponding to inputs that do not determine the range of another input. For each i and j for which elementary effects are computed, N input sets are sampled from Z using RS, with z j ≤ 1 − ∆ , and d ij computed. This gives N d ij values that form the sampling distribution. The mean and variance of these d ij values, M (d ij ) and V (d ij ) respectively, can then be used as unbiased estimates of the mean and variance of the true distribution of elementary effects (McKay et al. 1979). M (d ij ) measures the overall effect of an input on the output and V (d ij ) gives a measure of non-linearity in the output due solely to the input and/or due to interactions between the input and other inputs (Morris 1991). From Nystrom et al. (2000), mass bleaching and CoT outbreaks occur on a timescale of years to decades. From a survey of the literature, hurricanes/cyclones/typhoons can occur between once every 4 yrs and once every 69 yrs (Randall and Eldredge 1977, McClanahan 1997, Tanner 1997, Connell et al. 1997, Gardner et al. 2005, Mumby 2006, Lough 2007). Thus, the sensitivity of the outputs of the models after t = 5 , 25 and 50 yrs are computed, to represent 5, 25 and 50 yrs of reef dynamics without acute reef disturbances, i.e. short-term disturbances that last for less than a year, starting with a reef that is not necessarily near equilibrium. For each value 271
of t, three values of ∆ are tested, 0.1, 0.3 and 0.5, which represent a low, medium and high value of ∆ . The only exception to this is when convergence of M (d ij ) and V (d ij ) is tested (see section 15.1.3), where t = 10 yrs is chosen as a test value and ∆ = 0.6 ,
12 22 and 102 202 are tested. These three ∆ values are chosen for testing to make the results comparable to those obtained using two sampling methods other than RS, one of which requires ∆ > 0.5 (section 15.1.3).
15.1.2 Sobol’ Variance Decomposition
Sobol’ Variance Decomposition separates the variance of y into the contributions by each of the inputs and the contributions by the interactions between inputs. This allows the effect of the interaction between an input and other inputs on an output to be determined, relative to the effect of the input on its own on the same output. The Elementary Effects method is unable to do this. This technique relies on the fact that if all the inputs are mutually independent, yi (x ) = f (x ) , 1 ≤ i ≤ m , can be decomposed uniquely to (Helton and Davis 2003):
n
f ( x) = f 0 + ∑ f j ( x j ) + j =1
∑f
jk
( x j , x k ) + K + f 1, 2,Kn ( x1 ,K, x n ) .
(15.3)
1≤ j < k ≤ n
This allows the variance of y i , V ( y i ) , to be written as V ( yi ) =
∑V
j
+
1≤ j ≤ n
∑V
jk
+ K + V1, 2,K,n ,
(15.4)
1≤ j < k ≤ n
where V j is the part of V ( y i ) due solely to x j , V jk is the part of V ( y i ) due to the interaction between x j and x k , and so forth (Helton and Davis 2003). Sensitivity measures can then be defined by sij = V j V ( y i ) ,
(15.5a)
sijT = V j + Vkl + K + V1, 2,K,n V ( yi ) , ∑ 1≤ k
(15.5b)
272
where sij is the fraction of V ( y i ) due solely to x j and sijT is the fraction of V ( y i ) due solely to x j or its interactions with any of the other input variables (Helton and Davis 2003). When Sobol’ Variance Decomposition is applied in this chapter, sij and sijT are computed for all 1 ≤ i ≤ m and 1 ≤ j ≤ n . This is done by estimating V ( y i ) , V j and
Vj +
∑V
kl 1≤ k
+ K+ V1, 2,K, n using the Monte Carlo procedure in Saltelli (2002) together
with expressions in Homma and Saltelli (1996) and Helton and Davis (2003). This procedure involves, for each i and j, sampling 2N input sets using RS, which then gives N estimates each of sij and sijT . As in the application of the Elementary Effects method, the sensitivity of the outputs after t = 5 , 25 and 50 yrs is found when Sobol’ Variance Decomposition is applied to the models in this thesis, except when convergence is being tested, when t = 10 yrs is used (section 15.1.3). Due to the need for independence of inputs, Sobol’
Variance Decomposition is only applied to the simplest benthic model, the CTm.
15.1.3 Convergence of Results The results from the Elementary Effects method are M (d ij ) and V (d ij ) , and those from Sobol’ Variance Decomposition are sij and sijT (see section 15.1.2). For sensitivity analyses, a commonly used criterion for convergence is when the percentage change in a result (e.g., sij ) derived using L and L + 1 input sets is less than a small percentage for the first time. This was used, for example, in Borgonovo and Peccati (2007) for sij and sijT , with the percentage being 1%. To test whether this criterion is suitable for use with
the sensitivity analyses and models in this thesis, both of the sensitivity analyses described in section 15.1.2 were applied to the CTm with t = 10 yrs, a value within the range 5-50 yrs found using the literature (section 15.1.2). For the CTm, there are two initial conditions, the initial coral and turf algal covers, C 0 and T0 respectively. In the application of Sobol’ Variance Decomposition, T0 was fixed to half of 1 − C 0 ; this is necessary to keep all the inputs independent, as required (see section 15.1.2). T0 was fixed in this way in the test using the Elementary 273
Effects method too, as when generating results using this method in section 15.2.1; this is done such that results can be directly compared to those from Sobol’ Variance Decomposition. Thus, there are 10 inputs for the CTm, C 0 and the nine parameters for the CTm, and 2 outputs, coral cover C and turf algal cover T at time t. In these tests, RS was used to sample the input sets. In addition, two other sampling regimes were tested using the Elementary Effects method – Latin Hypercube Sampling (LHS) and the Morris Method (MM). These two regimes are described in more detail in McKay et al. (1979) and Morris (1991) respectively. However, results showed that using LHS gave results that do not stabilise as well as those when RS was used, and together with the fact that LHS gives biased estimates of variances (McKay et al. 1979), this meant that LHS was no longer used after the tests. Results obtained using MM stabilised as well as those from RS; however, the MM requires that
∆ = p [2( p − 1)] , with p an even integer, and that each input be drawn randomly from the set {0,1 ( p − 1) , 2 ( p − 1) ,K,1 − ∆} (see Morris (1991) for details). This means that ∆ > 0.5 and therefore, smaller changes cannot be explored. In addition, the sampling is
done from a discretised version of the input space, such that the space in between the discrete values cannot be investigated. Given these disadvantages, MM was no longer used after the tests. For the Elementary Effects method, the three values of ∆ tested are 0.6, 12 22 and 102 202 , which correspond to p values of 6, 12 and 102 when using the MM and give ∆ = 0.60 , 0.55 and 0.50 to two decimal places respectively. These values of ∆ are chosen to make results using RS and LHS comparable to those using the MM. Results using the RS were also used to test the criterion convergence described above. The number of input sets sampled, N, is 10,000, because graphically, results have clearly stabilised by N = 10,000 . It was found that, for all i and j, the difference between M (d ij ) values obtained using L and L + 1 input sets sampled using RS ( 2 ≤ L ≤ N − 1 ),
and the difference between V (d ij ) values obtained using L and L + 1 input sets were first less than 1% at L ≤ 34 . However, at L ≤ 34 , it can be shown graphically that not all the M (d ij ) and V (d ij ) values have stabilised; some values have far from stabilised. Similar results were found for ∆ = 12 22 and 102 202 , and for the sij and sijT values obtained using Sobol’ Variance Decomposition. Thus, the criterion for convergence used in this chapter is not when the percentage change in a result ( M (d ij ) , V (d ij ) , sij or sijT ) derived using L and L + 1 274
input sets is less than a small percentage for the first time, because the tests show that the results may be far from having stabilised. Instead, to determine whether M (d ij ) or V (d ij ) has converged in this chapter, for N input sets sampled using RS, the maximum
absolute percentage change between the value derived using N – 1000 input sets and the values derived using N – 1000 + k sets, 1 ≤ k ≤ 1000 , is computed. If this is less than 5%, then M (d ij ) or V (d ij ) is taken to have converged and the value derived using N input sets is used. Initially, for each input and output, N = 15,000 was used, which gives 15,000 elementary effects ( d ij ). If this did not give a converged value of M (d ij ) or V (d ij ) , then N = 20,000 is used, followed by 25,000, 50,000, 100,000, 150,000, 200,000 and 250,000 if necessary. N > 250,000 was never necessary. To determine whether sij or sijT has converged, the same criterion is used, except that instead of N, it is 2N, because 2N input sets are required to obtain one value of sij or sijT (section 15.1.2). Initially, for each input and output, 2 N = 1,000,000 was used. If this does not give a converged value of sij or sijT , with sij ≤ sijT and sij , s ijT ≥ 0 as required, then 2 N = 1,500,000 is used, followed by 2,000,000, 2,500,000, 5,000,000, 7,500,000 and
10,000,000 if necessary. 2 N > 10,000,000 was never necessary for convergence; however, with 2 N = 10,000,000 , sij ≤ sijT or sij , s ijT ≥ 0 does not hold in a few cases (see results in section 15.2.4). For the cases where sij ≤ sijT or sij , s ijT ≥ 0 , 2 N > 10,000,000 was not tested because results are still interpretable (section 15.2.4)
and further runs with more sampled input sets are computationally expensive. The 5% threshold used is chosen to strike a balance between accuracy and computational speed. Results for a lot of the inputs and outputs would have converged, with the values of N tested, using a lower threshold of 1%, but the higher 5% threshold is needed to take into account the slower converging results. Ideally, a lower threshold such as 1% would be used for all results, but this would have been computationally prohibitive. Nevertheless, the criterion used in this chapter is more stringent than that used in Borgonovo and Peccati (2007), and the study by Cariboni et al. (2007) did not even test for convergence for the Elementary Effects method example.
275
15.2 Sensitivity Analysis Results 15.2.1 Elementary Effects Method Results for Benthic Models
The Elementary Effects method is applied to each of the benthic models as described in section 15.1. For the CTm, as in the convergence tests detailed in section 15.1.3 and for the same reasons, the initial coral cover, C 0 , is chosen randomly between 0 and 1 but the initial turf algal cover, T0 , is fixed at a half of the chosen C 0 value. For the other benthic models, to be consistent with the analysis for the CTm, C 0 is chosen randomly as an input with the remaining initial cover split evenly between the other benthic groups. Table 9.1 in Chapter 9 gives ranges and definitions for all the parameters in the benthic models. Firstly, M (d ij ) and the standard deviation for d ij , s.d. (d ij ) , for the CTm are shown in Figures 15.1 and 15.2, for all three values of t and all three values of ∆ . The results show that at t = 5 yrs, C 0 (and hence T0 and the initial space cover, which are fixed by C 0 ) has the largest effect (on average) on coral cover C and the second largest effect on turf algal cover T. As t increases to 25 yrs, the effects of C 0 on both types of benthic cover decrease dramatically until at t = 50 yrs, its effects are very small. This shows that changing the initial covers has very little effect on average on the final covers after 50 yrs, suggesting that dynamics are usually at or close to equilibrium after a number of yrs that is in the range 25-50 yrs. Thus, for a deterministic reef system unaffected by acute disturbances, the results suggest that it usually takes about 25-50 yrs to reach equilibrium. For the other inputs, as t increases, M (d ij ) increases, which means that on average, the two trajectories of the benthic dynamics for before and after a parameter changes become further apart as time increases. This shows that the full impact of a change in parameters may only manifest after decades and warns against using benthic covers shortly after such a change as an indicator of future reef health. Also, s.d. (d ij ) increases with time, which means that there is greater variability in the difference between the two trajectories as time increases. For the output C (coral cover) and input d C , M (d ij ) is always large and negative, having the largest magnitude out of all inputs for t = 25 and 50 yrs (over 0.6). M (d ij ) for input ζ T is also negative, but its magnitude is much smaller. At t = 25 and
276
50 yrs, M (d ij ) for input θ has the largest positive effect (over 0.3), followed by M (d ij ) for inputs l Cb , rC and α C . s.d. (d ij ) for inputs d C and θ are the largest out of all the parameters. Also, s.d. (d ij ) for inputs ζ T , l Cb , rC and α C are smaller but moderate ( ≈ 0.1-0.2) to large for t = 25 and 50 yrs , reaching above 0.15 for t = 50 yrs. For output T (turf algal cover), d C and ζ T have large positive effects, whereas
θ has a large negative effect. l Cb , rC , α C and g T have moderate positive effects at t = 25 and 50 yrs, whereas l Cs and ε C only have small effects. Out of all inputs, θ has the largest effect. d C , ζ T and θ have a large s.d. (d ij ) whereas l Cb , rC , α C and g T have smaller s.d. (d ij ) values that are moderate for t = 25 and 50 yrs. l Cs and ε C have small s.d. (d ij ) values. These results show that C is very sensitive to fishing of herbivores, provided this decreases θ as described in section 13.2.1 in Chapter 13, and sedimentation, which increases d C and decreases l Cb and rC (Chapter 13, section 13.2.3). Also, corals are moderately sensitive to nutrification, which increases ζ T (Chapter 13, section 13.2.2). Furthermore, the results show that T is very sensitive to all three types of stress. In general, the three different values of ∆ give similar results for M (d ij ) and s.d. (d ij ) , with the main difference being that for some inputs, s.d. (d ij ) is greater for smaller ∆ . For the CAm, the range for g M is dependent on g T and hence the elementary effects for g T are not computed (see section 15.1.1). The results for t = 25 yrs for the 14 inputs excluding g T are given in Figures 15.3 and 15.4. Results for t = 5, 50 yrs are similar in that the graphs have the same shape, and are therefore not shown. The only exception is for the input C 0 , for which M (d ij ) and s.d. (d ij ) both decrease in magnitude from large to small values as t increases from 5 to 50 yrs, as for the CTm. For inputs other than C 0 , in general, M (d ij ) and s.d. (d ij ) both increase as t increases. Analogous to the CTm, C 0 has large positive and negative effects on C and algal cover A respectively at t = 5 yrs, with relatively little effects at the two higher t values. Figure 15.3 shows that (at t = 25 yrs), as for the CTm, d C has the largest negative effect (on average) on C and θ has the largest positive effect. l Cb has a moderate positive effect whereas ζ T and ν M have moderate negative effects. The other 277
(a)
C0
M (d ij )
θ
l Cb l Cs
rC α C
εC
gT
ζT
dC (b)
θ l Cb
M (d ij )
rC l Cs
αC
εC
C0
gT
ζT
dC (c)
θ M (d ij )
l Cb
rC l Cs
αC
εC
gT
C0
ζT
dC
278
(d)
C0
s.d. (d ij ) dC
θ
ζT
l Cb
rC l
s C
αC
εC
gT
(e) s.d. (d ij )
dC
θ
ζT
l Cb
rC l
αC
C0
gT
εC
s C
(f) s.d. (d ij )
dC
θ
ζT
l Cb
rC l
αC εC
s C
gT
C0
Figure 15.1. Graphs showing (a)-(c) M (d ij ) and (d)-(f) s.d. (d ij ) for each of the 10 inputs for the CTm. The output is C at (a), (d) t = 5 yrs , (b), (e), t = 25 yrs and (c), (f) t = 50 yrs , and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 ,
0.3 and 0.5 respectively. See Table 9.1 for definitions of the nine parameters.
279
(a) M (d ij )
ζT
dC
l
b C
l Cs
rC α C
εC gT
C0
θ (b) M (d ij )
ζT
dC
l Cs l
b C
rC
αC
εC
C0
gT
θ (c) M (d ij )
ζT
dC
l Cs l
b C
rC α C
εC
C0
gT
θ
280
(d)
s.d. (d ij )
θ
ζT
C0 dC
gT l
(e)
b C
l Cs
rC
αC
εC
s.d. (d ij )
θ
ζT dC
C0 l Cb
rC
αC εC
l Cs
(f)
s.d. (d ij )
gT
θ dC
ζT
l Cb
rC l
αC
gT
C0
εC
s C
Figure 15.2. Graphs showing (a)-(c) M (d ij ) and (d)-(f) s.d. (d ij ) for each of the 10 inputs for the CTm. The output is T at (a), (d) t = 5 yrs , (b), (e), t = 25 yrs and (c), (f) t = 50 yrs , and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 ,
0.3 and 0.5 respectively. See Table 9.1 for definitions of the nine parameters.
281
parameters have relatively little effect. d C , l Cb , ζ T , ν M and θ have a large s.d. (d ij ) , whereas rM and γ MC have a moderate s.d. (d ij ) . The other inputs have a relatively small s.d. (d ij ) . Figure 15.4 shows that for algal cover A, ζ T has the largest positive effect and
θ has the largest negative effect. d C and ν M have moderate positive effects whereas g M has a moderate negative effect. All other parameters have relatively little effect. As for C, d C , ζ T , ν M and θ have a large s.d. (d ij ) . g M , rM and γ MC have moderate s.d. (d ij ) values. All other parameters have a relatively small s.d. (d ij ) . These results show that C is very sensitive to fishing and sedimentation and moderately sensitive to nutrification (with effects of these stressors as described in Chapter 13, section 13.2), as for the CTm. Algal cover A is very sensitive to fishing and nutrification, and moderately so to sedimentation. As for the CTm, in general, the three different values of ∆ give similar results for M (d ij ) and s.d. (d ij ) , with the main difference being that for some inputs, s.d. (d ij ) is greater for smaller ∆ . For the CTMm, g T is not investigated for the same reason as for the CAm. The results for t = 25 yrs for the 14 inputs excluding g T are given in Figures 15.5-15.7. Results for t = 5, 50 yrs are similar in that the graphs have the same shape, and are therefore not shown. The only exception is for the input C 0 , for which M (d ij ) and s.d. (d ij ) both decrease in magnitude from large or moderate to small values as t increases from 5 to 50 yrs, as for the CTm and CAm. For inputs other than C 0 , in general, M (d ij ) and s.d. (d ij ) both increase as t increases. At t = 5 yrs, C 0 has a large positive effect on C, a large negative effect on T and a fairly small negative effect on M. C 0 has relatively small effects at t = 25 and 50 yrs. From Figure 15.5, C is affected most negatively (on average) by d C and most positively by θ . l Cb has a fairly large positive effect on coral cover whereas rC has a moderate positive effect, and ζ T has a moderate negative effect. d C , g M and θ have a very large s.d. (d ij ) , whereas l Cb , ζ T and rM have a fairly large s.d. (d ij ) . rC and γ MC have moderate s.d. (d ij ) values. From Figure 15.6, T responds most positively to d C and ζ T , and most 282
(a)
θ
M (d ij ) l Cb l Cs
rC
αC
C0
gM
εC
βM γ MC
rM
ζT
νM
dC (b)
θ
s.d. (d ij ) dC
ζT νM l
b C
rC l Cs
gM
αC
εC
γ MC
rM
C0
βM
Figure 15.3. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CAm. The output is C at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Table 9.1 for definitions of
the 13 parameters.
negatively to θ . g M has a moderate positive effect and l Cb has a moderate negative effect. d C , ζ T , rM and γ MT have fairly large s.d. (d ij ) values; θ has a very large s.d. (d ij ) whereas g M has a very large s.d. (d ij ) for ∆ = 0.1 and a fairly high s.d. (d ij ) for higher ∆ values. rM and γ MT have moderate s.d. (d ij ) values. From Figure 15.7, g M and θ both negatively affect macroalgal cover M by a large amount. In contrast, rM and 283
(a)
ζT
M (d ij )
νM
dC
rM
l
b C
l Cs
rC α C ε C
βM
γ MC C0
gM
θ (b)
θ
s.d. (d ij )
ζT
dC
νM γ MC
g M rM
l Cb
rC α C l Cs
C0
βM
εC
Figure 15.4. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CAm. The output is A at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Table 9.1 for definitions of
the 13 parameters.
γ MT have a moderate positive effect on M. Also, g M and θ have a very large s.d. (d ij ) , whereas rM and γ MT have a large s.d. (d ij ) . These results show that C is very sensitive to fishing and sedimentation, and moderately sensitive to nutrification, as for the CTm and the CAm. T is very sensitive to all three stressors, as for the CTm; M is very sensitive to fishing and moderately 284
(a)
θ
M (d ij ) l Cb
rC l
s C
αC
gM
C0
εC rM
β M γ γ MT MC
ζT
dC (b)
gM
s.d. (d ij )
θ dC
ζT
l Cb
C0
rM
γ MC rC α C l Cs
γ MT εC
βM
Figure 15.5. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CTMm. The output is C at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Table 9.1 for definitions of
the 13 parameters.
sensitive to nutrification, and not sensitive to sedimentation. Also, since T and M are moderately sensitive and very sensitive to increasing g M respectively, and increasing
g M tends to result in an increase in T and a decrease in M, this demonstrates the importance of grazing in mediating competition between turf and macroalgae, and the importance of this competition. In addition, M is moderately sensitive to increasing γ MT , 285
(a)
M (d ij )
ζT
dC
gM
l Cs l Cb
rC
αC
εC rM
β M γ MC
γ MT
C0
θ (b)
θ
s.d. (d ij )
gM
ζT
dC
rM rC α C
l Cb l Cs
γ MT C0
εC
βM
γ MC
Figure 15.6. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CTMm. The output is T at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Table 9.1 for definitions of
the 13 parameters.
which tends to lead to an increasing M. This again shows the importance of competition between turf and macroalgae. For each type of benthic cover, parameters not mentioned all have relatively little effect on that type of benthic cover on average and a small s.d. (d ij ) . As for the CTm, in general, the three different values of ∆ give similar results for M (d ij ) and 286
(a)
rM
M (d ij )
γ MT βM
dC l Cb
l Cs
γ MC
rC α C ε C ζ T
(b) s.d. (d ij )
C0
gM
θ
gM
θ
rM
γ MT dC
γ MC
ζT l
b C
l Cs
rC α C
εC
C0
βM
Figure 15.7. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 14 inputs for the CTMm. The output is M at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Table 9.1 for definitions of
the 13 parameters.
s.d. (d ij ) , with the main difference being that for some inputs, s.d. (d ij ) is greater or M (d ij ) and s.d. (d ij ) are greater for smaller ∆ , sometimes much greater (e.g., g M ).
287
15.2.2 Elementary Effects Method Results for Fish and Urchin Models
For each of the two fish and urchin models, the HPUm and the HSLUm (Chapter 6), the inputs are those parameters which can be tested and the initial biomasses for each modelled group. The range of C B used is from zero, its theoretical minimum (no algae), to its theoretical maximum, which is given by eqn. 7.15 in Chapter 7 with the maximum grazing rates, turf algal growth rate and µ parameters (from Tables 9.1, 10.1 and 13.1 in Chapters 9, 10 and 13 respectively). All other parameter ranges are given in Tables 10.1 and 11.1 in Chapters 10 and 11, together with the parameter definitions. The initial herbivorous fish, piscivorous fish, small to intermediate piscivorous (SIP) fish and large piscivorous (LP) fish biomasses, H 0 , P0 , Ps 0 and Pl 0 respectively, are taken to vary from zero to the maximum values reported from the literature, which are 0.282 kg m-2, 0.433 kg m-2, 0.070 kg m-2 and 0.433 kg m-2 respectively (see section 6.2.2 in Chapter 6 for sources used to derive the maximum values of H 0 and P0 ; maximum values of Ps 0 and Pl 0 were derived using data for piscivorous fish biomasses from the same sources). U 0 , the initial sea urchin biomass, is taken to vary from zero to a maximum value of 0.390 kg m-2 (Lessios 1988, Levitan 1988). Only the graphs for t = 25 yrs are shown for each fish and urchin model, because the graphs for t = 5 yrs and t = 50 yrs have a similar shape, except for the initial biomasses. As for the benthic models, the magnitudes of M (d ij ) and s.d. (d ij ) for each input apart from the initial biomasses tend to increase with t, whereas the magnitudes for the initial biomasses decrease with t. For both models, the three ∆ values tend to give similar results except that for some inputs, s.d. (d ij ) increases or M (d ij ) and s.d. (d ij ) increase noticeably as ∆ decreases from 0.5. The elementary effects for a number of parameters are not calculated for both models. For the HPUm, the elementary effects for C B , i H , l Hen , g P , l Pen , rP , iU , κ U are not calculated because the values of each of these parameters determine the ranges for at least one other parameter (see section 15.1.1). For the same reason, the elementary en effects for the parameters C B , i H , l Hen , g Ps , l Ps , rPs , ρ Ps , g Pl , l Plen , rPl , ψ Pl , iU , κ U are
not calculated for the HSLUm. Results for the HPUm are shown in Figures 15.8-15.10. At t = 25 yrs, changing the initial biomasses usually has little effect (on average) on any of the biomasses
288
(unlike at t = 5 yrs), which shows that the reef system is near or has reached equilibrium by this time. Also (at t = 25 yrs), the herbivorous fish biomass H is most sensitive to d H and ρ H (on average), which affect the biomass negatively, and l Hex , which affects the biomass positively. l Pex has a moderate ( ≈ 0.005-0.01 kg m-2) negative effect. These parameters also have the largest s.d. (d ij ) values, with ψ P and f having large values too. The other inputs have little effect on H and have a relatively small s.d. (d ij ) . l Pex has a moderate positive effect on piscivorous fish biomass P. ψ P has a large s.d. (d ij ) value and l Pex has a moderate s.d. (d ij ) value. In particular, ψ P has a very high s.d. (d ij ) reaching 0.015 kg m-2 for ∆ = 0.1 . The other inputs have little effect on P and have a small s.d. (d ij ) . Only two inputs have a large effect on sea urchin biomass U. d U has a large negative effect whereas lUex has a large positive effect. These two parameters also have a very large s.d. (d ij ) value. d H has a moderate positive effect whereas l Hex and λ H have moderate negative effects. These three parameters have large s.d. (d ij ) values. The other inputs have little effect on U and have a relatively small s.d. (d ij ) . These results show that the biomasses of all the functional groups are affected most positively by increases in recruitment, which shows that the model groups are predominantly recruitment-limited. However, increasing the piscivorous fish exogenous recruitment rate l Pex results in a significant increase in P on average and a significant decrease in H, due to increased predation. Thus, herbivorous fish are also predationlimited to a certain degree. The results also show that an increased total fishing effort f affects both fish groups negatively on average, which means that for herbivorous fish, decreased predation does not compensate for increased fishing on average. For urchins, the parameter that has the dominant negative effect on biomass is the death rate d U , which includes predation. The magnitude of the effect for changing d U is similar to that for changing lUex (the urchin exogenous recruitment rate) and thus, urchins could be limited by predation to a similar extent as by recruitment. Results for the HSLUm are shown in Figures 15.11-15.14. At t = 25 yrs, changing the initial biomasses has little effect on any of the biomasses (on average), which suggests that the system is near or has achieved equilibrium. For H, the dominant ex negative input is d H , followed by l Ps and then ρ H ; the dominant positive factor is l Hex .
289
(a)
l Hex
M (d ij )
ψP
λH d P
i FH i PH
H0
dU lUen lUex
i FP i PP
P0 U 0
f
l Pex
ρH
dH (b) s.d. (d ij )
dH
ρH l
i FH i PH
ex H
l Pex
λH
dP
ψP i FP i PP
f
dU
lUen
lUex H 0 P0
U0
Figure 15.8. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 18 inputs for the HPUm. The output is H at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 15 parameters.
ex All other inputs have little effect. d H , l Hex and l Ps exhibit the largest s.d. (d ij ) values,
with d H having the largest value, especially for ∆ = 0.1 . All other inputs have a
290
(a) l Pex
M (d ij )
ρH l Hex
i FH dH
i FP i PP
λH
dU
H 0 P0 en U
i PH
l
ex U
l
U0
dP f
ψP (b) s.d. (d ij )
ψP
l Pex
dP
ρH d H i i l Hex FH PH
f
λH
i FP i PP
d U lUen lUex H 0 P0 U 0
Figure 15.9. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 18 inputs for the HPUm. The output is P at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 15 parameters.
relatively small s.d. (d ij ) value. For SIP fish biomass Ps , d Ps is the dominant negative input, followed by f, φ Ps
291
(a) M (d ij )
lUex
ρH
dH i FH i PH
l Hex
f
i FP i PP l Pex
lUen
P0 U 0
ψP
λH d P
H0
dU (b) s.d. (d ij )
dU
lUex
dH
i FH i PH
l Hex ρ H λ H
d P i FP i PP
l Pex ψ P f
lUen
H 0 P0
U0
Figure 15.10. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 18 inputs for the HPUm. The output is U at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 15 parameters.
ex and l Hex , and the dominant positive input is l Ps . These five parameters also show large ex s.d. (d ij ) values, particularly d Ps and l Ps . All other inputs have little effect and have
292
relatively small values of s.d. (d ij ) . For LP fish biomass Pl , the dominant negative input ex is f, which has a moderate effect. The dominant positive input is l Ps , but this has a small
effect. The value of s.d. (d ij ) for f is high. The results for U for the HSLUm are similar to those for the HPUm, with d U being the dominant negative input and lUex being the dominant positive input, with both parameters having large values of s.d. (d ij ) . Also, d H has a large positive effect with l Hex and λ H having large negative effects. These three parameters also have large s.d. (d ij ) values. The other inputs have little effect on U and have a relatively small s.d. (d ij ) . As for the HPUm, these results show that all the functional groups are predominantly recruitment-limited. However, similar to the HPUm, increasing the ex piscivorous fish exogenous recruitment rate l Ps leads to a significant increase in Ps and
a smaller increase in Pl , on average, accompanied by a significant decrease in H on average. This suggests a degree of predation-limitation for herbivorous fish. As for the HPUm, an increased fishing effort affects all fish groups negatively on average, which means that for herbivorous fish and SIP fish, decreased predation tends not to compensate for the increased fishing effort. An interesting result is that Ps is affected negatively by an increase in the exogenous recruitment rate of herbivorous fish, l Hex , to a moderate degree. This is interesting because it suggests that an increasing l Hex increases H, which in turn increases predation by SIP fish. However, Ps does not increase because more piscivores become large and there is also increased predation by LP fish. This (upwards) trophiccascade effect is supported by the fact that H increases significantly with increasing l Hex and Pl increases by an amount that is close to moderate. However, LP fish parameters only have a small effect on H on average and herbivorous fish parameters only have a small to nearly moderate effect on Pl on average, so trophic cascade effects between more than two fish trophic levels seems limited. As for the HPUm, there is evidence that urchins are limited by predation, possibly to a similar extent as they are by recruitment.
293
(a) l Hex
M (d ij )
λH
i FH i PH
ρH
d Ps
φ Ps
d Pl
i FPs i PlPs
i FPl
dU f
H0 lUen lUex
Ps 0 Pl 0 U 0
ex l Ps
dH (b) s.d. (d ij )
dH
l Hex
i FH i PH
ρH
ex l Ps
d Ps
λH
i PlPs
i FPs
φ Ps
f
d Pl i FPl
H P d U lUen lUex 0 Ps 0 l 0 U 0
Figure 15.11. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is H at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 17 parameters.
294
(a) ex l Ps
M (d ij )
dH
i PH
d Pl
i FPs i PlPs
ρ H λH
i FH
lUen lUex
i FPl
dU
Ps 0 H0
U0 Pl 0
φ Ps
l Hex
f
d Ps (b) d Ps
s.d. (d ij )
ex l Ps
φ Ps
l Hex
dH i FH i PH
ρH
λH
i i FPs PlPs
f
d Pl i FPl
d U lUen lUex H 0 Ps 0
Pl 0
U0
Figure 15.12. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is Ps at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 17 parameters.
295
(a) M (d ij )
l
ex l Ps
ex H
φ Ps
ρH i PlPs i FPs
λH
i FH
Pl 0 H 0 Ps 0 U 0
d U lUen
i FPl
lUex
i PH dH
d Ps
d Pl f
(b) s.d. (d ij )
d Pl
f
l Hex ρ H
dH
ex l Ps
φ Ps
d Ps
i FH i PH
λH
Pl 0
i i FPsPlPs
i FPl
d U lUen lUex H 0 Ps 0
U0
Figure 15.13. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is Pl at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 17 parameters.
296
(a) lUex
M (d ij )
ρH
dH i FH i PH
l Hex
ex i FPs i PlPs l Ps
λH
lUen
i FPl f
Ps 0 Pl 0 U 0
φ Ps d Pl
d Ps
H0
dU (b) dU
s.d. (d ij )
lUex
dH
i FH
ex l Hex ρ λ H d Ps i PlPs l Ps φ Psd Pl i FPl f i PH H i FPs
lUen
H 0 Ps 0 Pl 0 U 0
Figure 15.14. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the 21 inputs for the HSLUm. The output is U at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 10.1 and 11.1 for definitions of the 17 parameters.
297
15.2.3 Elementary Effects Method Results for Integrated Model
For the most complex integrated model, the CTM-HSLUm (Chapter 7), the elementary effects for the seven parameters that represent fishing of herbivores, nutrification and sedimentation were calculated. These parameters are ρ H and f (fishing of herbivores),
rM (nutrification; section 13.2.2 in Chapter 13), and d C , l Cb , l Cs and rC (sedimentation; section 13.2.3 in Chapter 13). ζ T is also affected by nutrification, but elementary effects are not calculated for it because it determines the range of at least one other parameter. The results are shown in Figures 15.15-15.21. Only the graphs for t = 25 yrs are shown; graphs for t = 5 yrs and t = 50 yrs have similar shapes and the magnitudes of M (d ij ) and s.d. (d ij ) tend to increase with t. Also, the results are similar for all three ∆
values, with the magnitudes being greater at lower ∆ values for some of the inputs. The results show that d C has the greatest negative effect (on average) on C and the greatest positive effect on T. For C, rM has a moderate negative effect, whereas l Cb and rC have fairly small positive effects. For T, rM has a large negative effect. s.d. (d ij ) is large for d C and rM for both types of cover. For M, rM has by far the greatest (positive) effect and has a very large s.d. (d ij ) . Although the fishing parameters ρ H and f have little effect on the three benthic covers on average, changing either of these factors may have significant negative effects on T and M for a particular set of parameters. This is because of the moderate s.d. (d ij ) for these two types of cover, for both parameters. From the values of the elementary effects obtained, changing ρ H or f by an amount equivalent to their full ranges can change T or M by an amount greater than 0.2. For H, d C has the strongest positive effect, although the average effect is small. d C has a moderate value of s.d. (d ij ) , so the actual effect for a particular set of parameters may be significantly larger than the average. ρ H has the greatest negative effect, although this effect is fairly small, followed by f, which has a small effect; these two parameters have large s.d. (d ij ) values. For both types of piscivorous fish biomass, f has a moderate to large negative effect, with large s.d. (d ij ) values. ρ H has a small negative effect on Ps and a small positive effect on Pl . However, in both cases, it has a moderate s.d. (d ij ) , so the change in biomass may be significantly greater for a particular 298
set of parameters. For U, d C has a moderate positive effect with a large s.d. (d ij ) . l Cb , rC and rM have small negative effects on average. However, l Cb and rC have a moderate s.d. (d ij ) and rM has a large s.d. (d ij ) . Thus, as l Cb and rC change, the change in U may be significantly greater for a particular set of parameters and as rM changes, the change in U may be much greater. These results show that as for the CTMm, the coral mortality rate d C has a large negative effect on C and a large positive effect on T. However, the negative effect on C is about half the corresponding effect for the CTMm, and the positive effect on T is slightly lower than the corresponding effect for the CTMm. As for the CTMm, the macroalgal lateral growth rate rM has negative effects on C and T and a positive effect on macroalgal cover M, but the magnitudes of these effects are at least doubled relative to the corresponding effects for the CTMm. For C, the brooding coral recruitment rate l Cb and the coral lateral growth rate rC have moderate positive effects, similar to the
CTMm on its own, although the magnitudes of the effects are smaller. These results show that sedimentation and nutrification are still able to have large effects on benthic covers in the integrated model. However, the smaller effect of parameters affected by sedimentation on C and the larger effect of increasing rM on all types of cover suggests that the grazing pressure θ is lower on average than in the pure benthic models, so that C decreases by less with increasing sedimentation because its value before sedimentation is increased is lower than in the CTMm, and M is able to increase by more as rM increases because the lower value of θ is less able to suppress the proliferation of macroalgae. Fishing effort f, and the proportion of it directed to herbivorous fish, ρ H , have little effect on all benthic covers on average, and affect the fish biomasses in similar ways as in the HSLUm on its own. This shows that a decrease in herbivorous fish biomass and hence grazing does not usually lead to much change in benthic covers. Thus, there is not much forward feedback from the fish populations to the benthic components. A possible reason for this is that a decrease in herbivorous fish grazing may be compensated for by an increase in sea urchin grazing. Another possible reason is that sea urchins usually dominate grazing, by attaining a greater biomass. It may also be the case that, on average, fishing is not able to decrease herbivorous fish biomass by enough to have a significant effect on the grazing pressure. Nevertheless, due to the moderate s.d. in the elementary effects of f and ρ H for T and M, fishing may have 299
(a)
l Cb
M (d ij )
rC l Cs
ρH
f
ρH
f
rM
dC (b) s.d. (d ij )
dC
rM
rC
l Cb l Cs
Figure 15.15. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is C at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
moderate effects on T and M. The benthic parameters tested have little effect on fish biomasses on average, which shows that there is not much feedback from the benthic components to the fish populations on average. This could indicate that herbivorous fish are not usually limited by the amount of algae, but by other factors such as the accessibility of algae and recruitment. It could also indicate that sea urchins dominate grazing and are able to take advantage of the increased algae more effectively than fish. The latter hypothesis is 300
(a)
M (d ij )
dC
l Cs
l Cb
rC
ρH
f
ρH
f
rM (b)
rM
s.d. (d ij ) dC
rC
l Cb l
s C
Figure 15.16. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is T at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
supported by the fact that an increase in coral mortality d C increases U (but not H) by a moderate amount on average.
15.2.4 Sobol’ Variance Decomposition Results
Sobol’ Variance Decomposition is applied to the CTm as described in section 15.1, with
301
(a)
rM
M (d ij )
ρH
dC
l
s C
l Cb
f
rC
(b)
rM
s.d. (d ij )
dC
ρH l
b C
l Cs
f
rC
Figure 15.17. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is M at t = 25 yrs and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
the inputs being the nine parameters and C 0 . T0 is set to half of 1 − C 0 - this keeps all the inputs independent as required (see section 15.1.3). Figures 15.22-15.23 show sij and sijT for all outputs i and inputs j, when sijT > s ij , the indices are positive and there is convergence, as required. For t = 5 yrs,
all indices have converged, but for C, sij is still negative for l Cs and ε C , whereas for T,
302
(a)
M (d ij )
dC
l Cb
l Cs
rC
rM f
ρH (b) s.d. (d ij )
ρH
f
dC l Cb
rC
rM
l Cs
Figure 15.18. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is H at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
sij > s ijT for l Cs and ε C . The magnitudes of all sijT for l Cs and ε C are less than 0.001,
which means that these two parameters do not contribute significantly to the total variance and the corresponding sij and sijT indices do not contribute significantly to the sum of each type of index. The negative values for sij are likely to be a result of the indices having a true positive value close to 0. Similarly, the fact that sij > s ijT is probably a result of sij not being close enough to its true value. For t = 25 yrs, all the 303
(a)
l Cb
M (d ij )
l Cs
rC
rM
dC
ρH
f
(b)
f
s.d. (d ij )
ρH dC
l Cb
l Cs
rC
rM
Figure 15.19. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is Ps at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
indices have converged apart from sij for T and l Cs . However, since sijT has converged and is < 0.0005 , sij is very small, i.e. l Cs makes little contribution to the total variance and sij for l Cs contributes little to the sum of the sij indices for T. For t = 50 yrs, all the indices satisfy sijT > s ij , are positive and have converged. The results show that at t = 5 yrs, two inputs dominate the total variance of C, d C (contributing approx. 33% on its own and through interactions with other inputs) 304
(a)
ρH
M (d ij ) l Cs
dC l Cb
rC
rM
f
(b)
s.d. (d ij )
f
ρH
dC
l Cb
l Cs
rC
rM
Figure 15.20. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is Pl at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
and C 0 (contributing approx. 59%). θ dominates the total variance of T (contributing approx. 56%), and d C , ζ T and C 0 also contribute a substantial amount to total variance ( > 10% each). For C, the sum of the sij indices (taking the two negative index values as 0) is 0.915, which shows that most of the variance is due to the effects of inputs acting on their own. However, there is some interaction between inputs and this is most pronounced for d C and C 0 . For T, the sum of the sij indices (taking sij = s ijT for the 305
(a)
M (d ij )
dC
ρH l Cs
f
rC
l Cb
(b)
s.d. (d ij )
rM
dC
rM
l Cb
rC
ρH l
s C
f
Figure 15.21. Graphs showing (a) M (d ij ) and (b) s.d. (d ij ) for each of the seven inputs for the CTM-HSLUm, which are seven parameters representing fishing of herbivores, nutrification and sedimentation. The output is U at t = 25 yrs in kg m-2 and the red, green and blue bars represent M (d ij ) or s.d. (d ij ) for ∆ = 0.1 , 0.3 and 0.5 respectively. See Tables 9.1 and 10.1 for definitions of the seven parameters.
two sij indices that have sij > s ijT ) is also 0.915, which again shows that most of the variance is due to inputs acting on their own. At t = 25 yrs, the contribution of C 0 to either C or T drops to below 5%, which shows that dynamics are much nearer to equilibrium. This is consistent with the results for the CTm from the Elementary Effects method. For C, d C is now the sole dominant factor (contributing approx. 61% to the total variance), with substantial contributions by l Cb and θ ( > 10% each). For T, θ continues to dominate (contributing approx. 62% to
306
total variance), again with substantial contributions by d C and ζ T ( > 10% each). The sum of the sij indices (taking the non-convergent value as an estimate for the one index that has not converged) for C and T is 0.879 and 0.899 respectively. So as at t = 5 yrs, most of the variance is due to the effects of inputs acting on their own. Finally, at t = 50 yrs, the results are similar to those at t = 25 yrs. d C still dominates the total variance of C (contributing approx. 62%) with substantial contributions by l Cb and θ ( > 10% each); θ still dominates the total variance of T (contributing approx. 61%) with substantial contributions by d C and ζ T ( > 10% each). The sum of the sij indices for C and T is 0.880 and 0.920 respectively, which again shows that most of the variance in C and T is due to the effects of inputs acting alone.
307
(a)
C0
sij or sijT
dC
l Cb l Cs
(b) sij or
rC
αC
εC
gT
ζT
θ
dC
sijT
θ l Cb
rC l
(c) sij or
s C
αC
εC
gT
ζT
C0
dC
sijT
θ l
b C
rC l Cs
αC εC
gT
ζT
C0
Figure 15.22. Graphs showing sij and sijT for each of the 10 inputs for the CTm, for the output C at (a) t = 5 yr, (b) t = 25 yrs and (c) t = 50 yrs. The red and blue bars represent sij and sijT respectively. In (a), negative sij values for l Cs and ε C are not plotted. The sij values not plotted are small (see main text). See Table 9.1 for definitions of the nine parameters. 308
(a)
θ
sij or sijT
C0
ζT
dC l Cb
l Cs
rC
αC
εC
gT
(b)
θ
sij or sijT
dC
ζT l Cb
l Cs
rC
αC
εC
gT
C0
(c)
θ
sij or sijT
dC
ζT l Cb
l Cs
rC
αC
εC
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C0
Figure 15.23. Graphs showing sij and sijT for each of the 10 inputs for the CTm, for the output T at (a) t = 5 yrs, (b) t = 25 yrs and (c) t = 50 yrs. The red and blue bars represent sij and sijT respectively. In (a), sij values for l Cs and ε C , for which sij > s ijT , are not plotted and in (b), sij for l Cs has not converged and is not plotted. The sij values
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not plotted are small (see main text). See Table 9.1 for definitions of the nine parameters.
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Chapter 16. Discussion for Part IV In Part IV, the results for the three benthic models obtained using the Immune Inspired Algorithm (IIA; Chapter 13) show that macroalgae are necessary for multiple equilibria and hence alternative stable states (ASS). Since the Coral-Turf-Macroalgae model (CTMm), where turf algae and macroalgae are modelled as two independent groups (Chapter 5), can exhibit ASS, this shows that ASS are not an artefact of the Coral-Algae model (CAm), where the two algal groups are amalgamated into one group with turf and macroalgae in fixed proportions (Chapter 5). This result suggests that macroalgae generate ASS through positive feedbacks that are sufficiently strong. Macroalgae can generate these feedbacks by laterally overgrowing live corals (e.g., Nugues and Bak 2006), reducing coral growth rates (e.g., Jompa and McCook 2002a), preventing recruitment of corals onto them (Birrell et al. 2008) and preventing coral growth over them (Jompa and McCook 2002a). This result supports the conclusions of the modelling work by Mumby et al. (2007a) and the schematic representation of feedback mechanisms in coral reefs by Mumby and Steneck (2008). The presence of positive feedbacks on its own does not guarantee ASS, however, because feedbacks may be either too powerful or too weak relative to other mechanisms to give ASS (Scheffer et al. 2001). Results from the IIA searches of sub-regions of the full parameter space for the CAm show that for ASS to occur, the effect of macroalgal processes needs to be quite strong regardless of the strength of other processes. Thus, no ASS were found with the macroalgal growth rate over space ( rM ), the parameter measuring the reduction in coral growth by macroalgae ( β M ) and the macroalgal growth rate over corals relative to the rate over space ( γ MC ) taking values from the lowest 45% of their pristine ranges and with the maximum grazing rate on macroalgae ( g M ) taking values from the highest 45% of its pristine range. In addition, no ASS were found when ν M , which measures the fixed proportion of total algal cover that is macroalgae and therefore the magnitude of the effect of macroalgal processes on corals per unit of total algal cover, was less than 30%. However, IIA results for the CTMm show that there is the possibility of ASS even when the macroalgal parameters rM , β M , γ MC and γ MT (the macroalgal growth rate over turf relative to the rate over space) take values from the lowest 5% of their pristine ranges and with g M taking values from the highest 5% of its pristine range. 311
Thus, although both models give ASS in some circumstances, there appears to be more scope for ASS when turf algal and macroalgal dynamics are independent than when they are amalgamated, with the possibility of ASS even if the macroalgal processes are relatively weak. A likely reason is that when turf algal and macroalgal dynamics are amalgamated, there is less opportunity for positive feedbacks toward an algaldominated state because turf algal cover must increase alongside macroalgal cover. However, for the CTMm, a necessary (but not sufficient) condition for ASS is
g M θ < rM (the negation of condition 5.8 in Chapter 5), that is the grazing rate on macroalgae must still be low relative to the growth rate of macroalgae. The IIA searches of sub-regions of parameter space also show that, for both the CAm and CTMm, if macroalgae are only able to grow laterally over space, then ASS cannot occur. This suggests that reduction of coral growth and/or coral growth over live corals or turf algae is necessary for positive feedbacks to be strong enough for ASS to occur. Some coral species appear to be less prone to overgrowth by macroalgae, with some preventing macroalgal overgrowth possibly through the use of sweeper tentacles and/or mesenterial filaments (Nugues and Bak 2006). Thus, model results suggest that reefs on which these species are abundant are less likely to exhibit ASS. For both the CAm and the CTMm, for a pristine reef system, ASS are not possible if there are no parameter values that lie outside the middle 80% of the parameter ranges. This is also true when only sedimentation is applied to a pristine system, which shows that this stressor on its own is not effective at promoting ASS. When just nutrification is applied, for the CAm, there are still no ASS when no parameters take extreme values outside the middle 75% of their pristine ranges, whereas for the CTMm, the same qualitative result is found, except that the ‘safe’ range is the middle 55%. With nutrification and sedimentation, the same results as for the case with just nutrification are found. This suggests that nutrification is the dominant stressor of the two in promoting the existence of ASS, and that there is little synergy between these two stressors in giving rise to ASS. The biggest discrepancy in the results between the two models occurs when fishing is applied, either on its own or in combination with other stressors. For the CAm, when fishing is applied to a pristine reef area, on its own or in combination with nutrification and/or sedimentation, there are no ASS when all parameters for the pristine reef area take values within the middle 75% of their ranges. In contrast, for the CTMm, there is the possibility of ASS when parameters for the pristine reef area take values within the middle 15-30% of their ranges. This again suggests that when turf algal 312
dynamics are decoupled from macroalgal dynamics, the likelihood of ASS increases. Thus, overall, the IIA results suggest that fishing is a more potent stressor in promoting the existence of ASS than either sedimentation or nutrification, and that nutrification is a more potent stressor than sedimentation. A likely reason is that fishing is modelled as decreasing θ and nutrification increases rM , which increases the likelihood that the necessary condition g M θ < rM for the CTMm holds. Furthermore, nutrification can increase the θ value at which a discontinuous phase shift occurs with increasing fishing pressure, as well as increasing the hysteresis effect. The results of Chapter 13 show that a continuous phase shift can occur with either one of fishing, nutrification or sedimentation. As discussed above, if macroalgae are present, then the shifts can be discontinuous. However, discontinuous phase shifts are significantly less common than continuous shifts. This is because the Coral-Turf model (CTm) cannot show discontinuous shifts at all and the CAm and the CTMm cannot show discontinuous shifts for pristine reef areas with nutrification and/or sedimentation unless some parameter values take extreme values. In contrast, continuous phase shifts can occur for all three models for pristine reef areas taking parameter values within the middle of their ranges, with any one of the three stressors. Furthermore, for all three models, for pristine reef systems taking their mid-range values, the equilibrium coral and algal covers can show disproportionate changes due to the application of any combination of more than one stressor. Thus, the results show that synergy between the three stressors could be common. The parameter sweep results of Chapter 14 show that for the CAm, the percentage of parameter sets giving discontinuous phase shifts with changing θ was low, less than 0.2%, using any type of criteria devised and for the full parameter space which includes the effects of fishing, nutrification and sedimentation. Furthermore, even when there is a discontinuous shift, the range of possible grazing pressures θ for which there are multiple equilibria is very small, at most 0.03. Thus, the hysteresis effect is very small. However, since results from Chapter 13 suggest that the likelihood of ASS for the CTMm is greater than for the CAm, the potential of the CTMm to exhibit discontinuous shifts with changing θ is likely to be greater than for the CAm. In contrast to discontinuous shifts, continuous phase shifts with changing θ of any type (weak to severe) were far more common for all parameter spaces searched. These results therefore support those of Chapter 13 in that they provide evidence that continuous shifts are likely to be common, while discontinuous shifts are much rarer. Compared to the CTm, the percentage of parameter sets giving any type of 313
continuous shift is greater for the CAm, particularly if nutrification is considered. Also, if there is nutrification, the percentage of parameter sets giving the most severe type of continuous shift, as measured by changes in coral cover, is much greater. This shows that the presence of macroalgae leads to steeper changes in benthic covers at equilibrium with changing θ , particularly for coral cover. In general, the use of criteria that incorporate both coral and algal cover gives frequencies of shifts of increasing severity that are very similar to those obtained using just algal criteria. This suggests that the less complex and computationally expensive algal criteria can be used as a reliable proxy for the more complex set involving both types of cover (coral and algae). However, in general, the coral cover criteria give more continuous shifts in total than for the other types of criteria, suggesting that coral cover is more prone to sharp changes than algal cover. For the CTm, the percentage of parameter sets giving continuous shifts is less when the effects of nutrification and/or sedimentation are added. This shows that on average, the introduction of these two stressors decreases coral cover and increases algal cover at equilibrium in such a way that the response of these two covers to changing θ is more linear. This means that the maximum changes in these covers with changing θ decreases, resulting in a lower frequency of shifts. The frequency of shifts with nutrification and sedimentation is greater than when there is just nutrification or sedimentation, which shows that when both stressors are acting, there is some synergy that increases the non-linearity of the benthic covers with changing θ , compared to the cases with just nutrification or sedimentation. For the CAm, the frequency of continuous shifts of any type changes little with nutrification and/or sedimentation, when coral criteria are used. Using algal cover criteria, the frequency remains similar with nutrification, but decreases with sedimentation, for the same reason as for the CTm. Thus, a lower frequency of phase shifts does not necessarily mean healthier reefs with higher coral covers and lower algal covers; rather, it means that shifts from high to low coral cover and/or low to high algal covers are less likely with changing θ – and this could just be because reefs have lower coral covers and higher algal covers across the range of θ . The parameter sweeps results also show that nutrification is more important than sedimentation in determining the turf algal cover at equilibrium, for the CTm, or the algal cover at equilibrium, for the CAm, at which a continuous phase shift occurs. Sedimentation, on the other hand, is more important than nutrification in determining the coral cover at equilibrium at which such a phase shift occurs. This suggests that the 314
main effects of both of these stressors are on the organisms directly affected – nutrification affects algae directly and hence mainly affects turf algal cover or algal cover, whereas sedimentation affects corals directly, with the main impact being on coral cover. The sensitivity analysis using the Elementary Effects method (Chapter 15) shows that, for the benthic models and for a reef that is not affected by acute disturbances such as hurricanes, after 5 yrs the initial covers still have a large effect on the variables, whereas after 25 yrs the effect is small and after 50 yrs, the effect of initial conditions is very small. This shows that if a reef is hit with an acute disturbance about once in every 5 yrs to 25 yrs, then the effect that each disturbance will have on benthic covers can depend strongly on the effect of the previous disturbance and on how quickly the reef recovers in between, which supports the empirical study by Hughes and Connell (1999). Similar results were found for the initial biomasses for the fish and urchin models, with the same interpretation. The means of the elementary effects for all parameters tested for all models tends to increase with time. This shows that time trajectories that arise due to changing one or more of the parameters usually become further apart from the original trajectories (with parameters unchanged) with time, leading to greater long-term differences. Thus, the substantive effect of one or more parameters changing, with this change being maintained in the long-term, may only manifest itself fully over a timescale of decades. For example, this may be the case if fishing pressure increases rapidly and is maintained due to economic development, if algal growth rates increase quickly and are maintained due to the construction and continued use of a sewage plant that discharges sewage into nearby coral reef waters, or if the coral mortality rate decreases quickly and is maintained due to more effective management of nearby deforestation that leads to a reduction in runoff and hence sedimentation rates for nearby coral reefs. Importantly, this result suggests that snapshot measures of variables such as covers and biomasses over a few years are not good indicators of the future trajectory and hence health of a reef, and suggests that the management of chronic stressors requires a long-term focus to maximise reef health. The elementary effects for the three benthic models show that, for each model, corals are very sensitive to fishing of herbivores and sedimentation, and moderately sensitive to nutrification, if these three stressors have the effects as parameterised in Chapter 13, section 13.2. In addition, turf algae are very sensitive to all three stressors and macroalgae are very sensitive to fishing, moderately sensitive to nutrification and 315
not sensitive to sedimentation. These results demonstrate the large potential for these three stressors to degrade coral reefs by promoting algal cover at the expense of coral cover, and are consistent with the results from Chapters 13 and 14. However, unlike the results from Chapters 13 and 14, the elementary effects show that these stressors have a large potential to change non-equilibrium benthic covers as well as equilibrium covers. In addition, these results demonstrate that sedimentation affects corals more than nutrification, whereas the reverse is true for macroalgae. The likely reason is that sedimentation affects corals directly whereas nutrification affects macroalgae directly. Nutrification also affects turf algae directly, whereas sedimentation does not. However, turf algae is still very sensitive to sedimentation, and a possible reason is that since turf algae have a faster growth rate than macroalgae, they can more readily take advantage of the extra space that is available due to sedimentation-induced coral mortality. For each benthic model, after t = 25 yrs, the elementary effect of the grazing pressure θ has a high standard deviation for all types of cover. In addition, after t = 25 yrs, the elementary effect of at least one parameter affected by nutrification and
of at least one parameter affected by sedimentation has a high standard deviation, for all types of cover. The exception is macroalgal cover – the elementary effects of parameters affected by sedimentation have low standard deviations for this type of cover. These results demonstrate that benthic covers can show a high degree of nonlinearity when any one of the three stressors studied is applied, because of the parameters that are changed by the stressors each acting on their own and/or because of interactions between these parameters and other parameters. In particular, the standard deviation is always very high for θ after t = 25 yrs, sometimes exceeding 0.6. This means that changing θ , which may occur by changing fishing pressures, has a high potential for giving disproportionate changes in benthic covers. This supports the results from the parameter sweeps which show that continuous phase shifts are common on coral reefs with changing θ . The elementary effects for the fish and urchin models show that for both models, recruitment is the main factor in limiting the biomasses of the functional groups, but that predation-limitation is also operating for herbivorous fish and urchins. Thus, the models are able to account for both recruitment and predation as important factors in limiting fish and urchin populations, and are not strictly recruitment-limited or predation-limited. In addition, both models show that if the total fishing effort on all fish groups, f, increases, then the biomasses of all fish groups tend to respond negatively. This demonstrates that for herbivorous fish, increased fishing mortality is more 316
important than the reduced predation mortality. For piscivorous fish, the direct effect of increased fishing mortality may also be exacerbated by a decrease in herbivorous fish prey due to increased fishing mortality of herbivorous fish. For the fish and urchin models, the parameters that have moderate to large effects on biomasses on average tend to have moderate to large standard deviations for their elementary effects as well. This implies that changes in the exogenous recruitment parameter for each functional group tend to lead to non-linear changes in the biomass for that functional group, because of just the recruitment parameter and/or because of interactions between the recruitment parameter and other parameters. In addition, changing f tends to have non-linear changes in fish biomasses; this is consistent with results from the benthic model, which showed that benthic covers tend to change nonlinearly with changing grazing pressure, which could arise due to changing fishing pressures. The elementary effects for the integrated model, the CTM-HSLUm (Coral-TurfMacroalgae-Herbivorous Fish-Small to Intermediate Piscivorous Fish-Large Piscivorous Fish-Urchins model), show that if the benthos, fish and urchins all vary dynamically, then nutrification and sedimentation have large non-linear effects on the benthic covers. This is consistent with the CTMm, where only the benthos can vary. However, sedimentation seems to have less of an effect on benthic covers in the CTMHSLUm, whereas an increase in macroalgal growth has more of an effect. This suggests that the grazing pressure θ , which can now vary dynamically, tends to be lower compared with in the CTMm (where it could take values in the entire theoretical range 0 ≤ θ < 1 with equal probability), such that coral cover tends to be lower and
macroalgal cover has the potential to be higher with increasing macroalgal growth, resulting in changes in coral cover that are smaller and changes in macroalgal cover that are larger. Thus, results from the benthic models may overestimate the effects of sedimentation on benthic covers and for the models with macroalgae, underestimate the effects of nutrification on benthic covers. Results for the integrated model also show that although increasing fishing effort affects fish biomasses in a similar way to that seen for the HSLUm with fixed benthic dynamics, the effects on the benthic covers tend to be small. This suggests that sea urchins may be compensating for the decrease in herbivorous fish grazing or that sea urchin grazing dominates, such that a decrease in herbivorous fish grazing has little effect. Thus, the results suggest an important role for sea urchins in providing functional redundancy in the grazing function. These results could also indicate that, on average, 317
fishing is unable to change herbivorous fish biomass by a large enough amount to affect
θ significantly. Thus, it is possible that the effects of fishing for the benthic models, as parameterised in section 13.2.1 in Chapter 13, are overestimated. However, the results for the benthic models hold for any factor that changes θ significantly, whether this is fishing or not. Furthermore, for the integrated model, changing the benthic parameters tested had little effect on the fish biomasses, with the effect on the sea urchin biomass being greater. This again suggests that urchins dominate grazing, possibly because they are able to achieve a higher biomass in the model than herbivorous fish and are hence able to achieve a greater grazing pressure on algae, i.e. the urchin population is able to graze algae more effectively than the herbivorous fish population. The Sobol’ Variance Decomposition sensitivity analysis technique was also applied to the simplest benthic model, the CTm (Chapter 15). For each type of benthic cover (coral or turf algal), this technique ‘decomposes’ the variance of the benthic cover with changing input values to obtain the variances due solely to individual inputs and due to interactions between inputs. This allows the degree of interaction between inputs to be found, which cannot be done using the Elementary Effects method. First, consistent with the results obtained using elementary effects, the Sobol’ Variance Decomposition results show that at t = 5 yrs, the initial covers have a large effect in changing the benthic covers relative to all the parameters, but at t = 25, 50 yrs, they have relatively little effect. This supports the conclusion that the benthos takes about 25-50 yrs to reach equilibrium in the absence of acute disturbances. For coral cover, changes in the coral mortality rate d C explains most of the variance, followed by changes in the grazing pressure θ . For turf algal cover, θ is the dominant parameter followed by d C and the turf algal vertical growth rate ζ T . These results support the elementary effects results which showed that for the CTm, fishing, nutrification and sedimentation can have large effects on benthic covers, provided they have the effects as parameterised in section 13.2. The effect of sedimentation is mediated mainly through an increase in direct coral mortality. The Sobol’ Variance Decomposition results also show that at any of the three t values, the proportion of the variance of coral or turf algal cover that is not due to interaction between inputs is always high at 0.879-0.920. This shows the relative insignificance of parameter interactions in changing benthic covers, compared with the effects of parameters acting on their own. Thus, the large degree of non-linearity found 318
for θ and some of the parameters affected by nutrification and sedimentation, using the elementary effects for the CTm, is primarily due to parameters acting on their own, rather than interactions between parameters. Also, the non-linearity in benthic covers with changing θ found for the CTm in the parameter sweeps, which gave rise to a high frequency of continuous shifts, is due mainly to θ on its own, rather than interactions between θ and other parameters. These results suggest that on a reef with no macroalgae, although fishing, nutrification and sedimentation have the potential to give high non-linearity in benthic covers (from the elementary effects results), the synergistic effect when more than one stressor is acting is not large. This means that although results from Chapter 13 suggest that synergy between stressors is common on reefs without macroalgae, the magnitude of this synergistic effect tends to be small.
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Part V: Conclusions
Chapter 17. General Discussion for Whole Thesis Ecosystem models have become popular tools over the last 35 yrs (Fulton et al. 2003). This has been paralleled by a shift in resource management to a holistic paradigm that recognises the whole as well as the parts – an ecosystem management paradigm (Hatcher and Bradbury 2006). With this paradigm comes an emphasis on how an ecosystem responds to stress, for example how a marine ecosystem responds to fishing (Pikitch et al. 2004). In coastal marine ecology, the bridge between models and management has been aided by the increasing use of manipulative experiments to formally test hypotheses generated by models (Underwood et al. 2000). Despite the value of modelling in generating hypotheses on how an ecosystem behaves under anthropogenic stress and in pointing out the experiments that are needed to test these hypotheses, as recently as 10 yrs ago, the use of coral reef modelling was relatively rare compared to the use of modelling for other habitats, although it has increased since (Pennings et al. 1997; see review in Chapter 2). This thesis fits into the broad coral reef research framework by advancing the state of coral reef modelling, and by doing so, generating hypotheses/predictions on how reefs behave under anthropogenic stress at a local scale. This in turn highlights the critical experiments that need to be performed to test these hypotheses and validate predictions.
17.1 Thesis Results in Relation to Existing Studies and Management 17.1.1 Construction and Parameterisation of Models
Three mean-field, deterministic, differential equations models of a coral reef benthos with fixed fish and urchins, at a local scale, have been constructed. Following a stepwise refinement strategy (Bradbury et al. 2005), the models were of increasing complexity, beginning with a model having only coral and turf algal cover variables (and ‘space’ to represent the residue), but without macroalgae, to one in which turf and macroalgae were represented as a single dynamic group, with the two types of algae in 320
fixed proportions, to the most complex model in which turf and macroalgae are represented as separate dynamic groups. The advantage of the first two (twodimensional) benthic models was that the number of equlibria could be determined as well as the stability of these equilibria. This was not the case for the most complex (and realistic) model, but nonetheless, a useful condition was derived for this model (condition 5.8 in Chapter 5) which, if true, implies that there are no macroalgae at equilibrium, such that the model behaves exactly like the simplest model at equilibrium. In addition, two mean-field, deterministic, differential equations models of a coral reef fish and urchin community with a fixed benthos, at a local scale, were constructed. Again, the two models were of increasing complexity, with the more complex model splitting the piscivorous fish group into two size classes (small to intermediate and large), to allow the representation of different life-history traits of and different fishing rates on these two classes. These models were mathematically analyzed and sets of conditions were obtained which ensured that the models made biological sense – i.e. the dynamics had suitably bounded trajectories and fish or urchin populations could not be sustained by feeding-induced growth in the absence of recruitment. In addition, it was found that each model had at least one equilibrium, which suggests that in the absence of sustained disturbances, the system dynamics specified by each model can stabilise. Functional groups and interactions in all models were included only after a careful review of the literature, from which evidence was found that they could potentially be important in the coral-algal phase shifts that are characteristic of reef degradation, and hence be of major importance to reef managers. Following the stepwise refinement procedure, integrated models were then constructed that link a benthic model with a fish and urchin model, by representing the grazing pressure parameter in the benthic models as a function of herbivorous fish and urchin biomasses, and by representing the benthic state parameter (which characterises the amount of food available to herbivores in the fish and urchin models) as a function of the benthic algal cover variables. In particular, the grazing pressure function increases non-linearly with fish or urchin biomass, which makes it different to representations of grazing in other ODE models of coral reefs. In these other models, when grazing is explicitly represented as a function of grazer biomass, it is represented as a linear function (e.g., McClanahan 1995, Gribble 2003, Kramer 2007, Tsehaye and Nagelkerke 2008). However, this linear response is unlikely in reality for very large herbivore biomass, when accessibility to the algal food resource is limiting. In addition, the models in this thesis assume that the achieved predation rate of 321
piscivorous fish groups (per unit biomass of the predator) follows a Holling Type-III response, and hence is non-linear. In existing ODE models that model piscivorous fish predation explicitly, other than the ECOSIM models, the achieved predation rate is a linear function of the prey biomass (e.g., McClanahan 1995, Kramer 2007), which has the disadvantage that the predation rate is not bounded. The integrated models were mathematically analyzed and were found to support the same number of equilibria as the benthic models, provided that the grazing pressure at equilibrium does not change much with increased availability of the algal food resource to grazers – in this case, the effect of changes in the grazing pressure can be investigated using the corresponding benthic models. The first two benthic models (the Coral-Turf model, or CTm, and the CoralAlgae model, or CAm) are each amenable to a fairly complete mathematical analysis, but this was not achieved in this thesis; for all models, mathematical analyses were not carried out to their full extent in this thesis. More complete analyses than the ones presented in this thesis would involve analyzing the CTm to try to derive, and examine the biological relevance of, analytical conditions for the parameters which ensure that there is a unique equilibrium, and for all models, using non-dimensionalisation to reduce the number of parameters. In addition, there is the potential for model dynamics to be more complex than convergence to an equilibrium and this potential could be more fully investigated. For example, although the CTm cannot exhibit periodic orbits using the empirically derived parameter ranges (Chapter 4, section 4.2.4) and a condition was found for the CTm which ensured that there cannot be periodic cycles (eqn. A4.19 in Appendix A4, section A4.2.3), the possibility of periodic orbits such as limit cycles, when equilibrium is not attained, can be investigated further for the other models (to see if there are conditions for the parameters under which periodic orbits can or cannot exist). For the fish and urchin models and the integrated models, such an investigation could reveal whether predator-prey cycles for fish can occur, and if so, how likely they are to occur. However, it is not clear from existing studies that such cycles are ecologically significant in the context of coral reef degradation. More thorough analyses of the models could also involve looking in more detail at features such as the sign structures of the Jacobians. A more in-depth analysis of such features could reveal more about, for example, the possibility of multiple equilibria and the stability of equilibria for the fish and urchin models. All the models constructed were parameterised after a thorough search for available data from surveys, an online database (FishBase; Froese and Pauly 2007) and 322
numerous sources from the literature. A variety of parameterisation techniques were developed for translating the available data into estimates of ranges for the model parameters. Inevitably, assumptions had to be made during this process for simplification and in the absence of information, the possible consequences of which were discussed in Chapter 12. Because of the variability and uncertainty in model parameters, it is important to investigate appropriate ranges of these parameters, especially when the reef system considered is ‘generic’. A generic reef system is one which includes organisms, such as hard corals and herbivorous fish, and interactions, such as herbivorous grazing on algae, which are common to reefs worldwide. Such a generic system is not constrained to one particular geographic location and thus, will show variability in parameters that characterise the system. Ranges that reflect variability and uncertainty were derived for all the parameters in this thesis and model dynamics using these ranges were investigated in some detail using several different types of analyses. Thus, the analyses in this thesis are more comprehensive than existing coral reef modelling studies that do not explore parameter ranges in much detail. For example, the study of McClanahan (1995) used point estimates for all model parameters except for those representing fishing pressure, which were varied with a view to investigating optimal fish yields, and the study of McCook et al. (2001b) used point estimates apart from parameters affected by nutrients or sediments. Similarly, the ODE model of Mumby et al. (2007a) used only point estimates apart from the parameter representing algal overgrowth of live corals, for which two values were investigated (a base value and a higher value), and the grazing pressure, for which a wide range was used. Kramer (2007) used an ODE model to study the effects of changing many parameters one at a time from a baseline, but the baseline set of parameters used only point estimates.
17.1.2 Phase Shifts under Anthropogenic Stress
The analytical tractability of the benthic models meant that they could be analyzed for the potential to exhibit continuous and discontinuous phase shifts from coral- to algaldominance. A continuous shift involves equilibrium benthic covers that change smoothly as a parameter or parameters change. In contrast, a discontinuous shift involves a discontinuous change in equilibrium benthic covers, resulting in a hysteresis effect. An Immune-Inspired Algorithm (IIA; Kelsey and Timmis 2003), a type of 323
genetic algorithm, was used to search the parameter space of each benthic model for sets of parameters that gave rise to discontinuous phase shifts. In addition, comprehensive parameter sweeps were undertaken for the two simplest benthic models. These analyses provided theoretical evidence that continuous phase shifts under anthropogenic stress are common on reefs with or without macroalgae; however, they appear to be more common on reefs with macroalgae. In contrast, although discontinuous phase shifts can occur on reefs with macroalgae, they seem to be much rarer, and even when they do occur, the associated hysteresis effect is small, with the range of grazing pressures over which multiple equilibria exist being small. The high frequency with which continuous shifts seem to occur suggests that benthic covers have a high capacity for non-linear changes in response to anthropogenic (or other) stress. This is supported by sensitivity analyses for the benthic models, which showed that coral and algal covers exhibited a high degree of non-linearity with respect to parameters that are affected by fishing, nutrification or sedimentation. A sensitivity analysis of the integrated model also showed this, though the fishing parameters showed only a moderate degree of non-linearity. A manual search of the parameter spaces found synergistic effects of fishing, nutrification and sedimentation in changing benthic covers, when combinations of the stressors were applied to a pristine reef area with all parameters taking their mid-range values. This suggests that synergy is common. Darling and Côté (2008) did a meta-analysis of 112 studies on the effects of two stressors on animal mortality in marine, freshwater and terrestrial communities, and found that synergy occurred about a third of the time. Similarly, Crain et al. (2008) synthesised 171 studies on the effects of two stressors in marine and coastal communities, and found that synergy occurred about a third of the time as well. The results in this thesis are consistent with these studies because they suggest that synergy in coral reef communities could potentially occur a third of the time when two stressors act simultaneously, or perhaps more than a third of the time. However, the Sobol’ Variance Decomposition sensitivity analysis results for the simplest benthic model showed that, for reefs without macroalgae, the non-linearity is primarily due to parameters acting on their own rather than interactions between parameters. This suggests that although synergy may be common for reefs without macroalgae, the magnitude of the synergistic effect tends to be small. It also suggests that parameter interactions do not contribute greatly to the non-linearity in benthic covers. Sensitivity analyses were also performed for the fish and urchin models and these identified two processes that are important in changing fish and urchin biomasses: 324
recruitment and mortality. However, the effect of changing the size of the algal food resource could not be investigated due to limitations of the analysis techniques used. Thus, to gauge how important this could be, a sensitivity analysis of the integrated model was performed and it was found that the benthic parameters have little effect on fish biomasses on average. An increased coral mortality rate, indicative of an increase in algae, does increase urchin biomass by moderate amounts on average. These results suggest that changing the availability of the algal food resource affects only urchin biomass by moderate amounts on average. This suggests that grazing pressure at equilibrium usually changes little with the availability of algae, so that the benthos in an integrated model behaves like a benthic model with the grazing pressure fixed. Thus, phase shift results from the stand-alone benthic models can be used to give an indication of phase shift results for the integrated models, though the results for the integrated models may differ in detail, because the range of realised grazing pressures for the integrated models will be less than the entire theoretical range considered for the stand-alone benthic models. The sensitivity analyses for the fish and urchin models support the hypothesis that recruitment is the principal driving force behind coral reef fish and urchin dynamics. Recruitment limitation is a paradigm for reef fish that has been established by empirical studies (Doherty 1991, Russ 1991, Doherty and Fowler 1994, Ledlie et al. 2007), and there is also evidence of recruitment limitation for reef urchins (Karlson and Levitan 1990). However, there is some food-limitation as noted above and there is also some predation control in the models because the sensitivity analyses show that herbivorous fish biomass decreases by moderate amounts as recruitment of piscivorous fish increases. Thus, the results are consistent to some degree with a predation-limitation hypothesis (Hixon 1991). The fish and urchin groups are also recruitment-limited in the sense that each group cannot be sustained if there is no recruitment. This is a design of the models which reflects the biological reality that fish and urchins must die out if there is no supply of new recruits (Chapter 6). However, this does not mean that predation and food availability cannot limit their biomasses to some degree when there is recruitment. The results discussed above show that although the models are predominantly recruitment-limited, it is not purely so. Thus, due to the complex interplay of the ecological interactions modelled, fish and urchin biomasses can be limited to different degrees by different factors, and are not limited solely by a single factor. This is likely to be the case in reality as well. 325
The sensitivity analysis results showed that on average, decreasing the total fishing effort leads to an increase in the biomasses of all fish groups. In particular, if fishing pressure is decreased, then increased predation by more piscivorous fish is usually insufficient to stop the prey fish groups from increasing in biomass. Thus, the results are consistent with the experimental results of Mumby et al. (2006b), which showed greater biomasses of herbivorous fish and their predators within Exuma Cays Land and Sea Park (ECLSP) compared to the surrounding areas; the ECLSP had lower fishing pressure compared to the surrounding areas. The reason given was that the larger herbivorous fish escaped predation due to their size (Mumby et al. 2006b), but this effect is not incorporated into the models in this thesis because there is no size structure modelled for herbivorous fish. An increase in piscivorous fish biomass due to a decrease in fishing pressure is also consistent with the findings by Williamson et al. (2004) and Evans and Russ (2004) for Plectropomus spp. and Lutjanus carponotatus in the Great Barrier Reef (GBR), and with the study by Polunin and Roberts (1993) for lutjanids (snappers) for reefs in the Netherlands Antilles. Sensitivity analysis of the integrated model showed that fishing has little effect on benthic covers on average, although it can have moderate effects on algal cover. Mumby et al. (2007b) found that within the ECLSP, the grazing intensity of herbivorous fish was higher, and so was coral recruitment. Mumby et al. (2006b) also found that this increased grazing was correlated with a lower mean macroalgal proportional cover, which was 0.6 less than in areas outside the ECLSP. However, the increased grazing did not affect the size-frequency distribution or cover of corals (Mumby et al. 2007b). Furthermore, Newman et al. (2006) surveyed 14 reef sites across the Caribbean and found that herbivorous fish biomass was negatively correlated with macroalgal biomass, but was not correlated with coral biomass. Evans and Russ (2004) found that protected and fished zones in the GBR did not differ significantly in their benthic characteristics, including live coral cover. Thus, the model results are consistent with these findings in that decreased fishing can increase herbivorous fish biomass and this can lead to a significant decrease in macroalgal cover, with the effects on coral cover usually being negligible. Other studies have found evidence to support greater changes in coral cover due to fishing. For example, McClanahan et al. (1999) found that coral proportional cover at unprotected areas in Tanzanian coral reefs was 0.2 lower than in protected areas, although the difference was not statistically significant. For fished and unfished Kenyan reefs, McClanahan (2008) found that herbivorous fish grazing and coral cover were significantly higher for unfished reefs compared to fished reefs, with coral cover 326
being twice as high. However, the results from studies such as those by McClanahan et al. (1999) and McClanahan (2008) must be interpreted with caution because the differences in coral cover found do not necessarily imply that fishing is the cause. Thus, controlled experiments that explicitly test for causation, which minimise differences between treatments except in the factor(s) being tested, need to be performed to assess the degree to which fishing-induced trophic cascades can affect benthic covers. The results from the integrated model suggest that this degree is small on average. The overall message from these results is that coral reefs appear to be highly prone to (continuous) phase shifts from coral- to algal-dominance under anthropogenic stress at a local scale, and that this ‘instability’ is due to the inherent capacity of the reef benthos to undergo non-linear changes. This is important because the three stressors studied – fishing, nutrification and sedimentation – are projected to continue to impact reefs worldwide in the future (Bryant et al. 1998, Burke et al. 2002, Burke and Maidens 2004) and hence, the results suggest a high potential for non-linear ‘ecological surprises’ for these reefs in the future. These results are consistent with the frequent documentation of phase shifts in the literature for coral reefs under anthropogenic stress (see Chapter 1, section 1.3), as well as the more general declining trend in coral cover seen in both the western Atlantic and the Indo-Pacific (Gardner et al. 2003, Bruno and Selig 2007). Also, these results support the hypothesis that reefs usually do not recover from chronic human disturbances (Connell 1997), because they can effect a sustained alteration in environmental parameters and hence shift the underlying dynamics towards a new equilibrium. In contrast, reefs usually recover from low frequency, acute disturbances on their own (Connell 1997), probably because they tend to alter environmental parameters only temporarily. However, results from the integrated model suggest that phase shifts with increased fishing is not as common as results from the benthic models suggest, probably because fishing tends to decrease the grazing pressure by less than was assumed for the benthic models. To determine whether this is the result of urchins compensating for the decrease in herbivorous fish grazing, more simulations need to be done (see section 17.2). From a management perspective, the model results demonstrated the importance of keeping macroalgae to low levels because, not only can they alter the equilibrium landscape by allowing discontinuous phase shifts to occur, but they can also increase the likelihood and severity of continuous phase shifts. In particular, results for the most complex benthic model showed that, if the achieved grazing rate ( g M θ ) is greater than the macroalgal lateral growth rate ( rM ), then macroalgae cannot exist at equilibrium. 327
Also, g M and rM are two of the parameters that macroalgal cover is most sensitive to. Thus, to increase resilience to phase shifts, it is important to keep macroalgae to low levels by identifying and maintaining stocks of herbivorous fish that feed significantly on macroalgae (Bellwood et al. 2006a), and by managing nutrient loads from anthropogenic sources such as urban run-off, agricultural activities and aquaculture. The model results for macroalgae show the utility of stepwise refinement: the stepwise complexity of the benthic models allowed the effects of macroalgae to be unambiguously separated out and identified. Such an approach avoids model results that are too complex to interpret, due to the inclusion of extraneous detail (Fulton et al. 2003), or which are prone to over-interpretation. The relative importance of nutrification and a loss in herbivorous grazing in causing a phase shift in coral reefs has been argued at length, with some researchers arguing that loss of fish and urchin grazing is the main driver (Hughes et al. 1999, Miller et al. 1999, Jackson et al. 2001) and others arguing that increased nutrients is also a main driver (Delgado and Lapointe 1994, Lapointe 1997, Lapointe et al. 1997, Lapointe 1999, Lapointe et al. 2005a,b). Aronson and Precht (2006) also cautioned against taking a loss of herbivory as the driving force because coral mortality may be the result of factors other than algae, for example hurricanes or white-band disease. This debate on the relative importance of top-down versus bottom-up processes is recurrent in the field of ecology in general (Hunter and Price 1992). The model results from this thesis showed that for coral reefs, top-down or bottom-up processes on their own can be the main driver of a phase shift. This is because a phase shift, continuous or discontinuous, can occur with fishing, nutrification or sedimentation alone. Although several studies have shown that a phase shift occurs in reefs exposed to combinations of these three stressors, such as Smith et al. (1981) for nutrification and sedimentation and Cuet et al. (1988) for fishing and nutrification, no study has unambiguously shown that one of the three stressors alone has caused a phase shift. Koop et al. (2001) attempted to determine the effects of nutrification for pristine reefs in the GBR, but their results were ambiguous because of possible confounding factors in experimental design, such as a high natural level of nitrate in the study area (Szmant 2002). This shows the difficulty in isolating the effects of different stressors in the field and the value of modelling in this respect. Thus, the thesis results suggest that all three stressors need to be addressed simultaneously by reef management in order to maximise resilience to phase shifts. This is likely to lead to non-linear improvements in reef health for degraded reefs, because of the high potential for non-linearity of benthic covers arising from application of the 328
three stressors, some of which arises due to synergy between stressors. The model results in this thesis are also interesting in that they showed that coral reefs are able to exhibit alternative stable states (ASS) and that this requires the presence of macroalgae, probably to generate positive feedback towards an algaldominated state. This positive feedback may also be responsible more generally for the increased non-linearity seen in the presence of macroalgae. ASS in ecological systems have attracted much theoretical and empirical research over the past 40 yrs (e.g., Lewontin 1969, Holling 1973, May 1977, Schröder et al. 2005). For management, it is important to establish the existence and frequency of discontinuous shifts with ASS because such a shift exhibits hysteresis, which means that after a discontinuous collapse, the parameters have to be changed back beyond the threshold at which the collapse occurred (Beisner et al. 2003). However, for marine benthic communities dominated by sessile organisms, including coral reef benthic communities, there is no empirical evidence to support the existence of ASS (Petraitis and Dudgeon 2004, Schröder et al. 2005). In this context, the value of the results in this thesis is that they provide theoretical evidence that ASS and hence discontinuous shifts can occur on coral reefs with any one of the three stressors studied, and give an insight into the possible mechanisms that give rise to ASS. Nevertheless, the parameter sweeps suggest that ASS and hence discontinuous phase shifts are rare, which would be good news from a management perspective. However, this rarity was for the benthic model where turf and macroalgae are amalgamated into one group; the potential for discontinuous phase shifts could be greater if the benthic model where the two groups of algae are separate is considered, because the IIA results showed that this model can exhibit ASS over a far greater range of parameters when fishing, nutrification and sedimentation are considered. Thus, to reach a more definitive conclusion on the potential of reefs with macroalgae to exhibit discontinuous phase shifts, parameter sweeps need to be performed for the benthic model with separate turf algal and macroalgal dynamics (see section 17.2). These results support and builds on modelling work by Mumby et al. (2007a), which showed that discontinuous phase shifts can occur on a model Caribbean reef with a decrease in grazing. More generally, the thesis results provide evidence that human stressors can alter the competitive balance between organisms that are able to show positive feedback, in such a way as to favour ASS. Evidence for this phenomenon exists for other ecosystems, for example lakes and woodlands. In lakes, submerged vegetation and 329
phytoplankton compete for light, and the former can change the environment to favour its own growth by improving the water clarity. Nutrification can tip the competitive balance in favour of phytoplankton, thus promoting positive feedback towards a phytoplankton-dominated stable state: the less submerged vegetation there are, the worse the water clarity, which in turn favours the proliferation of more phytoplankton (Scheffer 2001). In African woodlands, trees and grass compete for space, and trees can shade out grass. Hunting can decrease the grazing pressure which prevents the tree saplings from growing large, and this could tip the competitive balance in favour of trees, especially if hunting occurs together with another factor that decreases grazing, such as a rinderpest epidemic. This would favour positive feedback to a tree-dominated stable state because the more trees there are, the more shading of grass there will be, which favours the proliferation of more trees (Scheffer 2001). However, for coral reefs, there is a need to determine the existence and potential for ASS using field experiments, to validate model results. This could be done according to the guidelines set out by Petraitis and Dudgeon (2004). The ‘characteristic length scale’ (CLS; Habeeb et al. 2005, Habeeb et al. 2007, Johnson 2009) is a system-level quantity which could be helpful in these experiments, because it can be calculated using field data and a change in the CLS could be used to determine whether dynamics indicate a shift towards an alternative stable state. As noted earlier, the results of this thesis provide theoretical evidence that there is high potential for non-linearity on reefs. Also, synergy between stressors can occur on reefs and may be common, although the synergistic effect tends to be small for reefs without macroalgae. These results are consistent with the modelling study by Mumby et al. (2007a), which showed that nutrification, if it increases the rate of coral overgrowth by algae, can increase the grazing pressure threshold at which a discontinuous phase shift occurs in response to increased fishing (as was found for the models with macroalgae in this thesis), resulting in a type of synergy. Mumby et al.’s study used the differential equations model described in section 2.1.2.4 of Chapter 2, which is parameterised for a Caribbean fore-reef. For this parameterised model, it was shown that discontinuous phase shifts could occur as the grazing pressure (proportion of the reef grazed) decreased, for different rates of coral overgrowth by algae. Here, a discontinuous phase shift is one where coral cover at equilibrium decreases discontinuously from a high value above 70% to a low value below 10% as the grazing pressure is decreased continuously, due to a saddle-node bifurcation. The way Mumby et al. identified a discontinuous phase shift was to use a bifurcation diagram that was 330
drawn using equilibrium covers derived by running simulations at different grazing pressures and with different initial conditions. In addition, the modelling results by McCook et al. (2001b) showed a non-linear decrease in coral cover as eutrophication levels increased, for a reef exposed to cyclones. This indicates synergy between the human stressor of eutrophication and cyclones. The modelling study by Kramer (2007) showed the possibility of discontinuous phase shifts with fishing, although non-linearity is hardwired into the model (see Chapter 2, section 2.1.2). Furthermore, modelling results from Mumby et al. (2006a) showed that for a spatial model of a coral reef, nonlinear increases in mean coral cover were found when the proportion of the substratum grazed increases. In general, however, there are not many modelling studies which focus on non-linearity or which demonstrate synergy between stressors. Many authors have suggested that there is synergy between stressors. For example, Hughes (1994) highlighted synergistic effects between overfishing, hurricanes and the disease-induced mass morality of Diadema antillarum that helped to precipitate phase shifts in Jamaica, Pandolfi et al. (2003) suggested that synergy between pollution and overfishing was responsible for historical coral reef decline and McClanahan et al. (2002b) suggested that synergy between disturbances may often be the cause of permanent ecological transitions on reefs. However, experimental studies that test for synergy between human stressors are uncommon. One such study by Jompa and McCook (2002b) found that increased nutrients only significantly increased growth of the macroalga Lobophora variegata when herbivory was decreased using a cage – this demonstrates that fishing of herbivorous fish and nutrification could have synergistic effects on macroalgal growth. Similarly, Smith et al. (2001) found that an interaction between herbivory and nutrients was significant in changing algal biomass. Some studies perform multi-factorial experiments to test for synergy but find that there is none for the stressors tested. For example, Belliveau and Paul (2002) studied the effects of different nutrient and grazing levels on fleshy reef macroalgal biomass and found that nutrients had no significant effect and did not interact with herbivory. Thacker et al. (2001) also tested the effects of different levels of these two factors on reef macroalgae and found that nutrients had no significant effect on total macroalgal cover or biomass, and did not interact with herbivory. Overall, there is a real need for more, and better controlled, experiments to validate model results from this thesis on non-linearity.
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17.2 Limitations and Further Research The results in this thesis need to be seen in light of the limitations of the models used. These limitations arise because of the modelling assumptions used and the scope of the models. In terms of modelling assumptions, first, and most important, the models assume that exogenous recruitment occurs at a constant rate. This is based on assumed ‘larval pools’ from which recruitment is drawn and to which the modelled reef area contribute. Other reef areas, not modelled, also recruit from and contribute to these larval pools, and it is assumed that, compared to the modelled reef area, these non-modelled reef areas dominate recruitment from and contributions to the larval pools. If this is not the case, then non-linearity in the model reef systems may be enhanced due to positive feedback between the local populations and the regional scale larval pools. For example, if there is significant feedback between the coral regional larval pool and the local spawning coral population, a sufficiently high (or low) local spawning rate may lead to increased non-linearity in the benthic dynamics. Thus, dominant contributions from one locality to a regional larval pool could increase the likelihood of ASS and discontinuous phase shifts at that locality, with knock-on effects at other, dependent localities. This would increase the sensitivity of the models to exogenous coral recruitment, which was found to be very low using the assumptions in this thesis. Wakeford et al. (2008) also found low sensitivity to coral recruitment, for their spatial model of a coral reef in the GBR. The authors found that major changes in recruitment rates, up to 10 times the mean observed recruitment rate of all coral groups, had small effects on the time trajectory of coral cover generated from the model. Second, for the benthic models, another assumption used was that grazers do not respond to changes in benthic covers. This is a fairly severe assumption which was necessary to make the models analytically tractable enough to apply the IIA and the parameter sweeps. Thus, the results from these analyses would overestimate the potential for phase shifts for situations where grazers can respond to increased algae by grazing more, leading to an increase in grazer biomasses and hence grazing pressure. Nonetheless, results from the benthic models form a baseline against which results from models for which grazing pressure can change with algal cover, such as the integrated models, can be compared. Also, the sensitivity analysis of the most complex integrated model showed that the biomass of grazers were only affected by changes in food
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availability to a small to moderate degree. This suggests that the assumption of constant grazing pressure used in the benthic models is a reasonable one. Third, due to lack of knowledge of supply-side dynamics for macroalgae (McCook 1999), the models do not include explicit macroalgal recruitment dynamics. There is evidence that macroalgal recruitment is highly localised, such that recruits establish themselves close to existing macroalgal patches (Diaz-Pulido and McCook 2004, Mumby 2006). If this is the case, then these recruitment dynamics could simply be modelled as an increase in the macroalgal lateral growth rate, although to quantify this effect, further experiments need to be performed. This enhanced lateral growth effect would increase positive feedback due to macroalgae and could increase nonlinearity, as well as the likelihood of ASS and associated discontinuous shifts. Fourth, invertivorous fish were not included in the models and were assumed to exert a constant predation pressure on the urchin group modelled. The possible effects of this assumption are that urchin biomass may be overestimated because they only suffer from a constant mortality rate and piscivorous fish biomass may be underestimated because they do not feed on invertivorous fish in the models. Planktivorous fish were not modelled because there is little evidence that they have a large role in coral-algal dynamics, but they may be important as prey for piscivorous fish. Thus, their exclusion could lead to the piscivorous fish biomass being underestimated. A useful extension to this thesis would be to add a dynamic invertivorous fish group to the models, and to use these extended models to investigate the effects on the other groups modelled, in particular sea urchins and piscivorous fish. Similarly, planktivorous fish dynamics could be added and the effects of doing so investigated. Fifth, the effects of sedimentation on algae are not modelled, due to a lack of data. Sedimentation can potentially decrease turf algal and macroalgal growth by smothering and/or reduction of light (McCook 1999). A reduction in turf algal and/or macroalgal growth due to sedimentation can be modelled and the effects on model dynamics investigated. However, Stamski and Field (2006) observed that areas on a Haiwaiian reef flat where levels of terrigenous sediment trapped by macroalgae were high also had greater macroalgal cover, suggesting that sediments could benefit macroalgal cover in some way, perhaps through the provision of sediment-associated nutrients (McCook 1999); theoretically, this may also be the case for turf algae. Thus, it is not clear that sediments have a negative effect on algae. Sixth, since the models are all deterministic and non-spatial, it is assumed that 333
local-scale effects arising from spatial heterogeneity are negligible, and that short timescale stochastic variations in parameters due to fluctuations in the environment and recruitment (Doherty 1991) are insignificant. Thus, the dynamics are smoother than would be expected in reality. Nevertheless, the models could be spatialised at a local level by splitting the benthos into local cells with each cell representing one type of cover. Then the fish and urchins can be modelled as mobile units moving among these cells. Stochasticity can be added to the models by assuming that some or all of the parameters fluctuate randomly within certain ranges. It could also be added by splitting the benthos and biomasses into discrete blocks and then having these blocks randomly partake in interactions with probabilities defined by a set of parameters. In either case, outputs for a single run would be distributions of the variables in the models. Results from spatialised and/or stochasticised models can then be compared to those obtained in this thesis. This approach would take the stepwise refinement approach further than has been done in this thesis. Seventh, the model does not have any size structure for corals. The effect of this is that different size-dependent parameters cannot be used to represent the demographic rates for coral colonies of different sizes. Coral size structure can be added to the models by splitting the coral group into several different size groups, although this results in a loss of analytical tractability and multiplies the (already large) number of parameters to estimate. An initial attempt at adding size structure to corals has been made by Knight (2007), although the resulting model has not been parameterised. Eighth, apart from coral-algal phase shifts, there are also other types of shifts from coral-dominance to dominance by other organisms such as soft corals, sponges and corallomorphians (Norstrom et al. 2009). Although these other phase shifts appear to be relatively rare, with many reports being anecdotal (Norstrom et al. 2009), the models in this thesis could be extended by adding one or more of these groups of organisms and then investigating the potential of reefs to undergo other types of shifts. Addition of these groups is likely to make the models too analytically intractable to apply the IIA or parameter sweeps, but sensitivity analyses could be performed. These other types of shifts could be important to management because added nutrients and sediments from humans may facilitate such shifts (Norstrom et al. 2009). Ninth, the three stressors investigated – fishing, nutrification and sedimentation – were assumed to act independently. However, these stressors could be significantly interdependent due to, for example, fish producing nutrients and sediments or the adsorption of nutrients by sediments – see Chapter 13, section 13.1.2.4. Thus, further 334
work could investigate whether including interdependencies for the effects of the three stressors affects results such as the likelihood of a phase shift or the degree of synergy between the three stressors in changing benthic covers. Interdependencies could be added by, for example, expressing the increase in algal growth due to nutrification as a positive function of the magnitude of the effects of sedimentation. A challenge would be to quantify the interdependencies for such an investigation. Finally, several assumptions were made in the parameterisation of the models. This was necessary to apply methods that translate available data into estimates for the model parameters. The implications of these assumptions have already been discussed in Chapter 12. The models in this thesis operate at a local level and focus on the biophysical aspects of coral reefs. This imposes limitations because regional dynamics that can interact with the local dynamics are not modelled (except via larval pool assumptions) and the dynamics of socioeconomic factors such as the number of fishers are not modelled (except passively through fixed fishing pressure parameters). The inclusion of regional dynamics and socioeconomics is far from trivial and is currently being investigated by other Modelling and Decision Support Working Group (MDSWG; see Chapter 3, section 3.2) researchers – see Melbourne-Thomas et al. (2007) for an introduction to a regional scale model which uses the local models in this thesis as building blocks. In addition to model elaboration, there is scope for further research using the models and analyses as they are presented in this thesis. A parameter sweep of the most complex benthic model (the Coral-Turf-Macroalgae model, or CTMm), which would give the likelihood of different types of coral-algal phase shifts with changing grazing pressure, could be performed to determine if parameter sweep results in this thesis still hold when turf algae and macroalgae are allowed to vary dynamically, although this would be computationally much more expensive. The Elementary Effects sensitivity analysis method could be applied to the integrated models for parameters that were not tested. This would shed more light on what other parameters the integrated models are sensitive to, if any, and hence give a better indication of the potential for trophic cascades from fish and urchins down to the benthos, and feedback from the benthos to the fish and urchins. Furthermore, the sensitivity of the integrated models without the urchins group can be tested to see if fishing has a greater effect on benthic covers, due to a lack of urchins to compensate for decreases in herbivorous fish grazing. Since corals provide a habitat for fish to live in and to hide from predators, 335
especially for juvenile fish and coral-dwellers (Jones et al. 2004, Bellwood et al. 2006b, Garpe et al. 2006, Wilson et al. 2006), it would also be interesting to include a positive relationship between coral cover and fish recruitment and/or a negative relationship between coral cover and fish mortality, and investigate the sensitivity of the models with these new interactions. The effects of stressors apart from fishing, nutrification and sedimentation on coral reef dynamics can also be investigated, provided that the effects of these stressors on model parameters and/or variables can be parameterised. Investigating other stressors is important because the three stressors studied in this thesis are far from being the only significant causative factors behind reef degradation, as discussed in section 13.1.2.4. For example, tropical storms can be modelled as a sharp decrease in coral and algal covers, whereas bleaching might be modelled as temporarily increasing coral mortality and temporarily decreasing coral growth and recruitment (Chapter 1, section 1.3.4); ocean acidification might be modelled as a chronic stressor which decreases the coral growth rate (Chapter 1, section 1.3.4; De’ath et al. 2009). Destructive fishing using sodium cyanide or dynamite, common in Southeast Asia (Burke et al. 2002), can also be modelled as a chronic stressor which increases coral mortality as well as fish mortality. Phenomena such as the frequency of hurricanes a generic reef can take before coral cover fails to recover between hurricanes could be investigated as well. In this thesis, eclectic data have been used to parameterise ‘generic’ models of a local coral reef system, which take parameter values in broad ranges that include values from reefs around the world. Hence, they are useful for investigating the general processes and trends of coral reef degradation and recovery. However, the focus on a generic reef imposes the limitation that the modelled reefs are not usually representative of a particular reef at a specific location. A reef at a specific location will likely take parameter values within narrower ranges than for the generic models and the effects of fishing, nutrification and sedimentation may also fall within narrower ranges. In addition, some specific reefs may take extreme parameter values that lie outside the generic ranges; for example, this may occur because a specific reef has a coral community composed of fast-growing species that give a growth rate above the upper limit of the generic range, or an algal community composed of slow-growing species and/or which is exposed to unfavourable conditions for algal growth, giving turf algal and macroalgal growth rates below the lower limits of the generic ranges. Thus, to investigate the potential for degradation at specific reefs, which is required for local management, it is best to parameterise the models in this thesis using data from specific 336
localities. For example, this has been done for Banco Chinchorro on the Mesoamerican barrier reef and for local reefs in the Linguyan Gulf/South China Sea area in the Philippines. Parameters which cannot be estimated using data from a specific location can be identified and the missing data collected through more experiments or surveys. Site-specialised models can then be analyzed using the methods described in this thesis and results compared to the generic results, which serve as a useful baseline. They can also be used for investigation of scenarios at a specific location, and scenario-testing results then need to be validated using survey data from that location; this requires monitoring programs to carry out surveys at regular intervals into the future. To be of most use in the field, the models in this thesis should be modifiable as necessary by local researchers, and a VENSIM implementation of these models, which has a userfriendly interface, has been programmed by other MDSWG members to facilitate this task. Hypotheses generated by the results in this thesis can also be tested for other ecosystems. For example, multiple human stressors often affect not only tropical coastal ecosystems such as coral reefs but temperate coastal ecosystems (Lotze et al. 2006) and other marine ecosystems (Halpern et al. 2008), and it would be interesting to see if models of these other systems are able to show the same degree of synergy between stressors as found in this thesis. Furthermore, the techniques used in this thesis can be applied to other ecological models. For example, the parameter space search technique using the IIA could be applied to other models that are sufficiently analytically tractable, to find sub-regions of parameter space that exhibit ASS.
17.3 Conclusion Coral reefs are diverse, complex ecosystems that are subject to a multitude of natural and human stressors at multiple scales. By virtue of this, the causes underlying reef dynamics are hard to identify and manage, and as with most ecosystems it is often necessary to simplify the system in order find interpretable results that aid in management decisions (Dayton 2003). In this thesis, this simplification is achieved through modelling. Using a simplified model does not mean detachment from reality because a good model is firmly based on the scientific knowledge we have gained of this reality. Rather, the model extracts the principal features of reality that appear to be relevant to the practical problems at hand. Models are also useful in that they can be 337
used to assess long-term trends and thus generate a focus on long-term management issues in addition to short-term ones. This helps to avoid an exclusively short-term outlook, which characterises most coral reef research (Hughes et al. 2003) and which contributes to a shifting baseline of increasingly lower management expectations and standards (Jackson 1997). In this spirit, the models in this thesis were constructed as scientific and management tools to investigate how reef areas at a local scale respond to anthropogenic stress, to assess the resilience of reefs through a consideration of key processes such as recruitment, to generate hypotheses of reef dynamics and to make long-term predictions on reef dynamics at particular locations. The aim is to use these models as part of a suite of models which represent coral reefs at multiple scales, and which can be used to provide holistic answers to scientific and management questions. Model results alone will not result in the improved reef management that is needed to slow down, stop or reverse global coral reef decline. They must be tested and validated using scientific experiments and refined repeatedly. A useful plan in this regard is to promote adaptive management, in which management programs double up as experiments to test model predictions (Sale et al. 2005, Sale 2008); this can save valuable time and money, and is being employed to manage the GBR (McCook et al. 2007). However, despite the substantial potential value of coral reef modelling in practical applications, and the increase in coral reef modelling studies over the past decade, there is a lack of studies that test modelling results experimentally, and that use models as a basis for management action. Such studies are needed to demonstrate to reef stakeholders clearly how modelling can be an integral part of management and policy-making that yields tangible results. It is important to see coral reef models as part of an integrated, interdisciplinary strategy to ensure sustainability of coral reefs. Such a strategy must take into account that reefs are social-ecological systems, with human societies having the potential to be the main drivers of reef health (Steneck 2009). This means that effective management involves not only managing reef organisms, but also tourists and people whose livelihoods depend on reefs. In addition, the entities through which change is enacted include businesses and governments, which can fundamentally alter the economics and culture within which reef managers and users operate. These entities must also be taken into consideration. An integrated strategy must tackle the root causes of reef decline such as human population growth, which are high for many countries with bordering reefs, and a growing sense of detachment from natural resources amongst people living in urban 338
areas (Birkeland 1997c, McManus 1997, Dayton 2003, Birkeland 2004). Jameson (2008) draws attention to the difficulty of attaining environmental sustainability on the large scales required for coral reefs because of competing interests that do not favour sustainability, and asks whether the human species has the genetic ability to achieve this sustainability. Since the thoughts and actions of people are formed not only by genes, but by their environment and the interaction between this environment and their genes, a parallel question is whether the societies people live in, which include the economic rules and cultural values people live by, have the adaptability to achieve this large scale sustainability. As for the related issue of human population growth, this question is very much about human values, or morality, as well as survival – see Hardin (1968) for the relevance of morality to the problem of human population growth in the context of the ‘tragedy of the commons’. Also, this is a more practical question than the one asked by Jameson (2008), because the genetic evolution of the human species towards a sustainable interaction with its environment is not feasible on a relevant time scale. Only if human societies can have and realise the capability for large scale sustainability will the practical potential of coral reef modelling results be fully realised as part of an integrated management strategy. These results include those from this thesis, because even though this thesis has focused on human stressors at a local scale, the severity of stressors acting at a local scale depend crucially on management and policies that operate at larger scales.
339
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