Calibration of the individual-based model MORPH for mussel-eating of the Exe Estuary. John Goss-Custard

2018

ExeMorph is the individual-based model for the Exe estuary oystercatchers that eat mussels and use upshore areas and fields for supplementary feeding when they cannot meet their requirements from mussels alone. The first version was that published by Stillman et al (2000). This version was set-up – or ‘calibrated’ - to predict the observed within-winter mortality rate of adults for the years 1976-80 when there was an estuary-wide density of oystercatchers on the mussel beds of 18/ha. It correctly predicted the increased mortality rate amongst adults that accompanied the increase in population density that occurred over the years 1980 to 1991. Over the two periods when the mortality rate was measured, the density of oystercatchers increased to 25/ha in 1980-83 and 31 in 1983-91. The three phases of population size are referred to here as the Calibration (Cal), Validation 1 (Val 1) and Validation 2 (Val 2) years. Although it predicted the increased adult mortality rate impressively in Vals 1 and 2, this model did not accurately predict the mortality rates of the sub-groups of birds: i.e. immature hammerers, immature stabbers, adult hammerers and adult stabbers. This was always a worry. Also unsatisfactory was the need to include the ‘Aggregation factor’ (AF) in order to calibrate the model successfully. The AF was included in ExeMorph to capture the widespread tendency of oystercatchers to aggregate in particular parts of each mussel bed, the standard habitat patch in ExeMorph. For example, with an AF of 8 and 100 birds on a mussel bed of 10ha, the density of oystercatchers in ExeMorph would be elevated by AF from 10 to 80 oystercatchers per ha. This frequently increased bird densities in the model well above the interference threshold density of about 100 oystercatchers/ha, and thus increased the difficulties the birds had in obtaining all their food requirements from (parasite-free) mussels alone. Our major concern about using the AF was that the degree of aggregation of oystercatchers probably varied a lot between mussel bed, through the exposure period and probably with population size, yet there was available only a very limited number of measurements that had only been made over low tide. The AF was retained for over a decade, however, because it acted as a calibration factor and without it no oystercatchers starved. Additionally, the model with AF included was very successful at predicting adult mortality in Val 1 and Val 2. Bowger (2016) devised an alternative parameter to capture the tendency of oystercatchers to aggregate in limited parts of many mussel beds which she called ‘Regulated Density (RD)’. The basic idea was that birds tend to aggregate - either by mutual attraction or by being drawn to the best localities within a patch – but would spread out if interference started significantly to reduce their intake rate. This ‘attraction-avoidance’ concept was attractive as it coincided with my own early findings in shorebirds (Goss-Custard 1970) and with recent work on waders in the Wadden Sea by Folmer, Olff & Piersma (2012). Research on the Exe suggested that, for the mussel-eating oystercatchers there, the value of RD is 45/ha: this is the maximum density typically occurring when the population size is high and there is a large area of mussels over which the birds can spread themselves if they wanted to. In the model, RD serves to spread the birds out more widely over the available area of mussel beds and, in Bowgen’s (2016) work in Poole Harbour, its inclusion improved the match between the predicted and observed distribution of birds at low tide.

A version of ExeMorph was developed in which AF was replaced by RD. But no oystercatchers starved: without AF increasing their foraging density and suppressing their intake rate, life was far too easy for them. However, RD had been measured at low tide when large areas of mussel beds were accessible whereas, in nature, RD would probably be much higher when oystercatcher densities were high, as at the beginning and end of the exposure period. I therefore gradually increased the value of RD in ExeMorph, and achieved the desired effect of increasing the mortality rate above 0% as soon as RD exceeded the interference threshold density. RD looked like being a promising calibration factor. But it wasn’t! Most deaths occurred in hammerers which are more interference-prone than stabbers yet have a lower mortality rate in nature! A number of very time-consuming attempts were made to find a way round this, but without success. For example, the 10 mussel bed patches in ExeMorph were sub-divided into 30 sub-patches, each with its own shore-related mussel fleshcontent and size-distribution but this had no effect on the mortality rate. Eventually it was realised that, without AF in ExeMorph, the only way to make birds starve was to change the values of the parameters that are specific to age-class and feeding-method groups. Although it is possible that stabbers and hammerers, adults and immatures differ in their postconsumption efficiency at assimilating energy (for example, through different gut parasite infestation loads), there was no reason to believe it. The only option seemed to be to reduce the gross intake rates or the intensity of interference and by different amounts in the different groups of oystercatchers. Interference is represented by two coefficients: the threshold density of oystercatchers at which interference starts to depress the intake rate and the subsequent rate at which intake rate declines as competitor density increases. The number of combinations which could be used across four ageclass and feeding-method groups seemed endless. Accordingly, I opted for the ‘simpler’ option of varying the ‘efficiency’ with which the average bird in one oystercatcher group consumed food. This is how it was done. A functional response in ExeMorph predicts the interference-free intake rate (IFIR) in mgAFDM/s of the average oystercatcher feeding in a given place at a given time from the numerical density, flesh-content and size-distribution of the mussels present. The predicted average intake rate is then multiplied in ExeMorph by an individual’s ‘foraging efficiency’ (FE), the value for which is drawn at random from a normal distribution with a standard deviation of 0.125 and a mean of 1 (Stillman et al. 2000). An individual’s FE remains constant throughout the winter and, in previous models, only applied to birds when feeding on mussels: i.e. all individuals had the same intake rate upshore and in the fields. To calibrate the model using average IFIR, a ‘Calibration Coefficient (CC)’ that could differ between immatures and adults and between hammerers and stabbers was added to the model. This was used to vary IFIR of the average bird in a given place at a given time, as predicted by the functional response. The predicted average IFIR on mussels in a given place at a given time was multiplied by the current value of the CC for the group of birds in question; for example, let’s say the functional response predicted an average intake rate of 500mgAFDM/s, and the CC for the immature stabbers was 0.6, the average immature stabber would get (0.6x500) 300mgAFDM/s. If an individual immature stabber’s FE was, say, 1.10, that individual would have an IFIR at that time and place of 1.1x300=330mgAFDM/s. Initially, I applied the CC to mussel-eating only. But, on the Exe, in normal winter weather, birds can survive by feeding only upshore and in the fields at the average rate. So the CC was made to apply

across all diets; i.e. as was done for mussel-eating, the intake rate of the average bird on the upshore flats and in fields was also multiplied by CC and also by an individual’s FE. This meant that a bird that was inefficient when feeding on mussels was equally inefficient when feeding on upshore prey (eg. clams and cockles) and earthworms in fields: there is no evidence that either is the case but at least the idea has the merit of being testable. And it had the desired outcome: now birds starved. As a result of all this, I was able to calibrate ExeMorph so that it replicated the observed and different winter mortality rates of immature hammerers, immature stabbers, adult hammerers and adult stabbers, as measured in Exe oystercatchers by Durell (2001) and Durell et al. (2007). But when I ran the validation tests at the higher bird densities, the predicted mortality rates were too high! Then I remembered that Sitters (2000) had very good evidence that interference might be much reduced, even absent, at night in mussel-eating oystercatchers. Accordingly, I reduced the intensity of night-time interference by raising the interference threshold and thus reducing the number of time-steps in which interference would have occurred. The predicted mortality duly fell in the validation years by a substantial amount – but not enough. So I then removed interference at night completely and the predicted and observed mortality rates, averaged over Val 1 and Val 2, coincided almost exactly. Across the four oystercatcher groups, CC varied between 0.53 and 0.71. Both represent a substantial reduction in the time-consuming estimates that I had obtained for IFIR over several tough fieldwork winters! I was not happy with this procedure for calibrating the model, because there was good evidence that the functional response equation used in ExeMorph does predict actual intake rates pretty well. However, I could think of no alternative that could be applied to the four sub-groups of birds separately. As the key feature of MORPH is individual variation, my Dutch colleague Kes Rappoldt suggested that we should use the coefficient of variation (CV) of the individual variation in foraging efficiency (FE) to calibrate the model. The idea was that, with larger CVs, there would be more inept birds at the low end of the distribution of FE and so more would starve. I explored this possibility over three months by varying the CV for each oystercatcher group separately: attempts to use a single value across all groups failed. Individuals had the same FE when feeding upshore and in fields as they did when feeding on mussels. Depending on the group, the CV had to be increased across all diets from its field-estimated value of 0.125 to as much as 0.3625. So, just as there had been when using the average IFIR as the calibrator, there was a substantial departure from the estimates made in the field. [I take this to mean that we still have a lot to learn about the processes and the magnitudes of parameter values that, in nature, determine how many of each group of oystercatchers starve. But this is one of the advantages of quantitative modelling: the magnitude of the calibration coefficients provide a measure of how much more we have yet to learn!] Sadly, however, using the CV as the calibration device gave predictions for the validation years that were less good than those I had obtained when using IFIR as the calibrator. This was disappointing as using CV is such a neat idea and would have avoided the need to change well-measured estimates of IFIR by as much as it proved necessary to do. The graphs etc which follow compare the two main methods – IFIR and CV – that I have explored as calibrators of ExeMorph. Reluctantl;y, I have concluded that I should continue using average IFIR across all diets for calibrating the model on the grounds that it is currently the best option available. However, I will still be open to alternative ideas, of course!!

Comparison of using average IFIR and CV as calibrators VALUES OF THE COEFFICIENTS Average IFIR – applied to all diets Fraction of IFIR predicted by the functional response equation: Immature hammerer

0.657

Immature stabbers

0.528

Adult hammerers

0.713

Adult stabbers

0.570

Coefficient of variation – applied to all diets Value of CV (baseline value was 0.125) Immature hammerer

0.2565

Immature stabbers

0.3625

Adult hammerers

0.2055

Adult stabbers

0.2635

OBSERVED AND PREDICTED MORTALITY RATES IN CALIBRATION YEARS Average IFIR – applied to all diets

Coefficient of variation

OBSERVED AND PREDICTED MORTALITY RATES IN THE VALIDATION YEARs The small dots show the predicted adult mortality rates (hammerers and stabbers combined) for Cal, Val 1 and Val 2, respectively going from left to right. The large dots show the average values for Val 1 and Val 2. These coincide very much more closely with the observed values when ExeMorph was calibrated using average IFIR (the blue dot) than when calibration was done by varying the CV (red dots).

References

Bowger, K.M. (2016). Predicting the effect of environmental change on wading birds: insights from individual-based models. Ph.D. thesis, Bournemouth University. Durell, S.E.A. Le V. dit (2007). Differential survival in adult Eurasian oystercatchers Haematopus ostralegus. Journal of Avian Biology, 38, 530-535. Durell, S.E.A. Le V. dit, Goss-Custard, J.D., Caldow, R.W.G. Malcolm, H.M. & Osborn, D. (2001). Sex, diet and feeding-method related differences in body condition in the Oystercatcher Haematopus ostralegus. Ibis, 143, 107-119 . Folmer, E.O., Olff, H. & Piersma, T. (2012). The spatial distribution of flocking foragers: disentangling the effects of food availability, interference and conspecific attraction by means of spatial autoregressive modelling. Oikos, 121, 551-561. Goss-Custard, J.D. (1970). Feeding dispersion in some overwintering wading birds. In: Social Behaviour in Birds and Mammals. (Ed. by J.H. Crook), pp 3-35. Academic Press, London. Sitters, H.P. (2000). The role of night-feeding in shorebirds in an estuarine environment with specific reference to mussel-feeding oystercatchers. D.Phil., Oxford University. Stillman, R.A., Goss-Custard, J.D., West, A.D., Durell, S.E.A. le V., Caldow, R.W.G., McGrorty, S. & Clarke, R.T. (2000). Predicting to novel environments: tests and sensitivity of a behaviour-based population model. Journal of applied Ecology, 37, 564-588.