ORIGINAL ARTICLE doi:10.1111/j.1558-5646.2007.00054.x

HOW BRIGHT AND HOW NASTY: EXPLAINING DIVERSITY IN WARNING SIGNAL STRENGTH Michael P. Speed1,2 and Graeme D. Ruxton3,4 1 School

of Biological Sciences, University of Liverpool, Liverpool L69 7ZB, United Kingdom

2 E-mail: 3 Division

[email protected]

of Environmental and Evolutionary Biology, Institute of Biomedical and Life Sciences, Graham Kerr Building,

University of Glasgow, Glasgow G12 8QQ, United Kingdom 4 E-mail:

[email protected]

Received May 17, 2006 Accepted November 20, 2006 The conspicuous displays that warn predators of defenses carried by potential prey have been of interest to evolutionary biologists from the time of Wallace and Darwin to the present day. Although most studies implicitly assume that these “aposematic” warning signals simply indicate the presence of some repellent defense such as a toxin, it has been speculated that the intensity of the signal might reliably indicate the strength of defense so that, for example, the nastiest prey might “shout loudest” about their unprofitability. Recent phylogenetic and empirical studies of Dendrobatid frogs provide contradictory views, in one instance showing a positive correlation between toxin levels and conspicuousness, in another showing a breakdown of this relationship. In this paper we present an optimization model, which can potentially account for these divergent results. Our model locates the optimal values of defensive traits that are influenced by a range of costs and benefits. We show that optimal aposematic conspicuousness can be positively correlated with optimal prey toxicity, especially where population sizes and season lengths vary between species. In other cases, optimal aposematic conspicuousness may be negatively correlated with toxicity; this is especially the case when the marginal costs of aposematic displays vary between members of different populations. Finally, when displays incur no allocation costs there may be no single optimum value for aposematic conspicuousness, rather a large array of alternative forms of a display may have equal fitness.

KEY WORDS:

Aposematism, coevolution, predation, secondary defense, signaling.

In prey species, aposematism is the combination of repellent antipredator defenses (such as toxins, stings, and spines) with some advertisement of this secondary defense that is often visually conspicuous to predators. Although the concept dates back to discussions between Wallace (1867) and Darwin (1867), aposematism remains an important and contentious area of evolutionary research. Over recent years the focus of much research has been placed on evaluating the early evolution of aposematic displays, because novel displays may be quickly snuffed out due to their rarity and unfamiliarity to predators (see discussions in Sword  C

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2002; Ruxton et al. 2004; Despland and Simpson 2005b; Mappes et al. 2005; Marples et al. 2005). Although there are now many potential solutions to this problem of initial origins (Ruxton et al. 2004), wider theoretical questions about the optimization of aposematic defenses have been largely ignored. This is true despite the fact that the seminal theoretical paper in the field (by Leimar et al. 1986) explicitly outlined an evolutionarily stable strategy (ESS) account of optimal aposematic conspicuousness (see development of this model in Broom et al. 2006). The key work by Endler and Mappes (2004)

2007 The Author(s) C 2007 The Society for the Study of Evolution Journal compilation 

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which examined the role of variation in predator tolerances for aposematic prey is one of the few recent theoretical treatments to consider variation in aposematism in detail. Variation in aposematic display matters because the possession of aposematic traits may be a crucial factor in determining the position of an animal’s niche within its wider ecology, affecting the evolution of other important characteristics, such as reproductive capacity and other life-history traits (Srygley and Chai 1990). One result of this overemphasis on the initial evolution of aposematic display is that when new ecologically relevant data are published, a theoretical framework within which such results can be interpreted and explained does not exist. Thus, for example, in a broadly based phylogenetically controlled study, Summers and Clough (2001) recently demonstrated a positive correlation between toxicity and aposematic conspicuousness in Dendrobatid frogs. This appears to support a suggestion that the brightness of a warning signal could be selected as a “wasteful, extravagant” handicap signal, in which the nastiest prey are those that are able to “shout loudest” about their defenses (Zahavi 1991; Guilford and Dawkins 1993). On a related theme, general models of the initial evolution of aposematic and mimetic conspicuousness point out similarly that because conspicuousness is costly in terms of attracting predator attention, it may be that only well protected prey can afford this kind of advertisement (Charlesworth and Charlesworth 1975; Sherratt 2002; Sherratt and Beatty 2003; Sherratt and Franks 2005). In contrast, when examining three selected Dendrobatid species Darst et al. (2006) recently found the converse to be the case: brighter species had the least toxic secondary defenses, so that those with the weakest defenses could be “shouting louder” than those with stronger defenses (cf. Leimar et al. 1986; Speed and Ruxton 2005b). At present there is no coherent theoretical treatment that can accommodate both patterns of result. Furthermore, it has been repeatedly argued (and recently demonstrated) that some larval insect prey combine both aposematic conspicuousness with some component of crypsis, so that they are cryptic at a distance, and readily apparent from close quarters (Tullberg et al. 2005). As it stands at the time of writing, only the work of Endler and Mappes (2004) has attempted to explain the general circumstances in which low levels of aposematic conspicuousness are indeed optimal. Endler and Mappes (2004) considered predator variation in the toleration of prey defenses as a cause of variation in signaling phenotypes. Here we take an alternative and complementary position: that the economics of defenses causes variation in aposematic traits. In this paper we identify a general framework within which the optimal values of both aposematic displays and their associated secondary defenses can be specified. Our model is a deterministic development of the stochastic individual-based models we recently used to explore the initial evolution of aposematism (Speed

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and Ruxton 2005b). The deterministic methodology is an adaptation of that recently described by Puurtinen and Kaitala (2006) and derived from that of Servedio (2000) and Engen et al. (1986). The general framework describes predator–prey interactions with a set of probabilistic statements. The focus of the model is economic, in the sense that we specify a set of different sources of fitness costs and benefits. We use this model to systematically vary the conditions under which aposematic traits can evolve, taking into account variation in threats from predation, in marginal costs of aposematic traits, and variation in the extent to which toxins (and other defenses) contribute to individual survival during an attack. We show that the novel application of an economic-optimization framework to aposematism theory generates important insights into the informational content of aposematic displays. In some cases the model predicts no correlation at all between secondary defense and conspicuousness: in others there are positive or negative correlations between optimized defenses.

Methods We first provide an outline of our models before mathematical details are spelled out. OUTLINE MODEL

This model is a development of that described by Speed and Ruxton (2005b). We describe a focal prey population (of size N t=0 , at the start of a foraging season) that can evolve repellent secondary defenses (described as a parameter, D, taking values greater than zero) such as toxins, and bright, conspicuous aposematic warning displays (described as a parameter A). Other nonfocal prey in the locality are assumed to be both cryptic in appearance and edible to predators. Encounter rates are determined by the conspicuousness of the focal prey (C) and their abundance. Conspicuousness of an individual, C i , is positively related to the value of its display, A i . We assume that when a prey is encountered, the probability of attack declines as the conspicuousness of the individual prey increases. This effect of conspicuousness on attack decisions could be considered to reflect growing caution of predators as aposematic traits increase in brightness. Alternatively, it could be thought to represent the possibility that a member of the focal prey is less likely to be confused with the nonfocal edible prey if its conspicuous coloration makes it bright and distinctive (Wallace 1867; Gamberale-Stille 2001; Lynn 2005; Lynn et al. 2005; Pie 2005). Furthermore, we assume that predators also reduce attack probabilities on individual prey as the average aversiveness (D∗ ) of the prey population increases (reflecting likely wariness by predators based on previous experiences with this prey type). If an attack takes place, the prey survives with a probability positively related to both the wariness provoked by the prey’s aposematic

EXPLAINING DIVERSITY IN WARNING SIGNAL STRENGTH

display (which increases with A i ) and its repellent secondary defenses (D i ). Aposematic traits can incur allocation costs, such that the relative fecundity value for each survivor decreases with raised values of its display, A i , and of its secondary defense, D i . Furthermore we modified fecundity costs for different prey systems by the inclusion of fecundity coefficients for D and A, respectively,  and  . High values of  and  imply high marginal fecundity costs to secondary defense and display. Within a generation, predators interact with the prey for 750 time intervals (unless stated otherwise), after which fitness is calculated as the product of proportionate survival and fecundity. For each fitness calculation, we assume that the population is monomorphic for values of A and D. We calculate fitness for a very large range of A and D values for systematic variation in key parameters, such as costs of display; values of A and D, which generate the highest fitness values, are taken as the optimal states for these traits.

DETAILS OF THE MODEL

We describe here the functions used to define components of predator–prey interaction and the mechanisms by which season duration and fitness are calculated.

Conspicuousness and encounter rate To describe encounter rates, we define the conspicuousness of an individual, C i , as the rate at which individual i encounters (and is detected by) the predator. We assume a nonzero bound to C i , such that maximally cryptic prey can be detected by the predator, albeit at a low rate, for example, C min = 0.001 (so that a specific prey is seen on average every 1000 time units, compared to a typical generation time of 750 units). More generally, C i will be a function of the conspicuousness of the colorful aposematic display, A i . We assume a negative exponential relationship C i = C min + (1 − exp (−α.A i )), with α = 0.5C min in all cases, such that conspicuousness, C i, increases with increased value of the aposematic signaling trait, A i . This functional form is in our view reasonable given that in psychophysics, perceived intensity is generally described using a power or specifically exponential function, rather than, for example, a sigmoidally shaped function (Smelser and Baltes 2001). Furthermore, though the function is nonlinear, the relationship between C i and A i is a near-linear, positive correlation over the typical range of A values used in this paper. Since conspicuousness is defined in terms of encounter rate, we can state that a predator encounters a prey every CN t time intervals (where N t is the average number of the focal prey in existence at time t) and we assume that attacking a prey takes up one time interval.

Attack decision Probability of attack on the detected individual i declines with increases in the value of C i /C min . We assume that predators have some stable knowledge state about the average level of secondary defense in the population (D∗ ) and hence probability of attack given detection declines with increases in the value of D∗ . In the case of recognition, a sigmoidal form is appropriate for the signal detection-like mechanism that we imply. In the case of knowledge of a prey’s unprofitability, we again assume a sigmoidal relationship between the prey’s investment in defense and its deterrence of attacks, as is likely for dosage effects in predator–prey systems (Mallet 1999). Furthermore, depending on the type of predator present, the minimum level of attack probability (Att min ) can vary from zero upwards. The conditional probability of attack given detection is now defined as P(attack given detection) = Att min + (1 − Att min ) × (1 − exp(− exp(−(cCi /Cmin )z + β))) × (1 − exp(− exp(−D ∗ z + β))),

(1)

where c is a scaling parameter with value 0.5, which prevents premature saturation of the function (increasing the value of c increases the marginal effect of increases of C i on the probability of attack given detection). Note that z (value 0.632) and β (value 2.5) are parameters that affect the shape and value of the y-axis intersection of the sigmoid curve (values taken here give similarly parameterized curves to those used in Speed and Ruxton 2005). Increases in z would move the inflexion point to lower values of A and D, generally benefiting the prey, by diminishing probability of attack given detection for set values of aposematic defenses. Conversely increasing the value of β would shift the inflexion point in the opposite direction to the prey’s detriment. Survival from attack There is empirical information that both the conspicuous appearance of an aposematic prey and the strength of secondary defenses that it presents can affect survival from attacks (Wiklund and J¨arvi 1982; Sill´en-Tullberg 1985; Skelhorn and Rowe 2006). Indeed, in the best systematic investigation of its kind Marples et al. (1994) demonstrated that multiple components of individual aposematic displays (including sight and taste of toxins) interacted nonadditively to enhance prey protection. We therefore assume that survival following an attack is proportionate to the product of A and D, again taking a sigmoidal form, with minimum value Kill min, such that P(death from attack) = Killmin + (1 − Killmin ) ×(1 − exp(−exp(−Ai Di z + β)).

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Although we show results using equation (2) in this paper, we also ran the model assuming an additive relationship of A and D with respect to changes in probability of death for selected scenarios (i.e., P(death from attack) = Kill min + (1 − Kill min )(1 − exp(−exp(−z(A i + D i ) + β)))). Increases in z would again move the inflexion point to lower values of A and D, generally benefiting prey. Conversely increasing the value of β would shift the inflexion point in the opposite direction, to the detriment of the prey. Implementing the season and fitness calculations Within a season of T time intervals (usually 750 intervals), our model iterates the following procedures: (1) Starting at time t = 0, time moves forward to the next prey encounter; the increase in time is defined by the value of the average time to the next prey encounter (1/CN t ). (2) At a specified predator–prey encounter, the average prey number killed is defined by the probability of prey death at that encounter (i.e., product of P[attack given detection] and P[death caused by an attack]). (3) Current time is updated subsequent to predator–prey encounter, assuming that handling a prey takes up some time so if predator decides to attack then time moves forward by one time interval. Time is therefore updated by 1 × P(attack given detection; if, for example, the average probability of attack is 0.5, half of one time unit is added to the current time value. (4) Surviving prey numbers at a specified time (N t ) are calculated after an encounter: we subtract the average prey number killed at the encounter (i.e., P(attack given detection). P(death caused by an attack)), from the average number of prey alive before the encounter. (5) The system repeats stages i–iv until time = T units, when the season ends. The model is implemented and run within Matlab (code available on request). After the season ends we calculate fitness as the product of proportionate survival (average number of focal prey surviving/number initially present at the start of the season) and fecundity. The reproductive ability of individual i, F i , decreases with its level of investment in defenses (A i and D i ) according to a simple negative exponential function F i = exp(−C D ( A i +  D i )), for the positive constants C D (“cost of defenses,” of value 0.05) , and fecundity coefficients  (display) and  (defense), which determine the marginal costs of investment in aposematic traits. We do not systematically vary C D in this paper; however, it is worth pointing out that increases in the value of this parameter lead to increased costs and generally lead to reduced optimal values for aposematic display and for secondary defense. Unless stated, we assume that a prey that invests sufficiently in aposematic defenses can completely avoid mortality costs of attack (i.e., Att min and Kill min both take values of zero). We use this basic model to investigate the optimal values for aposematic displays and secondary defenses for specified prey

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populations. In most cases we allow investment in display and secondary defenses to vary in value, so that we find optimal values for the pair of parameters {A i ·D i } which specifies the global maximum fitness value. To find optimal fitness values we increase values of A and D from zero by increments (generally of value 0.1) in a factorial design. We use this technique to evaluate the effects of variation in three ecological parameters on optimized aposematic defenses: first, marginal costs of defenses, second effects of different kinds of predator on prey survival, and third changes in predation pressure. In the latter two, we make minor developments to the basic model. For each use of the model we describe the scenario evaluated, and then the results which pertain.

Results MARGINAL COSTS OF APOSEMATIC DEFENSES

Scenario I: In the first set of results in this section we set D to a single, fixed value of 1 (and  = 1) and vary A from values between 0 and 100. Between runs we change the value of the fecundity coefficient ( ) for aposematic display between 1, 0.1, and zero (Fig. 1a–c). Both Att min and Kill min take values of zero, so that sufficiently well-defended prey suffer no mortality costs from predation. However, in the final graph (Fig. 1d) we modified this assumption and set both Att min and Kill min to a value of 0.01, so that on average very well-defended prey suffer a small mortality cost from an encounter with a predator. Result I: As the costs of display diminish (i.e.,  decreases from 1), the optimal value of investment in aposematic display A increases and the fitness value at the optimum also increases (e.g., compare Fig. 1a and 1b). However, as the marginal costs of display decrease, so the fitness curve begins to flatten around the optimum point, such that when  = 0, and displays are cost free in terms of fecundity, there is no single optimum value for aposematic display; instead all forms are equally fit beyond a critical value (cerca A = 21, Fig. 1c). Here the functions that determine attack decisions to which A contributes (probabilities of attack and survival from attack) saturate in their values with respect to variation in A such that the prey will not be attacked or killed when it is detected because Att min and Kill min are set to zero. Increases in conspicuousness beyond the critical value therefore impose no detriment to survival. Increasing the level of secondary defenses has the general effect of lowering the minimum value of A for cost free displays of maximum fitness, but does not change the essential result that cost free defenses specify a range of possible aposematic displays. For example, when D is increased to a value of 10 (not shown here), then the same qualitative pattern of results is seen, but the region of neutral variation in display begins at much lower values than when D = 1 (i.e., A = 3 rather than A = 21).

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Figure 1.

In the final graph (d) in Figure 1 we relaxed the assumption that prey with sufficiently strong defenses become immune from the threat of predation, to do this we assumed that the minimum probability of being attacked (Att min ) and of being killed, given an attack (Kill min ) are both 0.01. Now, when the survival functions (in eqs. 1 and 2) saturate, increases in the conspicuousness of an aposematic display can still impose small mortality costs on the prey. With no fecundity cost to the display (i.e.,  = 0), the model now predicts a single optimum value of aposematic display (of value cerca 56 units). Hence the model makes alternative predictions for prey that, on one hand, can arm themselves with

aposematic defenses that ensure survival from attack and, on the other hand, those for whom there is always some mortality cost from increases in conspicuousness. Scenario II: In the second investigation of variation in costs of aposematic defenses, we allowed the marginal costs of secondary defense or display to change and allow both traits to optimize. In Result II, we first kept the fecundity cost coefficient for secondary defense constant ( = 1), but varied the marginal fecundity costs of display ( = 0.01, 0.5, 1). Subsequently we held the cost of display constant ( = 1) and varied the costs of secondary defenses ( = 0.01, 0.5, 1). In addition we repeated

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the investigation, but used an additive relationship between A and D in equation (2), that is, P(death from attack) = Kill min + (1 − Kill min )(1 − exp(−exp(−z(A i + D i ) + β))). Result II: Increases in the fecundity costs of display causes a decrease in optimal values for aposematic display (A); investment in secondary defense (D) increases in value to compensate (Fig. 2a). The two defensive components of the aposematic ensemble would coevolve here in an antagonistic manner (see Leimar et al. 1986). Though we do not show it here, the same qualitative pattern of antagonistic optimization is seen if the cost of displays is held constant ( = 1) and the costs of secondary de628

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fenses vary ( = 0.01, 0.5, 1): relatively high levels of display are optimized with relatively low levels of secondary defenses. When we repeated this investigation with the additive (A + D) function for survival from attack, prey either invested in displays or in secondary defenses but not both for the parameters used. We found that displays (but not secondary defenses) were produced for values of  less than 0.6; secondary defenses, but not displays, were produced for values greater than and including 0.6. Failure to predict aposematism (i.e., the joint existence of display and secondary defense) makes this specific form of the model unlikely in our view.

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Figure 2. Variation in optimal levels of aposematic display and secondary defense for changes in marginal costs of defenses. (a)  = 0.01, 0.5, 1 and  = 1 throughout; Kill min = 0, Att min = 0, T = 750, z = 0.632, β = 2.5, C min = 0.001, α = 0.5C min , N (t=0) = 200. Top graph, optimal display (A) declines as the marginal cost of display increases. Bottom graph, optimal secondary defense (D) increases as the marginal costs of display increase. (b) Both defenses and displays increase in marginal costs (left to right)  =  = 0.01, 0.5, 1, other parameters as in (a).

Scenario III: Here we fixed the values of  and  , setting both of them to 0.01, 0.5, and 1, so that both aposematic traits become more costly simultaneously. Result III: When marginal costs of display and secondary defense are positively correlated (Fig. 2b), the model predicts that the optimal values for both defensive traits are similarly positively correlated. Thus depending on the economic regime imposed on the defensive traits, the model predicts that variation in marginal costs on one hand can lead to negative correlation between the values of displays and secondary defenses (Fig. 2a) or, on the other hand, can cause positive correlations between the traits (Fig. 2b).

EFFECTS OF VARIATION IN PREDATOR HANDLING OF PREY

We now assume that the value of Kill min can vary, while holding other parameters constant. Some predators may release very well-defended prey effectively unharmed (i.e., for high levels of A and D, Kill min is close to a value 0): in contrast other predators may always cause substantial injury to such prey, even though the prey have high values for aposematic traits (i.e., Kill min is closer to 1, even for high values of A and D). In this context we are interested in the qualities of the alternative forms of aposematic defense (display vs. secondary defenses) which attract investment. EVOLUTION MARCH 2007

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Scenario IV: We want to know how investment in the type of defense (display or predator repellence) is affected by predator handling during an attack. We therefore assume that the marginal costs of display and defense are equal and constant (i.e.,  = 1,  = 1). Furthermore, in our basic model, the marginal effects on attack probability (eq. 1) of an incremental change in either A or D are unequal because the component of attack probability describing the effects of recognition (i.e., [1 − exp(−exp(−(cC i /C min )z + β))]) saturates more slowly than the component describing the effect of average defensive levels (i.e., [1 − exp(−exp(−D∗ z + β))]). Although we ran the model with equation (1) as stated, we also ran a modified version in which A and D contributed to attack avoidance in a quantitatively equivalent manner; that is, P(attack given detection) = Attmin + (1 − Attmin ) ×(1 − exp(− exp(−Ai z + β))) ×(1 − exp(− exp(−D ∗ z + β))). (3) Both equations (1) and (3) generate the same qualitative pattern, and we show the results using equation (3) in Figure 3. If Kill min is large, and defenses do not offer much protection from attacks to individuals, then kin (or some other kind of group selection) may explain their existence. We do not include group effects in the model, hence we restrict the analysis to lower values of Kill min ranging from 0 to 0.3. Result IV: Conditions for the prey become less challenging as the value of Kill min falls from 0.3 to zero in (Fig. 3), because in these circumstances the probability of surviving an attack in-

creases. As this happens, the survival benefits to individuals increase and aposematic displays increase in intensity, whereas optimal secondary defenses decrease in value. Conversely, looking at the x axis in reverse (right to left), as conditions become more challenging to survival prey reduce investment in displays, but increase investment in secondary defenses. Though displays can be effective in preventing attacks, display confers an additional cost of conspicuousness. As the model is set up, prey can gain the same marginal reduction in the probability of attack given detection by increasing investment in secondary defenses as in aposematic displays; however, the net gain is larger for investment in secondary defenses, because these do not incur survival costs of added apparency to predators. Some investment in both forms of aposematic defense is often optimal though, because survival from attack is proportional to the product of the values of aposematic display (A i ) and secondary defense (D i ). A general prediction that the model makes is that variation in death rates caused by variable injury rates from attack leads to a negative correlation between aposematic display and aposematic conspicuousness. VARIABLE PRESSURE FROM PREDATION: SEASON LENGTH

It is possible to envisage similar prey species living in different geographical regions which vary in their season length. Alternatively we can consider “season” in terms of life span to reproduction. For example it is possible to consider prey species in similar geographical areas, but with alternative durations of life cycle, in some cases short, in the others longer.

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Optimal investment in aposematic displays and repellent secondary defenses with variation in the value of Kill min , the minimum probability of death given attack. Upper graph, investment in aposematic display (A) increases (left to right) as the minimum probability of death Kill min decreases toward zero. Lower graph, investment in secondary defense (D) decreases as the value of Kill min declines (left to right). Att min = 0, = 1  = 1,T = 750, C min = 0.001, α = 0.5C min , N (t=0) = 200, z = 0.632, β = 2.5.

Figure 3.

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Scenario V: We vary season length from an initial value of 250, by increments of 100 time units, up to 850 time units, assuming that the marginal costs of aposematic displays and secondary defenses are equal. Result V: When the season length is relatively short (250, 350 units), the optimal state is crypsis and an absence of secondary defense (Fig. 4). However, once season length becomes sufficiently long, and the prey is increasingly exposed to the threat of predation (between 350 and 450 time units), then some level of investment in both aposematic display and secondary defenses become optimal. Both traits increase slowly with season length after 450 time units.

Scenario VI: In this section we include a general description of density-dependent change in the predator’s behaviors. To do this we include a parameter, I, into the functions used to determine attack probability and survival from attack. I takes a value that increases with prey density by a negative exponential function, i.e., I = 1 − exp(−0.005 × N ( t =0) ), (e.g., such that I = c. 0.9, when initial prey density is c. 470 individuals). I replaces z in equation (1), that is, P(attack given detection) = Attmin + (1 − Attmin ) ×(1 − exp(− exp(−(cCi /Cmin )I + β))) ×(1 − exp(− exp(−D I + β))).

VARIABLE PRESSURE FROM PREDATION: PREY ABUNDANCE

As it stands, our basic model does not account for densitydependent changes in predator behavior. We assumed that the predator has constant knowledge of the prey’s aversiveness in making attack decisions, as such we envisage that our predator represents the average knowledge state of many predators in the locality (i.e., mixtures of na¨ıve and experienced animals, with variable states of forgetfulness). However, it seems likely that as unprofitable prey increase in abundance, predators will on average decrease attack probabilities upon encounter. This is because as prey become more abundant we assume that predators will recognize the prey more effectively and have greater knowledge of prey unprofitability. Furthermore it seems likely, as we describe below, that as average knowledge states of predators increase with prey abundance they handle prey more cautiously when they do choose to attack.

(4)

If prey density increases, and I increases as a result, there is a decrease in attack probability for given values of aposematic display and secondary defense, up to the limit where I approaches 1. In the previous sections N ( t =0) = 200, and z = 0.632. For N ( t =0) = 200, I = 0.632, so that the previously reported results are consistent with this density-dependent extension of the model. In equation (2), I is added to the terms in the index of the second exponent, that is, P(death from attack) = Killmin + (1 − Killmin ) ×1 − exp(− exp(−I Ai Di + 2.5), (5) such that survival from an attack increases with abundance. At the time of writing there are, to our knowledge, no published datasets that explicitly describe how predator handling of animal prey changes with absolute prey density. However, it is known

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that prey handling can become less injurious as predators become familiar with the aversiveness of a prey item. Notable examples are provided by Ritland (1991, 1998), who showed that, on average, handling by avian predators of unpalatable lepidopteron prey became less injurious to the insects as predator experience of their unpalatability increased. Where na¨ıve predators initially killed and ate a prey, experienced predators attacked carefully, often pecking but releasing prey almost unharmed. If, as seems likely, general familiarity increases with abundance, it seems reasonable to us to assume that prey handling should become less injurious to the prey as abundance increases. Result VI: Figure 5 shows changes in optimal aposematic display and secondary defense as the abundance of the focal prey increases from 10 to 104 . At either extreme of prey density (10 or 104 ), the optimal form for the prey is cryptic and lacking repellent secondary defenses. When prey are rare and very unfamiliar to predators, density-dependent components of aposematic prey defenses (mediated by I) are so ineffective that aposematic traits are not cost effective at all. When, in contrast, prey are very highly abundant, individuals benefit from high levels of densitydependent predator avoidance behaviors (again, mediated by I) and by a classical dilution effect, in which there are many more prey in the population than the predator could attack in the time

period (104 prey in a season of 750 time intervals). In this case, the risk from predation to an individual is negligible, so the optimal state is not to invest resources in either repellent secondary defenses or conspicuous displays. For the values chosen between these extremes, we see that some form of aposematism is optimal, but that there is a decline in investment in aposematic defenses, between N (t=0) = 100 and N (t=0) = 500. Here aposematism is a beneficial evolutionary response to the predatory threat, but the marginal benefits from investing in displays and repellent defenses decline as prey numbers increase, and the threat from predation declines. If we rerun the program, but use the original equations (1), (2), which lack density-dependent changes in predator behavior, the qualitative pattern of results is very similar to that shown in Figure 5, except that the very rare prey (i.e., N ( t =0) = 10) exhibits strong aposematic defenses because with these equations predator behavior is not adjusted for unfamiliarity of the rare prey.

Discussion In this paper we used a simple deterministic model to ask a specific question about prey defenses: Why do prey appearances vary from

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Figure 5.

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the highly cryptic to the very bright and conspicuous? Apart from the treatment by Endler and Mappes (2004) there has been very limited discussion of this question in the warning signals literature, which has directed much of its attention to the question of initial evolution of the signaling trait (recent discussions in Mappes et al. 2005; Marples et al. 2005) and the special responses of predators to aposematic displays (e.g., Rowe and Skelhorn 2005; Skelhorn and Rowe 2005). Clearly variation in environmental constraints such as dietary limitations may explain why some aposematic individuals are more conspicuous than others (Grill and Moore 1998). Similarly, variation in predator tolerances for toxins (Endler and Mappes 2004), or in the workings of predator visual systems, must surely explain some components of interspecific variation in prey displays. However, we argue that even discounting these environmental factors, the conspicuousness of aposematic displays should vary because the economics of display vary for different prey species. The results described here provide three general explanations for interspecific variation in the brightness of aposematic displays, thus providing explanations for the growing body of empirical evidence that the form of an aposematic display can be related in some way to the defense that it advertises (Summers and Clough 2001; Darst et al. 2006). We briefly consider these three explanations below. First, interspecific variation in the marginal costs of aposematic display can lead to several contrasting outcomes for different prey populations. If displays are costly in terms of fecundity, then our model predicts negative correlations between defenses and displays when the marginal costs of secondary defenses are fixed. When the cost of aposematic display increases, it may be optimal for prey to diminish investment in this trait, but increase investment in repellent secondary defenses to compensate (see Speed and Ruxton 2005b). An explanation for negative associations in aposematic display and toxins (or disassociation like that seen by Darst et al. 2006) is therefore simply that marginal costs of display vary between species, or conversely, that costs of display are constant, but marginal costs of toxins vary across species. However, the model can also predict the converse: if costs of displays and secondary defenses both increase together, positive correlations between display and defenses can be predicted (cf. Summers and Clough 2001). A second cause of aposematic variation is the extent to which aposematic defenses can protect an individual during attack. If the minimum probability of being killed (Kill min ) rises from zero investment in optimal displays decreases, whereas investment in optimal secondary defenses increases. One major reason for this is that aposematic displays incur costs of apparency to predators that secondary defenses do not, but investment in display is often maintained because of our assumption that survival from attack is proportional to the product of display (A i ) and secondary defense

(D i ). We should point out that our model does not include the contribution that kin (or generally some group-level) selection could make to the specification of optimized secondary defenses, and this will make some difference. Some complex analytical models have included both group and individual selection (Leimar et al. 1986; Broom et al. 2006), hence a possible extension to our approach outlined here is the inclusion of kin grouping within the model. A third explanation for variation in aposematic traits is variable exposure to predation. Animals for whom predation risk is not a problem have little inducement to invest in costly defenses; hence when prey life spans are very short, and exposure to predators is minimal, the model predicts an absence of secondary defenses and maximal crypsis. As life span increases beyond some threshold, and the threat from predation rises, the model predicts increased investment in aposematic traits. We have, of course, not exhaustively covered the complexities of animal life history and aposematism in this paper, but these predictions perhaps serve as one of several useful starting points for theories of aposematism and life history (see also Longson and Joss 2006). Similar questions about exposure to predation relate to prey density. When prey densities are very high (and predation threat is diluted) our model predicts very low, or nonexistent, levels of investment in aposematic defenses. Furthermore if prey exist at very low densities, predators will be generally ignorant of prey defenses and hence readily attack. Here again, the model predicts that aposematic displays are not cost effective. Furthermore, if there are no effective aposematic advertisements, the rare focal prey is predicted not to invest in toxins either. There are empirical data supporting these predictions. Notably desert locusts show density-dependent color changes consistent with the idea that edibility and crypsis is a better state for rare than for common prey (Sword 1999, 2002; Sword and Simpson 2000; Sword et al. 2000; Despland and Simpson 2005a, 2005b). It seems anecdotally true, on the other hand, that very highly abundant species, such as schooling fish, are not individually conspicuous or chemically defended. Sheer abundance itself provides costeffective defenses, so that aposematism is an unnecessary expense. For species for which some kind of aposematism is optimal, increases in abundance are predicted to cause a decline in the value of aposematic traits. Because per capita risk declines when population sizes increase, the optimal state requires decreased investment in secondary defenses (which incur fecundity-based allocation costs) and in aposematic displays (which incur costs of raised conspicuousness as well as allocation costs). Hence, other things being equal, variation in population size may cause a positive relationship between defense and conspicuousness (cf. Summers and Clough 2001).

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ARE DISPLAYS COSTLY TO GENERATE AND USE?

Although other theoretical models of aposematic evolution assumed that displays can be costly in terms of raised apparency to predators, we assumed in addition that they could potentially be costly in other ways, such as diversion of resources from reproduction. It may, for example, be costly to acquire, synthesize, and deposit certain showy pigments in the epidermis. Such costs of display may in some cases be greater than the ancestral, nonaposematic form. In an intriguing study, Ohsaki (2005) recently demonstrated that the addition of bright mimetic coloration in the Batesian mimic, Papilio polytes, is associated with a shortened life span. Alternatively, travel speeds may be modified to facilitate signal reception by predators; and this may lead to energetically suboptimal rates of locomotion (Srygley 2004). Furthermore, if particular dietary components are necessary for the generation of colorful pigments (as reported for elytral color in the aposematic ladybird beetle (Harmonia axyridis), Grill and Moore 1998) then there may be costs associated with locating and obtaining these food materials (see also, for an example of costly display, Holloway et al. 1995). Finally, deploying a colorful pigment for aposematic purposes could impose some thermoregulatory costs on the animal, diminishing the size of melanized areas below levels that maximizes thermoregulatory efficacy. At the time of writing, fitness costs of generating and using aposematic displays are relatively unknown (to the best of our knowledge) as are the relative costs of displays versus secondary defenses. EVOLUTIONARY HISTORY AND OPTIMAL PREY DEFENSES

We should make clear that our model predicts the optimum state for a monomorphic population of potentially aposematic prey; it therefore explains variation in aposematism between populations or species, but not within a population. If optimal forms can be specified then it should be noted that prey populations will not always be found at their theoretically optimal states. We have deliberately not sought to find localized evolutionarily stable strategy solutions for prey defenses. Thus it may be that the global optimum for a prey population is extreme aposematism—a bright display and a repellent warning coloration. However, the prey species may for historical reasons be “trapped” in the local optimal state of crypsis and edibility, or a suboptimal form of aposematic traits, with a major fitness tough between these states. Figure 1a, for example, shows a local cryptic optimum point separated from the global aposematic optimum by a fitness trough. This fitness trough is caused by the sigmoidal nature of predator responses to variation in prey defenses, in which predators show small changes in prey avoidance for modifications to low values of display and secondary defenses. Note, though, there is no density dependence shown against a novel rare form in this figure, so that the trough between crypsis and aposematism would be expected to be much deeper than shown. As has been pointed out several 634

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times before, aposematism is therefore an excellent case study for Sewell–Wrightian models of shifting balances in evolution (Mallet and Joron 1999).

Conclusions Few authors have considered how components of an aposematic defense might interact to cause variation in intensities of display; yet we contend that it is one of the most interesting remaining questions about the phenomenon. Here we provided an economicevolutionary framework for exploring questions of how aposematic signals of defenses and the defenses themselves should coevolve, and for addressing questions about the potential informational content of aposematic signals. This framework can accommodate both of the apparently contradictory empirical patterns in the relationship between signal and defense intensities, and demonstrate that there need be no contradiction at all. We have not attempted by any means to describe all factors which may lead to trends in aposematic conspicuousness. We suggest that future work considers some factors that have already been identified as important in the design of aposematic signals: variation in behavioral conspicuousness (Merilaita and Tullberg 2005; Speed and Ruxton 2005a), parasitic mimicry (Sherratt and Franks 2005), life history (Longson and Joss 2006), and community structure (Merilaita and Kaitala 2002). One key conclusion that we draw from our simulations is that when viewed between species aposematic displays are not likely to evolve as primarily selected “handicaps” in the sense that better defended prey always take the most costly route and always “shout loudest” to predators. The level of aposematic display can be informational in this sense, but we may equally predict the opposite result, in which those with the weakest secondary defense shout loudest to compensate for the poor quality of their repellent defenses. Whether the intensity of aposematic signals increase with intensity of secondary defenses depends on a number of interacting factors, and will vary between prey types. ACKNOWLEDGMENTS We are especially grateful to T. Sherratt for stimulating discussions, to M. Puurtinen for assistance with modeling issues, and to M. Servedio and two anonymous referees for very constructive comments on a previous version of the manuscript. MPS’s work on this paper was supported by a research fellowship from the Leverhulme Trust.

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