vol. 176, no. 1

the american naturalist

july 2010

E-Article

Imperfect Batesian Mimicry and the Conspicuousness Costs of Mimetic Resemblance Michael P. Speed1,* and Graeme D. Ruxton2 1. School of Biological Sciences, University of Liverpool, Liverpool L69 7ZB, United Kingdom; 2. Graham Kerr Building, University of Glasgow, Glasgow G12 8QQ, United Kingdom Submitted February 12, 2009; Accepted October 2, 2009; Electronically published May 24, 2010

abstract: We apply signal detection methodology to make predictions about the evolution of Batesian mimicry. Our approach is novel in three ways. First, we applied a deterministic evolutionary modeling system that allows a large number of alternative mimetic morphs to coexist and compete. Second, we considered that there may be natural boundaries to phenotypic expression. Finally, we allowed increasing conspicuousness to impose an increasing detection cost on mimics. In some instances, the model predicts widespread variation in mimetic forms at evolutionary stability. In other situations, rather than a polymorphism the model predicts dimorphisms in which some prey were maximally cryptic and had minimal resemblance to the model, whereas many others were more conspicuous than the model. The biological implications of these results, particularly for our understanding of imperfect mimicry, are discussed. Keywords: Batesian mimicry, signal detection, polymorphism, evolutionarily stable strategy (ESS).

Introduction In Batesian mimicry, a well-defended aposematic species (the “model”) has its warning displays copied by an essentially undefended “mimic” species. Theoretical models of mimicry look at a range of different evolutionary facets of Batesian mimicry systems, and, historically, particular interest has been placed on explaining mimetic polymorphisms and on model-mimic coevolution (Nur 1970; Charlesworth and Charlesworth 1975a, 1975b, 1975c; Turner 1987; Gavrilets and Hastings 1998; Holmgren and Enquist 1999; Speed 1999; Blows and Wimmer 2005). Most recently, attention has focused on the explanation of imperfect mimicry, where the resemblance between mimics and their models is inexact (Dittrich et al. 1993; Edmunds 2000; Holloway et al. 2002; Johnstone 2002; Sherratt 2002; Holen and Johnstone 2004). * Corresponding author; e-mail: [email protected]. Am. Nat. 2010. Vol. 176, pp. E1–E14. 䉷 2010 by The University of Chicago. 0003-0147/2010/17601-51072$15.00. All rights reserved. DOI: 10.1086/652990

One good reason for the interest in imperfect mimicry is that it appears to defy the prediction from a simple adaptive perspective (Sherratt 2002) that in order to avoid unnecessary costs of predation, all Batesian mimics will evolve toward a very close—almost perfect—resemblance of their models (Fisher 1930). In addition, there is a growing recognition that imperfection is common in examples of Batesian mimicry; indeed, some mimetic resemblances appear to human observers to be very imperfect (Dittrich et al. 1993; Edmunds 2000; Howarth and Edmunds 2000; Howarth et al. 2004). A number of adaptive explanations for an evolved state of imperfect mimicry have been proposed. Most obviously, perhaps, it has been suggested that mimics may have imperfect resemblance to their models simply because of an evolutionary lag (see review in Edmunds 2000). Edmunds (2000) also proposed that some mimics manifest a “jack-of-all-trades” phenotype that optimized imperfect “compromise resemblances” to several diverse models and thus had close resemblance to no single species. The plausibility of this idea was demonstrated with a rigorous analytical model by Sherratt (2002). An alternative economic explanation for imperfect mimicry, recently suggested by Holen and Johnstone (2004), is that modification of a prey phenotype to facilitate mimicry imposes one or more of a range of costs on an individual, perhaps deforming organs and limbs in order to make a mimic resemble its model or perhaps compromising its thermoregulatory or locomotor capacities. In fact, the idea that mimicry imposes nonpredatory costs on individual prey is a familiar one in the mimicry literature (e.g., Turner 1977, pp. 190–196, lists a set of likely constraints on mimetic perfection). In Holen and Johnstone’s (2004) models of Batesian mimicry, evolutionarily stable imperfect mimicry can result from a trade-off between costs of mimicry (through mechanisms other than predation) and antipredatory benefits of mimicry. Both these costs and these benefits are predicted to increase, although not necessarily at the same rates, with increasing perfection of mimicry. A theoretical framework used to evaluate these and other explanations of imperfect mimicry is signal detection

E2 The American Naturalist Table 1: Key terms used in the model Term

Definition

Dmim Dmod c b fmod(ij) fmim(ij) j Xmim(i) Xmim( j) Xmod(i) Xmod( j) ri Si f

Absolute abundance of the mimic Absolute abundance of the model Cost to predator of attack on a model Benefit to predator of attack on a mimic Probability density function for the model Probability density function for the mimic Standard deviation of probability density functions Phenotypic value of mimic i Perceived phenotypic value of mimic i Phenotypic value of model i Perceived phenotypic value of model i Measure of rate of encounter of predator and prey of phenotype i Probability that a mimic prey survives a season Scaling parameter for relating phenotype X to encounter rate r

theory (Oaten et al. 1975; Staddon and Gendron 1983; Greenwood 1986; Getty 1987; Sherratt 2002; Holen and Johnstone 2004; Lynn et al. 2005; McGuire et al. 2006). In this framework, predators are assumed to take an economic view of decision making, weighing up the fitness implications of correct responses (attacking a mimic, ignoring a model) and of incorrect responses (ignoring an edible mimic, attacking an aversive model). Signal detection theory provides a statistical mechanism, based on evaluation of the probability density functions of more or less overlapping Gaussian curves, that can be used to predict the optimal attack rate by predators on defined populations of models and their Batesian mimics. It is notable that all of the key articles that apply signal detection models to mimicry have assumed that mimetic phenotypes differ in appearance, without this leading to a difference in detection rate by predators. This seems unlikely for many new cases of Batesian mimicry in which, given the edibility of the prey, the ancestral state is often relatively more cryptic than conspicuously colored, like the putative model species (Charlesworth and Charlesworth 1975a). Copying a brightly colored, well-defended model species therefore seems likely to impose costs of raised detection rates (Sherratt and Franks 2005). Furthermore, signal detection models assume that probability density functions are defined by the area under full Gaussian curves. Nonetheless, it is likely that in many real situations, stimuli have natural boundaries beyond which they cannot be generalized, and as we show in this article, truncation of Gaussian curves can lead to important changes in predictions made for optimized Batesian mimicry. This article therefore reconsiders the predictions of signal detection models by incorporating costs of conspicuousness into the fitness evaluation of the prey and by allowing that stimulus dimensions have natural bound-

aries. To achieve this, we developed a numerical frequencydependent model of signal detection and Batesian mimicry. Our theoretical model can allow any number of alternative mimetic forms to coexist within a species at any given time and allows variable mutation rates between generations. We first outline how signal detection theory can be applied to Batesian mimicry, before developing and applying a numerical-optimization model. Methods Signal Detection Theory and Mimicry Suppose that an individual prey’s appearance can be summarized by a single variable X, with values between a minimum (Xmin) and a maximum (Xmax). We imagine an aposematic prey species of fixed appearance (e.g., Xmod), termed “model,” that has its aposematic appearance mimicked by an edible species termed “mimic.” We set X min p 0, X max p 10, and X mod p 5 (for a description of terms, see table 1). A common assumption made in the application of signal detection models to Batesian mimicry is that a predator’s perceptual-recognition systems are error prone, so that when it sees a prey i of trait value Xi, it has nonzero probabilities of misrecognizing it and assuming that the trait is actually higher or lower than Xi. We use the subscript j to refer to a perceived phenotype that originates objectively from some particular trait i, so that for a prey i of trait X i p 5, we calculate the probability that the prey is recognized as some phenotype value Xj, which can take any value between and including Xmin and Xmax. In signal detection models, it is usually assumed that the likelihood of such misperceptions declines as the distance in perceived phenotype space (Xj) deviates further from the actual phenotype value of the prey (Xi; Green and Swets

Conspicuous Batesian Mimicry E3 1966; Macmillan and Creelman 1991). This relationship is usually represented by a normal (or Gaussian) probability density function, by application of the central limit theorem (though other distributions, such as Poisson, can be assumed; Green and Swets 1966). In this article fmod(ij) refers to the Gaussian probability density function for the model of trait value Xmod(i) and fmim(ij) to that for the mimic. Hence, there is a probability density function fmim(ij) that describes the probability that a mimic of phenotype Xmim(i) will actually be recognized (or, when i ( j, misrecognized) as having a phenotype Xmim(j). Similarly, for a model of phenotype Xmod(i) the probability that it is (mis)recognized as having phenotype Xmod(j) is fmod(ij), and we assume that, like actual phenotypes, perceived phenotypes are bounded between Xmin and Xmax. Though it is standard in signal detection models of mimicry to assume that prey phenotypes are continuously distributed, for the numerical system we develop in this article, we needed to make prey phenotypes (and hence their probability distributions) discrete, separated into many small phenotype categories. Thus, for the model species, we calculate the probability density function by dividing the phenotype space between Xmin and Xmax into 101 discrete phenotype categories (of values 0–10 by increments of 0.1, though any large number of categories is appropriate). For a model phenotype of value Xmod(i), the probability that the prey will be recognized as Xmod(j) is fmod(ij), where fmod(ij) p

F

(ij) 101

冘

jp1

F(ij)

,

Macmillan and Creelman 1991). In our numerical model, however, we allow that stimulus dimensions have physically or biologically constrained boundaries so that generalization curves cannot extend indefinitely. Suppose that, for simplicity, the signal describes grayscale change over the range of black to white. Stimuli cannot be generalized beyond the most extreme values along the stimulus gradient (there is nothing darker than black), so all our generalization curves are, to some extent, truncated, removing “impossible” stimulus values (see “General Discussion”). As described above, we divided the frequency of each point on the remaining Gaussian curve by the cumulative total of possible stimulus values so that the area of the remaining Gaussian curve summed to 1. When stimuli lie close to the absolute boundaries, nearly half of their potential frequency distribution is missing; consequently, the points on the remaining curve have higher values than those on curves whose stimulus values are farther away from the boundary (see illustration in fig. 1A). Alternatively, we could consider that the parts of the normal distribution that fall beyond the stimulus boundary are added on to the most extreme allowed values, Xmin or Xmax (fig. 1B). Either assumption leads to qualitatively similar effects, and we show results using the truncation method in figure 1A. In “General Discussion” we consider how truncation may arise.

(1)

⫺(X mod(j) ⫺ X mod(i)) 1 F(ij) p exp , 2(j 2) j冑2p 2

(2)

describing a Gaussian distribution around Xmod(i). Furthermore,

冘 101

fmod(i,j) p 1

jp1

so that the total probability density function sums to 1. Calculation of fmim(ij) is analogous. Stimulus Boundaries and Generalization Functions It is standard in treatments of signal detection based on Gaussian distributions to consider that perception extends from some specified point j to infinity, and, hence, cumulative normal distributions can be used in calculating the areas of phenotype space that are necessary for determining optimal decision making (Green and Swets 1966;

Figure 1: Generalization curves for a mimic (solid curve) and its model (dashed curve), assuming that generalization distributions become truncated at absolute stimulus boundaries. Dmim p 2,000, Dmod p 2,000, j p 0.8, Xmod p 5, Xmim p 0.0, c p 1, and b p 1. Y-axes are deliberately not equal across graphs. A, Values beyond the boundaries are ignored; in the case of both model and mimic, the area of the curves sums to 1. Because the mimic’s phenotype is closer to the stimulus boundary of X p 0, the relative frequencies of existing mimic phenotypes is raised. The red line indicates the phenotype space over which prey are avoided. B, Values beyond the boundaries are added to the nearest extreme value (in the case of Xmim p 0.5, the additional values are added to X p 0); in both cases, the area of the curve sums to 1.

E4 The American Naturalist

冘 101

The Attack Decision When there are no differences in conspicuousness between model and mimic color patterns, what matters to the predator in deciding which phenotype values to attack and which to reject is the additional information of the relative abundance of the mimic (p) compared to the model (1 ⫺ p) and the costs of attacks on models (c) and the benefits of attacks on mimics (b). Assuming that the real mimic phenotype is Xmim(i) and the real model phenotype is Xmod(i), then a prey of perceived phenotype Xj is accepted if, on average, the outcome is positive, that is, if bfmim(ij) p 1 cfmod(ij)(1 ⫺ p)

(3)

(see Oaten et al. 1975; McGuire et al. 2006); otherwise, it is rejected. We can illustrate how phenotypic zones of acceptance or rejection are determined using the illustration in figure 1A (assuming that the cost of an attack on a model is c p 1 and the benefit from an attack on a mimic is b p 1 and that model and mimic are equally abundant; that is p p 0.5 and 1 ⫺ p p 0.5). In figure 1A the horizontal red arrow illustrates the phenotype space within which prey are rejected by the predator. Now given that a prey has been detected, the probability of attack on a mimic form i (Amim(i)) in a given time period can be calculated as the proportion of its probability density function (fmim(ij)) that falls inside the phenotypic zone in which prey are attacked; in this case, the value is ∼0.99 because less than 0.01% of the probability distribution lies within the zone of rejection (fig. 1A).

Cij p cfmod(ij)(1 ⫺ p(i) );

(5)

here (1 ⫺ p(i) ) refers to the frequency of the model only in relation to the abundance of mimics of phenotype Xmim(i). Similarly, we calculate the average benefit of an attack for each perceived phenotype value Xj if the prey turns out to be a mimic of form Xmim(i), termed Bij, defined as Bij p bfmim(ij) p(i) .

(6)

Hence, as we did for the two-party model, it is simple to calculate the phenotype space in which a predator avoids each mimetic form in isolation of the others. However, we assume that the predator takes into account all mimic forms (i p 1–101) when determining the phenotype space in which it accepts or rejects prey. Across each mimetic form i p 1–101, we calculate the average of the costbenefit sums for all the available phenotype space Xmin to Xmax, weighted according to the relative frequency of each mimetic form compared to the mimic population as a whole. Thus, for a perceived phenotype of value Xj, the predator accepts the prey if

冘 ip1

We are interested in using signal detection theory in a numerical framework, in which mimics can evolve toward their optimal form of mimicry. To achieve this, we need to extend the traditional two-party model of signal detection theory such that at any given point in time, mimic abundance can be divided over many phenotypes between (and including) Xmin and Xmax. We do this using the Matlab programming environment (sample code by request from M.P.S.). In the simulations represented here, there are 101 possible mimic phenotypes, evenly spaced between the maximum and minimum values (e.g., X1 p 0 … X101 p 10; though, again, any sufficiently large number of phenotype classes is appropriate). The total absolute abundance of mimics (Dmim) can then be distributed over a large number of possible phenotypes such that

(4)

For each pairing of a model and a particular form of mimic (Xmim(i)), we calculate the average cost to the predator of an attack for each perceived phenotype value Xj if the prey turns out to be a model, termed Cij, where

101

Development and Implementation of the Basic Signal Detection Model

Dmim(i) p Dmim.

ip1

( )

(Cij ⫺ Bij )

Dmin (i) 1 0. Dmin

(7)

By weighting the cost-benefit comparison according to relative frequency in this way, we allow that predators, for example, pay little attention to very rare mimetic forms when determining which prey to attack. Having calculated the phenotypic zone within which prey are attacked, we next calculate the probability of attack given detection (Ai) for all mimic forms i p 1–101. Because an aim of the article is to consider the cost of conspicuousness on mimicry, we need a mechanism that enables us to include consideration of encounter rate ri. In the figures shown, we use a negative exponential term to define ri: ri p rmin ⫹ g ⫺ g exp (⫺Z),

(8a)

where rmin is the minimum level (and g p 1 ⫺ rmin so that the maximum value is ri p 1) and Z is a parameter by

Conspicuous Batesian Mimicry E5 which variation in the signaling phenotype causes variation in the encounter rate. In our first set of simulations, mimicry has no effect on encounter rate, in which case Z p 100 and ri p 1 for all prey. When prey appearance does affect encounter rate, we set Z p X i /f, where f scales the value of mimicry in relation to ri. Our results are fundamentally unchanged if we adopt some other function, such as a sigmoidal one, so that, for example, ri p rmin ⫹ g ⫺ g exp [⫺ exp (⫺0.3X i ⫹ 2)].

(8b)

Our figures show examples using equation (8a). We now calculate the probability that mimetic prey of phenotype Xi survives a time interval, such as a season, as Si p exp (⫺ri A i).

(9)

We use this framework to generate evolutionary simulations in which the 101 morphs of the mimic species compete by asexual and discrete reproduction. Densities are initially fixed at the start of the first “season” (e.g., with 19.8 individuals in each of m1 to m101, giving a grand total of 2,000 individuals). Subsequently, we simulate asexual reproduction by the mimetic morphs (m1 … m101); to do this, we find the total remaining abundance of all mimic forms at the end of a given season (number of a form at the start of the season multiplied by the probability of survival) and calculate the proportionate composition of this total from each of the 101 mimic prey forms. These proportions are multiplied by 2,000, to scale the prey abundances for each of m1 … m101 up to the next generation. In most of our simulations, we allow a rate of mutation between prey forms in which a small proportion of each morph’s offspring (unless stated, 1 # 10⫺6) are assigned different phenotypes by following a normal distribution function N(X i , jmut ). When jmut p 1, then most mutations are of relatively small effect with respect to the originating phenotype. When jmut p 100, mutations have effects that are widely dispersed across all phenotypes, and in this case, we can be sure that when an evolutionary equilibrium is reached, it cannot be invaded by any phenotypes within the range that we use. We again excluded mutations that fell outside the phenotype values Xmin and Xmax and adjusted the frequencies accordingly so that the same total proportion of each trait value was distributed to other values. In virtually all cases, variation in the distribution of mutations did not change the results observed, and we illustrate the case of near-uniform mutational distribution (or jmut p 100). We also considered the possibility that mutation to phenotype forms becomes less likely near the

boundaries. As this made no qualitative difference to the predicted outcomes, we describe this work in the appendix. Evolutionary Stability Unless stated otherwise, we used three starting distributions for each set of conditions simulated: (i) mimics evenly distributed across the 101 phenotypic forms, (ii) all mimics located at X p 0, and (iii) all mimics located at X p 5 (which we class as “perfect mimicry” because here mimics perfectly resemble the model). In a number of cases, the simulations rapidly approached stability, in the sense that after a few thousand generations, mimetic forms reached stable abundance levels, after which their frequencies did not vary except because of the mutation process. We judge that this is an evolutionarily stable state because no rare mutants invade at this point. In other cases the simulations also approached stability but maintained very small oscillations around the mean frequencies (e.g., on a mean mimic frequency of 502.01, 1 SD was 9.8 # 10⫺3). Crucially, in these cases there was no longer any directional change in morph frequencies over longer timescales (we examined the correlation coefficient between generation time and the frequencies of mimetic morphs for periods of 5 # 10 4 generations; when the value of the coefficient was less than 0.001, we determined that stability had been reached; see examples in the appendix). In one case, in which we wish to illustrate the ecological importance of shallow selection gradients, we deliberately did not run the simulations to stability. We first explore some general properties of the model before introducing the possibility that mimicry affects encounter rates with predators. We show that the model can replicate the general predictions of earlier models and makes interesting predictions for the case in which predators generalize broadly and there are absolute boundaries to the stimulus. General Properties of the Model We first consider how mimicry would evolve when it has no effects on encounter rate (so that Z p 100 in eqq. [8] and the probability of detection for any prey in the time interval is 1). Except where explicitly stated, starting conditions made no difference to the outcome. We set Dmim p 2,000, Dmod p 2,000, and b p 1 and used three values of c (2, 5, and 10; increase in c is equivalent to increases in Dmod or reductions in Dmim and/or b). We first set j p 0.01, a small level of generalization, such that predators are very accurate in their recognition of models and mimics. With very narrow generalization curves, monomorphism at perfect mimicry was observed from any of our three starting points (furthermore, if we

E6 The American Naturalist move the population from X p 5 to X p 4.9 or X p 5.1, it returns to X p 5). We next set a wider zone of generalization with j p 0.5. Prey tended to converge around the point of perfect mimicry (see, e.g., fig. 2A) whether they started from a point of equal distribution across phenotypes or whether they were exclusively located at X p 0. However, the rate of evolution rapidly became very slow, so that after 5 # 10 4 generations, the total change in frequencies was less than 10⫺5 per generation (as a proportion of the total 2,000 prey) and the population remained widely polymorphic (the figure shows prey frequencies after 106 generations). There is still some directional selection in favor of close mimics, but it is very weak. Repeating, but now without mutation, led to very slow selection for prey congregated around perfect mimicry (and a result like that in fig. 2A). Properties of the Model When Predators Show Broad Generalization We increased j to 1, so that there was relatively wide generalization by predators, and found that the outcomes were now more sensitive to the starting distribution of the mimics and to the presence of recurrent mutation. For example, when all phenotypes were initially equally abundant and no mutation was allowed, the simulations rapidly came to an equilibrium state in which there was no further change in frequencies and all prey that remained with abundance greater than 0 had equal fitness (fig. 2B). Stable abundance levels diminished toward 0 as a mimic’s phenotype got further from that of the model. The stable result here was susceptible to starting conditions, so that, for example, if we put 1,900 of the prey at X p 0 and the remaining 100 were uniformly distributed between X p 1 and X p 100, then the simulations rapidly came to stability, but the peak frequency was higher than perfect mimicry, located at X p 5.85 (fig. 2C; with c p 5, 10, the result is bimodal, with a small number of prey remaining at X p 0). Alternatively, with 1,900 of the prey initially at X p 5 and the remaining 100 prey uniformly distributed, no further evolution took place, and the prey remained at their starting frequencies. If, after stability was reached, we allowed mutation for one generation (using a normal distribution of mutants, with jmut p 100), then the popFigure 2: Frequencies of evolved mimetic forms and survival probabilities of mimetic prey. Dmim p 2,000; Dmo p 2,000; b p 1; and c p 2 (blue), 5 (black), 10 (red). For each of A–D, upper panels represent mimic abundance versus phenotype (X), and lower panels represent survival probability versus X. A, j p 0.5. The figure shows 5 # 106 generations (where proportionate rate of change in mimic frequencies is very small, !2 # 10⫺7). jmut p 100. B, j p 1. The initial distribution was even across

X p 0 to X p 10. At evolutionary stability, the mimetic form is highly polymorphic, and all extant forms gain protection from predation at their stable frequencies. jmut p 0 ; there is no mutation. C, As in B but the initial point is nearly all at 1. jmut p 0; again there is no mutation. D, As in B and C, but now mutation is run at jmut p 100 (and 10⫺6 of each prey phenotype abundance is distributed by mutation).

Conspicuous Batesian Mimicry E7 ulation returned to stability subsequently but at slightly modified values. Hence, the simulations come to a form of equilibrium in which all existing prey have equal fitness, but this can be undermined by perturbation. When we ran the models with recurrent mutation, they all settled to the same stable distribution shown in figure 2D, regardless of the initial prey distribution (we used a relatively high mutation rate of 10⫺3 to achieve relatively rapid evolution here, though similar results were reached when mutation was set to 10⫺6).

we may expect stable variation in mimicry, including perfect and imperfect forms. If generalization is at some intermediate level, we predict that although there is directional selection toward perfect mimicry, the rate of evolution may become so slow that, for all intents and purposes, polymorphisms are essentially stable (Sherratt 2002).

Discussion of General Properties of the Model

We next examined how the predictions of the model would change if variation along the phenotype spectrum X causes an increase in the value of encounter rate ri. We set Z p X i /f, where f is a scaling parameter evaluated for variation between 10 and 20; figures show f p 10 and rmin p 0.01. In addition to this change, we need to modify the decision rule described by equation (3), replacing p and (1 ⫺ p) with amended terms. In the standard formulation of signal detection and mimicry, these terms describe the relative frequency of mimic and model, respectively, indicating that the probability of a predator encountering either prey is proportional to their relative frequency. If, however, there is variation in conspicuousness between the prey forms, then probability that an encountered prey is a mimic or a model depends on their relative frequencies and their relative conspicuousness values. We therefore amended equation (3) to

If generalization is wide in relation to available parameter space (but not excessively so; j p 0.5), then our model predicts that the mimic population reaches a state in which there is very slow directional selection toward perfect mimicry. This result agrees with predictions from the signal detection model described by Sherratt (2002), who found a “nearly neutral” zone of mimetic protection (with a very slight advantage to perfect mimics) under similar conditions. In this area of phenotype space, the proportion of the imperfect mimic’s probability density function that lies in an area where attacks take place can be very small. The fitness difference between perfect and some imperfect mimics is then itself very small, and hence the rate of evolution is miniscule. A novel result from our simulations is that in cases in which generalization curves are wide relative to the available phenotype space, widespread polymorphisms can be stable (fig. 2B, 2C). With j p 1, a relatively wide generalization parameter, polymorphisms including imperfect mimics could remain stable because the tails of their probability distributions, where attacks could theoretically take place, are not manifest between Xmin and Xmax; rather, they are truncated by the stimulus boundaries imposed. In this situation, then, the attack probability on a prey can be 0, even though mimicry is imperfect. We found two kinds of stable outcome. In the first, where all prey existed initially, evolution rapidly ceased and the equilibrium state was strongly influenced by the starting conditions. In the second kind of outcome with recurrent mutation, the prey evolved toward an equilibrium state that reflected mutation-selection balance, and this was not strongly influenced by starting conditions. One conclusion that we can reach, then, is that the predicted outcome is sensitive to the scaling of the maximum range of perceived phenotypes (Xmin to Xmax) and the broadness with which predators generalize (j). If generalization is narrow relative to the phenotype space within which generalization occurs, a simple monomorphism of perfect mimicry is predicted. If generalization is relatively wide and stimulus dimensions have natural boundaries,

Variation in the Phenotype Incurs Costs of Conspicuousness

bfmim(ij) p  1 cfmod(ij)(1 ⫺ p  ),

(3a)

where p  p p 7 r(mim)i /[p 7 r(mim)i ⫹ (1 ⫺ p) 7 r(mod)i], r(mim)i is the conspicuousness value of a mimic prey of phenotype Xi, and r(mod)i is the conspicuousness value of the model prey. To keep the theoretical model tractable, we assumed that the encounter rate but not the probability of erroneous identification is affected by conspicuousness of the prey. Thus, the conspicuousness of a prey’s appearance is evaluated relative to its background, but recognition accuracy (defined as the relationship between Xi and Xj) depends on inspection of the prey’s appearance alone. Because variation in prey color (Xi) now affects encounter rate, effectively increasing the conspicuousness costs of the phenotype, it might be expected that the optimal phenotype for mimics is lower than that of perfect mimicry. However, this is not the case. In fact, resimulations of the graphs shown in figure 2 but with these revised parameter values showed that the results were fundamentally unchanged. When j p 0.01, perfect mimicry resulted; when j p 0.5, prey evolved (slowly) toward perfect mimicry; and when j p 1, a range of stable monoand polymorphisms resulted. The reason that conspicuousness has no effect on the

E8 The American Naturalist dict a greater phenotypic distance between the mimics and their model if there are increases in one or all of (i) the cost to predators of attacks on models, c; (ii) the abundance of the model, Dmod; and (iii) the breadth of the generalization curve, j. There will be a similar reduction in mimic phenotype value if there are reductions in the benefits of attacks on mimics to predators, b, or in the abundance of the mimic, Dmim. Conspicuousness Can Lead to a Bimodal Distribution of Mimetic Forms

Figure 3: Frequencies of evolved mimetic forms and survival probabilities of mimetic prey when there is a cost of conspicuousness to mimicry. Dmim p 2,000; Dmo p 2,000; b p 1; j p 0.5; a p 0.01; rmin p 0.01; f p 10; and c p 2 (blue), 5 (black), 10 (red). Upper panel represents mimic abundance versus phenotype (X); lower panel represents survival probability versus X.

predictions is that, like other models of signal detection theory and mimicry, we assumed that when the average cost of an attack outweighs the benefits to predators, all predation ceases. For mimic prey that are protected from attacks, encounter rates can have no effect on survival. Hence, the standard approach to signal detection theory and mimicry presents us with a paradoxical situation in which increasing perfection of mimicry, from X p 0 to X p 5, does increase encounter rates with predators, but this often has no effect on survival. One way to resolve this paradox is to assume that predators do have a small probability of attack on prey, even when average costs are higher than average benefits. We could then rewrite equation (9) as Si p exp {⫺ri[a ⫹ (1 ⫺ a)A i]},

So far, we have mainly demonstrated monomodal populations, where the mimic population converges around one phenotype value. We can use the system to examine what kinds of ecological conditions may tend to generate dimorphisms (and see Holen and Johnstone 2004 for an investigation of dimorphisms not specifically related to crypsis). We investigated whether such dimorphisms could be generated using simulations in which the mimicry phenotype can incur costs of raised encounter rates and all prey face a small nonzero probability of attack on encounter (i.e., a 1 0; eq. [9a]). We first considered the condition in which predator generalization was at some intermediate level (j p 0.5). Stable bimodal distributions could be generated, for example, by setting Dmod p 2,000; Dmim p 2,000; c p 1; b p 1; a p 0.01; and rmin p 0.0038, 0.0037, or 0.0036 (fig. 4). One form of mimicry, a “cryptic form,” is located at the point of minimal conspicuousness, X p 0; the other is a “bright form,” located at a position of close but imperfect mimicry. Hence, both forms of the mimic that coexist are imperfect and have lower conspicuousness val-

(9a)

where a sets the minimum actual value of attack on any prey form. There are a number of biological interpretations for an assumption of a 1 0; for example, predators may continue attacks on prey that they expect to be unprofitable at a very low rate in order to gain information about the existence of mimicry. The result of setting a 1 0 is that the optimal form of mimicry now can tend toward values lower than perfect mimicry (here X ! 5). With a p 0.01, rmin p 0.01, j p 0.5, and b p 1 (and c p 2, 5, or 10), the population evolves so that existing mimetic forms have phenotypes lower than perfect mimicry (see fig. 3). Numerical interrogation of the model showed that factors that increase the phenotypic space over which mimicry is beneficial will tend to push the position of the optimal point toward lower phenotype values. Specifically, the simulations pre-

Figure 4: Frequencies of evolved mimetic forms and survival probabilities of mimetic prey when there is a cost of conspicuousness to mimicry: bimodal results. Dmod p 2,000; Dmim p 2,000; f p 10; c p 1; b p 1; a p 0.01; j p 0.5; and rmin p 0.0036 (blue), 0.0038 (black), 0.0037 (red). Upper panel represents mimic abundance versus phenotype (X); lower panel represents survival probability versus X.

Conspicuous Batesian Mimicry E9 ues than does the model. If a increases in value, the survival benefit that is due to mimicry of the model is degraded, and the cryptic form increases in abundance at the expense of the bright mimic. If rmin increases, then the stable value of abundance for the cryptic mimic form decreases (fig. 4), perhaps to the point of extinction, because the protection afforded by crypsis cannot compete with that provided by close mimicry. Hence, one important prediction is that dimorphisms are not easily predicted in prey that have a high minimum rate of detection by predators. If c or Dmod increases in value, the benefits of close mimicry increase compared to protection from crypsis, and so the bright form tends to increase in abundance (and conversely, if b or Dmim increases in value, the protective benefits of mimicry are degraded, so that the cryptic form tends to increase). Furthermore, if changes in the parameters lead to an increase of the bright form of the mimic species, it also tends to move them to a higher optimal value of X because in order to reduce predation, more abundant mimics need the greater protection offered by close mimicry (e.g., we set rmin p 0.0039, and the location of optimized bright mimicry increased to 4.8). General Discussion Our approach to signal detection and Batesian mimicry is, in our view, novel in three ways. First, we applied a deterministic evolutionary modeling system that can allow a large number of alternative mimetic morphs to coexist, mutate, and reach evolutionary stability over a number of generations, from almost any initial distribution of mimic forms. A second novel aspect is that in setting up the fundamentals of our signal detection model, we considered that there may be natural boundaries to phenotypic expression that affect the shapes of the probability distributions used to determine attack probabilities. This can have important and, to our knowledge, previously unrecognized consequences for the stable forms of mimicry that emerge from signal-detection-based simulations. Finally, we allowed conspicuousness to impose an increasing cost on mimics as they change along a phenotype axis, and we were able to predict several alternative outcomes. We discuss the implications of these aspects of our model below, starting with the question of stimulus boundaries. Stimulus Boundaries and Variation in Mimetic Forms In some instances, we can predict widespread variation in mimetic forms at evolutionary stability. This prediction emerged from our model because we imposed stimulus boundaries on the system beyond which prey phenotypes cannot vary. For some prey, this led to a substantially truncated Gaussian probability density function, with the

result that they were not attacked even though their mimicry was imperfect. In constructing the signal detection model, our assumption has been that though the correct visual information is presented to the predator, its nervous system is error prone. Sampling of this error, through the application of the central limit theorem, leads to a normal distribution of errors around the stimulus value (see Green and Swets 1966). By assuming that Gaussian distributions are truncated, we assume that recognition errors do not occur in perceptual space that cannot exist: a stimulus that is completely black can be misrecognized for one that is less black but not for one that is more black than black. Furthermore, we can consider at least two mechanisms of perceptual error around stimulus boundaries. First, we can assume that the part of the nervous system that reports sensory input is only error prone within the perceptual limits within which it operates. A perceptual error that may have taken place beyond the stimulus boundary (Xmin or Xmax in our model) is effectively ignored, so that the relative frequency of the remaining events is raised (fig. 1A). Alternatively, it may be that errors that could occur beyond the stimulus boundary are effectively classed as having the most extreme possible values (fig. 1B), so that, for example, perceptual errors that would be “blacker than black” if they were possible are instead all classed as black. The general qualitative pattern of results is not changed by either of these assumptions, at least for the evaluations described in this article. There are, of course, other ways that receivers may become error prone close to stimulus boundaries. It seems clear to us that the general question of how to model perceptual error near boundaries is an important one, worthy of further consideration. In addition, if we had assumed Poisson probability distributions instead, by their nature they do not have long tails at low mean values, and many of our more novel results would stand. We could, in contrast, assume that predators have perfect recognition accuracy but that there is variation in the phenotypic expression of the prey. In this interpretation of the model, we again can assume that stimulus boundaries will be truncated, and again we could assume that “impossible” phenotypes are not realized within prey (so that relative frequencies of remaining phenotypes are higher; fig. 1A) or that they accumulate at the boundaries (fig. 1B). Our claim is that, in nature, stimulus boundaries must exist, so that truncation of some probability density functions is actually an important part of a signal detection model that deals with mimicry. When the standard deviation of generalization curves is large relative to the available phenotype space, we found some unexpected results, such as stability with widespread polymorphism (or, more

E10 The American Naturalist generally, with wide phenotypic variation) and dimorphisms in which some prey had minimal resemblance to the model (X p 0) whereas many others were brighter than the model (X 1 5). The extent to which these predictions have any relation to real systems depends crucially on the scaling of phenotype space and the width of probability density functions that apply.

applies whether or not a prey is completely protected from predation by mimicry. In contrast, when the cost of mimicry is one of raised encounter rate with predators caused by visual conspicuousness, it cannot affect fitness when predators always decline to attack.

Conspicuousness Costs and Optimal Mimicry It seems likely that for many species that evolve Batesian mimicry, the ancestral mimetic state was, in fact, relatively cryptic (see arguments in Charlesworth and Charlesworth 1975a, 1975b, 1975c), so that conspicuousness is a major cost of Batesian mimicry. Hence, the application of conspicuousness costs to our model are realistic and the predictions therefore potentially important. When generalization takes relatively moderate values (j p 0.5), the most abundant mimic form is often predicted to be imperfect (i.e., not equal to that of the model) and less conspicuous than the model it copies (fig. 3). In optimizing mimetic phenotypes, the prey seek to gain the maximum level of protection for the smallest conspicuousness cost incurred. If a change in conditions increased the survival benefits from mimicry (e.g., increased costs of attacks on or increased density of the model), so the optimal form of the mimic declines in value, taking it further away from the state of perfect mimicry (X p 5), to a phenotype that has reduced costs of conspicuousness. Conspicuousness is, of course, merely one form of cost that mimicry can impose. Turner (1977) and Holen and Johnstone (2004), for example, provide excellent discussions of the range of costs to which mimicry may pertain. However, conspicuousness is arguably a special case in the sense that it affects the same component of fitness as mimicry, namely, survival from the threat posed by a set of predators. Thus, for example, we found that if there were cases in which prey were completely protected from predation by mimicry, they were not affected by the conspicuousness costs that mimicry imposed. We had to assume that the minimum attack rate on models without mimics was greater than 0 for conspicuousness to have a noticeable effect on optimized levels of mimicry. Having to assume that even defended prey are subject to some small level of predation is not, in our view, unreasonable, given the need for predators to occasionally attack prey to gain information about edibility and mimicry. However, this does, in our view, mark out conspicuousness as kind of cost that acts differently with the general costs envisaged by Holen and Johnstone (2004) in their signal detection model of mimicry. Their “generalized” cost of mimicry is decoupled from survival probability and

Mimetic Dimorphisms and Conspicuousness Conspicuousness enables us to predict evolutionary stability, with some prey being highly cryptic and having a very poor resemblance to the model and others being conspicuous and showing much closer mimetic resemblance. The idea of stable, frequency-dependent mimic-nonmimic dimorphisms within species of Batesian mimicry is, of course, not new (see Turner 1977). Stable equilibria are possible whenever the morphs within a population suffer sufficiently reduced fitness with increased frequency such that a balanced polymorphism is possible (Turner 1978). In Batesian mimicry we might, for example, expect to see stable dimorphisms between a mimic and its ancestral, presumably relatively cryptic, nonmimetic form. The well-known examples in which mimics are clearly and commonly dimorphic in this way is through sex limitation, in which Batesian mimicry is often shown only in females, though the causes of sex-biased mimicry are still uncertain (Wallace 1889; Hespenheide 1975; Turner 1977, 1978, 1980; Ohsaki 1995, 2005). However, crypsis-mimicry dimorphisms are seen in the rare examples in which female-limited mimicry is itself expressed in only a proportion of the females within a population. In Papilio polytes, for example, brightly colored mimetic females coexist with relatively dull-colored nonmimetic females (Kunte 2008). The costs of mimicry here may well include conspicuousness because the mimetic forms are a lot brighter than the nonmimics, though reduction in the physiological life span of mimetic females has been suggested as one cause of the dimorphism (Ohsaki 2005). Nonetheless, this mimic-nonmimic dimorphism is rarely seen in the natural world. Why is this? One explanation is that mimicry often evolves because it protects prey that through their behavior have sufficient foraging incentives to become behaviorally conspicuous anyway. Just this scenario has been used by Gilbert (1991) to explain how pollen feeding and mimicry evolved in Heliconius and Laparus species. We demonstrated this idea in our model, by increasing rmin and showing that this could lead to the removal of prey with low phenotype values (X near 0), so that the mimic-nonmimic dimorphism is reduced to a mimetic monomorphism.

Conspicuous Batesian Mimicry E11 Methods of Modeling Recently published analyses of signal detection theory and mimicry use adaptive-dynamics equations (Sherratt 2002; Holen and Johnstone 2004, 2006) to evaluate the evolutionary stability of mimicry under signal detection assumptions. This involves looking at the invasion probabilities of single mimetic morphs in the context of morphs that already exist. Our modeling approach is arguably conceptually simpler than that used in adaptive dynamics. An important property of our evolutionary approach is that it allows any distribution of mimics to be easily considered, so that it is possible to evaluate the effects of alternative and complex starting distributions on the resulting evolutionarily stable state; as we have seen in the preceding sections, initial conditions can sometimes have profound effects on the predicted evolutionarily stable outcomes. We have deliberately avoided modeling the specifics of particular mimicry systems, as has characterized some of the seminal theoretical work on mimicry in the past (Charlesworth and Charlesworth 1975a, 1975b, 1975c). In these articles Charlesworth and Charlesworth evaluated how the evolutionary genetics of specified butterfly species such as Papilio dardanus may have evolved. Some of their simulations suggested that the particular genomic struc-

ture of the butterfly species may explain the persistence of imperfect mimicry. Our aim, by contrast, was to construct a much more general model that attempts to find the theoretical optimum state for mimicry in the absence of genetic constraints such as linkage disequilibrium and allelic dominance. As in other recent models of signal detection and mimicry, ours could be adapted to specific situations if the data were available. Finally, we argue that because our method is relatively simple both conceptually and computationally, it would be easy to extend it, for example, to look at other aspects of mimicry that may not be easily approached by analytical methods, such as modelmimic coevolution and multidimensional description of mimetic patterns.

Acknowledgments This work was initiated during a visit by M.P.S. to the lab of T. Sherratt, funded by a Leverhulme Trust research fellowship and by Natural Environment Research Council (NERC) grant NE/D010667/1. M.P.S. would like to express his gratitude to T. Sherratt, the NERC, and the Leverhulme Trust for their support. We thank two referees for very helpful advice on earlier drafts.

APPENDIX Alternative Methods of Simulation and Illustrations of Stability Method for Alternative Truncation of Distribution Boundaries Here the “missing” parts of the Gaussian probability distribution are added to the nearest extreme boundary value (fig. 1B). To calculate the value to add to the boundary in a standardized manner, we first calculated the total value of the curve around the model (between Xmin and Xmax) for a given value of j and took this to be the maximum (the model’s probability density values beyond the boundaries are trivially small). We then subtracted the area of each mimic curve from this maximum and added half of the difference to the most extreme values. The only real difference in the results is that when polymorphisms are stable, the variance of mimic frequencies around the mean X p 5 is smaller and the curve higher than when we run with the alternative truncation method (fig. 1A).

Alternative Method for Mutation We considered an alternative in which there is (1) a nonuniform distribution of mutations over the available phenotype range, in which mutations to more extreme values are less probable than mutations to less extreme values, and (2) no simple boundary but instead a reduction in the probability of a mutation occurring until some point at which we consider the probability so small so as not to matter. We imagined a Gaussian function of mean p 5 (midpoint of the phenotype range), which reaches very low values at Xmin, Xmax. We standardized heights so that the sum of frequencies is 1. For each prey, we next took a proportion (e.g., 10⫺7) of its abundance and distributed this across the whole phenotype range in proportion to the relative height of the Gaussian function for that X value. Now mutation toward the extremes is much less frequent compared to mutation toward the midpoint of the phenotypic range, and with the variance used in the curve (below), the proportion of a prey’s mutants that change into the most extreme phenotype is now very small (1.07 # 10⫺5).

E12 The American Naturalist

Figure A1: Comparison of stable frequencies for two truncation methods. j p 1 , Dmim p 2,000 , Dmod p 2,000 , c p 2 , and b p 1 . There is no mutation in this simulation. The black curve represents results with this alternative method of truncation, and the blue curve uses the method shown in figures 1–4.

Figure A2: Probability of mutation to phenotypes across the range.

Conspicuous Batesian Mimicry E13

Figure A3: Re-representation of figure 3 showing small cyclical variation in the frequencies of the most abundant form of the mimic. In all cases correlation of generations against frequencies is !0.0005. Dmim p 2,000 , Dmod p 2,000 , b p 1 , j p 0.5 , a p 0.01 , rmin p 0.01 , and f p 10 . A, Stable frequencies for c p 2 (blue), 5 (black), and 10 (red). B–D, Oscillations of the frequency of the most abundant forms for c p 2 , 5, and 10, respectively.

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