The Evolution of Mimicry under Constraints€¦ · mimicry is presented that incorporates such...

vol. 164, no. 5 the american naturalist november 2004 �

The Evolution of Mimicry under Constraints

Øistein Haugsten Holen1,* and Rufus A. Johnstone2,†

1. Centre for Ecological and Evolutionary Synthesis, Departmentof Biology, University of Oslo, P.O. Box 1050 Blindern, N-0316Oslo, Norway;2. Department of Zoology, University of Cambridge, DowningStreet, Cambridge CB2 3EJ, United Kingdom

Submitted March 24, 2004; Accepted July 21, 2004;Electronically published September 29, 2004

Online enhancement: appendix.

abstract: The resemblance between mimetic organisms and theirmodels varies from near perfect to very crude. One possible expla-nation, which has received surprisingly little attention, is that evo-lution can improve mimicry only at some cost to the mimetic or-ganism. In this article, an evolutionary game theory model ofmimicry is presented that incorporates such constraints. The modelgenerates novel and testable predictions. First, Batesian mimics thatare very common and/or mimic very weakly defended models shouldevolve either inaccurate mimicry (by stabilizing selection) or mimeticpolymorphism. Second, Batesian mimics that are very common and/or mimic very weakly defended models are more likely to evolvemimetic polymorphism if they encounter predators at high rates and/or are bad at evading predator attacks. The model also examineshow cognitive constraints acting on signal receivers may help deter-mine evolutionarily stable levels of mimicry. Surprisingly, improveddiscrimination abilities among signal receivers may sometimes selectfor less accurate mimicry.

Keywords: arms race, evolutionarily stable strategy (ESS), signal de-tection, polymorphism, aggressive mimicry, Batesian mimicry.

Many predators and parasites use aggressive mimicry todeceive their victims. Some examples are avian brood par-asites that lay mimetic eggs in the nests of other birds(Wickler 1968; Rothstein and Robinson 1998b; Davies2000), spiders that vibrate the webs of others in imitationof captured prey, thus luring the owners toward them tobe consumed (Jackson and Pollard 1996; Tarsitano et al.

* E-mail: [email protected].

† E-mail: [email protected].

Am. Nat. 2004. Vol. 164, pp. 598–613. � 2004 by The University of Chicago.0003-0147/2004/16405-40367$15.00. All rights reserved.

2000), and the Ophrys orchids, which have rewardlessflowers that mimic female wasps and bees in order toattract the males as pollinators (Kullenberg 1961; Wickler1968; Nilsson 1992). All of these aggressive mimics gainby exploiting behavioral patterns that the victims usuallydirect toward the model.

Prey organisms may also use mimicry in order to de-ceive their predators. In classical Batesian mimicry, pal-atable prey mimic poisonous or otherwise defended modelprey in order to deter attack. In this group we find hov-erflies that mimic wasps and bees, palatable butterflies thatmimic unpalatable butterflies (e.g., Wickler 1968), andmany myrmecomorphic (ant-mimicking) spiders (McIverand Stonedahl 1993; Cushing 1997).

Systems of aggressive mimicry and Batesian mimicrymay be understood as evolutionary arms races (Dawkinsand Krebs 1979) in which signal receivers (operators) areunder selection to improve discrimination between modelsand mimics and the mimics are under selection to appearmore similar to the models and thus more difficult todiscriminate against. Note that classical Mullerian mimicryis quite different because both model and mimic prey aredefended and benefit from sharing a warning signal, andthere is no selection on the predators to discriminate be-tween models and mimics.

The degree to which mimetic organisms resemble theirmodels varies quite a lot. Among avian brood parasites,the level of egg mimicry ranges from near perfect to non-mimetic (Wickler 1968; Rothstein and Robinson 1998a;Davies 2000). Many common hoverflies that are Batesianmimics show only a crude resemblance to their models,while other, less common hoverflies resemble their modelsvery closely (Azmeh et al. 1998; Edmunds 2000; Howarthand Edmunds 2000; Gilbert in press). Edmunds (2000)suggests that a similar pattern may be found among myr-mecomorphic spiders.

Several hypotheses have been put forward to explain theexistence of inaccurate mimics. For instance, it has beenproposed that inaccurate mimics may be “general mimics”that resemble several model species crudely but none veryclosely (e.g., Edmunds 2000; Pekar and Kral 2002; Sherratt2002), or that mimics may be kept from evolving accuratemimicry by kin selection (Johnstone 2002). Other expla-

Evolution of Mimicry under Constraints 599

nations have focused on the cognitive apparatus of theoperator: Dittrich et al. (1993) suggested that what is onlya crude resemblance to human observers may appear quiteconvincing to a predator. Moreover, the combination ofcognitive constraints and environmental noise can makeit impossible for an operator to simultaneously maximizeits chances of correctly identifying both model signals andmimic signals, and a trade-off between the two may exist.Thus, if it is very costly to erroneously classify a model asa mimic, and there is only a small benefit to be gainedfrom correctly identifying a mimic, operators may benefitfrom being susceptible to deceit by mimics (Wiley 1994),which again may weaken selection for accurate mimicry.For instance, predators may avoid attacking inaccurateBatesian mimics if they resemble very noxious models(e.g., Edmunds 2000; Holloway et al. 2002; Sherratt 2002).Similarly, it could benefit male bees and wasps to be sus-ceptible to deceit by sexually deceptive orchids if this re-duced their chance of erroneously rejecting real females;being duped by an orchid may only cost some time andenergy (Kullenberg 1961; Peakall 1990), while the cost ofmissing a real mating opportunity could be greater. If dis-crimination per se carries a cost, for instance, related toincreased vigilance or to the maintenance of a specializedcognitive apparatus, then operators with only a little togain from discrimination may be selected to give up dis-crimination entirely. Thus, there will be no selection foraccurate mimicry.

The evolutionary lag hypothesis provides a nonadaptiveexplanation for inaccurate mimicry (e.g., Rothstein andRobinson 1998a; Edmunds 2000; Holloway et al. 2002).It suggests that the mimics simply lag behind in the evo-lutionary arms race; more accurate mimicry is adaptivebut has not yet appeared and increased in frequency inthe population due to lack of suitable genetic variation ortime for natural selection to work. The amount of evo-lutionary lag may also be affected by evolution of themodel population. In systems of Batesian mimicry, thepresence of mimics is generally thought to weaken theprotection from predation given by the model signal.Therefore, the model organisms may in fact be under se-lection to become more easily distinguishable from themimics, giving rise to a separate coevolutionary race(Vane-Wright 1976; Gavrilets and Hastings 1998; Holm-gren and Enquist 1998). However, if models benefit fromthe presence of mimics, as when an aggressive mimic sim-ulates the signal properties of a prey item that the operatorconsumes, the model may in fact be under selection tobecome more similar to the mimic (Vane-Wright 1976).

Typically, predictions about equilibrium levels of mim-icry are made under the implicit assumption that evolutionmay improve mimicry at virtually no cost, that is, withoutaffecting other activities or functions of the mimic in a

negative way (but see Servedio and Lande 2003). This maybe a realistic assumption in some cases, as when mimicstrick operators by using emitted signals of short duration,such as the light flashes used by Photuris fireflies (e.g.,Haynes and Yeargan 1999) or the mechanosensory stimuliused by web-invading spiders. Although the mimic mightpay an energetic cost per emitted signal, and there maybe costs related to the maintenance of the signaling ap-paratus, the difference in costs between a well-tuned mi-metic signal and a badly matched signal may still be neg-ligible. It is hard to see why a small evolutionary tweakof, say, the temporal properties of a signal must be costlyto the mimic even if it could greatly improve the chancesof deceiving the operator.

However, there are also many ways in which traits thatimprove mimicry may confer significant costs on theircarriers. Selection for mimicry can entail radical pheno-typic changes in the mimic, including changes in body sizeand shape, modifications and loss of body parts, andchanges in color, scent, and behavior (e.g., Wickler 1968;McIver and Stonedahl 1993), all of which may potentiallyinterfere with other activities or functions of the mimeticorganism. For instance, selection for mimicry has lead toa reduction or loss of wings in many myrmecomorphicinsects and to a more narrow body shape and reducedfecundity in many myrmecomorphic spiders (McIver andStonedahl 1993; Cushing 1997). Mimetic traits may po-tentially reduce mating success, as in butterflies, wherecolor patterns are important both for courtship and mim-icry (e.g., Turner 1978). Eggs of the common cuckoo Cu-culus canorus often closely match the colorations andmarkings of host eggs but nevertheless tend to be largerthan host eggs (Davies and Brooke 1988; Rothstein 1990).Good mimicry of egg size may be costly to the parasite,because reductions of egg size in birds may reduce hatch-ing probability, size at hatching, and survival in the earlynestling stage (Wilson 1991; Williams 1994). In small or-ganisms with a high surface-to-volume ratio, changes inbody color and brightness may significantly affect the ab-sorption of heat radiation from the environment. Bodycolor and brightness have been found to influence ther-moregulation in spiders (Oxford and Gillespie 1998), but-terflies (Shreeve 1992), beetles (Brakefield 1985), and hov-erflies (Holloway et al. 1997; Ottenheim et al. 1999), allof which are groups well known for containing many mi-metic species.

Costs of mimicry are not necessarily restricted tochanges in morphological traits. Behavioral mimicry mayrequire a mimic to spend more time performing certaintasks than it otherwise would. For instance, some hov-erflies mimic the flight behavior of their models whileforaging, possibly compromising their superior flight abil-ities (Golding and Edmunds 2000; Golding et al. 2001).

600 The American Naturalist

Figure 1: Operators estimate the trait values of encountered models andmimics and classify all those with perceived trait values above their re-sponse threshold t as mimics and all below as models. The true traitvalue of the models is standardized to 0, and the true trait value of themimics is denoted m. Mimic trait values closer to 0 represent moreaccurate mimicry. Trait value estimates follow a normal distribution cen-tered on the true value (as shown), with a standard deviation j thatreflects perceptual errors. The dark gray areas give the probability oferroneously classifying a mimic as a model, while the light gray areasgive the probability of erroneously classifying a model as a mimic. Asillustrated by a and b, setting a lower threshold increases the probabilityof correctly identifying mimics but simultaneously decreases the prob-ability of correctly identifying models. As illustrated by a and c, if op-erators evolve a better cognitive machinery (i.e., if j decreases), they maybe able to increase the probability of correctly identifying both modelsand mimics.

Myrmecomorphic spiders often walk with a more erratic,ant-like gait, which they abandon when disturbed, prob-ably in order to escape predators after being discovered(Cushing 1997).

In this article, evolutionary game theory (MaynardSmith 1982) is used to explore the equilibrium level ofmimetic resemblance in several different contexts (includ-ing both aggressive mimicry and Batesian mimicry), as-suming that closer similarity to the model entails fitnesscosts for the mimic.

The Model

We present a general model of mimetic evolution appli-cable to both aggressive and Batesian mimicry. Severaldifferent model variants will be considered for each typeof mimicry, characterized by different assumptions abouthow the fitness of a mimic depends on its ability to deceivesignal receivers (operators). We start by outlining the basicassumptions of the model that are common to all variants.

The discrimination task of the operators is modeledusing signal detection theory (e.g., Egan 1975; Wiley 1994).Operators encounter models and mimics at fixed relativefrequencies ( ) and P, respectively, and attempt to1 � Pdiscriminate between them based on a continuous traitsuch as size, width, or brightness. Perception is assumedto be imperfect due to environmental noise and limits tothe cognitive abilities (e.g., resolution of the sense organs)of the operators. Formally, it is assumed that operatorestimates of individual model and mimic trait values fol-low a normal distribution centered on the true values withstandard deviation j, which represents the degree of per-ceptual error. The operators classify all individuals withperceived trait values higher than some threshold t as mim-ics and the rest as models (fig. 1). Setting a high thresholdincreases the chance of correctly classifying models butreduces the chance of correctly classifying mimics. A lowthreshold has the opposite effect (fig. 1).

The trait value of the model organisms is standardizedto 0 (with no loss of generality) and is assumed fixed inthe model. The latter assumption, although common inmodels of mimicry (e.g., Rodrıguez-Girones and Lotem1999; Johnstone 2002; Sherratt 2002), is simplistic. Nev-ertheless, there is some biological justification for it; themodel organism may be constrained by its need to becorrectly identified by operators (Nur 1970).

Mimic trait values (denoted by m) are by conventionpositive and are measured relative to the trait value of themodels. Thus, the closer a mimic trait value is to 0, thebetter the mimicry is. However, the evolution of bettermimicry is assumed to interfere with other functions ofthe mimetic organism; a more accurate mimic will be bet-ter at deception but will have reduced fitness in other

contexts. The exact relation between mimetic trait and costwill become clear later.

The mimic trait value is assumed to be determined bygenotype; thus, the population level of mimicry changeson an evolutionary timescale. In contrast, operators areassumed to adjust their discrimination strategies on amuch faster timescale; they adjust their thresholds behav-iorally in an optimal manner given the currently typical


level of mimicry. Such strong separation of timescales isappropriate when operator discrimination is based onlearning from direct encounters. However, if predatorymimics are very efficient at capturing duped victims orBatesian mimics simulate models that are extremely dan-gerous to a predator (e.g., poisonous coral snakes, Wickler1968), learning may be difficult, and innate recognitionmay evolve instead (e.g., Smith 1975), which falls outsidethe scope of this article.

Operator Fitness Functions: Aggressive versusBatesian Mimicry

Our model variants fall into two main categories, aggres-sive mimicry and Batesian mimicry, which are character-ized by different operator fitness functions.

Consider, first, the case of aggressive mimicry. Aggres-sive mimics take advantage of operators by simulatingmodel organisms that the operators benefit from inter-acting with. Hence, we assume that interacting with amodel organism yields a benefit B to the operator and thatinteracting with a mimic entails a fitness cost C. If a modelor mimic is rejected, the payoff is 0. The operator interactswith all models and mimics whose perceived trait valuefalls below t, and the others are rejected. The probabilitiesthat a model and that a mimic are perceived to have atrait value below t are equal to Z(t) and , re-Z(t � m)spectively, where Z denotes the cumulative distribution ofN(0, j2). We follow earlier approaches (e.g., Greenwood1986; Johnstone 2002; Sherratt 2002) and assume that theoperator maximizes expected fitness gain per encounter.This may be a reasonable assumption when search timeis long relative to handling time or when future benefitsare strongly discounted (shortsighted decision rules mayalso be more psychologically realistic; Stephens [2002] sug-gests that shortsighted rules may be optimal when animalssuffer information-processing constraints). The expectedfitness gain per encounter (our measure of fitness) is thengiven by

w p (1 � P)Z(t)B � PZ(t � m)C. (1a)op

By setting , checking second-order conditions,�w /�t p 0op

and introducing the “mimetic load” K, the optimal thresh-old as a function of mimic trait value m is found as∗t

2m ln (K)j PC∗t p � , where K p . (1b)2 m (1 � P)B

The mimetic load is 0 in the absence of mimics and in-creases with mimic encounter rate P and the cost C ofresponding to mimics. It may be interpreted as follows: ifwe think of model organisms as emitting—possibly un-

intentionally—a signal that informs operators of the ben-efits to be gained by responding, then the mimetic loadrepresents the extent to which the operators should distrustthat information.

Now consider the case of Batesian mimicry. Batesianmimics simulate noxious prey in order to deter attack bypredators. We thus assume that attacking a mimic yieldsa fitness benefit B to the operator and attacking a modelyields a fitness cost C, while ignoring an encountered preyitem yields a zero payoff. The predator attacks all modeland mimic prey with a perceived trait value 1t (fig. 1),and the rest are left in peace. The probabilities of attackingencountered model prey and mimic prey are, respectively,

and . The expected fitness gain to1 � Z(t) 1 � Z(t � m)the operator per prey encounter is

w p P[1 � Z(t � m)]B � (1 � P)[1 � Z(t)]C. (2a)op

The optimal threshold as a function of mimic trait value∗tm can now be found as

2m ln (K)j PB∗t p � , where K p . (2b)2 m (1 � P)C

This equation is identical to (1b) except that B and C haveexchanged places in the expression for the mimetic loadK. Here, K increases with the mimic encounter rate P andthe benefit B of attacking mimics and decreases when thecost of attacking models increases. Again, the mimetic loadmay be understood as the extent to which the informationprovided by the model’s signal (here a warning signal)should be trusted.

Mimic Fitness Functions

In all model variants, constraints on the evolution of mim-icry are incorporated in the following way. A function S(m)gives the fraction of the maximum possible fitness gain,given by the constant b, that can be obtained by a mimicwith trait value m (fig. 2). When the mimic trait valueequals mopt, which, for instance, could be a physiologicaloptimum, the mimic may obtain the full fitness gain b(i.e., ). Formally, the trait value mopt is a uniqueS(m ) p 1opt

global maximum point of S(m), and S(m) falls off on eachside of mopt, enabling only smaller portions of the benefitb to be realized. For trait values lower than or equal tosome threshold mmin, no benefit at all may be realized. Itis assumed that S(m) is differentiable on (mmin, �) and logconcave on (mmin, mopt). Furthermore, in our explorationof the model, we have restricted the range of parametersso that and .�2j ≤ m ! m ≤ 6j (m � m ) ≤ 6jmin opt opt min

Additional assumptions about the mimic fitness func-tion differ between model variants. We consider three ver-


Figure 2: The mimics are assumed to pay a fitness cost that increasesthe more closely the trait value of the models is mimicked. Mimic traitvalues closer to 0 represent more accurate mimicry. In the absence ofoperator discrimination, the trait value mopt is optimal for the mimics.The function S(m) represents the fraction of some benefit b that may beobtained by a mimic with trait value m. If , no benefit may bem ! mmin

obtained. Some possible shapes of the function S(m) are shown, andmmin may be either smaller or larger than the model trait value (whichis 0). The dashed line shows a second-order polynomial; we have usedsecond-order polynomials for the function S(m) in figures 3–6.

Table 1: Mimic fitness functions

Scenario Function

Aggressive mimicry:

A1 (obligate aggressive mimicry, multiple victims) w p Z(t � m)S(m)bmA1

A2 (obligate aggressive mimicry, single victim)�nZ(t�m)w p [1 � e ]S(m)bmA2

A3 (facultative aggressive mimicry) w p (1 � r)Z(t � m)S(m)b � rS(m)bmA3

Batesian mimicry:

B1 (single reproductive bout)�nq[1�Z(t�m)]w p e S(m)bmB1

B2 (many reproductive bouts)S(m)b

w pmB2 1�l�lq[1�Z(t�m)]

sions of the model that deal with aggressive mimicry: Sce-nario A1 (multiple victims) focuses on an obligateaggressive mimic that obtains an additive fitness benefitfrom each operator it deceives. The resources obtainedthrough deception are necessary for survival and/or re-production and cannot be gained by other means. ScenarioA2 (single victim) focuses on an obligate aggressive mimicthat need only trick one victim before some fixed periodof time has passed in order to obtain a fitness benefit. Theaverage number of encounters with operators in the periodis denoted n. This scenario is an illustrative contrast toscenario A1 and is motivated by the fact that in manycases it would be unrealistic to assume that the total fitnessgain obtained increases proportionally with the numberof victims deceived. For example, the male function (i.e.,pollinia removal) of the flower of a deceptive orchid maybe completed after one or two pollinator visits (e.g.,

O’Connell and Johnston 1998; Johnson and Edwards2000); similarly, the orchid may gain little from receivingpollen more than once on a single flower (Nilsson 1992;O’Connell and Johnston 1998; Schiestl and Ayasse 2001).Finally, scenario A3 (facultative) focuses on a facultativeaggressive mimic that faces the same challenge as in sce-nario A1 but that is sometimes able to obtain fitness ben-efits without deceiving victims. For instance, it may obtainthe benefit by force in a proportion r of encounters, or itmay exploit a second victim species that does not discrim-inate between models and mimics and which has a relativeabundance equal to r (scenario A1 is thus a special caseof A3, with r set equal to 0; we shall treat the two separately,however, because they yield very different results).

We consider two versions of the model that deal withBatesian mimicry. Scenario B1 (single reproductive bout)focuses on a mimic that must avoid being killed by pred-ators for some fixed period of time before it obtains afitness benefit. This may apply to mimics that must surviveuntil the end of the season or until reaching another lifestage before starting to reproduce. The average number ofpredators encountered in the period is denoted n. ScenarioB2 (multiple reproductive bouts) focuses on a mimic thatgains time-discounted benefits as long as it continues toescape predators (the discount factor is denoted l). Thismay apply to mimics that reproduce in many small boutsunder continuous risk of predation. In both scenarios, amimic has a probability of evading capture even(1 � q)if it fails to dupe a predator.

The precise mimic fitness functions for each scenarioare summarized in table 1; further explanation of the der-ivations are given in the appendix in the online edition ofthe American Naturalist.

Analysis

First, consider the optimal discrimination strategy of theoperators. The optimal rejection threshold is given by∗t(1b) and (2b) for aggressive and Batesian mimicry, re-spectively. For any fixed level of mimetic resemblance, thethreshold decreases as mimetic load K increases. More-∗t


Figure 3: Mimic nullclines (a–c) and operator nullclines (b, c) are shownfor different choices of parameters, detailed below. The mimic nullclinesgive the optimal trait value corresponding to each operator threshold∗mt, while the operator nullclines give the optimal threshold correspond-∗ting to each mimic trait value m. a, Mimic nullclines in scenario A1 (solidlines) and scenario A2 (dotted lines) are shown for comparison. b, c,Operator nullclines (dashed-dotted lines) are shown superimposed onmimic nullclines (solid lines). Due to strong separation of timescales(evolutionary versus behavioral timescale), the dynamics are simplified,and the trajectories follow the operator nullcline (see main text for furtherexplanation). The mimic trait values at the intersections of operator andmimic nullclines are monomorphic evolutionarily stable strategies, andall except the one marked with a circle in c are also attainable by evolution.Parameter values: . a, Solid lines (scenario A1), top to bottom:j p 1

, ; , . Dotted lines (scenario A2),m p 1 m p 4 m p �0.5 m p 1.5min opt min opt

top to bottom: , , ; , ,m p 1 m p 4 n p 12 m p 1 m p 4 n pmin opt min opt

; , , ; , , .4 m p �0.5 m p 1.5 n p 12 m p �0.5 m p 1.5 n p 4min opt min opt

b, Scenario A1: , . Dash-dotted lines, left to right:m p 1 m p 4min opt

, , , . c, Scenario A2:ln (K) p 5 ln (K) p 1 ln (K) p �5 ln (K) p �10, , . Dash-dotted lines, left to right:m p �1 m p 3 n p 10 ln (K) pmin opt

, .2 ln (K) p 0.2

over, if K11, then will also decrease if mimicry improves.∗tIf , then will decrease as mimicry improves until∗K ! 1 t

, below which will start to increase2 1/2 ∗m p [�2j ln (K)] tas mimicry improves further.

Now consider the mimics. A mimic trait value is anevolutionarily stable strategy (ESS) if it cannot be invadedby a mutant with another trait value when adopted by allthe members of a population (Maynard Smith 1982). Weassume that a mutant will have a negligible effect on theoptimal operator threshold when rare. Thus, for a mimic∗ttrait to be an ESS, it is necessary (but not sufficient)∗mthat it satisfies

�wm ∗ ∗(m, t (m )) p 0, (3)F�m ∗mpm

where wm is the mimic fitness function in question (table1). Any strategy pair ( , ) that simultaneously solves∗ ∗m t

and also satisfies equation (3).�w /�m p 0 �w /�t p 0m op

In order to be an ESS for a given mimetic load, a mimictrait must both satisfy equation (3) and be a global∗mmaximum point of wm(m) when operators set the thresh-old ( ). Since for all , S(m) de-∗ ∗t m S(m) p 0 m ≤ mmin

creases for , and strictly decreases for allm 1 m Z(t � m)opt

m, it is easy to see from the fitness functions that anymimic ESS m∗ must satisfy .∗m ! m ! mmin opt

In the following analyses, we will identify the endpointsof the evolutionary process (i.e., the mimic ESSs) for thedifferent model variants. We will check evolutionary at-tainability by assuming adaptive dynamics, that is, that therate of change in a trait is proportional to the slope of theassociated fitness function (Hofbauer and Sigmund 1998).In the (t, m) plane, we will refer to as the�w /�t p 0op

operator nullcline (i.e., t nullcline) and as�w /�m p 0m

the mimic nullcline (i.e., m nullcline) because they definewhere the rate of change in the relevant trait is 0. Becauseoperators are assumed to adjust their threshold optimallyon a very fast timescale, we can simplify the dynamics andassume that the pair of population strategies (t, m) willalways be positioned on the operator nullcline. Assumingthat mutations in the mimic population have only smallphenotypic effects and that K is constant, we may thinkof the populations of operators and mimics as movingalong the operator nullcline, with the operators constantlyadjusting their thresholds as the mimic trait value evolves.

For mathematical convenience, obligate aggressivemimicry (scenarios A1 and A2) will now be analyzed sep-arately from facultative aggressive mimicry and Batesianmimicry (scenarios A3, B1, and B2).

Scenarios A1 and A2: Obligate Aggressive Mimicry. In


Figure 4: The mimic nullclines of scenarios A3, B1, and B2 are shownfor different parameter choices. The solid line sections represent mimictrait values that (locally) maximize fitness at different operator thresh-∗molds t, while dotted sections represent trait values that (locally) minimizefitness. Selection for accurate mimicry is strongest at intermediate rejec-tion thresholds and when the “incentive for deception” is high. Parametervalues: , , . a, Scenario A3, top to bottom:j p 1 m p �0.5 m p 3min opt

, , , . b, Scenario(1 � r)/r p 3/7 (1 � r)/r p 1 (1 � r)/r p 4 (1 � r)/r p 9B1, top to bottom: , , , . c, Scenario B2,nq p 0.5 nq p 1.2 nq p 2 nq p 3top to bottom: , , ,ql/(1 � l) p 0.5 ql/(1 � l) p 1 ql/(1 � l) p 10/3

.ql/(1 � l) p 20

scenarios A1 and A2, the mimic nullclines (i.e.,and ) give the unique op-�w /�m p 0 �w /�m p 0mA1 mA2

timal mimic strategy for each operator threshold t, and∗mit can be shown that in both scenarios strictly increases∗mwith t (see appendix). Figure 3a plots as a function of∗mt for some illustrative parameter values. As we should ex-pect, as n tends to 0, the nullcline becomes�w /�m p 0mA2

increasingly similar to the nullcline , im-�w /�m p 0mA1

plying that the optimal mimic strategies for the two sce-narios converge. This makes sense because the mimics inscenario A2 (“single victim”) are under increasingly strongselection pressure to trick each operator as n decreases andencounters with operators become rarer.

In figure 3b and 3c, operator nullclines �w /�t p 0op

given by equation (1b) are plotted for several differentmimetic loads K superimposed on a plot of mimic null-clines (fig. 3b) and (fig. 3c).�w /�m p 0 �w /�m p 0mA1 mA2

Each intersection of the nullclines represents a mono-morphic ESS for the mimic population at the correspond-ing mimetic load. Above the mimic nullcline, there is se-lection for lower mimic trait values (more accuratemimicry), so the populations will move downward alongthe operator nullcline. Below the mimic nullcline there isselection for higher trait values (less accurate mimicry),and the populations will move upward along the operatornullcline. Thus, all the ESSs in fig. 3b and 3c are attainableby natural selection except the one marked by a circle.When and , there will often be one suchm ! 0 K 1 1min

unattainable ESS in addition to an attainable ESS. How-ever, if the mimic and operator nullcline do not intersectfor any positive m, the mimics will instead race towardperfect mimicry, and the operators will respond by low-ering their thresholds toward minus infinity. The biologicalrelevance and interpretation of this outcome will dependon the biological context: For instance, if operators cannotraise offspring without interacting with models, the pop-ulation will crash if t becomes too low.

Scenarios A3, B1, and B2: Facultative Aggressive Mimicryand Batesian Mimicry. In scenarios A3, B1, and B2, themimic nullclines (i.e., , , and�w /�m p 0 �w /�m p 0mA3 mB1

) may represent both local fitness maxima�w /�m p 0mB2

and local fitness minima. It can be shown that there is aunique optimal mimic strategy close to mopt if the∗moperator threshold t is sufficiently high or low (see ap-pendix). The level of mimicry is most accurate when in-termediate operator thresholds are adopted.

We will refer to the expressions (scenario A3),(1 � r)/rnq (scenario B1), and (scenario B2) as the “in-ql/(1 � l)centive for deception,” because for any fixed operatorthreshold, the optimal level of mimicry is more accuratewhen these expressions are high. This is illustrated infigure 4, where the mimic nullclines ,�w /�m p 0mA3

, and are plotted in the (t,�w /�m p 0 �w /�m p 0mB1 mB2

m) plane for some different parameter values. The incen-tive for deception is a measure of how tightly the mimicsare locked into the ecological interaction with the oper-ators and plays a key role in determining the qualitativebehavior of the model. When the incentive for deceptionis low, the qualitative results are easily summarized. Thereis a unique optimal mimic strategy corresponding to eachoperator threshold (fig. 4), and each intersection of mimicand operator nullclines corresponds to a monomorphicESS, as in scenarios A1 and A2.


A high incentive for deception, by contrast, may leadto an S-shaped mimic nullcline, which for intermediateoperator thresholds has two local fitness maxima with alocal fitness minimum somewhere between them (fig. 4).Such S-shaped mimic nullclines may allow for dimorphicESSs (featuring two morphs that differ in their mimeticaccuracy), which will be investigated in the next section.(Exactly how strong the incentive for deception must bebefore the mimic nullcline becomes S-shaped depends onthe function S(m). If mmin is close to mopt, so that fitnessfalls off very fast as mimicry improves, then the mimicnullcline becomes S-shaped only at a very high incentivefor deception, or it may never do so; scenario B2 is par-ticularly sensitive to this effect. If the distance betweenmmin and mopt is greater, then an S-shaped nullcline arisesmore easily.) Our implementation of signal discriminationin the model presupposes positive mimic trait values; thus,our explorations of dimorphic ESSs are restricted to S-shaped nullclines that feature only positive mimic traitvalues.

Dimorphic ESSs. In the following, we will use m1 and m2

to denote any two mimic types in a dimorphic population(by convention, we will let ). We will assume thatm ! m1 2

evolution takes place on two timescales: selection causesgenotype frequencies to change fast in the population,while mutations arise only rarely. Thus, we may assumethat the genotype frequencies in a population always areat equilibrium when new mutations are introduced. Con-sequently, in order for a pair of mimic strategies (m1, m2)to constitute a dimorphic ESS, we require that the mimictypes should coexist at a stable equilibrium and be un-invadable by mutant strategies at that equilibrium.

Let r denote the proportion of m1 mimics and (1 �denote the proportion of m2 mimics. We will use (m1,

∗r) tm2, r) to denote the optimal operator threshold against adimorphic mimic population (as before, (m) will denote∗toptimal operator thresholds in monomorphic mimic pop-ulations). The optimal threshold (m1, m2, r) changes∗tmonotonically from (m2) to ( m1) as r changes from∗ ∗t t0 to 1 (see appendix). In the following, we will refer to apopulation state (m1, m2, r) as stationary if m1 and m2

have equal payoffs when operators adopt the optimalthreshold (m1, m2, r). A stationary population state (m1,

∗tm2, ) is stable if it also satisfies∗r

� �∗ ∗w (m , t (m , m , r)) ! w (m , t (m , m , r)) . (4)m 1 1 2 m 2 1 2[ ] [ ]∗ ∗�r �rrpr rpr

This condition says that the fitness of the two morphs isfrequency dependent in the sense that each morph has aslightly higher fitness than the other when present in amarginally smaller proportion than at the equilibrium r∗.

In order to constitute a dimorphic ESS, a dimorphicstrategy pair must coexist at a stable population state andbe uninvadable by mutants at this population state. If thestrategy pair has a unique stable population state, the di-morphic ESS can be characterized solely by its constituentstrategies m1 and m2. In scenarios B1 and B2, a stationarypopulation state is stable and unique if ∗ ∗t (m ) ! t (m )1 2

(see appendix). In scenario A3, a stationary populationstate is stable and unique if and m1 and∗ ∗t (m ) ! t (m )1 2

m2 are mutually invasible (see appendix). The two strat-egies m1 and m2 are mutually invasible when both

and∗ ∗ ∗w (m , t (m )) 1 w (m , t (m )) w (m , t (m )) 1m 1 2 m 2 2 m 2 1

.∗w (m , t (m ))m 1 1

Existence of Dimorphic ESSs. In the appendix, it is shownthat each S-shaped mimic nullcline features at least oneoperator threshold at which the corresponding pair of′tmimic strategies ( , ) lying on the upper and lower∗ ∗m m1 2

parts of the nullcline obtain equal payoffs. Since we arenot able to solve for m analytically, such�w /�m p 0m

strategy pairs must be found numerically; our explorationsof scenarios A3, B1, and B2 always resulted in a uniquesolution. Given that the upper and lower part of the mimicnullcline represent fitness maxima, such pairs are, whenstable, also uninvadable by mutants. Because (m1, m2,

∗tr) changes monotonically from (m2) to (m1) as r∗ ∗t tchanges from 0 to 1, a necessary condition for the stabilityof ( , ) is that must lie between ( ) and∗ ∗ ′ ∗ ∗m m t t m1 2 1

( ). By applying the additional stability conditions∗ ∗t m 2

given above for scenario B1 and B2, it is easy to see that( , ) is a dimorphic ESS if ; in∗ ∗ ∗ ∗ ′ ∗ ∗m m t (m ) ! t ! t (m )1 2 1 2

scenario A3 ( , ) is a dimorphic ESS if∗ ∗ ∗ ∗m m t (m ) !1 2 1

and and are mutually invasible.′ ∗ ∗ ∗ ∗t ! t (m ) m m2 1 2

In our exploration of the model, we found that S-shaped mimic nullclines intersected with operator null-clines once (the most typical case) or three times (thishappened most often when both ( ) and moptm � mopt min

were high). Due to the shape of the nullclines, it is clearthat the requirement must always∗ ∗ ′ ∗ ∗t (m ) ! t ! t (m )1 2

hold when the operator nullcline intersects with the S-shaped mimic nullcline only at a single point, and thispoint lies on the section of the mimic nullcline that runsfrom ( , ) to ( , ). Moreover, in each case where′ ∗ ′ ∗t m t m1 2

the nullclines intersected three times and was between′t( ) and ( ), we found that . In∗ ∗ ∗ ∗ ∗ ∗ ′ ∗ ∗t m t m t (m ) ! t ! t (m )1 2 1 2

our explorations of scenario A3, the strategy pair ( ,∗m1

) was almost always mutually invasible when∗m 2

. In the very few cases where mutual∗ ∗ ′ ∗ ∗t (m ) ! t ! t (m )1 2

invasibility did not hold, stability was confirmed usingcondition (20) in the appendix.

The Attainability of Dimorphic ESSs. In many cases, op-erator nullclines may intersect with S-shaped mimic null-


Figure 5: a, An S-shaped mimic nullcline is shown; dashed line sectionsgive mimic trait values m∗ that are global fitness maxima for differentoperator thresholds t, solid line sections give trait values that are localfitness maxima, and dotted line sections give trait values that are localfitness minima. Two operator nullclines are also shown (dashed-dottedlines); the upper intersects the mimic nullcline at a local fitness maximum,while the lower intersects at a local fitness minimum, representing anevolutionary branching point. Due to strong separation of timescales(evolutionary vs. behavioral timescale), the dynamics are simplified, andthe trajectories follow the operator nullcline (see main text for furtherexplanation). The two circles represent the two mimic types in the di-morphic evolutionarily stable strategy (ESS). b, c, The white areas rep-resent pairs of trait values (m1, m2) that are mutually invasible. Thesuperimposed vectors give the relative rate and direction of evolution inthe two morphs. b, The dimorphic ESS is attainable from a monomorphicstate using small mutational steps. This corresponds to the case in awhere the operator nullcline intersects the mimic nullcline at a localfitness minimum. c, The dimorphic ESS is not attainable from a mono-

morphic state using small mutational steps but requires an initial mu-tation with a large phenotypic effect. This corresponds to the case in awhere the operator nullcline intersects the mimic nullcline at a localfitness maximum. Parameter values: , , ,nq p 2 m p �0.5 m p 3min opt

. a, Upper operator nullcline: . Lower operator null-j p 1 ln (K) p 3.2cline: . b, . c, .ln (K) p 1.5 ln (K) p 1.5 ln (K) p 3.2

clines only at a point (t, m) that is a local fitness minimumfor the mimic population, that is, where (fig.2 2� w /�m 1 0m

5a). Monomorphic mimic populations will evolve closerand closer to this point, which is an evolutionary branch-ing point sensu Geritz et al. (1997, 1998). If the residentpopulation is close to the branching point and a mutanttype arises with a phenotypic value just on the other sideof the branching point, the mutant will invade but cannotreplace the resident population. The two mimic types willbe mutually invasible, each having higher fitness than theother when sufficiently rare, and will coexist in a “pro-tected dimorphism” sensu Geritz et al. (1998). When

, each protected dimorphism has a unique∗ ∗t (m ) ! t (m )1 2

stable population state.The evolutionary attainability of dimorphic ESSs has

been explored (numerically) for different sets of param-eters by using adaptive dynamics in which each morph ina protected dimorphism evolves at a rate proportional toits fitness gradient at the unique stable population state.The results when nullclines intersect only once may besummarized as follows. Dimorphic ESSs are attractors forsufficiently close strategy pairs and are thus attainable bynatural selection. The size of the basin of attraction de-pends on the mimetic load K. If the operator nullclineintersects the mimic nullcline at a fitness minimum, amonomorphic population will first evolve toward the evo-lutionary branching point, at which the population willdiverge into a protected dimorphism. The protected di-morphism will then evolve toward the dimorphic ESS. Thisis illustrated in figure 5b, which show mutually invasiblemimic strategy pairs for the same parameter choices as infigure 5a, with vectors superimposed that give the relativerate and direction of evolution in the two morphs. If theoperator nullcline instead intersects the mimic nullclineat a local fitness maximum, the monomorphic populationwill evolve toward it and remain there. In this case, thedimorphic ESS has a smaller basin of attraction, and onlymutations with large phenotypic effects can take the pop-ulation away from the local fitness maximum and give riseto a protected dimorphism that evolves toward the di-morphic ESS (fig. 5c).

By choosing the right parameter values and mimeticloads, one can find operator and mimic nullclines thatintersect more than once (not shown). Partial explorationof such cases indicates that dimorphic ESSs will still be


Figure 6: Evolutionarily stable levels of mimicry are plotted against, where K is the mimetic load. a, Scenarios A1 and A2 (aggressiveln (K)

mimicry): Solid line sections represent evolutionarily stable strategies(ESSs) that are attainable by natural selection, while dotted line sectionsrepresent unattainable ESSs. Increasing mimetic loads select for bettermimicry. If is negative (the three bottom lines) and K is sufficientlymmin

high, no ESS exists, and the mimics will race toward better and bettermimicry, forcing the operators to set lower and lower thresholds. b,Scenario A3 (aggressive mimicry) and scenario B1 and B2 (Batesian mim-icry), low incentive for deception: The lines represent ESSs that are at-tainable by natural selection. Intermediate mimetic loads select for moreaccurate mimicry, while high and low mimetic loads select for inaccuratemimicry. c, Scenario B1, high incentive for deception: Solid, curved linesections represent mimic trait values that are monomorphic ESSs, whilethe solid, straight line sections represent the two mimic types in thedimorphic ESS. All ESSs are attainable by natural selection. The dashedline sections represent mimic trait values that locally maximize fitness.Parameter values: . a, Top to bottom: scenario A2, ,j p 1 m p 1min

, ; scenario A2, , , ; scenario A1,m p 4 n p 12 m p 1 m p 4 n p 4opt min opt

, ; scenario A2, , , ; sce-m p 1 m p 4 m p �0.5 m p 1.5 n p 12min opt min opt

nario A2, , , ; scenario A1, ,m p �0.5 m p 1.5 n p 4 m p �0.5min opt min

. b, , . Dashed line, scenario A3,m p 1.5 m p �0.5 m p 3 (1 �opt min opt

. Dotted line, scenario B1, ; solid line, scenario B2,r)/r p 1 nq p 0.5. c, Scenario B1, , , ,ql/(1 � l) p 1 nq p 2 m p �0.5 m p 3 j pmin opt

(as in fig. 5).1

attainable, in the sense that strategy pairs that are suffi-ciently close to a dimorphic ESS will evolve closer to it.

The Relationship between Cognitive Abilities and MimeticAccuracy. Operators may in the long run evolve a fun-damentally better cognitive machinery that makes themperceive model and mimic trait values as more distinct.In the model, we may represent the evolution of a bettercognitive apparatus by reducing the standard deviation j

of the operators’ trait value estimates. When j decreases,the probability of correct model classification will increasefor a given probability of correct mimic classification.Thus, a better cognitive apparatus will always be beneficialfor an operator.

If operators evolve a fundamentally better cognitive ma-chinery, the optimal level of mimicry may change in∗mthe mimic population. In the appendix, we show (for sce-nario B1) that for positive , satisfying∗ ∗�m /�j 1 0 m

∗ 2 4 2� �m ! 2j � 2 j (ln (K)) � 1). (5)

For higher , by contrast, . Thus, at any given∗ ∗m �m /�j ! 0mimetic load K, a lower j will select for better mimicryonly if is sufficiently low.∗m

Results

The main results of the model are summarized in figure6, which shows evolutionarily stable levels of mimicry cor-responding to different mimetic loads K. In the case ofaggressive mimicry (scenarios A1–A3), the mimetic loadis low when mimics are rare relative to models and whenmimics impose low costs on operators. Under these cir-cumstances, the operators adopt a high threshold and willoften interact with crude mimics (“adaptive gullibility”;Wiley 1994). This favors inaccurate mimicry, which haslow cost. When the mimetic load is high, by contrast, theoperators adopt a low threshold and tend to be deceivedonly by very good mimicry (“adaptive fastidiousness”; Wi-ley 1994).

In the case of obligate aggressive mimicry (scenarios A1and A2), the only way that an aggressive mimic can obtaina fitness gain is by deceiving an operator. As a result, highermimetic loads and adaptive fastidiousness on the part ofoperators always select for better mimicry (fig. 6a). When

and the mimetic load is sufficiently high, no ESSm ! 0min

exists, and the mimics will race toward better and bettermimicry. However, such steady improvements in mimicrywill force the operator population to adopt lower andlower rejection thresholds (fig. 3c), and the mimicry systemwill at some point break down, because virtually no op-erators will respond to either models or mimics.

In the case of Batesian mimicry (scenarios B1 and B2),


the mimetic load is low when mimic prey is rare relativeto model prey and when attacks on mimics yield a benefitthat is small relative to the cost of attacking defendedmodels. Under these circumstances, the operators set ahigh attack threshold so that few prey will be attacked.Thus, costly mimicry is not necessary, and crude mimicsare favored (as in scenarios A1, A2, and A3). As the mi-metic load increases, there is at first selection for moreaccurate mimicry (fig. 6b), but at sufficiently high mimeticloads, crude mimics have greater fitness again. At thesehigh mimetic loads, predators are very likely to attack, andit may simply become too costly to evolve the level ofmimicry that is necessary for protection. Instead, the mim-ics profit from having an inaccurate but cheap level ofmimicry that will enable them, if by chance they escapethe attention of predators, to realize as much fitness benefitas possible.

The fact that the aggressive mimics in scenarios A1 andA2 must encounter and deceive operators in order to ob-tain fitness gains and the Batesian mimics in scenarios B1and B2 may obtain fitness gains without encountering asingle operator leads to opposite predictions for the levelof mimicry at high mimetic loads. However, if we allowfor the possibility that aggressive mimics can sometimesreceive a fitness benefit without deceiving the focal op-erator species, as in scenario A3, high mimetic loads willfavor crude mimics, just as in the case of Batesian mimicry(fig. 6b).

When mimics may potentially receive a fitness benefitwithout deceiving any operators, as in scenarios A3, B1,and B2, but the incentive for deception is high, the mimicpopulation may also evolve mimetic dimorphism at someintermediate mimetic loads. One mimic morph will beinaccurate (trait values close to mopt), while the other willbe much more accurate. Figure 6c summarizes the mosttypical case (when the nullclines only intersect once forany given K) for scenario B1: When the mimetic load islow or high, only a monomorphic ESS exists in the mimicpopulation (curved solid lines). When the mimetic load isintermediate, a unique dimorphic ESS exists (straight solidlines) but no monomorphic ESS. However, at some in-termediate mimetic loads, the mimic population may alsoreach a locally stable monomorphic level of mimicry(dashed lines); this can only be invaded by mutants withlarge phenotypic effects. Scenarios A3 and B2 lead to thesame qualitative results as shown for scenario B1 in figure6c.

For a given S-shaped mimic nullcline, the mimic traitvalues at the dimorphic ESSs are exactly the same overthe whole range of mimetic loads that give rise to di-morphism (fig. 6c). Thus, the trait values at a dimorphicESS are stable against small perturbations of mimetic load;no mutant may replace one of the pure strategies in the

support of the dimorphic ESS. However, the proportionof inaccurate mimics that will be present at a dimorphicESS will change when mimetic load is perturbed; the pro-portion rises from 0 to 1 as mimetic load increases. Thefitness values of the two morphs are constant over thisrange, which follows from the fact that the operators al-ways adopt the same threshold at the dimorphic ESSs. Therange of mimetic loads that give rise to dimorphic ESSsmay be very wide for a given mimic nullcline; in figure6c, K varies over an order of magnitude.

In order to illustrate what goes on at a dimorphic ESS,we have done some calculations for the dimorphic ESSsassociated with each of the seven S-shaped nullclinesshown in figure 4: The mimics’ expected payoffs are inthe range of 5%–31% of the expected payoff obtained bya hypothetical mimic that pays no costs of mimicry andalways succeeds in deception. Moreover, an accuratemimic is in each encounter much more likely to deceivean operator than an inaccurate mimic (eight to 1,009 timesmore likely). However, in the absence of discrimination(if all mimics were classified as models), the accuratemorph would, due to costs of mimicry, only obtain be-tween 42%–75% of the expected benefit obtained by theinaccurate morph.

Evolutionary improvements in cognitive abilities amongthe operators will affect the degree of mimicry at equilib-rium. This has been investigated in scenario B1. Supposethat a mimic population is monomorphic for a trait value

that is a local fitness maximum. If the operators be-∗mcome marginally better at discriminating between modelsand mimics (i.e., if j decreases slightly) and is suffi-∗mciently low, the mimics will come under selection for moreaccurate mimicry. In other words, better discriminationabilities select for better mimicry at equilibrium. On theother hand, if is high, better discrimination abilities∗mmay in fact select for less accurate mimicry at equilibrium.

Discussion

We have used evolutionary game theory and signal de-tection theory to analyze the evolution of mimicry underthe assumption that mimicry-enhancing traits confer costson their carriers. Two factors have turned out to be crucialfor predicting the degree of mimicry at equilibrium; bothreflect ecological conditions and emerge naturally as ag-gregated parameters in the model. The first predictive fac-tor, which we have called “mimetic load,” reflects the se-lection pressure imposed by mimics on operators. Thesecond predictive factor, called the “incentive for decep-tion,” reflects the extent to which the fitness of the mimicsdepends on their ability to deceive operators.

In Batesian mimicry, the mimetic load is low whenmimic prey are unprofitable and/or rare and when model


prey are very nasty or dangerous. In aggressive mimicry,the mimetic load is low when mimics are rare and/orimpose only low costs on their victims. The importanceof mimetic load for understanding mimicry systems isgenerally acknowledged. To be sure, the consensus view isthat Batesian mimics should be more successful at decep-tion (and be attacked less) at low mimetic loads (e.g.,Sheppard 1959; Nur 1970; Turner 1978; Huheey 1988;Lindstrom et al. 1997; Sherratt 2002). Likewise, aggressivemimics should be more successful at duping victims atlow mimetic loads (e.g., Dawkins and Krebs 1979; Wiley1994; Davies et al. 1996; Rodrıguez-Girones and Lotem1999; Holen et al. 2001). As mimetic load increases, how-ever, operators should become less gullible, which is ingeneral thought to increase selection pressure for bettermimicry, at least as long as the mimetic load does notreach a critical level (see below). In accordance with this,our model predicts inaccurate mimicry as the only evo-lutionary equilibrium at low mimetic loads, due to theassumed trade-offs between mimicry-enhancing traits andother aspects of fitness. As the mimetic load increases fromlow to intermediate, operators become more discriminat-ing, resulting in increasingly accurate mimicry at evolu-tionary equilibrium. However, mimicry will never be per-fect at evolutionary equilibrium: as in the model ofServedio and Lande (2003), the optimal mimic trait valuewill represent a compromise between the need to deceiveoperators and other aspects of fitness.

The second predictive factor—the incentive for decep-tion—has received less attention in mimicry theory butmay be crucial for understanding the selection pressuresacting on mimics at high mimetic loads. In Batesian mim-icry, the incentive for deception is high if the mimics oftenencounter operators and/or are easily captured by them.Thus, this factor is related to the amount of enemy-freespace that the mimetic prey has. In aggressive mimicry,the incentive for deception is high when mimics arestrongly dependent on deceiving victims in order to obtainresources necessary for survival and/or reproduction. Forinstance, a specialist avian brood parasite may be com-pletely locked into a coevolutionary arms race with its host,thus having a high incentive for deception, while a spider-hunting spider may use mimicry as only one of severalhunting techniques (e.g., Jackson and Pollard 1996), re-sulting in a lower incentive for deception. The modelshows that the level of mimicry at equilibrium increaseswhen the incentive for deception increases.

Batesian Mimicry at High Mimetic Loads

In the context of Batesian mimicry, it is thought that ahigh abundance of mimics relative to models may makeit difficult for predators to associate the warning pattern

with unpalatability, which could cause the protection frompredation to disappear completely both for models andmimics (e.g., Sheppard 1959; Nur 1970; Holloway et al.2002). It has therefore been proposed that a high abun-dance of mimics may lead to the evolution of mimeticpolymorphism because this will spread the parasitic loadover several defended prey species, making each warningsignal less diluted; alternatively, a high abundance maycause relaxed selection on mimetic perfectionism, leadingto inaccurate mimicry and/or high levels of phenotypicvariation in the mimetic trait (Sheppard 1959; Nur 1970;Huheey 1988; Holloway et al. 2002). Our model enablesus to predict the conditions that would favor one or theother of these two alternatives.

When the incentive for deception is low (i.e., mimicsrarely encounter predators and/or are difficult to capture),high mimetic loads should select for monomorphic in-accurate mimicry. When the incentive for deception is high(i.e., mimics often encounter predators and/or are easilycaptured), there may be selection for mimetic dimorphismwhen mimetic loads are high and monomorphic inaccu-rate mimicry when mimetic loads are very high. A mimeticdimorphism will consist of one inaccurate and one ac-curate morph (a so-called mimic-nonmimic polymor-phism; Turner 1978). The accurate morph maximizes itsprobability of deceiving predators, while the inaccuratemorph maximizes its fitness if by chance it should escapepredators. This result is in accordance with Turner (1978),who argued that stability of mimic-nonmimic polymor-phisms would depend on nonmimics being fitter thanmimics in the absence of models or predators, for instance,due to having better flight abilities or advantages in matechoice. Mimic-nonmimic polymorphisms are well-knownamong butterflies but typically in a sex-limited form withmimetic females and nonmimetic males (e.g., Wickler1968; Turner 1978). Sex-limited cases with mimetic malesand nonmimetic females have been reported in moths(Turner 1978) and beetles (Hespenheide 1975). Bothfemale-limited and male-limited mimicry are knownamong bumblebee mimics (Gilbert, in press).

To summarize, our model predicts that (1) Batesianmimics that are very common and/or mimic very weaklydefended models should evolve either inaccurate mimicry(by stabilizing selection) or mimetic polymorphism, and(2) Batesian mimics that are very common and/or mimicvery weakly defended models are more likely to evolvemimetic polymorphism if they encounter predators at highrates and/or are bad at evading predator attacks.

Note that these predictions concern patterns of mimicrythat are expected at evolutionary equilibrium but do notimply anything about the origin of mimetic interactions.Moreover, our model does not explicitly incorporateswitching to mimicry of other models than the focal


model. Nevertheless, there are good reasons to believe thatconditions that select for mimic-nonmimic polymor-phisms could also lead to mimicry of different models.Because the inaccurate morph in a sense has “given up”mimicry of the focal model, it may benefit strongly fromevolving similarity to other defended model species. Itwould be very interesting to extend our modelling ap-proach to multimodel mimicry; we suspect that prediction(1) and (2) may also hold for other types of mimeticpolymorphisms.

It has recently been pointed out that many commonBatesian mimetic hoverflies are rather poor mimics, whilerare mimics tend to have a closer resemblance to theirmodels (Azmeh et al. 1998; Edmunds 2000; Howarth andEdmunds 2000). Proposed explanations are that commonmimics are less precise because they are “generalist” mim-ics that resemble several model species but none veryclosely (Edmunds 2000; Sherratt 2002) and that commonmimics may be less precise due to kin selection effects(Johnstone 2002). Our model provides an alternative andquite general explanation for why common Batesian mim-ics should be inaccurate, though the different explanationsare not necessarily mutually exclusive.

A Batesian mimic that mimics an aposematic model maybe more conspicuous and be more often detected by pred-ators. Consequently, it will have a higher incentive fordeception. Therefore, we may expect polymorphisms tobe more widespread in Batesian systems where aposematicpatterns are mimicked (e.g., many butterflies and hover-flies) than in other Batesian systems (ant-mimicking spi-ders). Moreover, if accurate mimicry interferes with escapeabilities, for instance due to behavioral mimicry (e.g.,Golding and Edmunds 2000) or changes in body shape(e.g., Srygley 1994), polymorphisms may be more likely.A very intriguing pattern is present in mimetic hoverflies:bumblebee mimics are often polymorphic, while bee andwasp mimics are not (Gilbert, in press). Could it be thatbumblebee mimics are worse at evading predators whenattacked? Bumblebee mimics are also larger than bee andwasp mimics (Gilbert, in press); perhaps bumblebee mim-ics, due to their large size, are easier to detect and thusneed to deceive predators more often, which again mayselect for polymorphism.

Aggressive Mimicry at High Mimetic Loads

Obligate aggressive mimics cannot gain benefits in any wayother than through the deception of an operator. The pre-dictions for obligate aggressive mimicry are easily sum-marized: the mimetic accuracy at equilibrium always im-proves as mimetic load increases. However, note thatevolutionary equilibria are sometimes lacking when mi-metic load is sufficiently high; this gives rise to evolu-

tionary chases toward better mimicry and strongerdiscrimination.

In the facultative case, aggressive mimics may some-times obtain benefits without deceiving operators. Thisleads to essentially the same predictions at high mimeticloads as for Batesian mimicry: a low incentive for decep-tion selects for inaccurate mimicry, and a high incentivefor deception may select for mimetic dimorphism whenmimetic loads are high and monomorphic inaccuratemimicry when mimetic loads are very high.

At high mimetic loads, aggressive mimics impose a verystrong ecological pressure on operators. Many systems ofaggressive mimicry could probably not be maintained ata constant mimetic load sufficiently high to cause oper-ators to adopt the strict thresholds that make inaccuratemimicry and dimorphisms possible. Consequently, themodel predictions associated with high mimetic loads mayseem to be of limited value for aggressive mimicry. Thisline of argument, however, overlooks the possibility thatthe relative rate at which operators encounter mimics mayvary temporally. Operators would then benefit from ac-cepting most signalers (and reducing the risk of failing torespond to model organisms) when mimetic load is lowand discriminating more strongly (and reducing the riskof parasitism or predation) when mimetic load is high.Such flexible discrimination may cause mimics to mostlyencounter operators that adopt strict thresholds, thus mak-ing inaccurate mimicry and dimorphisms feasible evenwhen the mimetic load averaged over all periods is low.Flexible discrimination may easily follow from a contin-uous learning process or could be due to a phenotypicallyplastic response induced by environmental cues. Evidencefor cue-induced flexible discrimination is known fromavian brood parasitism: some hosts of the common cuckooCuculus canorus show stronger egg discrimination whenadult cuckoos have been spotted near the nest (e.g., Daviesand Brooke 1988; Moksnes et al. 1993).

Cognitive Abilities and Mimetic Accuracy

Although our model does not explicitly predict evolution-ary changes in cognitive abilities in the operator popu-lation, it does predict some effects that this would haveon the equilibrium level of mimicry. From an evolutionaryarms race perspective, one would expect improved dis-crimination abilities (e.g., better visual acuity) to select forbetter mimicry. Our model confirms this for Batesianmimics (scenario B1) that reside at an accurate evolu-tionary equilibrium, which certainly will be selected foreven better mimicry when discrimination abilities im-prove. However, improved discrimination abilities selectfor less accurate mimicry in populations that already resideat a sufficiently inaccurate evolutionary equilibrium. For


strongly constrained mimics, it may simply be too costlyto improve mimicry further when the discrimination abil-ities of the operators improve; instead, they benefit fromhaving an even cheaper level of mimicry. Thus, the op-erators may “win” this round of the evolutionary armsrace, forcing the mimics to shift to other strategies thanmimicry. This kind of collapse in investment by one sideas the other side’s investment increases has been predictedin several models of coevolutionary arms races (e.g.,Abrams 1986; Greeff and Parker 2000).

Conclusion

The explicit incorporation of costs of mimicry in ourmodel leads to several new predictions that have not beenanticipated beforehand. Most surprisingly, the model in-dicates that costs of mimicry may sometimes lead to theevolution of mimetic dimorphism. Thus, the effects ofsuch costs are not trivial. Our analysis shows that theevolution of mimicry under high mimetic loads dependsvery strongly on how tightly the mimics are locked intothe ecological interaction with the operators (i.e., on the“incentive for deception”). It suggests that enemy-freespace and escape abilities of mimetic prey are crucial fac-tors for understanding the evolution of inaccurate mimicryand polymorphisms among common Batesian mimics.

Acknowledgments

We thank C. Brinch, N. Davies, M. Enquist, G.-P. Saetre,T. Slagsvold, N. C. Stenseth, and J. O. Vik for helpfuldiscussions and comments on the manuscript; F. Schiestlfor valuable advice regarding deceptive orchids; and twoanonymous reviewers for their constructive comments andsuggestions. Financial support was provided by the Nor-wegian Research Council (to Ø.H.H.).

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