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Behavioral Ecology Vol. 13 No. 6: 821-826
© 2002 International Society for Behavioral Ecology
The evolution of imperfect mimicry
School of Biological and Biomedical Sciences, University of Durham, South Road, Durham DH1 3LE, UK
Address correspondence to T.N. Sherratt, who is now at the Department of Biology, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada. E-mail: sherratt{at}ccs.carleton.ca.
Received 2 October 2001; revised 18 March 2002; accepted 4 April 2002.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONAnalytical modelDiscussionREFERENCES |
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Examples of imperfect resemblance between Batesian mimics and their models appear widespread in the natural world, but so far few quantitative models have been proposed to explain the phenomenon. I used a simple signal detection model to show that the relationship between model—mimic similarity and mimic effectiveness is typically nonlinear. In particular, I found that there will be little or no further selection to improve model—mimic resemblance beyond a certain level if the model species is costly to attack, if the mimic species is not particularly profitable (e.g., hard to catch), or if the mimic is relatively rare. When there are two different sympatric model species, then mimics should usually evolve a phenotypic similarity to one or the other model species, but not to both. In contrast, when several model species occur in different areas (or emerge at different times) and individual mimics use each of these areas, then the optimal phenotype should be a "jack-of-all-trades" intermediate phenotype that does not closely resemble any particular model species. Somewhat surprisingly, the theory predicts that if mimics spend an equal amount of time with each model species, then the optimal intermediate phenotype should more closely resemble the least numerous and least noxious model. This phenomenon arises because a vague similarity to an extremely noxious species is usually sufficient to guarantee significant protection, whereas a much closer resemblance to a mildly noxious model species is necessary to afford a similar level of benefit.
Key words: Batesian mimicry, hoverflies, receiver sensitivity, signal detection.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONAnalytical modelDiscussionREFERENCES |
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Batesian mimicry, in which individuals of a more palatable species (the mimic) gain advantage by resembling members of another, less palatable species (the model), is now considered a classical example of adaptation through natural selection. Over the past few decades, mathematical models have helped highlight some intuitively reasonable properties of this type of mimicry system (e.g., Emlen, 1968
; Estabrook and Jespersen, 1974; Huheey,
1964,
1988;
Luedeman et al., 1981).
However, almost all of these analyses have focused on aspects of the stability
of established model—mimic interactions, rather than on the evolution of
similarity between the mimic and model.
Perhaps one reason for the relative neglect of the evolution of resemblance
between mimic and model is the apparent simplicity of the evolutionary
process. In particular, one widely held belief is that there should always be
strong selection pressure on mimics to resemble their models as closely as
possible. Thus, it has been argued that close Batesian mimicry will tend to
evolve when the mimic evolves faster toward the model phenotype than the model
evolves away from the mimic (Fisher,
1930), a requirement that will almost inevitably be met (e.g.,
Holmgren and Enquist, 1999;
Nur, 1970). Yet these views
are somewhat at odds with the fact that there are many cases in nature in
which potential Batesian mimics do not appear to resemble their models
particularly closely. For example, many species of hoverflies are generally
regarded as Batesian mimics of wasps and bees, yet to the human eye at least,
they do not resemble their models particularly well (e.g.,
Dittrich et al., 1993;
Edmunds, 2000). The few
theoretical studies that do allow for the possibility of imperfect mimicry
(e.g., Getty, 1985,
1987;
Greenwood, 1986;
Staddon and Gendron, 1983)
have all suggested that imperfect mimics could persist through a form of
frequency-dependent selection, but even these studies did not address the
question of why an imperfect resemblance should not be improved through
selection.
The widespread occurrence of apparently imperfect Batesian mimics has so
far been explained in a variety of different ways (for a short review, see
Edmunds, 2000). For instance,
mimics that seem imperfect to humans may actually appear as good mimics to
predators (Cuthill and Bennet, 1993;
Dittrich et al., 1993), or at
least they may provoke a similar avoidance response in predators to closer
mimics (Duncan and Sheppard,
1965; Schmidt,
1958). Alternatively, imperfect mimics may temporarily confuse
predators (Howse and Allen,
1994), they may be at an intermediate evolutionary stage due to a
change in ecological conditions (Azmeh et
al., 1998), or they may arise as a consequence of selection to
resemble simultaneously more than one model species living in separate
subareas (Edmunds, 2000). To
support the latter theory, Edmunds
(2000) presented some numerical
arguments that indicated that a poor mimic of several different models could
have a higher total population density than a good mimic of any individual
model species. This is an attractive explanation for imperfect mimicry, but as
Edmunds (2000) acknowledged,
the particular case he put forward had some important limitations. Not only
was Edmunds's approach limited to non-overlapping distributions of different
model species, but the stable densities of good and poor mimics in a given
area were arbitrarily chosen, rather than derived from some explicit function
relating the degree of mimicry to equilibrium population density. Furthermore,
no interaction (direct or indirect) between good mimics and poor mimics was
assumed, and the approach took no account of the psychology of individual
predators.
In this article, I present a series of models designed to identify the optimal degree of similarity between a mimic and a model under a variety of conditions. These models have been used to test the validity of several of the above hypotheses and to identify further potential explanations. In particular, I have sought to (1) identify some quantitative conditions under which one would expect imperfect mimicry to evolve, (2) evaluate how many mimics compared to models might be supported at equilibrium, and (3) determine the optimal degree of similarity of a mimic to several different model species when the models differ both in number and in unprofitability.
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Analytical model |
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TOPABSTRACTINTRODUCTIONAnalytical modelDiscussionREFERENCES |
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Basic structure
My starting point is the signal detection model first proposed by Staddon and Gendron (1983
) to identify
optimal predatory strategies when dealing with cryptic prey, which was
modified by Greenwood (1986)
to apply to Batesian mimicry (see also
Getty, 1985;
Sherratt, 2001). In the first
model, I consider a single imperfect mimetic species (such as a hoverfly) and
a single model species (such as a wasp). Given that mimicry is imperfect, then
predators may be able to differentiate mimics from models, but only
probabilistically. I assume that if a predator attacks a mimic on encounter,
then it gains a mean benefit, b, but if it attacks a model on
encounter, then it incurs a mean cost, c.
Let pmimic represent the probability that a predator
attacks a mimic on encounter, and pmodel represent the
probability that a predator attacks a model on encounter. If the densities of
mimics and models are dmimic and
dmodel, respectively, and prey are encountered at random,
then the net payoff to a predator adopting a given
(pmimic, pmodel) strategy will be
proportional to:
 | 1 |
Clearly, pmimic and pmodel are
derived variables (a predator will not know for certain what type of prey it
has encountered). How different pmimic and
pmodel are depends on the perfection of the mimicry. A
simple approximation to the relationship between pmimic
and pmodel, which captures several intuitively reasonable
properties, is given by a classical receiver operation characteristic (ROC;
Staddon and Gendron, 1983);
namely, pmimic =
(pmodel)s, (0 < s < 1).
If s 
1, then predators will behave the same toward mimics and
models (perfect mimicry), but if s 0, then predators will always
attack mimics (mimicry is poor). If we begin by quantifying phenotype in a
single dimension (x), then a convenient alternative representation
for this mimicry index, s, with the same general properties is
s = exp[—k(xmimic —
xmodel)2], where xmimic and
xmodel are the mean phenotypic characteristics of the
mimic and model, respectively. The constant k (> 0) simply
determines the rate at which s converges to 0 as
|xmimic — xmodel|
increases. Substituting for pmodel, we have:
 | 2 |
 | 3 |
By taking a cost—benefit analysis to understand predator behavior,
several properties of this mimetic system are immediately highlighted. For
instance, if we rearrange Equation 3, it is easy to show that the relative
density of mimics to models
(dmimic/dmodel) that can be supported
before pmimic* reaches a critically high value
is directly proportional to (c/b). Although Azmeh et al.
(1998: 2285) noted that
"poor mimics often outnumber their supposed models... by much larger
numbers than are allowable by any theoretical model," here we have one
example of a model that predicts no upper asymptote to the relative number of
mimics to models.
Hypothesis 1: an imperfect resemblance is sufficient
Figure 1 shows the optimal
attack probability of mimics on encounter for different similarities between
model and mimic as predicted by Equation 3. As a consequence of the power
relationship embodied in the ROC, the optimal attack probabilities of
predators consistently decrease with increasing model—mimic similarity,
but in a nonlinear fashion. The important thing to note from each of these
figures is that they tend to bottom out, reminiscent of the "curve of
protection" described by Turner
(1977). Not surprisingly, the
lower the relative equilibrium density of mimics, the greater the range of
mimic phenotypes that will be afforded complete protection
(pmimic * 0;
Figure 1a). Similarly, the
higher the cost of attacking the model, the wider the range of mimetic
phenotypes that are protected from predation
(Figure 1b).
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Equation 3 characterizes the optimal attack rates of predators on mimics
under certain fixed conditions, but it does not show how
xmimic, or indeed xmodel, are likely
to evolve. Nur (1970) argued
that the rate of evolution of the model phenotype was likely to be very low
compared to the mimic because, in contrast to the mimic, any model mutant that
did not look like the standard model would be less likely to be protected from
predation. I have therefore assumed for simplicity that
xmodel is fixed. To evaluate how
xmimic will evolve under these conditions, we need to
consider formally the relative success of rare mutations of the mimic. If the
density of a given mimic mutant with phenotype xmutant is
dmutant and this mutant form is attacked with probability
pmutant, then the net payoff to the predator from
attacking the standard mimic, mimic mutants, and models becomes:
 | 4 |
 | 5 |
 | 6 |
Clearly, an attack rate of 0 cannot be improved on by selection, but if the above relationships hold, then it is easy to show that pmutant < pmimic if and only if |xmutant - xmodel| < |xmimic - xmodel|. Therefore, any mutant mimic will always be attacked less or equally frequently compared to its conspecifics if it resembles the model more closely. Furthermore, any mimic that is initially selected from rarity (i.e., a closer mimic to the model) will inevitably rise to fixation in the absence of further mutations.
As we have seen, the precise optimal attack rates of predators on mimics at
equilibrium (and consequently models) for a given xmimic
and xmodel will depend on the equilibrium population sizes
of both dmimic (= dmimic*)
and dmodel (= dmodel*).
These equilibrium population sizes are straightforward to calculate if we (1)
assume that selectively advantageous mutations are so rare that they always
reach fixation in a population to xmutant before further
mutations arise and (2) adopt a suitable set of functions to relate
equilibrium prey densities (dmimic* and
dmodel*) to model—mimic similarity. Of
course, it is possible that the stable population sizes of both the model and
mimic are not determined by their interaction at all. Such conditions might
arise, for instance, if there were alternative prey species for the predator
and the mimic were limited by other factors, such as the availability of its
food resources. If this were the case, then xmimic would
simply evolve toward the range of values of xmimic that
minimize pmimic* in Equation 3 for particular
values of dmimic (
dmimic*) and dmodel
( dmodel*).
Alternatively, dmodel may be fixed (
dmodel*), but
dmimic* may be dependent on model—mimic
similarity. A standard function (see
Nisbet et al., 1991) for the
rate of change of a prey population that involves density-dependent growth and
a type II predator functional response is:
 | 7 |
dmimic/t = 0 yields a unique,
nontrivial equilibrium value dmimic*. Equation
3, in turn, provides the optimal probability of attacking mimics at this
combination of equilibrium densities (dmimic*,
dmodel*). Not surprisingly, the closer a mimic
resembles the model, the greater its equilibrium density. However, extensive
numerical analyses of this system of equations indicate that, despite their
higher densities at equilibrium, any better mimic that invades to fixation
will always be attacked less frequently than the population of poorer mimics
it replaced (Figure 2).
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In sum, the above analyses provide good support for the contention that mimics will tend to evolve a closer similarity to their model. However, once the mimics achieve a certain degree of resemblance to the model, further improvements in similarity are selectively neutral with respect to predation. This result is robust even if we allow for a degree of feedback in the system, such that better mimics have higher equilibrium population sizes than poorer mimics.
Hypothesis 2: jack of all trades, but master of none
Here I consider the multiple model theory of Edmunds
(2000) in more depth. I first
consider the case of the evolution of mimetic phenotypes when there are two
potential models living sympatrically, and then consider the case in which the
two model species occur in different areas or emerge at different times of the
season (these results are easily generalized for more than two model
species).
All species are sympatric. For ease of display, and to show that
phenotypes can be represented in any number of dimensions, in this set of
examples I allow phenotypes to vary continuously in two independent ways
(x and y, cf. Holmgren
and Enquist, 1999). When two model species occur sympatrically and
have phenotypes (xmodel1, ymodel1) and
(xmodel2, ymodel2), then the payoff to
the predator from attacking mimics with phenotype (xmimic,
ymimic) with probability pmimic
becomes:
 | 8 |
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Models occur in different places or at different times. When different models occur in different areas (or emerge at different times in the season) and individual mimics use each of these areas (or are present at all of these times), then rather different model predictions arise. If mimics spend a proportion, q1, of their time with model 1, and proportion (1 - q1) with model 2 and the models are equal in number and noxiousness, then it is easy to show that the mimic phenotype that would be attacked least frequently by predators should always be closer to the model with which the mimic coexists for longer (Figure 4). Yet if the mimic coexists an equal time with each model (q1 = 0.5), then the mimic phenotype that is attacked least frequently will always be closer to the model with the lowest dc. This somewhat surprising result occurs because the more unprofitable (or numerous) a given model, the less a mimic needs to look like it to gain complete protection.
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As before, it is easy to envisage scenarios in which the mimic phenotype consistently evolves toward these fitness peaks (minimum attack rates). Moreover, this result is robust even if we consider different equilibrium population sizes of the mimic according to its similarity to the model (by adding an additional predation term that allows for attacks according to the similarity to a given model in each of two areas; Figure 5). In this case the equilibrium population size of a jack-of-all trades mimic will always be higher than that of a close mimic of any one of these models that is evolving toward this compromise solution. Perhaps more important, if u (the upper level of a mimic population before it experiences intrinsic negative growth) is significantly lower for a mimic that keeps to one area (and is thereby selected to resemble one model only) than that for a generalist that uses several habitats, then one would expect the equilibrium density of the specialist to be much lower than that of the generalist.
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In sum, the complementary theory of multiple models can also satisfactorily explain imperfect mimicry. In cases where the different models occur in the same area and at the same time, then mimics should either center on one of the model phenotypes or some intermediate phenotype (if the models themselves are similar to one another). In cases where models occur in distinct areas or at distinct times, there should be selection on mimics that use all of these areas (or times) to develop an intermediate phenotype. The optimal intermediate mimic phenotype should more closely resemble the model with which the mimic spends most time, but all else being equal, the mimic should more closely resemble the less noxious and less numerous model.
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Discussion |
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TOPABSTRACTINTRODUCTIONAnalytical modelDiscussionREFERENCES |
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Hypothesis 1: receiver sensitivity
The analysis presented here strongly suggests that mimicry without perfect resemblance will readily evolve in systems containing a single model. This arises because the relationship between model—mimic similarity (|xmimic — xmodel|) and optimal predator attack rate on mimics (pmimic*) is typically nonlinear, leveling out over a broad range of phenotypic similarities. Duncan and Sheppard (1963
) also raised the
possibility of a nonlinear response when they proposed that information loss
(certain prey items not attacked) could in theory lead to "artificial
quanta" in receiver sensitivity (see also
O'Donald, 1969;
Turner, 1977). Several
experiments have indeed found that the relationship between model—mimic
similarity and the effectiveness of the mimic is nonlinear (e.g.,
Dittrich et al., 1993;
Goodale and Sneddon, 1977).
Recent work (Holloway et al.,
2002) has also found that hoverflies that were relatively good
Batesian mimics of vespid wasps exhibited high phenotypic variation, which is
consistent with the theory of relaxed selection. Of course, one obvious
question is what happens once the mimic phenotype reaches a level of
similarity with the model beyond which further changes are selectively neutral
with respect to predation. At this point (or even before), several additional
factors may influence the evolution of the mimic phenotype. For instance,
mimics may need to be recognized by mates, which might render an even closer
similarity selectively disadvantageous. Alternatively, mimic phenotype may be
influenced to an extent by a need to regulate abdominal and thoracic
temperature (Holloway et al.,
1997).
As this study highlights, imperfect mimicry will be more likely to persist
when the model species is costly to attack and when the mimic species is
relatively rare. In support of this, Pilecki and O'Donald
(1971) showed that neither
poor mimics (palatable mealworms) nor their models experienced high attack
rates when the poor mimics were relatively rare. Similarly, Lindström et
al. (1997) showed that
imperfect Batesian mimics had the lowest mortality when models were common and
when models were unprofitable. In contrast, other workers (e.g.,
Sheppard, 1959) investigating
the field distributions of mimetic African butterflies found that the
proportion of individuals of a given species with a poor resemblance to the
model was actually higher when mimics were relatively common. However, in each
of these studies, the phenomenon was attributed a local breakdown of mimicry
(Brower, 1960), which is
equivalent to pmimic* = 1 for all
xmimic when dmodel is low, rather than
a case of pmimic* 0 for a particular range
of phenotypes.
Hypothesis 2: multiple models
The possibility that a mimic species may have several different model
species to chose from has been recognized for some time (see
Mallet and Joron, 1999). For
example, in southern Florida where unpalatable monarchs are rare, the viceroy
resembles the queen (see Waldbauer,
2000). Similarly, O'Donnel and Joyce
(1999) reported that the wasp
Mischocyttarus mastigophorus was dimorphic and suggested that the
morphs resembled two species of swarming wasp in the genus Agelaia,
which were locally abundant but at predominantly different elevations. More
recently, Norman et al. (2001)
described a species of Indo-Malayan octopus in which individuals impersonate a
range of venomous animals that co-occur in its habitat.
The analyses in this study show how such mimetic polymorphism can arise,
for instance, if one population of a mimic coexists with one model species and
another population coexists with an alternative model species. However, the
analyses also indicate that when the models are similar to one another in
appearance (such as vespid wasps, perhaps) then mimetic polymorphism is
unlikely; rather, the mimetic phenotype will tend to have an intermediate
form. Such a jack-of-all-trades mimetic phenotype has only recently been
considered as an explanation for imperfect mimicry
(Edmunds, 2000), and the
analyses presented here support the view that such a phenomenon is plausible.
As this study shows, intermediate phenotypes are even more likely to arise if
the models are separated in space or time, whereas the mimic is not. Under
these conditions, the intermediate mimics should more closely resemble the
model with which they coexist for longer but, all else being equal, they
should resemble the less noxious and less numerous model. Some of these
predictions may well be amenable to experimental testing.
Alternative explanations
This study focused on quantitatively examining the validity of two
important explanations for why imperfect mimicry can persist in natural
populations, but there are others. Of the established explanations, I find it
difficult to accept the "satyric mimicry" hypothesis
(Howse and Allen, 1994)
because it appears to be based on frequency-independent constraints (an
ambiguous mixture of palatable and noxious signals temporarily confuse the
predator), rather than on frequency-dependent costs and benefits. In
particular, while Howse and Allen
(1994: 113) suggest that
imperfect mimicry may be explained because "small departures from
optimal ambiguity can destroy the paradoxical nature of the image," the
experimental results of Dittrich et al.
(1993) indicate that closer
resemblance of hoverflies to their model are at best selectively neutral, and
are never selectively disadvantageous. Perhaps more likely, imperfect mimics
(such as certain species of hoverflies) may not be Batesian mimics at all but
may be signaling their own unprofitability (e.g., their ability to escape
predation) (see Azmeh et al.,
1998; Edmunds,
2000). This is an interesting theory, but it should be noted that
some species of hoverflies also engage in behavioral mimicry, for instance by
waving their legs to resemble Hymenopteran antennae (see
Golding and Edmunds, 2000;
Waldbauer, 1988). Perhaps more
likely, the higher flight speeds of hoverflies might reduce their mean
profitability on pusuit (b). Lower b will allow a greater
density of mimics at equilibrium and generate a broader range of selectively
neutral phenotypes.
Alternative explanations for imperfect mimicry that were not identified in
Edmunds (2000) short summary
include the possibility that resemblance does not have to be close when the
mimic is so mobile that predators are rarely given a clear view and that there
are phylogenetic constraints on optimization that restrict closer resemblance
(see Gould, 1980). Although
earlier researchers rightly recognized imperfect mimics as a neglected problem
of Batesian mimicry (Dittrich et al.,
1993), and mathematics has an important role to play in formally
evaluating the validity of available verbal theories, it is already becoming
clear that this problem may have many solutions.
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ACKNOWLEDGEMENTS |
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I sincerely thank Professor Malcolm Edmunds for comments on an earlier draft of this paper. This work was motivated by empirical research funded by the Leverhulme Trust (F/00128/M).
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