Reciprocal mimicry: kin selection can drive defended prey to resemble their Batesian mimics

Abstract

Established mimicry theory predicts that Batesian mimics are selected to resemble their defended models, while models are selected to become dissimilar from their mimics. However, this theory has mainly considered individual selection acting on solitary organisms such as adult butterflies. Although Batesian mimicry of social insects is common, the few existing applications of kin selection theory to mimicry have emphasized relatedness among mimics rather than among models. Here, we present a signal detection model of Batesian mimicry in which the population of defended model prey is kin structured. Our analysis shows for most of parameter space that increased average dissimilarity from mimics has a twofold group-level cost for the model prey: it attracts more predators and these adopt more aggressive attack strategies. When mimetic resemblance and local relatedness are sufficiently high, such costs acting in the local neighbourhood may outweigh the individual benefits of dissimilarity, causing kin selection to drive the models to resemble their mimics. This requires model prey to be more common than mimics and/or well-defended, the conditions under which Batesian mimicry is thought most successful. Local relatedness makes defended prey easier targets for Batesian mimicry and is likely to stabilize the mimetic relationship over time.

1. Introduction

One of the classic defining distinctions between Müllerian and Batesian mimicry is their different evolutionary dynamics [1–3]. In Batesian mimicry, palatable prey obtain protection from predation by resembling defended model prey and are under selection to resemble the defended models more closely. Mimics usually obtain better protection from predation the more they resemble the models [4–6]. The presence of mimics dilutes the warning signal of the models, thus harming the models, and mimicry theory predicts that this creates selection on models to become more dissimilar from mimics [2,7–11]. In Müllerian mimicry, by contrast, two or more defended prey species benefit from sharing the same warning pattern. This allows the prey species to share the cost of predator education, and the classical prediction is selection (possibly of unequal strength) for mutual resemblance (but see discussions of unilateral ‘advergence’ in [12–14]).

Mimicry theory has been mostly concerned with solitary organisms, and with individual selection as the central evolutionary force. This is not surprising, since much of our knowledge of mimicry is based on the study of mimetic butterflies, a tradition dating back to the classical papers by Bates [15] and Müller [16]. However, Batesian mimicry can be found in a wide range of taxa and many examples involve social insects as models [17–19]. It is well known that factors such as kin structure and family grouping have the potential to facilitate the evolution of stronger secondary defences (i.e. prey defences that act after attack is initiated, such as toxins and distastefulness) and aposematism in defended prey [1,20–23]. Surprisingly, the consequences for mimicry evolution remain largely unexplored (but see [24–26]).

In Batesian mimicry, the predator and the defended models can be said to have partly overlapping interests, since they both benefit when the predator correctly identifies the models. Indeed, many defended models have very distinct warning coloration that sets them apart from other prey and facilitates recognition by the predator. However, traits that confer benefits to models at the expense of the predators may also be beneficial, even if they benefit mimics as a side-effect. An obvious example is a better secondary defence. This raises an intriguing question: Could a model ever benefit from becoming less distinct from a Batesian mimic, so as to make it harder for the predator to discriminate between the two? According to the classical view of Batesian mimicry, the answer is no, since a lone individual would never stand to gain from being more likely to be mistaken for a palatable mimic by a predator. This may change, however, when predators encounter model prey that are related.

Batesian mimicry is an effective defence because it changes predator behaviour. Theory predicts (e.g. [2,26–29]) that Batesian mimicry should be more effective and predators have less incentive to attack when the abundance of mimics relative to models is low, models are highly defended and/or mimics are only weakly profitable (i.e. when the ‘mimetic load’ sensu [26] is low). Likewise, Batesian mimicry should be less effective and predators have a higher incentive to attack when the abundance of mimics relative to models is high, models are weakly defended and/or mimics are highly profitable (i.e. when the mimetic load is high). These predictions are generally supported by experiments [5,6,30]. Theory also predicts that improvements in the average (population) level of mimicry will—as long as mimicry is not too accurate—benefit mimics but not models [26,27,31]. However, if the average level of mimicry evolves to become highly accurate, predators will be virtually unable to discriminate, and should either avoid the mimicry complex entirely (when the mimetic load is low) or attack indiscriminately (when the mimetic load is high) [26,27,31]. Thus, when the mimetic load is low, predators that encounter mimics and models of highly accurate resemblance should be less inclined to attack models than predators that encounter slightly less accurate ones [26,27,31], an effect that has also been demonstrated experimentally [31]. At the population level, at least, models may benefit from being less distinct from mimics, but such benefits do not matter from the perspective of individual selection.

We propose that a similar benefit of being less distinct from mimics may be enjoyed by kin-groups when the defended models are related to others in their local neighbourhood, and the predators adjust their strategies according to the local average level of mimicry. In such cases, rare mutant models with a higher-than-population-average similarity to the Batesian mimics will tend to be surrounded by other mutant models with a higher than average similarity. Across the population, mutant models will on average be more likely than non-carriers to encounter predators that are already familiar with the mutant type and have adjusted their foraging strategy accordingly, causing the mutation to be favoured through kin selection. The strength of the effect should depend on the relatedness among the models as encountered by the predators.

To formally explore this idea, we study the effects of relatedness on mimicry dynamics using a model of predator attack behaviour that is based on signal detection theory (SDT), a common framework for modelling predator discrimination (e.g. [24,27,28,29]). This framework has earlier been used to investigate the effects of kin selection on mimicry evolution when mimics rather than models are closely related [24–26]. In this paper, we explore the effect of relatedness among model prey, and consider both colony-living and free-living model prey.

2. The model

We follow previous approaches (e.g. [24,26]) and use a normal-normal equal-variance signal detection model [32] as our starting point. The signal detection approach focuses on discrimination strategies at learning asymptote, and is reasonable here given that accurate mimicry is likely to evolve mainly in response to fine-tuned discrimination by experienced predators rather than trial-and-error by naive predators. The predators encounter mimics and models sequentially and randomly at relative frequencies p and (1 − p), respectively, and discriminate between mimics and models on the basis of some continuous warning trait. Let the warning trait of the (monomorphic) model population be denoted and that of the (monomorphic) mimic population be denoted (by convention, ). Signals are not perceived perfectly by the predator: a noise term following a standard normal distribution is added to each signal. Let denote the dissimilarity between the resident mimic and model populations. We follow earlier approaches and make the standard assumption that predators maximize expected fitness gain per encounter (e.g. [24,26,29,31]). We assume the predator obtains the average benefit b when attacking a mimic, pays the average cost c when attacking a defended model, and obtains zero payoff when refraining from attacking [24,27,29,33,34].

(a). The basic signal detection theory model

As is well known for this type of SDT model, an optimal threshold exists above and below which prey should be classified as mimics and models respectively. A standard derivation of the optimal threshold is as follows: the conditional probability that a prey perceived as having trait value m is a mimic is

where the prime denotes the derivative and Z(x) denotes the cumulative function of the standard normal distribution. The predator should attack such a prey if the expected payoff from attacking it is higher than the payoff from ignoring it, or

2.1a

which is equivalent to

2.1b

Taking the natural logarithm on each side and solving with respect to m, we can show that the predator should adopt the attack threshold

2.2

and attack only those prey perceived to have trait values above the threshold. On encounter, a rare mutant prey with trait value m will be attacked with probability . The function increases with m; a mutant model prey that is slightly more dissimilar from the mimics than the resident models will be attacked less often. This is an individual benefit to models of increased dissimilarity, and is illustrated in figure 1a.

Figure 1.

(a) The perceived trait values of resident models (black, dashed curve) and mimics (black, solid curve) are normally distributed with unit variance. In this example, the peaks of the two distributions are three standard deviations apart (i.e. , with d being mimetic dissimilarity, the model trait and the mimic trait). The predators attack all prey having trait values above some optimal threshold t and may commit two types of error: they may attack some models (light grey area) and miss some mimics (dark grey area). If a mutant model arises that is more dissimilar from the mimics (grey, dashed curve) it will obtain the (individual) benefit of being less likely to be attacked upon encounter than the resident models. (b) The probability that a predator in a given encounter attacks a resident mimic (solid lines) and a resident model (dashed lines) as a function of the (population) level of mimetic dissimilarity is shown. As the dissimilarity approaches zero, attacks on models and mimics cease. Grey lines: the mimetic load K, or the negative impact of mimics on models, is low (i.e. mimics are rare and/or models are well defended, K = 0.2). Black lines: the mimetic load K is higher (i.e. mimics are more common and/or models are less well defended, K = 0.5).

The properties of the above basic SDT model are well known (e.g. [24,26,27]): the predator should generally attack fewer prey when the ‘mimetic load’ K is low, i.e. when the proportion of mimics is low (p is low), when the mimics have low profitability (b is low), and when models are strongly defended (c is high).

We have and , and the probabilities that mimics and models are attacked on encounter are and , respectively. It is of particular interest here that models benefit (at the population level) from improved mimicry when dissimilarity d is low and mimetic load K is low (figure 1b; see also [26,27,31]). Differentiating the attack probabilities with respect to d, we can determine the full parameter regions in which small improvements in the population level of mimicry increase and decrease the probability that mimics and models are attacked on encounter (figure 2a).

Figure 2.

(a) Small improvements in the population level of mimicry may increase the probability of attack on encounter for both mimics and models (white region, ), reduce it for mimics and increase it for models (grey region), and reduce it for both (black region, ). Here, K is the mimetic load, or the negative impact of mimics on models (see main text for explanation). (b) A constant number of predators is assumed across patches. Models will converge towards their Batesian mimics if the mimetic resemblance is already sufficiently close. The threshold of dissimilarity below which models will converge on mimics depends on the model relatedness in the patches: (solid line), (dotted-dashed line), (dashed line) and (dotted line). The solid line corresponds to the boundary of the black region in (a). (c) The same as (b), except predators have an ideal free distribution; there are more predators in the more profitable patches. The regions in which the models will converge towards their Batesian mimics (the regions below the curves) expand relative to those in (b). See main text for explanation. Panels (b) and (c) apply to solitary models with family grouping. For colony evolution, the same figure can be produced with , the average fraction of workers within a patch that is from the focal colony, exchanged for (assuming that f is uncorrelated with within-colony relatedness, see main text).

(b). A kin selection model

We study the effects of relatedness on the selection acting on the warning signal of the model prey by considering the invasion fitness of a rare mutant allele of small effect in a monomorphic model population. For simplicity, we assume discrete, non-overlapping generations and a population consisting of an infinite number of equally sized patches. Prey live most of their life in their natal patch but disperse at the end of their life to mate randomly, and then mothers produce all of their offspring in a new, randomly located patch. We assume for diploid and haplodiploid carriers that the heterozygous phenotype is intermediate to that of the homozygotes, so we may take invasion to imply substitution except at singular points where the local fitness gradient is zero. Let r denote the relatedness of the model individuals that grow up within a patch. For a sufficiently rare mutant allele, the probability that a patch cohabitant of a mutant selected at random in the population carries the same mutation will equal the patch relatedness coefficient r. By contrast, the probability that a patch cohabitant of a resident prey drawn at random in the population is a mutant will be virtually zero, since almost no patches will contain the mutant allele. We assume that predation pressure (the average number of predator encounters per prey per unit time) is the same across all prey patches (this assumption will be relaxed later), and that individual predators search for prey randomly within prey patches and use strategies that are optimal within the patch.

Because of the kin structure, a rare novel mutant may—in those patches where it is found—be present in significant numbers, and the standard signal detection model must therefore be modified to take this into account. In a patch with mutant model prey, predators encounter a fraction p of mimics , a fraction (1 – p)(1 – r) of resident models , and a fraction (1 – p)r of mutant models . The optimal attack strategy for a prey with perceived trait value m is then determined by taking the following conditional probability,

and substituting it into inequality (2.1a). Some rearranging shows that the predator should attack the prey if

We restrict our attention to the case where both and hold. It is then straightforward to show by differentiation that the left-hand side of the inequality is strictly decreasing with m from infinity to zero. When encountering a mixture of two types of model prey, the predator thus has an optimal attack threshold, , which is the unique solution to

2.3

There is no analytical solution, but is implicitly given by (2.3) and can be found numerically; and for , r cancels out and the threshold evaluates to (see equation (2.2)). The arguments of will for convenience often be suppressed in notation.

(c). Costs and benefits affecting kin-groups

Model prey always enjoy an individual survival benefit of increased dissimilarity: a mutant model that is slightly more dissimilar from the mimics than the resident models will less often be classified as a mimic than the residents (figure 1a). When the model population is kin structured, however, a (globally) rare mutant with a slightly different appearance will be surrounded by similar kin in the local patch, causing the average level of mimicry to change in the patch. This, in turn, causes predators in the local patch to decrease or increase their attack threshold, creating patch-level costs or benefits, respectively (in a later section, we explore additional mechanisms).

Let the probability of attack on a prey with trait m when the mimicry complex locally consists of non-negligible numbers of model types and and mimics be denoted

2.4

In the electronic supplementary material, section A1, we show that

2.5

This expression gives the direction of change in the attack threshold in a patch with a mutant model that is slightly more similar to the mimics than the resident, as compared to a patch with no such mutant. When (2.5) is negative, a mutant that is more dissimilar from the mimics than the resident models will enjoy the kin-mediated group-level benefit of the predator applying a more wary attack strategy (a higher attack threshold). Expression (2.5) is negative only when (white region in figure 2a), the same parameter space in which Johnstone [24] showed that related mimics may come under selection for inaccuracy. In most of parameter space, however, (2.5) is positive and there is no such kin-mediated benefit (i.e. grey and black region in figure 2a). Here, by being more dissimilar from the mimics than the resident models the mutants will actually suffer a kin-mediated cost, by causing predators to adopt more aggressive attack strategies (a lower attack threshold), which detracts from the individual benefit of increased dissimilarity. Model relatedness thus reduces the selective benefit of any small mutation that causes models to become more dissimilar from mimics; whether such mutations will at all be advantageous depends on whether the individual benefit outweighs the kin-mediated cost. Vice versa, a mutation that takes the model closer to the mimics will, in contrast, suffer an individual cost but a kin-mediated benefit. It is clear that the magnitude of the group effect depends on r (equation (2.5)). The balance between individual- and group-level costs and benefits will be explored in the following two scenarios, first considering free-living but related model prey and then colony-living model prey.

(d). Scenario 1: evolution of warning signals in solitary models with family grouping

Suppose that the absolute fitness of a model prey is a smooth, strictly decreasing function F of the probability of being attacked in an encounter with a predator (cf. [24]). Allowing there to be some variation in r across patches, with mean , the expected (absolute) fitness of rare mutant models with trait value in a population with trait value is where E_r denotes expectation taken over r (note that A depends on r). Using that reduces to when , and that does not depend on r, we can write the local fitness gradient near as

We see that the local fitness gradient has the same sign as

2.6

which has the same sign as

Thus, the local fitness gradient is positive when

When the local fitness gradient is positive, a rare mutant allele arising in the model population that causes carriers to have a slightly closer resemblance to the mimics will be able to invade (figure 2b).

(e). Scenario 2: evolution of warning signals in colony-living models

Here, we consider a colony-living species with one queen and a sterile worker caste. Each queen lands in a random patch and founds a colony, and the number of dispersing sexuals from the colony at the end of the season is taken as the measure of fitness; mating is random with respect to natal patch. We assume that increased colony production does not affect the expected representation of various parental alleles among sexuals (i.e. it does not change the sex ratio). Moreover, the number of dispersing sexuals produced by the colony, denoted G, is a smooth and strictly increasing function of the average lifetime contribution of its individual workers, and in turn, a worker's expected lifetime contribution, L, is a smooth and strictly decreasing function of the probability of being attacked upon encounter when foraging. Since not every worker may carry the mutant allele, the effect of the allele on non-mutant worker productivity must also be taken into account. When the mutant allele is rare, two colonies founded in the same patch are unlikely to both have the mutation present. Thus, if the fraction of the workers encountered by a predator within a patch that are from a focal colony is f (and a fraction ρ of these carry the mutation), then the fraction of encountered workers that are from other (non-related) colonies is 1−f, and the mutation will be carried by the fraction r = fρ of the encountered workers. Note that ρ may differ between colonies, in the haplodiploid case depending on whether the allele is maternally or paternally inherited. Thus, the expected number of sexuals produced by a colony is given by

Using that does not depend on r (= fρ), the local fitness gradient is given by:

Observing that G' is positive and L' negative, the fitness gradient has the same sign as

2.7

Calculating this and using (2.5), that r = fρ, and that , we find the fitness gradient has the same sign as

where is the average over all patches and over all colonies. Assuming f and ρ are uncorrelated, the local fitness gradient is positive when , which (assuming ) is positive when

This is the same result as in scenario 1, except that for a colony, takes the place of as regards the direction of selection ( is the average fraction of the models encountered by a predator that comes from the focal colony). When f and ρ are uncorrelated, the average within-colony relatedness affects the strength of selection but not the direction; we would expect the strongest kin effects when both and are high (i.e. when is high).

3. Predator payoff and attack rates

So far we have assumed that predation pressure is equal across patches. However, as pointed out by Reinhold & Engqvist [25], different levels of mimicry may not only affect the optimal attack thresholds adopted by individual predators within patches but also the predators' distribution over patches, in the sense that there will be more predators foraging in more profitable patches. Reinhold & Engqvist [25] showed that this could reduce the potential for kin selection to oppose individual selection acting on mimics, and it is therefore of interest to see how such a mechanism affects the results obtained here regarding defended model prey. We follow their approach and assume that the number of encounters experienced by prey is proportional to the net profitability of the mimicry complex. Using the shorthand for , the expected benefit per encounter for an optimal discriminator is

Following Reinhold & Engqvist [25], the rate of attack is given by

3.1

For scenario 1, we can calculate the sign of the local fitness gradient by substituting the function A_alt for A in (2.6). We show in the electronic supplementary material, section A2, that the fitness gradient, in this case, has the same sign as:

3.2

For scenario 2, we can calculate the sign of the local fitness gradient by substituting the function A_alt for A in (2.7). Assuming as before that and r are uncorrelated, we show in the electronic supplementary material, section A2, that the fitness gradient in this case has the same sign as the expression obtained when substituting for in (3.2), and that the gradient is also proportional to .

When the predators preferably forage in more profitable patches, the models may come under selection for closer mimicry at a higher phenotypic distance between resident mimic and model populations (compare figure 2b,c). As the models come to resemble the mimics more closely, the mimicry complex becomes less profitable, causing fewer predators to forage on it. Increased dissimilarity would in contrast increase the profitability of the mimicry complex and the number of predators that forage on it. This predator response comprises a second group cost of dissimilarity, in addition to the group cost of causing individual predators to set lower, more aggressive attack thresholds and thus become more prone to attack. Note that when both costs of dissimilarity are considered, there can be selection for mutual convergence when lnK > 0, although in a very restricted parameter region and also limited to an intermediate range of d (figure 2c).

4. Discussion

Given that so many defended prey have Batesian mimics, that mimicry research has a long history, and that kin selection is frequently invoked in discussions of secondary defences and aposematism, one cannot help but wonder why kin selection has received so little attention in the context of mimicry. One reason is probably that the context for discussing kin selection has tended to be aggregations of juvenile stages of defended insects [1,21–23,35], and Batesian mimicry of larvae and nymphs is not well studied. However, kin selection does not require aggregations to operate, but only that models are related to others in their local neighbourhood, a fact that was already recognized by Fisher [1, pp. 158–160]. As our analysis above shows, it is important to track relatedness among models when studying Batesian mimicry, since it can fundamentally alter the dynamics of mimicry evolution.

The harmful effect that Batesian mimics have on their models is commonly thought to create selection on mimics to resemble models more closely, and selection on models to become more dissimilar from mimics (e.g. [2,7,8,9]). The picture becomes more complicated when prey are related to others in their local neighbourhood so that kin selection operates. Although there is always an individual-level benefit to be reaped by novel model types that are more distinct from the palatable mimics, a benefit consisting of the mutant models being less predated than the resident models, in kin-structured populations increased dissimilarity will in addition typically incur a cost to the kin-group that may or may not outweigh the individual benefit. This effect arises because the defended prey are surrounded by kin, which causes models in patches with mutant models to encounter predators that set different thresholds than the predators encountered by most of the model prey in the population. The predators take advantage of a slightly higher average dissimilarity by adopting attack thresholds that are more aggressive in the sense that they allow for capture of a higher fraction of the mimics. In much of parameter space, our model shows that this will weaken rather than change the direction of selection on model appearance, reducing the benefit to the models of increased dissimilarity.

More intriguingly, however, there are also conditions under which the kin-mediated group cost outweighs the individual benefit and changes the direction of selection, turning the coevolutionary dynamics from the chase characteristic of Batesian mimicry to the mutual convergence characteristic of Müllerian mimicry. In addition to a high model relatedness r, this requires the mimetic load K to be low, i.e. that models are common relative to mimics and/or well defended, which are those conditions under which Batesian mimicry is assumed to be most successful and thus arises most easily (e.g. [6,30]). In this case, mutual convergence will typically occur if the mimics are first able to evolve a sufficiently close resemblance to the models, reducing the mimetic dissimilarity d below a threshold that is dependent on K.

Mutual convergence makes intuitive sense if one considers the profitability of the mimicry complex to the individual predator: When the mimetic resemblance is imperfect, the predator will be able to discriminate between mimics and models to some extent and will profit from attacking prey in the mimicry complex that are perceived as being sufficiently distinct from the models. However, the predator will occasionally attack a model by mistake. In contrast, if the mimetic resemblance is perfect then the predator cannot discriminate and should avoid all prey. In this way, the models may benefit (as a group) from having a very close resemblance to their mimics. Such a population benefit to models of close mimicry has been predicted for a long while [26,27,31] and has also recently been demonstrated experimentally in an operant conditioning setting [31].

Reinhold & Engqvist [25] in addition argued that more predators will forage on a mimicry complex when it is highly profitable. Following their example and making the 'ideal free distribution' assumption that the number of predators is proportional to patch profitability, the kin-mediated group effect becomes stronger, and the mimics will attract the models from a larger phenotypic distance, everything else being equal. Improved mimicry makes the mimicry complex less profitable and subject to predation by fewer predators, while increased dissimilarity makes it more profitable and subject to more predators. The latter constitutes a second group-level cost of dissimilarity that makes it harder for model prey that are locally related to evolve away from their mimics. If predators deviate from the ideal free distribution, this second cost may change quantitatively; over-matching (disproportionally more predators in profitable patches) would strengthen the effect while under-matching would temper it.

Taking a step back to summarize our findings, when models are kin structured we predict: (i) that accurate mimicry will evolve more easily or faster, since there typically will be a twofold kin-mediated group-level cost to models evolving increased dissimilarity; (ii) that the mimetic relationship may become more and more stable as mimicry becomes more fine-tuned, since mutual convergence (or at least no gradual escape) is expected at close range; (iii) that we in some cases may expect models to express traits that are more similar to their harmless imposters than to closely related species, i.e. that the models also to an extent mimic the deceptive harmless prey; and (iv) that all these effects increase with the level of model prey relatedness (as encountered by the predator).

(a). Testing the model

Any convergence by a model towards a mimic is likely to be fine-tuning of very accurate mimicry, and we expect it to be neither dramatic nor very costly: likely candidate traits would be adjustments in intensity or loss of certain warning signal components etc. that the mimics have failed to copy. We may think of the model prey as effectively being under selection to conceal the presence of mimics from the predators, so as to make the mimicry complex less profitable. It should in principle be possible to test for mutual convergence by investigating whether there are character shifts in models towards mimics in areas where mimics are present compared to areas where mimics are absent.

Our model should apply to larger aggregations (i.e. kin-groups) of defended, juvenile insect larvae or nymphs, but to our knowledge Batesian mimicry involving such models is not well known. One important group of defended models, social insects, might be the best candidate for investigation. Social insects can be numerous, and the workers of a single colony could potentially constitute a significant fraction of the model individuals that predators sample and generalize over. A novel mutation in a colony that increases the workers’ average resemblance to a mimetic species may conceivably have a strong, immediate effect on the local level of mimetic dissimilarity. Among the Batesian mimics that resemble social insects we find spiders and insects that mimic ants (reviewed in [17,19]) and insects that mimic bees and wasps (with hoverflies being a well-known example of the latter [18]). Batesian mimicry of social insects has been much less studied than Batesian mimicry of aposematic butterflies, which often disperse over long distances and are less likely candidates for kin selection to be a major evolutionary force, at least at the adult stage [23].

Although mimicry of social insects can be highly accurate [17–19] and suggestive of very fine-tuned discrimination by predators, much of the research on mimicry involving social insects has in fact focused on explaining crude mimicry, a subject of much debate for which many competing hypotheses have been put forward [18,19,24,26,29,36,37]. Penney et al. found that large species of hoverflies were more accurate in their mimicry of social insects than were small ones, possibly because larger species are more profitable to predators and thus under stronger selection for accurate mimicry [36]. We predict that another key factor influencing mimetic accuracy may be the local relatedness among models as experienced by the predators. When local relatedness is high, the kin-mediated cost of dissimilarity should be strong and models less likely to evolve away. This may be tested by correlating model relatedness (at the scale set by predators' foraging behaviour) with mimetic accuracy; we predict a positive relationship.

A key assumption underlying our model is that predators use attack strategies that are adjusted to local levels of mimetic accuracy. This is to be expected if the size of the foraging range of individual predators is not too large in scale compared to the size of the neighbourhoods over which mimetic accuracy varies. If this is not the case, however, adaptive matching to local mimicry levels could nevertheless occur through behavioural flexibility: one possibility is temporal tracking, i.e. that predators adjust their strategies as they move from one location to another and learn that the discriminability of the mimicry complex changes. Laboratory studies of stimulus discrimination show that birds can learn to adjust their discrimination strategies as dissimilarity d is manipulated [31,38,39]. Another possibility is context-dependent learning: In the ‘novel world’ experimental set-up, individual great tits (Parus major) were able to learn and maintain contrasting sets of prey preferences for use when foraging in different locations (aviaries) [40]. This involved a complete reversal of preferences, and smaller adjustments in discrimination between locations may be comparatively easier to learn, although such a task could also be less salient.

We are aware of no study of Batesian mimicry that has tried to quantify relatedness among model prey as experienced by predators. In our model, we assumed long-distance dispersal before reproduction, which means that the offspring of a single mother (solitary offspring or the workers of a colony) must have a substantial effect on the patch level of mimetic resemblance for kin selection to play an important role. Whether this is realistic for a given species will depend on the degree of overlap in foraging range by neighbouring conspecific colonies. Many species of ants defend territories around their nests, sometimes with very clearly defined boundaries [41], which ensures that perceived relatedness within a patch at a scale similar to a colony territory will be high. By contrast, foraging bees may be patchily distributed in wide areas around their colonies: Foraging ranges for wild eusocial bee species are positively associated with body size and has been found to vary over two orders of magnitude (from approx. 0.1 km to 10 km in feeder training experiments, cf. fig. 1b in [42]), but typical foraging distances may be substantially lower than maximum distances [43,44]. To determine overlap, it is the foraging range relative to colony density that matters, both of which are factors likely to vary with environmental conditions.

If mothers disperse only a limited distance before reproducing, however, mothers (solitary or queens) carrying mutations that affect mimicry will produce their offspring in the vicinity of their sisters that also are likely to carry the mutation. A viscous population structure of this kind will amplify the relatedness of model prey as perceived by predators, and will be conducive to the kind of kin selection effects we have studied in this paper. Relatedness arising through population viscosity relaxes the condition that the offspring of a single mother by themselves should have a substantial effect on the local level of mimetic accuracy. However, this mechanism requires that the kin-selected costs or benefits are not completely negated by local competition between relatives or colonies; a requirement that is likely to hold when there is local capacity for population growth [45]. We hope our study will stimulate research on predator behaviour and prey distributions that allows for estimates to be made of model relatedness as experienced by predators in the wild.

In conclusion, we have shown that local relatedness among defended prey reduces the selective benefit of being more dissimilar from Batesian mimics. At close mimetic range, local relatedness may qualitatively change the coevolutionary dynamics of Batesian mimicry from classical chase-away dynamics to reciprocal mimicry. We predict that defended prey should be easier targets for Batesian mimicry when they are locally related, and that such relatedness may also make the mimetic relationship more stable over time.

Data accessibility

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Authors' contributions

Ø.H.H. and R.A.J. conceived of the study together. Ø.H.H. did the model analysis and drafted the manuscript. R.A.J. gave input on the modelling and interpretation of results, and commented on the manuscript. Both authors gave their final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

Ø.H.H. has been funded by the Research Council of Norway (project no. 249987/F20).