Evolutionary Ecology 13: 755±776, 1999. Ó 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Evolutionary perspective

Batesian, quasi-Batesian or MuÈllerian mimicry? Theory and data in mimicry research M.P. SPEED

Environmental and Biological Studies Department, Liverpool Hope University College, Hope Park, Liverpool, L16 9JD, UK (tel.: +44 151 291 3097; fax: +44 151 291 3172; e-mail: [email protected]) Received 1 June 2000; accepted 27 November 2000 Co-ordinating editor: C. Rowe Abstract. In this paper I argue that the nature of mimetic relationships remains contentious because there are insucient data to enable full evaluation of theoretical models. There is, however, a growing appreciation of the need to draw together empirical studies to provide foundations for theoretical work. I review some recent data that considers the responses of predators to changing numbers of defended prey items and the nature of mimicry along a palatability spectrum. A simple model of predator behaviour is constructed which combines assumptions from Pavlovian learning studies with traditional `number dependent' learning models. This model has two important properties. First it shows that Pavlovian assumptions can be represented in a simple model which generates interesting predictions. Second it indicates some areas that still need detailed empirical study ± most importantly perhaps is the way that predators respond to prey with di€erent levels of edibility. Key words: Batesian, evolution, learning, mimicry, MuÈllerian, predators

Introduction The classical view of mimetic relationships is elegant, simple and intuitive. In Batesian mimicry, an edible species copies the warning signal of a defended, aposematic species, known as the model. Batesian mimics may be parasitic because if they become suciently abundant predators learn about their presence and raise attack rates on models and mimics. An evolutionary solution to the problem of Batesian abundance is the diversi®cation of mimicry patterns to match the aposematic signals of other defended species in the ecology. By diversifying mimicry patterns, individuals of an edible species thus share the parasitic burden out over several model species, raising the ®tness of each mimetic individual. Where there are alternative model species and the right mutational events, Batesian mimics are therefore expected to be polymorphic in their mimicry (see review in Turner, 1987).

756 Classical MuÈllerian theory holds, in contrast, that defended species all bene®t if they share a warning signal. This prediction follows since it is assumed that (i) there is a ®xed number of prey killed during the education of naive predators and (ii) if two or more defended species share a warning signal then the ®xed costs from predator education can be shared between mimetic species. The per capita mortality costs of predator education are therefore minimised by maximising the number of defended species that share a warning signal. MuÈllerian mimics thus have a strength in numbers and any variation from the common MuÈllerian warning signal is expected to sharply raise the costs of education for deviant signallers. `Universal monomorphism' is thus the predicted state for MuÈllerian mimics (again see review in Turner, 1987). Given the intuitive appeal of these de®nitions it seems surprising that there have been such lengthy and lively arguments about the similarities and differences between Batesian and MuÈllerian mimicries (see reviews in Turner, 1987; Huheey, 1988; Mallet and Joron, 1999). The duration of the debate surely arises from the lack of decisive experimental data: without such data it has been possible for a number of researchers to formulate a considerable array of more or less plausible hypotheses (e.g. Huheey, 1976; Owen and Owen, 1984; Turner et al., 1984; Endler, 1991; Speed, 1993; Gavrilets and Hastings, 1998; MacDougall and Dawkins, 1998; Joron and Mallet, 1998; Mallet and Joron, 1999). The lively nature of the discussion surely re¯ects the importance which evolutionary and behavioural ecologists have attached to the study of crypsis, aposematism and mimicry. Mimicry not only represents a celebrated example of Darwinian adaptation, it is also a tool for the investigation of a number of important biological and ecological phenomena. Thus mimicry impinges at least on: studies of signalling and communication, predator±prey relationships, coevolution, insect ¯ight, evolutionary history, shifting-balance (and other evolutionary processes), ecological heterogeneity, speciation and sexual selection. To greater or lesser extents the question of whether a mimic is a (Batesian) parasite or a (MuÈllerian) collaborator is thus of importance to a large number of biological ®elds. For those purely interested in aposematism, the question of mimetic de®nition has added importance because, if as seems likely, the origin of genuinely new aposematic patterns is a very rare event, then many of the warning signals that we see today may well have their origins in some sort of mimicry.

Problems with mimicry theory Despite the appeal of the classical Batesian±MuÈllerian theory, there are good reasons to suspect that this dichotomy in itself is too simplistic. Studies at the population level do not con®rm the prediction that mimetic polymor-

757 phism should be limited to edible, Batesian mimics (see, e.g. Turner, 1971; Joron and Mallet, 1998; Mallet and Joron, 1999). Indeed some of the most spectacular polymorphisms are found in apparently defended species (see Joron et al., this issue). There have been two major approaches to explain mimetic polymorphism in defended species. The ®rst approach examines the ecological structure of a prey's environment (e.g. Beccaloni, 1997; Mallet and Joron, 1999). Thus, structural heterogeneity could promote apparent polymorphism either by tight segregation of microhabitats (e.g. Beccaloni, 1997) or by temporal variation in the abundance of co-mimics with varied warning signals (Joron et al., this issue). Mallet and Joron (1999) have recently added the intriguing suggestion that if populations of defended prey are very large then the per capita costs of predator education will be very small. If this is the case then selection against polymorphisms within a defended species will be weak, which may explain the persistence of polymorphisms in large populations of aposematic prey. These ecological, population level explanations usually maintain the classical Batesian±MuÈllerian dichotomy, but add special case ecological factors to explain polymorphisms. A second approach works by establishing behavioural features of individual predators and then predicting the e€ects that groups of predators can have on populations of prey (e.g. Huheey, 1976; Owen and Owen, 1984, Speed, 1993; Gavrilets and Hastings, 1998; MacDougall and Dawkins, 1998). These `predator level' explanations of MuÈllerian polymorphism often assert that there are key assumptions in the classical MuÈllerian framework that require re-evaluation and reformulation. Common to these hypotheses is an assumption that if there are some levels of discrepancy in protection between two mimetic species, that the less well defended species could act in a Batesian manner, diluting the protection of a better defended species. This has been termed quasi-Batesian mimicry (Speed, 1993) and in proposing Batesian characteristics between two unequally defended prey, it o€ers an explanation for the origin of some cases of polymorphism in defended mimics. Huheey (1976) ®rst proposed a mathematical model that encapsulated this idea. However, Huheey's formulation generated considerable criticism, perhaps because ¯aws in its construction led to the radical prediction that collaborative MuÈllerian mimicry never occurs, even if co-mimics are equally defended (see Owen and Owen, 1984; Turner, 1987; Speed and Turner, 1999). Subsequent to Huheey, Owen and Owen, (1984); Speed (1993) and latterly Gavrilets and Hastings (1998), MacDougall and Dawkins (1998), suggested that moderately defended species could be either collaborative MuÈllerian mimics or parasitic Batesian-like mimics, depending on ecological circumstances. This position is rather less radical than that of Huheey (1976) and, albeit with di€erent theoretical underpinnings, has its precedents in the mainstream mimicry literature. Thus to quote Philip Sheppard (1975, p. 183)

758 `.. the edibility of an object is determined in part by the degree of starvation of the predator. Consequently, edibility is only a relative term. It follows that it is not always possible to determine with any certainty whether every particular association is MuÈllerian or Batesian. . . .a Batesian mimic of one species could act as a MuÈllerian mimic with another. . . '

However, unconventional theoretical models of Owen and Owen (1984) and Speed (1993) have themselves recently been criticised by Mallet and Joron (Joron and Mallet, 1998; Mallet and Joron, 1999; henceforth when I cite `Mallet and Joron' I refer speci®cally this pair of review papers). Mallet and Joron claim that on the grounds of inadequacy of theoretical models the traditional view of mimicry is likely to hold in all, or virtually all, cases. Since MuÈllerian mimicry is a form of aposematism, many of the contentious areas that a€ect de®nitions of mimicry also impinge on our understanding of warning signals in general. Given the importance of these critiques to studies of mimicry and aposematism and the recent publication of relevant data, now seems like a good time to evaluate both sides of the argument. In the ®rst part of this paper, I therefore attempt to identify the strengths and weaknesses of conventional and unconventional mimicry theory by speci®c reference to experimental data. In the second part I attempt a theoretical synthesis by combining components of traditional and unconventional theories mimicry. Predator behaviour: theories and data Predator behaviour and de®nitions of defence One reason to doubt that mutualistic MuÈllerian mimicry necessarily extends to all situations in which defended prey share a warning signal is found when the nature of animal learning is considered. In particular, characteristics of Pavlovian learning (encapsulated in simple models, e.g. Rescorla and Wagner, 1972; Pearce and Hall, 1980) suggest that, if a mimic is less well defended than its comimic, the predator may learn about the inequality in defences and to some extent raise attack rates (Speed, 1993). In their critique, Mallet and Joron contrast unconventional Pavlovian models with what they call the `natural history number dependent view', essentially Muller's original model extended to take into account inequalities in levels of defence (this extension appears to have been informed by reference to the theoretical work of John Turner (e.g. Turner, 1984, 1987; Turner et al., 1984). The `natural history number dependent' view described by Mallet and Joron has two important components, these are (1) an assumption that there is a simple, ®xed number of prey killed during predator education or more generally in the course of a foraging season and (2) an assumption that mimicry between defended species is mutually bene®cial even if there are large discrepancies between their levels of defence.

759 This assumption is important to conventional MuÈllerian theory since it implies that (1) attacks on moderately and highly defended prey will both lower the probability of further attacks toward zero, and therefore (2) mimicry by the less well defended of a pair of mimics can never be parasitic (and will usually be mutually bene®cial and MuÈllerian, Turner et al., 1984). Orange query: J. Mallet (this issue) has recently developed his views by considering dosage intake by predators per unit time. In the number dependent view a ®xed number (or a `®xed killing quota') of defended prey, nk, is assumed to be attacked and killed during a speci®ed time interval. If the population size, N, increases then the proportion of prey attacked must then decrease (i.e. since nk is a constant, the term nk/N decreases as N increases). In contrast a Pavlovian predator was constructed that learns and forgets over time. In constructing the Pavlovian predator it was assumed that when prey were met suciently often that the e€ects of forgetting were minimal then a certain proportion of prey (a `®xed percentage') would be attacked during a given time interval (Speed, 1993). As Mallet and Joron point out, however, if the population size (N) of a prey increases beyond this point, then the number of prey attacked must increase (since number attacked = proportion attacked. N). This makes the unlikely prediction that if N tends to in®nity, so the number of prey attacked increases as a ®xed proportion. Although the presence of forgetting in the models means that densities must rise to some critical level before prey are accepted at the asymptototic percentage, it is indeed likely that after this critical density the percentage attacked does fall as N increases. Thus the `®xed percentage' assumption used in constructing the Pavlovian predator needs modi®cation at higher densities. Whilst I accept this criticism of the `®xed percentage' assumption in the Pavlovian predator it is not fatal to the general hypothesis that inequalities between levels of defence in mimetic prey can lead to a quasi-Batesian outcome. Elsewhere, however, Mallet and Joron provide a di€erent de®nition of unpalatability. They suggest that we consider `unpalatable' any prey item whose asymptotic attack fraction is zero (Mallet and Joron, 1999, p. 221). This is a number dependent de®nition of defence which implies that if N, the population size increases toward in®nity then the ratio of prey killed (nk/N, the asymptotic attack fraction) decreases toward an asymptotic value of zero. Using this de®nition I propose that we consider two broad classes of defended prey: (i) highly defended prey, which always reduce attack probabilities to zero after learning is complete (e.g. Morell and Turner, 1970). An extreme example is the lubber grasshopper which `is large, aposematic, and extremely toxic. In feeding trials with 21 bird and lizard species, none were able to consume this chemically defended prey. Predators that attempted to eat lubbers, often gagged, regurgitated, and sometimes died' (Yosef and Whitman, 1992, p. 527).

760 Here since nk is likely to be small, nk/N will also reach values close to zero fairly easily as N rises (unless of course there are a large number of naive predators). A second class of prey are (ii) moderately defended prey which are attacked after learning is complete but in reduced numbers (see Brower et al., 1963; Lea and Turner, 1971; Platt et al., 1971; Speed et al., 2000). If a moderately defended prey had very low levels of defence (perhaps something very close to the hypothetical `neutrally palatable prey' of Turner et al., (1984) then nk/N would approach zero only if N, the population size, becomes large and/or if the number of predators is very small. Clearly it is members of this second class, the moderately defended prey, that are candidate species for a putative quasiBatesian mimicry. Highly defended prey are just that and are never likely to remove protection from other defended species. Four empirical questions The critique of Mallet and Joron suggests four interesting and related empirical questions that are likely to be important to studies of aposematism and mimicry. These questions are: (1) Is nk, the killing quota, ®xed with density? (2) Is there any evidence that moderately defended prey, as de®ned above, actually exist? If so (3) can moderately defended prey actually dilute the protection of better defended co-mimics? And (4) how are prey likely to be distributed along a spectrum of edibilities? I deal with each of these in turn. Are killing quotas ®xed? Following Muller's (1879) original mathematical model, Mallet and Joron assume that the number of a prey killed during predator education should be approximately ®xed, so long as the edibility of the prey and the number of predators remain constant. There is therefore an hypothetical `killing quota' for each edibility which is constant for a speci®ed time interval, irrespective of the prey's actual density. However the assumption of a `killing quota' may in itself be a simpli®cation since predators may to greater or lesser extents show frequency dependent predation modifying the nk, the number killed during a foraging period according to changes in prey densities. The data that are available suggest, surprisingly perhaps, that neither the killing quota nor the ®xed percentage models (e.g. Speed, 1993) are precisely accurate. There is some tentative evidence emerging that (i) contrary to Mallet and Joron, the killing quota, nk, increases as the size of a population of a defended prey increases but that (ii) contrary to Owen and Owen (1984) and Speed (1993) in the absence of forgetting the proportion of a prey attacked decreases as its population size increases. LindstroÈm et al. (2000, Fig. 3a) showed that, if the number of unpalatable food items presented to captive great

761 tits increases (from i. 8, to ii. 24 to iii. 64) so the number of unpalatable items attacked increased (from i. 4 to ii. 9 to iii. 18). However, the percentage of unpalatable food items attacked decreases as the population size increases (i.e. i. 4/8 ˆ 50%; ii. 9/24 ˆ 37.5% and iii. 18/64 ˆ 28%). Similarly, when Speed et al. (2000, Experiment 1, Fig. 2) presented garden visiting birds with ®ve moderately unpalatable prey items a day for 20 days, on average 4.2 items were attacked (i.e. less than the ®ve put out); when 20 were put out each day 8.4 items were attacked. Thus, nk may not increase as a constant fraction of prey attacked as the unconventional models assume, however, contrary to the assumptions made in classical MuÈllerian theory (see Discussions in Mallet and Joron), it does seem to increase systematically with the prey's population size. Thus, in as much as we have any data to test the predictions of Speed (1993) and Mallet and Joron (1999), the truth seems to lie somewhere between the two positions. If nk varies considerably with N then it follows that the zone for a putative quasi-Batesian mimicry situation could be large. If on the other hand nk varies very little then any quasi-Batesian zone would be smaller. This is obviously an area in which relatively simple empirical investigations will be of considerable value.

Learning about prey edibilities: all or nothing? Turner et al. (1984), have suggested that, given a constant density of prey, each attack on an unpalatable prey must lower the likelihood of a future attack toward zero. In this view, experience with moderately defended prey still reduce the likelihood of future attacks, but this happens more slowly than with highly defended prey (see Joron and Mallet, 1998, p. 463). If this assumption about animal learning is correct then quasi-Batesian mimicry is not feasible even as a theoretical creation: attacks on any defended prey, no matter how weak the defences, will always add protection to remaining prey. However it is very easy to demonstrate empirically that this view of learning is not correct: predators that have learnt about defended prey do not always show complete avoidance and indeed may continue to attack them in low, but fairly constant numbers. For example, in a laboratory experiment Speed (1990) trained 12, 3week old male chickens (Gallus gallus) to drink from a small container. In the ®rst part of the investigation the birds learnt that blue water was acceptable (Fig. 1, experimental details in legend), and ®nally accepted it on a high proportion of occasions. The birds were next presented with green water which had been made distasteful (a 0.0025% solution of quinine dihydrochloride). As Figure 1 shows the percentage of birds which attacked these `water prey' fell quickly to a steady value at around 36 percent. With learning apparently complete, the `water prey' was neither completely accepted nor completely

762

Figure 1. Twelve 3-week old male broiler chickens were put into an experimental arena (an illuminated skinner box) for one minute. Water was presented in a small white cylinder (3 cm diameter, 2 cm height). An attack is de®ned as an event in which the birds' beak makes contact with the water. The percentage of birds that `attacked' the water on each trial are shown above. In the ®rst 20 trials, water was coloured light blue by the addition of a small amount of food dye (i.e. 3 ml L)1). In the second set of 20 trials green water was made unpalatable by addition of a small amount of quinine dihydrochloride (a 0.0025% solution was added). Birds showed no intrinsic aversions to blue water.

rejected; instead some stable, intermediate number of attacks per unit time resulted. Demonstration of a stable attack number greater than zero for a defended item is actually not unusual. A number of other studies show this qualitative result with real and arti®cial defended prey (e.g. Brower 1958 ± pooled across birds E1-4, p. 37; Alcock, 1970; Platt et al., 1971; Gittleman et al., 1980; GriegSmith, 1987; Speed et al., 2000). These studies generally provide rapid conditioning procedures which demonstrate stable nonzero attack numbers uncontaminated by e€ects of forgetting. Examples in which birds attack apparently defended prey have been noted in wild situations in a number of ecological circumstances (e.g. Swynnerton, 1915; Srygley and Kingsolver, 1998 show how complex the notion of palatability actually is). Thus it does seem that there are both arti®cial and real prey that do not generate complete rejection by predators. The Pavlovian model of learning (in the sense of Rescorla and Wagner, 1972; Pearce and Hall, 1980, etc.) allows animals to vary not just their rates of learning, but also the intensity of their behavioural outputs to match the importance of the reinforcer. Thus having learnt about the aversive qualities of

763 the `water prey' the chickens in my experiment appear on average to have decided to take the `prey' on 30±40 percent of presentations. By doing so they gain a resource (water) at some cost (quinine) and appear to balance out the costs and bene®ts. Thus the ability to vary the level of behavioural output to match the hedonic qualities of the reinforcer must be an essential component of a model of learning (see use of the term in Rescorla and Wagner, 1972; Pearce and Hall, 1980). What matters is that a predator can ingest a defended prey at a suciently low rate to avoid toxicosis and to gain an overall nutritional (and hedonic) pro®t. Defended prey therefore represent complex trade-o€s between nutrition and toxicity which make it dicult to state a priori that virtually all defended prey should be avoided by cognisant predators. By continuing attacks on arti®cial and real prey items, predators seem to indicate that they can manage the trade-o€ between toxins and nutrition such that in all they make a net pro®t. Can moderately defended prey actually dilute the protection of highly defended species? Although it is easy to establish that some apparently `defended' food items do not generate complete rejection by predators, this is not the same as demonstrating a Batesian relationship between two unequally defended food items. In the only published experiment designed to test the contrasting predictions of conventional and unconventional models (Speed et al., 2000), garden bird predators raised the attack rate on a `highly aversive' arti®cial prey (a pastry worm, made distasteful by the addition of small amounts of quinine and mustard) if the highly aversive prey was mimicked by a moderately defended co-mimic. The experiment was replicated twice: in the ®rst replication mimicry raised the percentage of the well defended prey that were attacked from 15 to 40 percent. In the second replication mimicry raised the attacked percentage from 37 to 54 percent. The moderately defended prey was thus a quasi-Batesian mimic of the better defended prey. Up to this point, a conventional mimicry theorist could claim that the moderately defended mimic was in fact not defended at all, but instead was edible to the bird predators. In this view the experiment merely demonstrates Batesian mimicry. However, when a common moderately defended prey mimicked another moderately defended but very rare prey, it provided substantial protection to the rare prey in a MuÈllerian fashion. In the ®rst replication the proportion attacked fell from 85 down to 41 percent; in the second replication it fell from 95 to 45 percent. In contrast a common truly edible bait (without quinine and mustard) was accepted at something very near a 100 percent level. The moderately defended pastry prey acts thus as quasi-Batesian mimic in one situation and as a MuÈllerian mimic in another. This is precisely what unconventional

764 theories predict (e.g. Speed, 1993 and later in this paper). In contrast the classical view, predicts a mutualistic MuÈllerian e€ect in both cases. There are however some limitations to the data presented in Speed et al. (2000). First it was not possible to count the number of visiting birds and a further replication with systematic data on individually colour-ringed birds be very useful. Second, arti®cial prey made of pastry, quinine and mustard represent rather different trade-o€s to naturally occurring insect prey. Other laboratory experiments designed test the characteristics of the `conventional mimicry' learning rule (Turner et al., 1984) vs. the `unconventional mimicry' Pavlovian learning rule (Speed, 1993) so far support the unconventional rule unambiguously (S. Hannah et al., in preparation). They show that in laboratory conditions moderately defended arti®cial mimics can indeed dilute the protection of highly defended co-mimics. In summary then the data that we have at this point in time support the view that some arti®cial prey can be both (quasi-) Batesian and MuÈllerian mimics in di€erent circumstances. However there is an obvious need for further experimental work, especially testing the responses of individually identi®able wild birds to unequally defended prey items. A key question, as yet unanswered, is whether birds exercise their judgements after capture, releasing their highly defended prey but continuing to ingest the less nasty species. Mimicry and the geography of the palatability spectrum Much of the modern theoretical development of mimicry theory depends on an assumption that the acceptability of prey to predators is distributed along a spectrum, from the utterly edible and nutritious to the utterly toxic and dangerous (Turner, 1984). However, there has been relatively little discussion about the way that defended prey might distribute themselves along a continuous spectrum of defence. Suppose that we were to plot a histogram with numbers of species on the Y-axis and a continuous spectrum of defence on the X-axis. Defence would range from `neutral palatability' (Turner et al., 1984) to the very highly defended, and could be de®ned by numbers killed during a given period. Given this scenario, the key question is what shape would such a histogram have? On the one hand prey could be evenly distributed along this unpalatability spectrum, the histogram would be ¯at, in which case a zone for putative quasi-Batesian mimicry might be large. On the other hand, it might be the case that all defended prey are really very well defended and then the modal level of protection would be near the nasty end of the spectrum. In this scenario, putative quasi-Batesian mimics would be virtually non-existent. Although the truth is probably somewhere between these two positions I suspect that it is in some way closer to the second than the ®rst. The reason is that the psychology of perception has demonstrated repeatedly that perceived

765 intensity (e.g. perceived aversiveness) is approximately related to actual intensity according to a simple formula, known as Weber's Law: Perceived intensity ˆ k
log(actual intensity) where k is a scaling constant. If Weber's law holds for the relationship between the degree of prey avoidance (i.e. nk per unit time) and toxin content, then it would seem that ± for an even distribution of toxin content along a range from very low to very high doses ± a disproportionate number of prey might distribute toward the aversive end of the spectrum of acceptabilities (see Speed, 1990). We might then expect to see a world in which a majority but by no means all prey with chemical defences are well defended. If quasi-Batesian mimicry does exist as a real phenomenon then for reasons of psychophysics (and indeed the biochemistry of toxicity) I expect that it is rarer than mutualistic MuÈllerian mimicry (see Discussion in Speed, 1990; Mallet & Joron 1999). That said the empirical evidence is that `moderate defences' are easy to generate in arti®cial prey (see Speed et al., 2000) and easy to identify in real prey. Thus in a major study of prey acceptabilities, Sargent (1995) found that Lepidoptera were distributed along a spectrum rather than bunched at extremes. Sargent (1995) classed 11 percent of prey as being of moderate/high unpalatability and 33 percent as being slightly unacceptable/acceptable. Certainly, laboratory studies of prey acceptabilities have shown that while some prey are very well defended, others generate rather lower intermediate levels of deterrance that are likely to have higher rather than lower nk values (e.g. Brower et al., 1963; Bowers and Farley, 1990, Rowell-Rahier et al., 1995). The geography of the palatability spectrum, like so much in mimicry studies, needs further, systematic empirical study.

`Pavlovian' and number-dependent models: is a synthesis possible? I have argued that (1) when learning is complete predators will completely avoid some defended prey items and continue to attack others at diminished rates (i.e. in diminished numbers per unit time); (2) moderately defended prey may dilute the protection of highly defended co-mimics (see Speed et al., 2000; Hannah et al., in preparation) but that (3) earlier theoretical models (e.g. Speed, 1993) lack components that would take into account the e€ects of number dependence on survival. Mallet and Joron have argued that quasi-Batesian theories (Owen and Owen, 1984; Speed, 1993) rely on an assumption that attack fractions can asymptote at some intermediate level between one and zero. If this assumption is unlikely, it follows, theories of quasi-Batesian mimicry must be correspondingly unlikely. However, I do not believe that quasi-Batesian theories

766 must rely on an assumption of constant asymptotic attack fractions for their very existence. What is required to generate quasi-Batesian mimicry is that at some density levels, mimicry by a moderately defended prey can raise attack probabilities from the levels of an unmimicked highly defended prey (see Speed et al., 2000; Hannah et al., in preparation). This can be modelled to di€erent e€ects using either constant percentage assumptions (Speed, 1993) or by assuming number dependent e€ects on predation. In order to formulate a simple number dependent model that incorporates features of Pavlovian learning, I consider a scenario in which a common highly defended prey, species A, that is mimicked by a prey with very mild levels of defence (see Turner et al., 1984), species B. The attack fraction nk =N for both species will tend to zero as N increases and thus both species are defended by Mallet and Joron's second de®nition (see above). However, since species A is much nastier than species B, the attack fraction nk =N it comes close to zero at lower prey densities than for species B. Now the key question is, will species B add to or subtract from species A's levels of protection? It is possible to envisage two scenarios: ®rst, if predators increasingly reject all defended prey (after Turner et al., 1984; and see Joron and Mallet, 1998, p. 463) then the costs of educating predators toward complete avoidance are shared between species A and B (see Appendix to J. Mallet's paper, this issue). A classical MuÈllerian relationship results. Alternatively, in a quasi-Batesian scenario the addition of the less well defended species B raises A's attack rate (i.e. the number of A attacked per unit time). The mixture is, overall, less aversive than A on its own and the predator raises its attack probability accordingly (see Results in Speed et al., 2000 and Hannah et al., in preparation). At higher densities of species B, however, the less well defended prey may add protection by a classical dilution e€ect. This scenario can be modelled in the following manner. Suppose that a single predator inhabits a locality in which it meets a well defended prey (species A) and a moderately defended comimic (species B). A and B are equally abundant, with densities of, e.g. 100 each. Now suppose that in the absence of mimicry, over a given time period, 10 of A and 40 of B would be attacked (50 in all). In the presence of mimicry, species A may lose protection because some prey which look like A are not as nasty as expected. Species B may gain protection because of its resemblance to the nastier species A. This situation can be represented in a number dependent manner by assuming that with mimicry between species A and B, the number attacked will be some intermediate value between 10 and 40. Since A and B are equally abundant, I assume for simplicity that the total number of the AB mimicry complex attacked is 50/2 or 25. Again, since species A and B are equally abundant and identical, I assume that about 25/2 or 12.5 individuals of A are attacked and 12.5 of B are attacked. Thus the less well defended species B

767 bene®ts from mimicry (number attacked falls from 40 to 12.5), but the better defended species A loses (numbers attacked increase from 10 to 12.5). Mimicry by the less well defended species is thus quasi-Batesian in a number dependent model. An interesting result pertains, however, when in this scenario the moderately defended species B becomes more common. If there are 300 of species B present and all other conditions are unchanged then B shoulders a larger burden of predator education. By a classical dilution e€ect B now adds protection to A. If the population of B reaches 300 then 8.125 of species A are attacked `saving' 1.875 A individuals from attack. Thus, the addition of number dependence does not in itself remove the possibility of quasi-Batesian mimicry, but it does change the dynamics so that at lower density levels a moderately defended mimic may be parasitic, but at higher levels it will be a collaborative MuÈllerian mimic. Joron and Mallet (1998) make a similar argument about the abundances of edible prey. In order to explore the dynamics of a number dependent-Pavlovian system in more detail the model can be stated more generally like this: I assume that: (1) There are two species ± A and B. A is well defended, B is variable in its level of defence. (2) A and B have absolute numbers speci®ed. Call these NA and NB respectively. (3) N, is the total number of prey under consideration (N ˆ NA ‡ NB ). (4) Levels of defence are de®ned by the number of prey killed during a foraging season (for A and B these are, nA and nB , respectively ± the k subscript is not included for A and B to avoid complication). nk values represent the number of prey killed provided that N >ˆ nk . (If N < nk then predators satiate and all individuals are killed.) I consider nk values with and without mimicry such that: (5) When there is no mimicry, numbers of prey attacked are nA and nB . (6) When there is mimicry between A and B I assume that the total number attacked, nAB is de®ned by reference to the proportions of each present in the population, thus: nAB ˆ …nA
NA =N† ‡ …nB
NB =N† Some simple scenarios: Suppose that i. nA ˆ 10 and nB ˆ 40, and that there are 100 each of A and B. nAB ˆ …10  0:5† ‡ …40  0:5† ˆ 25 Likewise, if ii. nA ˆ nB ˆ 10 individuals, and A and B are equally common then nAB ˆ …10  0:5† ‡ …10  0:5† ˆ 10

…1†

768

as

and any increases in numbers of either A or B are exactly equivalent in terms of nAB . The proportion of Model and Mimic attacked k can be calculated simply k ˆ nAB =…NA ‡ NB †

…2†

Thus I assume that protection is a mean of n values weighted according to the proportion of each species in the mimicry complex. Note that cases in which NA ‡ NB < nAB represent situations in which demand exceeds supply. The proportion attacked, is thus a measure of the total prey number that would be attacked relative to the number that exist. Values of >1 indicates that all available prey will be attacked. In the diagrams below, I plot the proportion of available prey that are predicted to be attacked, hence the maximum value is 1. General Predictions Scenario 1. Mimicry through a spectrum of edibilities In a foraging season of ®xed duration, I assume that within a given locality there is a population of 1000 individuals of species A and there are 10 predators. Without mimicry each naive predator would attack 10 models. Hence, in total, nA ˆ 100. Species B is allowed to increase its density by increments of 100 prey. Figure 2 shows the e€ect of increases in B's abundance and variation in its edibility on the proportion of the AB mimicry complex that are attacked. In all cases, species B is less well defended than species A. B's edibility varies through a spectrum from the weakly defended (nB ˆ 1000) down to the well defended (nB ˆ 200). In all cases except that in which species B is well defended (i.e. nk ˆ 200) species A loses protection from mimicry (Fig. 2). However, as anticipated in the discussion above, the results are nonmonotonic. Increases in the population of species B does initially increase the proportion of both A and B attacked. However, at some threshold level of abundance (indicated with a star *) the direction of density dependence reverses and the protection of both A and B now increases with further increases in B's abundance. Before this point, rarity is the optimal state and diversity in mimicry will be selectively advantageous. After this point, however, the less well defended mimic (B) begins to experience strength in numbers and from this point onward monomorphic mimicry becomes favoured. Note, however, that mimicry is only MuÈllerian when the proportion of A attacked falls beneath the dotted horizontal line which represents the proportion of species A attacked when it is not mimicked (in constant

769

Figure 2. E€ects of density of species B through a spectrum of prey edibilities. The Y-axis shows the proportion of species A and B that are attacked, the X-axis shows the size of the population for species B. Species A, NA = 1000, nA = 100 Species B defensieve levels vary. In the curves, from topdown, nB = 1000, 750, 500, 200. The horizontal dotted line indicates the proportion of species A that are attacked when there is no mimicry (i.e. NB = 0). If the curves of predation fall below this value then mimicry is mutualistic and Mullerian. This is the case for the curve at the bottom in which nB = 200. *Indicates the value for species B's density at which the direction of density dependence changes.

asymptote models this pattern of up-down density dependent predation can only be generated if the less well defended of a pair of mimics evokes a higher forgetting rate than a better defended mimic: see Speed and Tomer, 1999). Scenario 2. Moderately defended mimics may be quasi-Batesian or MuÈllerian according to the abundance of a better defended comimic In Figure 3, the scenario changes both in terms of edibilities (nA ˆ 10; nB ˆ 80) and in terms of species A abundance. If species A is very rare (NA ˆ 20), Mimic is generally MuÈllerian; the addition of more than 20 moderately defended mimics adds net protection to species A. When species A is more common (NA ˆ 100), the same Mimic becomes more clearly quasi-Batesian; removing protection at low densities, enhancing it at higher densities.

770

Figure 3. E€ects of variation in Model density on the direction of density dependent predation. The Y-axis shows the proportion of species A and B that are attacked, the X-axis shows the size of the population for species B. Top curve, species A is very rare (NA = 20, nA = 10), bottom curve species A is more common (NA = 100).

Scenario 3. Moderately defended mimics can act as models for edible Batesian mimics In Figure 4, the scenario is that species B is now edible and nutritious: nB ˆ 10; 000 and NB = 9000); species A is moderately well defended (nA ˆ 500), as numbers of species A increase, attack probability falls and the moderately defended species A adds protection to its edible Batesian mimic. Summary. It is thus possible to construct a simple theoretical model that incorporates number dependence and predicts unconventional quasi-Batesain mimicry at lower density levels, and conventional MuÈllerian mimicry at higher density levels. The crucial assumption which leads to unconventional mimicry is encapsulated in Equation (1). Here I assume that with mimicry between a moderately and a highly defended prey the number of prey attacked is an average of the numbers that would be attacked without mimicry, but that the average is weighted according to the relative frequencies of the species within a mimicry complex. There seems to be some data to suggest that this is a reasonable strategy for modelling (e.g. Speed et al., 2000; Hannah et al., in

771

Figure 4. The moderately de®ned species A acts as a model for an edible species. B. Here species B which is entirely edible (nB = 10,000, N = 9000) mimics species A which is moderately defended (nA = 500). As species A becomes more common (X-axis) it adds to the protection for both A and B when its density >500.

preparation), but obviously we need further experiments to help test and re®ne the assumptions made in these sort of models (see also J. Mallet's paper in this issue). Scenario 4. Extension to Batesian mimicry: satiation and polymorphism Since a set of predators will eventually satiate on a given prey type, it is technically true that the asymptotic attack fraction for any prey must be zero: once N increases beyond the point of satiation (i.e. N > nk) then the attack fraction nk/N must decrease toward zero. It is of course an interesting question whether and how often truly edible prey really can satiate a set of predators, but this will happen at least when some prey populations undergo temporary `explosions' to very high levels. The number dependent models can be extended into the edible half of the palatability spectrum in as much as it is possible to: (i) specify a notional satiation value (i.e. a high value of nk) assuming that predator numbers remain constant and (ii) it is possible to assume a spectrum of edibilities (see Discussion in Turner and Speed, this issue). Edible prey may

772 vary in their satiation values because: (i) some edible prey are intrinsically of low net nutritional value (perhaps because in small prey there is a relatively high proportion of cuticle compared to fat/muscle content), or (ii) they contain nutrients that become aversive in high dosages, or (iii) nutritionally similar prey vary in their costs of detection and capture (see Discussion in Turner and Speed, this issue). However, if we assume at least for the sake of argument that there may be an edibility spectrum then using the equations above (1, 2), it is thus possible to model mimicry of a well defended prey (species A: nA =100, NA = 1000) by more or less edible prey (species B: nB = 3000, 5000, 10,000, and NB varies). The results of these scenarios are shown in Figure 5. It can be seen that in each case, the dynamics of predation are not monotonic, initially rising up to a critical point and then falling. The point of saturation for the unmimicked edible prey (i.e. nB ˆ N) is shown by the vertical dotted line for each edibility level. In the absence of mimicry, these points would represent population sizes beyond which additional prey start to increase protection. When there is mimicry however, the positions of the turning points are moved toward the left, to lower densities (indicated by *). In evolutionary terms once the density of the mimic reaches the critical point (*) then further increases in the density of the edible species increase protection of the Batesian mimic and reduce the losses of its model. From this density point onwards we would expect mimetic monomorphism to be favoured. Note that as the proportion attacked falls after the turning point (*) predators have not necessarily satiated on the edible species. From the turning point onwards, the numbers of the edible mimic that are attacked do increase with each increment of its density, however they increase more slowly than numbers that are not attacked and hence, a higher number of edible prey are attacked but in lower and lower proportions. If there are moderately edible mimetic species (in the order nk = 3000, 5000 for a locality with a de®ned number of predators), then this number dependent model makes an interesting prediction that if densities increase beyond some point, the prey populations develop strength in numbers, and `MuÈllerian monomorphisms' are predicted. Batesian mimics may outnumber their models not just because their models are very nasty, but also if they themselves are of limited edibility. This seems to contradict the traditional idea that Batesian mimics lose ®tness in a monotonic manner with each increment to their density. This unconventional result is caused, most obviously from the assumption that predators satiate on a de®ned number of edible prey. However, the exact shapes of the curve also depend on the assumption (in Equation (1)) that in a mimicry complex, the total number of models and mimics attacked is de®ned by an average of the numbers that would be attacked if the species were unmimicked weighted according to the relative proportions of each in the modelmimic complex. In reality weightings may di€er from this simple assumption

773

Figure 5. Predation on a model-mimic pair when a Batesian mimic varies in density. Species A is defended (NA = 1000, nA = 100, species B has varying edibilities. Top-down it is highly edible (nB = 10,000, 5000, 3000). The Y-axis shows the proportion of species A and B attacked as the density of species B increases (X-axis).

and this would change the shapes of the curves shown in the results in Figures 2±5. Thus if greater weighting were given to the nk value of the defended prey then the pivotal point (*) in Figures 2 and 5 at which the direction of density dependence changes would occur at lower densities. Conversely if greater weighting were given to the edible (or moderately defended) prey then the pivotal point (*) would be moved to higher densities since in this scenario predators value the bene®ts of nutrition higher than the costs of toxins.

4. Conclusions One of the most important outcomes from arguments about mimicry has been a growing recognition of the importance of data in generating and testing theoretical frameworks. By considering empirical studies in the ®rst part of this

774 paper I have attempted to evaluate some of the assumptions made in the theoretical literature. One clear message from this is the need for further well targeted empirical research. In the second part of this paper I have outlined a model that takes into account empirical ®ndings and attempts to put Pavlovian assumptions as I understand them into a number dependent model. This should be seen not as an attempt at a de®nitive description of predator behaviour, but instead as a `what if '? model. It asks `what will happen if we make the following assumptions about predators and prey? ' and tells us (i) whether anything interesting is predicted and (ii) what aspects of predator behaviour are crucial and in further need of empirical study. I contend that the results of such a model are indeed interesting and that they point, as ever, to the need for more data.

Acknowledgements I thank R. Alatalo, J. Mappes, L. LindstroÈm and Anne Lyytinen for their generous hospitality at `Aposematism: past, present & future'. I also thank Jim Mallet, Simon Hannah and John Turner for many helpful and enjoyable discussions.

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