Evolutionary Ecology 13: 807±827, 1999. Ó 2001 Kluwer Academic Publishers. Printed in the Netherlands.

How weird can mimicry get? Dedicated to Miriam Rothschild

JOHN R.G. TURNER1* and MICHAEL P. SPEED2 1

School of Biology, University of Leeds, Leeds LS1 9JT, UK; 2Department of Environmental and Biological Studies, Liverpool Hope University College, Hope Park, Liverpool L16 9JD, UK (*author for correspondence, tel.: +113 233 2828; fax: +113 233 3091; e-mail: [email protected]) Received 14 September 2000; accepted 11 January 2000 Co-ordinating editor: C. Rowe Abstract. Classical mimicry theory distinguishes clearly between the mutualistic resemblance between two or more defended species (muellerian mimicry), and the parasitic resemblance of a palatable species to a defended species (batesian mimicry). Modelling the behaviour of predators, without initially taking ecological complications into account, is a good strategy for exploring whether this division is valid. Two such behavioural models are described: conditioning theory, which simulates changes in motivational attack levels according to the norms of current learning theory; and saturation theory, which considers how a predator may become saturated with a particular toxic compound, and then cease feeding on the prey species that delivers it. This e€ect is to be clearly distinguished from simple satiation. Most formulations of the conditioning model allow the direction of reinforcement produced by a particular prey to change according the predator's current state of motivation: this leads to the existence of quasi-batesian mimicry, a parasitic mimicry between two species that could both be described as defended. At high densities, two prey species that share a chemical defense will be `muellerian mutualists', mutually protecting each other against predators that have been saturated with the defensive compound. This mutualism may be accompanied by true muellerian mimicry of the colour patterns, or the patterns may be completely di€erent. This can therefore be regarded as a form of mimicry in a non-visual communication channel. Even an apparently palatable prey species may be e€ectively unavailable to predators if its density is such as to deliver a particular nutrient in excess of the predator's need for a balanced diet. Such a nutrient in e€ect becomes a toxin, and such an abundant prey species would be partly defended and potentially able to act as the model in a mimicry system. Thus there might be protective mimicry between `palatable' species, and a `palatable' species might even function as the model for a `defended' mimic. These unorthodox kinds of mimicry probably exist transiently during ¯uctuations of prey populations. It is less likely that these conditions persist for long enough to induce the evolution of mimicry, and the relationships perhaps usually occur when mimicry already exists for other reasons. Mimicry rings may be mutually stabilised by a combination of toxic mutualism and the exchange of species between the rings. Colour polymorphism in a defended species is strictly neutral whenever the population is dense enough to saturate the predator. This, as well as quasi-batesian mimicry, may help to explain the minority of warningly coloured species that are polymorphic. Key words: conditioning, dietary complementation, mimicry, mutualism, nutrition, palatability spectrum, toxicity

808 Two ways of modelling predator behaviour It is a long time since Rothschild (1981) pleaded that mimicry theory should be brought up to date with the rich ®ndings of empirical biological research. Crucial to such an undertaking is a good strategy of model building. In classical mimicry, by which we mean the adaptive resemblance between protected and unprotected species ®rst noted by Bates and MuÈller (there are of course many other kinds of mimicry), there has been a sometimes unstated assumption that mimicry could easily be divided into mutualistic, muellerian mimicry between protected species, and the parasitic, batesian resemblance between a palatable and a protected species: we could call this division the classical mimicry spectrum (Gilbert, 1983; Turner, 1984, 1995). It has long been obvious that this distinction cannot be applied rigidly. Changes in the ecological circumstances with place and time could mean that what was a muellerian relationship at one point could well be a batesian relationship at another. The predators for instance might become very hungry and be willing to accept prey that in times of abundance they would reject. There was however a current of opinion (Benson, 1977; Sheppard and Turner, 1977) that the appropriate strategy for modelling this problem was to consider ®rst the relationships that would be generated between two species at one place and one time ± by modelling the psychology of the predator ± and only after that to consider what would happen to this relationship should the ecological circumstances change. The exciting development of the last ten years has been that when we probe predator behaviour in depth, we ®nd that the old clear distinction between batesian and muellerian mimicry still does not strictly apply. In this paper we review some aspects of this question, using two very di€erent models of predator behaviour and psychology: the conditioning model and the saturation feeding (or just saturation) model.

Conditioning model Conditioning theory uses equations or algorithms to model the behaviour of a predator that encounters a mixture of variously mimetic prey, using what is known, or believed, about vertebrate psychology from the rich experimental and observational literature since Pavlov (e.g. Huheey, 1964; Pough et al., 1973; Charlesworth and Charlesworth, 1976; Turner et al., 1984). The predator has an attack motivation toward prey whose level is modi®ed according to the predator's experience: palatable prey are de®ned as those which increase the attack motivation, unpalatable prey as those that decrease it. This is readily modelled by Monte Carlo methods (Turner et al., 1984; Speed, 1990; Turner and Speed, 1996; Speed and Turner, 1999). The most recent analysis (Speed,

809 1993; Speed and Turner, 1999) has challenged the classical view of mimicry as clearly batesian or muellerian: mimicry between two defended species may sometimes be not mutualistic, but parasitic like batesian mimicry. This view itself has been challenged by Joron and Mallet (1998). The model is appropriate to the behaviour of a polyphagous predator that is encountering the mimetic prey as a minor component of a mixed diet. Saturation theory on the other hand is appropriate to a predator that is exploiting a mimetic or defended prey as a major component of its diet. The prey being at high density, the predator may cease to feed because it can eat no more without risking its health. This behaviour has for instance been observed by Brower and Calvert (1985) among grosbeaks Pheuctitus spp. and orioles Icterus spp. feasting on over-wintering Monarch Danaus plexippus butter¯ies in Michoacan: the birds can individually eat only a limited number of Monarchs because they are in danger of overdosing on cardenolides. Although the earliest mathematical models of mimicry (MuÈller, 1879) are e€ectively based on this concept, and the recent challenge to the unorthodox ®ndings of conditioning theory (Mallet and Joron, 2000) uses a theory of this mathematical form to argue that mimicry between distasteful species will always be mutualistic, it is distinctly under-explored. We will suggest that its implications are even more startling than those of conditioning theory.

Conditioning theory The ®rst attempts (Turner et al., 1984) at imitating the behaviour of a predator encountering a mixture of prey which contained palatable and unpalatable mimics, or unpalatable mimics with varying degrees of protection, con®rmed the classical view of the mimicry spectrum: that it divided cleanly into parasitic batesian mimicry when one of the prey was palatable, and mutualistic muellerian mimicry when both were defended. A more sophisticated view of predator psychology on the other hand predicts what we will call the quasibatesian spectrum: between batesian and muellerian mimicry there is a third category, quasi-batesian mimicry, in which both the prey are defended, but defended to di€erent degrees. In this case, the less defended species can according to the relative densities of the two species, act as a parasitic mimic of the better defended (i.e. parasitic in the sense that the better defended of the species will be harmed by the mimicry), and have dynamic properties closely akin to those of a classical batesian mimic (hence quasi-batesian). The algorithms (Turner et al., 1984; Turner and Speed, 1996) are designed to imitate the likely behaviour of a vertebrate which is responding to a sequence of paired conditioned and unconditioned stimuli (respectively the colour pattern or other signal and the defensive ¯avour or toxin): they are clearly not

810 in any way an attempt to copy the neural mechanisms involved. Thus they cannot give precise predictions about the way predators will behave. What they can do rather well however is to show how robust particular outcomes are against changes in the assumptions built in to the algorithms. Thus they give us some indication of how precisely the real behaviour of real vertebrates would have to imitate the algorithms in order to generate particular kinds of mimicry spectra. The rather startling conclusion (Table 1) is that the classical mimicry spectrum is not robust against relatively minor alterations in the algorithm (Speed and Turner, 1999). Or to put it another way, if we think of the algorithm being tuned in various ways, rather precise tuning is required to produce this kind of spectrum. Over the broad range of tunings that are available, the majority produce spectra which contain quasi-batesian mimicry as an intermediate condition between muellerian and batesian mimicry. Rather precisely matched learning and forgetting mechanisms would have to be in place in the vertebrate brain to produce the classical spectrum, making it at the least plausible that this spectrum does not occur in reality. These conditions are that forgetting depends at least in part on the passage of time, and that the motivational state of the predator is purely a dynamic equilibrium between learning (acquisition) and forgetting (loss or reversal). An alternative way of stating the second condition is that there is no tendency for the brain itself to `set' or maintain a particular motivation or strategy based on stored information; or that sucient conditioning with a negative stimulus, if forgetting is reduced toward zero, will result in total aversion (or totally appetitive behaviour in the event of positive conditioning). That is to say, that all

Table 1. Types of mimicry spectrum produced by combinations of learning and forgetting rules in simulations of conditioning theory (simpli®ed from Speed and Turner, 1999) Learning rule

Generalised Bush±Mosteller Sheppard±Turner Owen±Owen Huheey

Forgetting rule Huheeyevent

Huheey-timed

Sheppard± Turner

Owen±Owen

Combined

Huheey Huheey Huheey Huheey Huheey

Quasi-batesian Quasi-batesian Quasi-batesian Quasi-batesian Classical

Quasi-batesian Quasi-batesian Classical Quasi-batesian ±

Quasi-batesian Quasi-batesian Quasi-batesian Quasi-batesian ±

Quasi-batesian Quasi-batesian Quasi-batesian Quasi-batesian ±

The learning and forgetting rules are fully explained in Turner and Speed (1996). Huheey-event is a rule in which forgetting is triggered only by the observation of prey; in Huheey-timed and the other rules forgetting occurs over time; and in the combined rule both events and the passage of time trigger forgetting. The classical mimicry spectrum contains only batesian and muellerian mimicry, the quasi-batesian spectrum adds quasi-batesian mimicry between muellerian and batesian mimicry, and the Huheey spectrum (not discussed in this paper) has only batesian and quasibatesian mimicry.

811 negative stimuli can eventually result in total aversion, the di€erence between negative stimuli being expressed only in di€erent rates of approach to this zero asymptote of motivation, but not in the asymptote itself. This dynamic motivational model was proposed by Turner et al. (1984), was implicit in the thinking of Benson (1977) and of Sheppard and Turner (1977), and has recently been advocated by Joron and Mallet (1998). In the context of modern cognitive and learning theory the model may seem somewhat eccentric, but it does not fail on one important criterion where at ®rst thought it seems to be de®cient (Huheey, 1988): because of the dynamic relationship between learning and forgetting, which sets the moment to moment level of motivation, it is the case that this type of model causes longer periods of avoidance of more unpleasant stimuli. In this respect it behaves very similarly to models in which there is a positive `setting' of the motivational level. Where it di€ers from these models is in failing to produce one outstanding aspect of classical conditioning behaviour, the `extinction' of a conditioned response by the continued presentation of a neutral unconditioned stimulus (the ¯avour) alongside the conditioned stimulus (the pattern) (Speed, 1993, 2001). The model which generates the classical spectrum makes the assumption that, given certain ecological and other constants, such as a constant level of hunger, the unconditioned stimulus (prey palatability) divides prey into two categories: the positively reinforcing and the negatively reinforcing (conventionally `palatable' and `unpalatable') and that any one prey species is always treated as being in the same category. (The strength of reinforcement varies of course with the properties of the prey, so that those very close to the neutral borderline between the two categories are only very weakly reinforcing.) Thus a prey which negatively reinforces a naive predator is also invariably a negative reinforcer for an experienced predator, and in the same amount. The cognitive algorithms (Speed, 1990, 1993) on the other hand, which generate quasi-batesian mimicry, assume that palatability is not a constant property of the interaction of the prey with the central nervous system of the predator, but is treated by the predator as context-dependent: the direction of reinforcement is in some way dependent on the experience of the predator, so that a prey which negatively reinforces when the predator is naive may under some circumstances be a positive reinforcer for predators which have become experienced. The way in which this is built into the algorithm is illustrated in Figure 1, which shows what may happen if a predator is encountering a pair of muellerian mimics, both `unpalatable' and sharing the same pattern (CS), so that the predator cannot distinguish them before it attacks. When the predator is naive, encountering either of the prey will produce negative conditioning, so that the probability or motivation of attack decreases, cumulatively in the illustration we are using. At point x the predator now attacks another prey. This is not the highly unacceptable species to which it has so far been

812

Figure 1. To illustrate the question of positive vs. negative conditioning. A predator is, as is shown by its declining attack motivation, being repeatedly conditioned to avoid a strongly defended prey (asymptotic level of attack Q), but at x encounters a mimic which is only moderately defended. The attack motivation might then increase or decrease. The two currently used null assumptions are that it always decreases (Turner et al., 1984; Mallet and Joron, 2000), or that it increases provided the current attack probability (at x) lies in zone B (that is below the asymptotic level of attack for the moderately defended species at P), but decreases if it lies in zone A (Speed, 1993; Speed and Turner, 1999). Other assumptions are possible: see text for discussion of the implications if the motivation increases only in zone B2, or increases in zone A2 as well as throughout zone B.

conditioned, but the less unpleasant mimic. In general, the attack probability may now further decrease, or may increase (or in the boundary case may remain unaltered). If under these circumstances the probability always decreases, then the mimicry is truly mutualistic and muellerian (Turner et al., 1984; Mallet and Joron, 2000). If on the other hand it increases at least under some circumstances, then there will be conditions (depending on relative palatabilities and densities) when the mimicry will be parasitic, and hence quasi-batesian. Our way of modelling this system has been to assume that each of the two species would generate its own asymptotic level of attack, much lower for the

813 less acceptable species, and that once the current probability of attack fell below the asymptote, an encounter would increase the probability toward that asymptote (Speed, 1990, 1993; Speed and Turner, 1999). The probability never falls below the asymptote for the better defended species; when the current level is below the asymptote for the less defended species, an encounter with it produces positive reinforcement. When the predator is naive, or when the current level is for any other reason above this asymptote, the reinforcement will be negative. It is this model which generates quasi-batesian mimicry. This method of carrying out the computation produces a convenient algorithm, but may be a simpli®cation of what could actually happen: in our algorithm, a moderately defended mimic whose asymptote lies at P (Fig. 1) will be negatively reinforcing when the current level of attack probability is in zone A (i.e. above this asymptote). When the current attack probability is in zone B this prey will be positively reinforcing. Such a mimic will be quasi-batesian. However we might imagine that the crucial dividing line is not the asymptote for the less defended species, but at some other point in zone B. Given that the predator cannot know what the asymptote of attack for a particular prey actually is, it may merely compare the mix of negative and positive sensations that it receives from the two kinds of prey, in which case it may be that positive reinforcement does not occur until the current attack probability is well below the asymptote for the less defended prey, so that negative reinforcement occurs in zone B1, and positive reinforcement only in zone B2. This will cause the circumstances under which quasi-batesian mimicry occurs to be more restricted. Once the point dividing positive from negative reinforcement reaches the lower axis (so that zone B2 disappears) then of course we have the classical view, and there is no quasi-batesian mimicry. Or it might be that the predator is positively reinforced when the current probability is still above the less defended asymptote (zone A2), in which case we have a situation in which quasibatesian mimicry is more widespread than initially predicted. Because questions involving mimicry have been uninteresting to experimental psychologists, the empirical evidence on this question is still very limited (reviewed by Speed, 2001). Both ®eld and laboratory studies have however shown that moderately defended species can indeed act as quasi-batesian mimics (Speed et al., 2000; S.L. Hannah, J.M. Forbes and J.R.G. Turner, in preparation). Mallet and Joron (2000) have suggested that what we are predominantly dealing with is a matter of de®nition. Even accepting that some prey that are negative reinforcers to naive predators may become positive reinforcers to experienced predators, they suggest, we could largely make the `problem' disappear by de®ning such prey as `palatable' or `acceptable', thus rede®ning quasi-batesian mimicry simply as batesian mimicry. Indeed, while devising an optimal terminology for a set of phenomena is useful, such word usages are

814 historically very labile, and the mere rede®nition of words would not constitute a particularly valuable exercise scienti®cally. However it is easy to see that we are here dealing not simply with matters of de®nition, but with potentially new and interesting phenomena (Speed and Turner, 1999). If the conditional positive reinforcement which produced our quasi-batesian mimicry does actually occur, then a mildly unacceptable species has the simultaneous properties that: 1. When it is neither a mimic nor mimicked it has regular aposematic properties (e.g. it is increasingly protected with increasing density). 2. It can act as a model for a palatable batesian mimic. 3. It can be a muellerian co-mimic of a species as unpalatable as itself. 4. It can behave like a batesian mimic of a species more unpalatable than itself.

Saturation theory If we are in doubt about the kind of behaviour shown by a predator operating with a dynamic conditioned memory, we are even more in doubt about the behaviour of predators which adopt saturation strategies with respect to mixtures of models and mimics. The model proposed by Mallet and Joron (2000) after MuÈller (1879) ± as a ®rst approximation to conditioning theory ± is particularly suited to this situation. For the sake of initial modelling, forgetting can be ignored, and it is assumed instead either: (1) that the predator eats a certain number n of the prey, after which it achieves total permanent aversion (prey of di€erent palatabilities being represented by di€ering values of n) (Brower et al., 1970), a model suitable to describing the development of a true phobic reaction ± a highly interesting matter relevant for instance to the mimicry of snakes, which we will nonetheless not explore here ± , or (2) that within a period of time the predator can eat n prey, but then ceases feeding on this prey completely because it has taken a full dose of toxin and can proceed no further without a risk to its health (Brower and Calvert, 1985; Forbes and Provenza, 2000). It may commence feeding again after a detoxi®cation break, and in that case n is the number eaten in a full time cycle. Clearly n is likely to be ¯exible, the acceptable risk of the toxin increasing as alternative sources of food decrease in availability, and the cessation of feeding in the region of n is likely to be manifest as a statistically but rapidly increasing aversion: but for simple initial modelling a constant value of n for any prey species, and a simple threshold cut-o€ will be adequate.

815 While saturation of this kind with a toxin is inevitable if predators need or choose to exploit a defended poisonous prey, the same sort of saturation could also occur, surprisingly, with palatable prey. Research in animal nutrition has amply shown (Forbes, 1995) that, because most food items do not in themselves constitute a balanced diet for the consumer, individual acceptable food species may saturate before the predator or herbivore is fully satiated. The predator or herbivore, feeding on, say, a source of food that is rich in protein but poor in carbohydrate, may have to switch to another source that is rich in carbohydrate but poor in protein in order to achieve a balanced diet, and failure to do so can produce physiological e€ects that are broadly similar to those of toxic poisoning (Forbes and Shariatmadari, 1994; Pearcy and Murphy, 1997; Forbes and Provenza, 2000). From this point of view, there is no clear distinction between a toxin and a nutrient: any `nutrient' taken in an excess which generates a de®ciency of some other nutrient, has toxic e€ects. In practice the predator learns rapidly that a particular food source is not balanced, and will try to switch repeatedly between this and another food source that restores the balance, thus keeping its diet nutritionally sound on a running basis (Forbes and Shariatmadari, 1994; Forbes and Provenza, 2000). In the wild, this will leave the excess of any hyper-abundant species unconsumed: as a ®rst approximation one can think of this as a situation in which a maximum of n of the hyper-abundant species can be eaten. This dietary balancing behaviour is shown by mammals, birds and even locusts and is well-recognised not only in captivity but also in wild birds (Murphy, 1994; Forbes, 1995; Whelan et al., 1998). As a simple way of conceiving the e€ect, imagine that there are several prey species which if consumed in equal numbers will produce a properly balanced diet. If the prey are of equal population sizes, all are predated equally. However if one of the species is extremely abundant, any attempt by the predator to exploit this apparent largesse would result in a dietary imbalance with quasitoxic e€ects. Thus a prey species which has a population size (relative to other prey) in excess of the number required to balance the predator's diet is in e€ect somewhat protected from predation. Such a species therefore, potentially, could act as a model in a mimicry system. Or as Miriam Rothschild put it long ago, in one of those insights that was well ahead of its time ``Personally, I think that all prey are somewhat unpalatable''. A `palatable' prey can indeed become toxic when consumed in excess, unless it happens to provide a perfectly balanced and complete diet in itself. Acceptability is simply a continuous scale, and (as a ®rst approximation to the full description of dietary balancing) palatable and unpalatable prey are distinguished only by very large versus very small values of n, the number of prey that can be consumed before the predator must cease feeding. There is no clear boundary between palatable and unpalatable prey, and no neutral point.

816 A potential source of confusion must be eliminated at this stage. The e€ect in question is of saturation with one particular substance, not of overall satiation ± the physical and psychological process by which the consumer ceases to feed when its dietary needs are suced. Simple satiation is a much less interesting e€ect from the point of view of mimicry, as all it does is to put defensive adaptations such as mimicry on hold until the predator becomes hungry again. Saturation by contrast is capable of a€ecting the mimetic relationships between individual species. To emphasise this, we have adopted the term `saturation' rather than using the term current in the nutritional literature: `sensory speci®c satiety.' The implications of this system have hardly yet been explored. An idea of the unorthodox results that may be expected from it can be gained by examining, rather than all the theoretical possibilities, merely two relatively simple and limiting cases 1. the species share the same toxin, but di€er in their patterns or other signals. 2. the species share the same pattern or signal, but di€er in their toxins. In both cases `toxin' embraces a nutrient in excess (Forbes, 1995; Forbes and Provenza, 2000). When the mimics (or potential mimics) share the same toxin, both potentially share the ability to saturate the predator jointly; when the mimics have di€erent toxins, they saturate the predator independently. Obvious further cases (not explored here) are partial interaction between the toxins, giving an intermediate situation to the above, and positive synergism between the toxins so that taking on both is worse than the sum of taking either separately; negative synergism, such that one toxin acts as an antidote for the other, will be treated as a matter of maintaining a balanced diet. A further possibly di€erent case which we will note but not discuss, is the substance which the predator needs in a de®ned low dosage, in that it is essential to metabolism but which becomes toxic in the traditional sense at higher levels: various metal ions are clear examples. In this case the predator might regulate its intake to n individuals over a rather wide range of densities, and in addition to saturating might actively seek out prey that o€ered this substance when they became rare. Such a system generates some extremely surprising results: 1. Two entirely `palatable' species may be mutualistic muellerian mimics, or one may be a batesian mimic of the other. 2. Two defended species which share the same toxic defense mechanism, but which have no mutual resemblance in their colour pattern (or for that matter in any other exterior signal) may nonetheless be muellerian mutualists. This could apply also to palatable species. 3. A rare but heavily defended species might be a batesian mimic of a common `palatable' species.

817 Same toxin in both species, di€erent patterns Toxic mutualism The two prey species share the same toxin, but may di€er in the dosage that they deliver. As the predator's physiology must react identically to the toxin regardless of the source, eating either of the prey raises the toxic load towards saturation, and once saturation is reached, the predator must cease feeding on either of the species, and take a detoxi®cation break (in vernacular terms, if you are taking the maximum dose of regular paracetamol, and then add a dose of a proprietary medicine containing paracetamol and codeine, you risk overdosing on paracetamol). Thus, provided that between them the species are common enough to saturate the predator, the species mutually protect each other, in the sense that attacks on either can protect individuals of the other species. We therefore come to the rather surprising conclusion that if two species share the same anti-predator toxin, they will mutually protect each other even if there is no visual resemblance whatever between them. All that is required is that the predator associate the appearance of each of them with the toxic e€ects (which might be assisted initially by recognising a similarity of ¯avour), should cease feeding once it has taken a toxic load, and have no tendency to become confused and continue to feed on one species after it has reached a toxic load from the other. This behaviour has been observed in laboratory experiments with mammals, which will change their feeding preference appropriately in response to an excess of a toxic substance introduced directly into the digestive tract (Forbes and Provenza, 2000). Such mutual defense could be widespread, as it does not require that the defended prey have an identical biochemical pro®le, but only that they share a toxin, or perhaps only that they deliver the same toxic radical, say ±CN, irrespective of the con®guration of the carrier molecule. Sharing of defensive compounds is likely to be rife in some groups by common ancestry ± it is likely for example that most heliconids share the ability to deliver cyanide and that most ithomiids can deliver pyrrolizidine alkaloids ± and will also occur for some substances at least across much wider taxonomic groupings, especially as defensive compounds tend to be related to the defensive compounds of the host plant. It is possible that most species of insect that feed on Passi¯ora are mutually protected in this way: butter¯ies and beetles, with no visual resemblance whatever, having a muellerian relationship. Mutualism even between di€erent phyla cannot be ruled out: it may be a coincidence, or may indicate a shared defense compound, that the defensive odour of the European Grass Snake Natrix natrix is identical with that of some burying beetles Necrophorus spp.

818 What then could we call this kind of mutualism? Within the current broad de®nition of mimicry ± the modi®cation of the behaviour of a `signal receiver' that has been given identical, or apparently identical signals by two transmitters, to the bene®t of at least one of the transmitters ± this toxic mutualism strictly is a form of mimicry, the signals merely being delivered through the proprioreceptors rather than the eyes. As it is a mutualism, it could therefore legitimately be called a form of muellerian mimicry. On the other hand this is widely taken to mean only `visual' mimicry, and to avoid confusion and for the purpose of having a distinct term, we will provisionally call this kind of toxic resemblance `muellerian mutualism.' Will muellerian mutualism of this kind tend to evolve into true muellerian mimicry? Clearly the conditions for two such mutually protected species to become visual muellerian mimics are broadly the same as those required for any two defended species to become muellerian mimics (for example, Turner, 1984; Sheppard et al., 1985). Normally one species must be better protected than the other. The bene®ts to a pair of muellerian mutualists are clearly mutual, in that either species gains from predation on the other, and the bene®ts will accrue provided that between them (but not necessarily individually) the prey are common enough to saturate their predators. The factor tending to make protection unequal and the evolution of true muellerian mimicry possible, is inequality of the cost:bene®t ratio. If the patterns are initially very di€erent as postulated here, the evolution of muellerian mimicry will be initiated when and if one of the species can generate a mutation of the pattern which produces an approximate resemblance to the other species. This will lead to the evolution of true muellerian mimicry only if the mutating species is less protected than the other species, and only if the patterns are similar enough for the `gap' between them to be bridged by a single mutation. If their patterns are already rather similar (low grade muellerian mimicry) they will undergo gradual mutual convergence and potentially re®ne the mimicry to a very close resemblance but again only if the cost:bene®t ratios are unequal. If the ratios are equal, then prey of both species are treated identically by a fully experienced predator, and those variants of the one which bear some additional resemblance to the other species are at no increased selective advantage (this is in contrast with the situation when the predators are operating via conditioning). If the two species are equal in individual body weight, and present the same doses of toxin and nutrient, then the predators will attack them with equal probability. The bene®t will then be equal (in terms of predation per individual) if the population sizes are the same. Even if the population sizes are unequal, but assuming that the species are still equal in apparency to predators,

819 they will be attacked in proportion to the population sizes. The protection will then still be equal and muellerian mimicry will not evolve. Deviations from these conditions, except in the unlikely event that they all cancel one another out, will make the bene®ts unequal. In general the species with the better cost:bene®t ratio in terms of the dose of toxin compared with its load of nutrients, will be preferentially attacked, as will the species that is more apparent or has the slower rate of escape. The e€ect of a di€erence in weight (given equal cost:bene®t) is not intuitively obvious. At any density the predator may be expected to attack the two species still in proportion to their population densities, as it would neither gain in terms of nutriment nor lose in terms of toxic load by preferentially attacking either species. In this case a di€erence in weight will not alter the equality of bene®t to the two species. If the predator preferentially attacks the lighter species so as to achieve a slower build up of toxin (perhaps perceiving this as more than o€setting the lower ingestion of nutrients), then the lighter species might be at a disadvantage unless its population size was greater (in which case equality might be achieved as the species tended to equality of biomass rather than of population size). On the other hand, the predators might take weight into account, along with escape potential, when deciding whether to expend energy attacking: this will cause predation to weigh more on the heavier species (Rothschild, 1971). Taking predator learning into account complicates this picture. If one of a pair of mutualists is much rarer than the other, it may be attacked more because the predators do not fully associate its pattern with presence of the toxin. In that case population sizes as well as cost:bene®t relations have to be weighed when considering the relative bene®ts to the partners, and the way in which true muellerian mimicry will evolve. Under this consideration the diculty that predators may have in recognising rare variants of either species will probably result in low grade mimicry evolving to high grade mimicry even when the cost:bene®t ratios are equal, and this will be especially true if the population sizes are not equal. Surprisingly, if one of the partners is so very rare that the predators may never properly associate the pattern with the toxin, then this species will confer some bene®t on its common partner (as the predators' sense of increased toxic load will reduce their predation rate) but will receive no bene®t itself from the common partner (as even a predator with a full toxic load will still attack the rarer species): this is a complete reversal of the situation with conditioning and true muellerian mimicry, in which it is the rarer of the partners that receives the greater bene®t. Overall we might expect that muellerian mutualists will tend to become true muellerian mimics at about the same rate that any pair of defended species tend to become muellerian mimics.

820 Mutualism between `palatable' species Further, as has been already pointed out, any species that has a relative abundance that is greater than the proportion to other species required to balance the predators' diet is to some extent protected. Given that there is no qualitative distinction between a toxin and a nutrient in excess, the above argument could apply not simply to species which are `toxic' or defended, but to any two species which are nutritionally substantially the same. The di€erence will be quantitative, in that `toxic' species will be able to saturate predators at relatively low densities, whereas highly nutritious species will need to be at very high density. If two species with say the same protein:carbohydrate composition jointly saturate the predator with, say an excess of protein, they mutually bene®t from each others' presence. (As a thought experiment, if you remove one of them, the remaining species takes the full load of predation that is required to saturate the predator.) Thus there is a form of muellerian mutualism between any species which, from the point of view of predator nutrition or toxicity, present the same chemical substance in excess of requirement, and this mutualism operates even if there is no resemblance between them in their pattern or other exterior signal properties. Stability of equilibria in polymorphisms It has been traditionally supposed that natural selection on defended species, being positively number dependent, tends toward uniformity of pattern, and that polymorphisms in warningly coloured species will be unstable (e.g. Turner, 1987). Some such species do however exhibit all kinds of variation, from polymorphism to racial di€erentiation. Mallet and Joron (2000) have pointed out that the equilibrium points of genetic polymorphism in warningly coloured species, although technically unstable, are only very weakly so when population densities are high. They argue that such polymorphisms could persist in this condition for long periods, and that only at extremes of gene frequency or low population densities would such polymorphisms be destroyed by natural selection. This result follows directly, and in a more extreme form, from the saturation model. The surprising case of the two species that are muellerian mutualists purely by chemical resemblance, applies with extreme precision to all the forms of a polymorphic, defended species. Two species which share the same toxin are, we have argued, mutually protected even if they have a di€erent pattern, and the bene®ts are equal between the species if they present the same cost:bene®t ratio to the predators. And polymorphic forms, while necessarily di€ering in pattern, must necessarily share the same toxin and present the same cost:bene®t ratio to the predators. The forms are thus muellerian mutualists and of equal Darwinian ®tness. Therefore the polymorphism is neutrally unstable (or if one prefers, neutrally stable): the frequencies of the alleles will

821 change only as a result of genetic drift. The polymorphism will tend to persist inde®nitely. This result is more extreme than that derived by Mallet and Joron (2000) ± who treat the forms as independently saturating the predator ± in that provided the species has reached a high enough density to saturate the predators, the selective neutrality of the forms is total and will persist at all frequencies, except perhaps at very extreme low frequencies when individual predators never encounter enough of the rarer form to learn that it presents them with part of a toxic load, or at slightly less extreme frequencies where the rates of learning may be slightly di€erent. Conclusion The surprising conclusion then is that if there are two independent species, with di€erent patterns, but delivering the same toxin or nutrient (at least one of them being at high density), then provided the predators can perceive that they are being jointly poisoned (in the broadest sense, including nutrient saturation) by both of them, these will be muellerian mutualists. If the protection although mutual, is unequal, then these species will tend, if they can, to evolve a shared pattern, and so become true muellerian mimics. Same pattern in both species, di€erent toxins Suppose that the species are nutritionally di€erent, but share the same pattern (a situation which uncontroversially includes classical batesian and muellerian mimicry). Imagine that a very common species is rich in carbohydrate, and a rare perfect mimic is rich in protein. The predator would, if it could, turn to the rare species as a source of protein once it sensed that its relative consumption of carbohydrate was excessive (e.g. Forbes and Provenza, 2000). However it cannot detect the protein-rich mimic, which is therefore protected by its resemblance to the common one. In short mimicry between two `palatable' species can be advantageous because it prevents the predator from balancing its diet by switching between them. Clearly the relationship is parasitic and therefore batesian, in that random encounters with the mimic may encourage the predator to continue attacking the other species. It is easy enough to ®nd a situation in which the mimetic resemblance confers no advantage. Suppose two chemically di€erent species provide a balanced diet when consumed in proportions 1:x. If their populations, both large, are in the ratio 1:x, then a randomly feeding predator consumes a balanced diet, irrespective of any resemblance or di€erence in their colour patterns. It follows that when the ratios are not so perfectly matched, one of the species would gain from the patterns being matched, one of the species gains if the patterns are the same: the mimetic advantage is always to the species whose population size is below the required ratio ± e.g. if the predator needs three of

822 A for every one of B, then the species which is below this ratio (for instance species A, if the population sizes are equal) is the one that is advantaged and is the batesian mimic. We now also encounter what we have called in the title a `weird' situation: this batesian mimicry will operate for two species which are chemically different in any way, and that includes any di€erence in good old-fashioned toxicity. Therefore a rare defended, and perhaps highly `toxic' species in the conventional sense that it is full of `poison', could be a true batesian mimic of a highly palatable species that was abundant enough to saturate the predator. This would be the situation provided that the toxic species also di€ered from the common palatable species in its balance of nutrients, so that the predator would, if it could tell the species apart, switch between them in order to balance its diet (trading o€ the bene®ts of balanced nutrition against the costs of a subtoxic dose of the poison). Clearly, if we think of a quasi-batesian mimic as a prey which is avoided when present by itself, but which can be parasitic on a better defended species, then in this case the distasteful partner has the properties of a quasi-batesian mimic. Obviously another special case is conventional batesian mimicry between a defended species that saturates its predator, and a `palatable' species that is below its own saturation density. Other cases will come to mind: a saturating defended species and a saturating `palatable' species would be muellerian mimics. In general then, when two species di€er chemically, and one of them can saturate the predator, whether by the toxic e€ects of dietary imbalance, or by the toxic e€ects of conventionally conceived `poisons', then mimicry between them will be e€ective. If both species saturate their predator (by whatever means) the mimicry is mutualistic or muellerian; if one is below saturation density then the mimicry is parasitic and batesian. Combining the conclusions of this and the previous subsection, relationship is mutualistic if the toxins (or nutrients) in the two species are the same or if both species saturate the predator; and parasitic if the toxins (or nutrients) are di€erent and one of the species does not saturate the predator. It may be useful at this point to remember that throughout this discussion we are dealing with saturation by particular substances, not with satiation from an overall suciency of prey.

Envoy The application of nutritional theory to mimicry produces startling results. It also emphasises that there is no clear distinction between a poison and a nutrient, and that the only di€erence between defended and palatable prey is in the number that can be consumed before the predator must stop feeding.

823 Although this blurring of traditional distinctions is not so obvious in conditioning theory, we are nonetheless hard put to it to say exactly what is palatable and what is unpalatable, in that besides changing with such factors as hunger levels and the availability of alternative prey, the tendency of prey to reinforce attack motivation positively or negatively may change according to the predators' previous experience (Fig. 1). It must be the case that predators will change their behaviour from patterns that can be described by the conditioning model toward patterns that are appropriately modelled by saturation theory, according to the density of the prey species. The models have somewhat disparate predictions about the nature of mimetic relationships and the evolution of mimicry, and the maintenance of polymorphisms. There is now a need to explore the consequences of predators changing from behaviour that is best described by conditioning theory to behaviour that is best described by saturation theory, as the density of their prey ¯uctuates, and mimicry theory will be incomplete until this is done. Saturation theory is potentially alarming in its unorthodox implications. It helps to keep three questions separate: 1. Could such relationships occur adventitiously if the `mimetic' species already resembled one another for other reasons, perhaps by accident or common ancestry, or because mimicry had evolved already by more orthodox routes? The various e€ects of predator saturation postulated here require that the populations of the various prey species ¯uctuate, sometimes reaching large numbers. This seems likely, at the very least occasionally and in the long term. Saturation e€ects with toxins require that predators sometimes exploit a defended species as a major food source, trading the costs of poisoning against the bene®ts of nutrition, but keeping below a lethal dose of poison. This has been observed (Brower and Calvert, 1985) but may perhaps be uncommon. Saturation with palatable species requires that predators switch between prey, in order to maintain a balanced diet. While this behaviour is known for wild frugivores (Murphy, 1994; Whelan et al., 1998) it might not occur with insectivorous species if they used physiological rather than behavioural mechanisms to cope with imbalances in their diet. We also need to know how much variation occurs in nutrient balance between the di€erent species of prey that the predators might switch between. It seems rather likely that toxic mutualism is quite widespread, at least adventitiously as populations ¯uctuate. Further, even if adventitious mimicry between highly palatable species was very rare, given that there is a continuum between poisons and nutrients, and between unpalatability and palatability, it is possible that saturation mimicry could occur between species that we do not

824 think of as defended, as a result of saturation with compounds that we do not normally regard as poisonous. A similar but probably invalid theory of `numerical mimicry' (van Someren and Jackson, 1959) was long ago proposed to explain apparently mimetic similarities between the patterns of what were reasonably surmised to be cryptically coloured, palatable species. 2. Will relationships of this kind lead to the evolution of mimicry de novo? There is likely to be a considerable time delay before a mimetic pattern can establish itself in a species, waiting as it does on the occurrence of a suitable mutation, and the rather unlikely escape of that mutation from the risks of stochastic loss from the population while it is still rare. We have to ask ourselves, but cannot yet answer, how many species can remain at saturation densities for long enough to produce new mimicries according to the predictions of saturation theory, or to maintain unstable polymorphisms in a state of total neutrality. Toxicity is a persistent condition, likely to generate the evolution of mimicry at all population densities; saturation e€ects for palatable species will be more transient, and much less likely to drive the evolution of mimetic patterns. 3. Will such mimicry be stable over evolutionary time? Toxic muellerian mutualists, like any other defended species, are likely to evolve into true muellerian mimics. However this does not destroy the muellerian mutualism: the species simply add one adaptation to the other, being defended by muellerian mimicry at all times, and by toxic mutualism if and when their densities saturate the predators. Toxic mutualists will exist both with and without the cover of classical muellerian mimicry. The puzzling coexistence of multiple muellerian mimicry rings (e.g Turner, 1984; Beccaloni, 1997) could perhaps sometimes be explained by saturation. If all the ithomiine species of the Neotropical rain forests are toxically very similar, it might be that several abundant mimicry rings are frequently numerous enough to saturate the local predators, and that, although the mimicry rings have very di€erent patterns, all the rings are mimics of one another in the proprioreceptor and toxin channel (being an extended example of the case discussed of `same toxin, di€erent patterns'). However it is obviously not the case that all the ithomiids and heliconids of the rain forests are equally defended and mutually neutral: if they were, there would be no mimicry rings, but merely a diverse spread of warning patterns. It is also very well known that ithomiids and heliconids use di€erent defense compounds ± which nevertheless could still lead to mutual protection if the poisons had an additive or synergistic e€ect on the predator. There also is a seductive possibility (which is not dependent on the saturation model) that the mimicry rings exchange species through `ring-swapping' (e.g. Turner and Mallet, 1996) until each ring, in

825 aggregate, is as well protected as every other. If the rings are prevented from gradual mutual convergence by the `cognitive gap' (see e.g. Turner, 1984 ± this point has been lost in much of the recent debate), a ring can only disappear if all its species switch into another ring, and while this may be expected to happen in due time to a weakly defended ring, it is likely that sometimes the species which switch are themselves its most weakly defended members: in this case the ring which loses them gains in overall defense, and the capturing ring, which by de®nition is better defended, loses some of its defense. Thus while some rings will wither away from successive captures, other rings will at least approximately stabilise, and co-exist almost inde®nitely. The polymorphisms of `defended' warningly coloured species now have an embarrassment of explanations: besides extraneous factors such as habitat heterogeneity (Brown and Benson, 1974; Joron et al., 2001) they may be explained by quasi-batesian mimicry (which has gene-dynamics like batesian mimicry) (Speed and Turner, 1999), near-neutrality at high density by predator conditioning (Mallet and Joron, 2000), and high-density full-neutrality by predator saturation (above). Against the last two explanations it can be argued that if populations ¯uctuate, then during periods of low population size selection against the rarer form may be intense, so that the polymorphism would gradually be eroded by repeated ¯uctuations in population size. Perhaps the most promising species as an example of a neutral polymorphism in an aposematic species is Danaus chrysippus, thought to be polymorphic as the result of massive introgression between previously isolated races (Smith et al., 1993; Owen et al., 1994), but perhaps now kept polymorphic by the fact that it is normally so abundant as to saturate its predators: to quote from Rothschild (1971) ``Danaus chrysippus was described by [E.B.] Poulton as the commonest butter¯y in the world''.

Acknowledgements We are greatly indebted to Prof. Michael Forbes and Simon Hannah of the Centre for Animal Sciences in the University of Leeds for introducing us to the implications for this work of concepts in animal nutrition and dietary strategies.

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