Biological Journal of the Linnean Society (1999), 67: 281–312. With 9 figures Article ID: bijl.1998.0310, available online at http://www.idealibrary.com on

Learning and memory in mimicry: II. Do we understand the mimicry spectrum? MICHAEL P. SPEED1,2∗ AND JOHN R. G. TURNER1† 1

Department of Genetics, University of Leeds, Leeds LS1 9JT Department of Environmental and Biological Studies, Liverpool Hope University College, Hope Park, Childwall, Liverpool L16 9JD 2

Received 18 June 1998; accepted for publication 20 December 1998

The evolution of mimicry is driven by the behaviour of predators. However, there has been little systematic testing of the sensitivity of evolutionary predictions to variations in assumptions about predator learning and forgetting. To test how robust mimicry theory is to such behavioural modifications we combined sets of rules describing ways in which learning and forgetting might operate in vertebrate predators into 29 computer predator behaviour systems. These systems were applied in simulations of simplified natural mimicry situations, particularly investigating the nature of density-dependence and the benefits and losses conferred by mimicry across a spectrum of palatabilities. The classical Batesian-Muellerian spectrum was generated only by two of our 29 predator behaviour systems. Both of these ‘classical predators’ had extreme asymptotes of learning and fixed rate, time dependent forgetting. All edible mimics were treated by them as Batesian in that they parasitized their model’s protection and had positive monotonic effects of density on model-mimic attack rates. All defended mimics were treated as Muellerian (Mu¨llerian) in that their presence benefited their Model’s protection, and showed negative monotonic density effects on attack rates. With the remaining 27 systems Batesian or Muellerian relationships extended beyond their conventional edibility boundaries. In some cases, Muellerian mimicry extended into the edible region of the ‘palatability spectrum’ (we term this quasi-Muellerian mimicry), and in others Batesian mimicry extended into the ‘unpalatable’, defended half of the spectrum (quasi-Batesian mimicry). Although most of the 29 behaviour systems included at least some regions of true Batesian and Muellerian mimicries, if forgetting was triggered by avoidance events (as suggested by J.E. Huheey) rather than by the passage of time then the mimicry spectrum excluded Mullerian mimicry altogether, and was composed of Batesian and quasiBatesian mimicries. In addition the classical prediction of monotonic density-dependent predation was shown not to be robust against variations in the forgetting algorithm. Time based forgetting which is retarded by observations of prey, or which varies its rate according to the degree of pleasantness or unpleasantness of a prey generates non-monotonic results. At low mimic densities there is a positive effect on attack rates and at higher densities a negative effect. Overall, the mode of forgetting has a more significant effect on mimetic relationships than the rate of learning. It seems to matter little whether learning and forgetting are switched or gradual functions. Predictions about mimetic evolution are therefore sensitive to assumptions about predator behaviour, though more so to variations in forgetting than learning rate. Based on findings from animal psychology and mimetic populations, we are able to rule out a number of predator behaviour systems. We suggest that the most credible ∗ Corresponding author. Email: [email protected] † Present address: Department of Biology, University of Leeds, Leeds LS1 9JT 0024-4066/99/070281+32 $30.00/0

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 1999 The Linnean Society of London

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of our 29 predators are those which generate results which incorporate Batesian, quasiBatesian and Muellerian mimicries across the ‘palatability spectrum’.  1999 The Linnean Society of London

ADDITIONAL KEY WORDS: — evolution – behaviour – Batesian – Muellerian – quasiBatesian – computer simulation – polymorphism – predation – forgetting. CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . Terminology of prey defences . . . . . . . . . . . . . . . . Modelling of learning and forgetting . . . . . . . . . . . . . . Memory jogging rules . . . . . . . . . . . . . . . . . Behavioural rules and virtual predators . . . . . . . . . . . Simulation routines . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . General mimetic relationships . . . . . . . . . . . . . . Spectrum 1. The ‘classical’ spectrum with Batesian and Muellerian mimicries only . . . . . . . . . . . . . . . . . . . Spectrum 2. ‘The quasi-Muellerian spectrum’ with Batesian, quasiMuellerian and Muellerian mimicry . . . . . . . . . . . . Spectrum 3. ‘The Huheey spectrum’ with quasi-Batesian, but no true Muellerian mimicry . . . . . . . . . . . . . . . . . Spectrum 4. ‘The quasi-Batesian spectrum’ with Batesian, quasi-Batesian and Muellerian mimicry . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . Evaluating the behaviour systems 1: psychology and behaviour . . . . Evaluating the behaviour systems 2: conditions favouring polymorphism . Dynamics of quasi-Batesian mimicry . . . . . . . . . . . . Conclusions: how well do we understand the mimicry spectrum? . . . . . Acknowledgements: . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . .

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282 284 284 285 287 290 292 292

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INTRODUCTION

Protective mimicry in its most intensely studied form (Sheppard, 1958; Ford, 1964; Turner, 1977) involves one species gaining relief from predation by taking on the appearance of another species which has an active defence against a shared predator. Traditionally such mimicry has been classed as Batesian when the mimicking species is edible, or as Muellerian (Mu¨llerian) when the mimicking species also has its own defence (Fig. 1: ‘classical spectrum’). These two kinds of mimicry are supposed to have their own properties, such as Batesian mimicry being essentially parasitic, benefiting the mimic but harming the model, and Muellerian mimicry being mutualistic, with benefits to both species (e.g. Mu¨ller, 1879; Wickler, 1968; Turner, 1995). It is supposed also that there are major differences in the dynamics of the two systems, with considerable implications for their ecology, long term evolution and genetic architecture (Turner, 1984a, 1987, Holmgren & Enquist, 1999). It has been a matter for debate just how far the supposed properties of Batesian and Muellerian mimicry may overlap, or how far some of the properties of one

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highly

moderately

Mimic Edible

highly

283

moderately

Mimic

Defended

Batesian

Spectrum 1: 'Classical Spectrum' Batesian

Spectrum 2: 'Quasi- Muellerian Spectrum' Batesian

quasi-Batesian

Spectrum 3: 'Huheey Spectrum' Batesian

quasi-Batesian

Spectrum 4: 'Quasi- Batesian Spectrum'

Figure 1. A general summary of the four types of mimicry spectrum generated in the simulations. Black bars: Model benefits; white bars: Model suffers (after Turner 1995). The four spectra vary in the distributions of Batesian/quasi-Batesian and Muellerian/quasi-Muellerian relationships along the ‘palatability spectrum’. In particular, Batesian or Muellerian like relationships intrude beyond their conventional boundaries in three of the four types of spectra simulated (spectra 2–4). When mimicry is Batesian or quasi-Batesian, then some tendency to mimetic polymorphism can be expected. When mimicry is Muellerian or quasi-Muellerian then strict monomorphism is predicted.

may apply to the other. Indeed the division into two types is perhaps invalid, there being either only one kind of mimicry, a smooth continuum (a ‘mimicry spectrum’ of which Batesian and Muellerian mimicry are only the extreme limiting cases), or other further definable kinds of mimicry—e.g. Huheey (1976, 1988), Sbordoni et al. (1979), Rothschild (1980, 1981), Owen & Owen (1984), Speed (1993a), Gavrilets & Hastings (1998). Some of the challenges to the traditional view have come from empirical observation (Bullini et al., 1969; Sbordoni et al., 1979). Others have arisen from a priori reasonable models of predator behaviour: Huheey (1976), Owen & Owen (1984) and later Speed (1993a, b) made predictions which challenged the traditional view; Turner et al. (1984, also Turner, 1984a) countered by producing a model of predator behaviour which supported it (see also Benson, 1977; Sheppard & Turner, 1977; Huheey, 1980. Explicit models make clear the assumptions which underlie the traditional (or maverick) views of mimicry. We examine the sorts of dynamics which are predicted in simplified natural mimicry systems from different assumptions about predator

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behaviour. There are only slender empirical or theoretical reasons for choosing among a wide variety of ways of modelling the behaviour of predators (Turner & Speed, 1996): therefore we now explore a wide range of models of predator learning and forgetting, including all those proposed during the debate, and ask, given that the mimic may vary from being highly nutritious and edible to being highly defended, what sorts of dynamics will mimicry show in the natural environment? The simulations are designed to examine certain crucial aspects of predator behaviour, and therefore simplify both the realities of psychology and the complexities of natural ecology. We focus on the general dynamics of predator learning and memory. We assume that only two species are involved in the mimicry, that there is only one species of predator, and that there are no effects resulting from predator generalization between the mimetic pattern and other mimicry rings. Further, we assume that the mimicry is perfect: this is never strictly true, except in the case of automimicry (Brower et al., 1968). Hence we assume that perceptual variations for example in the receiver psychology of discrimination learning (Guilford & Dawkins, 1991; Pearce, 1994) are not in need of modelling at this stage. We also exclude here the complexities of predator recognition errors, though this is clearly an interesting area which needs further investigation (see MacDougall & Dawkins, 1998).

TERMINOLOGY OF PREY DEFENCES

This paper is concerned with the kinds of mimetic relationship which might occur along a continuous spectrum of prey acceptabilities, conventionally known as the ‘palatability spectrum’. Terminology about levels of prey acceptability and defence is varied and is not a simple matter (Brower, 1984; Marples et al., 1994). The most commonly used terms such as ‘palatable’ and ‘unpalatable’ can be ambiguous as they are used to describe aversive qualities which are purely gustatory (i.e. in pure bad taste) and/or potentially toxic. As a result we avoid the use of words like ‘unpalatable’ and ‘distasteful’ (other than keeping the general term ‘palatability spectrum’). We consider that the sort of prey pertinent to the discussion are those which are defended (chemically or otherwise) at least in the sense that they ‘are noxious by virtue of their capacity to irritate, hurt, poison and/or drug an individual predator’ (Class I of chemical defence: Brower, 1984). We consider then a ‘palatability spectrum’ which divides a continuous spectrum of prey acceptabilities according to the presence or absence of a chemical (or other) defence. Along the spectrum of prey acceptabilities a theoretical knife edge of neutral edibility divides edible, nutritious prey from defended prey (cf. neutral palatability, Turner et al., 1984). Whether innocuous but foul tasting or smelling prey can confer similar or persistent levels of protection as those which have toxic qualities is doubted (Class II of chemical defence: Brower, 1984, though see Marples et al., 1994 and Speed, 1999). The present simulations represent only persistent defences to which the predator can make no accommodation.

MODELLING OF LEARNING AND FORGETTING

Our simulations use a variety of ‘virtual predators’ which forage in highly simplified ‘virtual ecologies’. Virtual predators are capable of modifying attack probabilities

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on different species of prey according to simple rules for learning and forgetting. We generally refer to learning and forgetting rules by the initials of their creators, hence the learning rule of Owen & Owen (1984) becomes OO, of Sheppard & Turner (1984) becomes ST, of Huheey (1976) becomes HH and so on for forgetting rules. Learning and forgetting can be described by two algorithms of identical form, based on modelling by Bush & Mosteller (1955) and Rescorla & Wagner (1972, although see also Wagner & Rescorla, 1972): P2 = P1 + a(k − P1)

eq. 1

P3 = P2 + φ(P0 − P2)

eq. 2

P1 is the probability of attack at the start of an encounter, P2 is the attack probability after an attack (if there is no attack P2 = P1), and P3 is the probability after forgetting has occurred. a is the learning rate variable; it can take any value within the limits 0 (no learning) and 1 (single-trial learning); φ is the analogous forgetting rate variable. k is the stable attack probability brought about by learning alone (the asymptote of learning), with limits 0 and 1. The naive attack probability P0 is identically the asymptote for forgetting. These algorithms are applied during each time interval in our simulations, causing the attack probability to change in response to an attack on a prey item, and then to return towards its naive value as time passes and the predator ‘forgets’. All predators are initially naive with respect to observed prey and the attack probability P0 is 0.5. Learning about edible and inedible prey therefore moves attack probability from this level toward some stable, asymptotic, level of attack. Forgetting subsequently returns disturbed attack probabilities back to the naive value of 0.5. Variations in levels of prey edibility can be represented as variations in learning rate (a in eq. 1), stable asymptotic level of attack (k in eq. 1) or both. Figure 2 provides a summary and pictorial representation of the different features of learning rules (see table 1, Turner & Speed, 1996 for details). Forgetting rules vary according to whether forgetting is triggered by the passage of time, avoidance events or both. In all except one forgetting rule, rates of forgetting are constant. In the OO forgetting rule (Owen & Owen, 1984), forgetting rates vary with the degree of edibility/ inedibility of the prey encountered. Figure 3 summarizes the major forgetting rules used (and see Table 2 for details). The HH and HT forgetting rules follow the procedure invented by Huheey (1964), that after an aversive encounter the prey are completely avoided over a period of n avoidance units, after which attack resumes at a probability of 0.5. In the HH rule n is the number of similar prey observed but not attacked; in the HT rule n is a fixed number of time intervals, which are counted whether or not a prey is observed. Variations in levels of defence are reflected in the value of n.

Memory jogging rules A number of workers (e.g. reviewed in Guilford, 1990) have suggested that merely seeing an item of prey again, without attacking it, causes a predator to be ‘reminded’ and thus modifies the forgetting process. To investigate this we have incorporated further procedures (not shown in Table 2).

Attack probability

Attack probability

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

1

D

A

2

2

3

3

8

4 5 6 7 Learning events

4 5 6 7 Learning events

8

9

9

10

Attack probability Attack probability

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

1

E

B

2

2

3

3

4 5 6 7 Learning events

4 5 6 7 Learning events

8

8

9

9

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

C

Learning events

286 M. P. SPEED AND J. R. G. TURNER

Attack probability

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Rule 1: Fixed rate memory jogging (FJ) Observing but not attacking a prey partly counteracts any forgetting of previous attack(s) on an individual of the same appearance. It does so by moving the attack probability toward its value immediately after the most recent attack (i.e. before any forgetting has taken place). The ‘jogging’ process is described by an algorithm with the standard form P3 = P1 + f(Pc − P1)

eq.3

where f controls the rate of jogging; its value is fixed at 0.5. Pc is the asymptote of jogging, i.e. the value of the attack probability immediately after the previous attack. Observing prey therefore gradually returns the attack probability to the previous post-attack value (before the application of the forgetting routine) by a constant proportion on each observation. Rule 2: Variable rate memory jogging (VJ) Jogging is described by eq. 3; f takes the value held by a (in eq. 1) at the last encounter with a prey of a particular appearance. The rate of jogging therefore depends on strength of edibility of the most recently experienced prey.

Behavioural rules and virtual predators Permutations of learning, forgetting and memory jogging rules can be assembled into a number of behaviour systems which represent ‘virtual predators’. Each behaviour system is identified according to its learning rule, forgetting rule and, if present, memory jogging rule in that order. For instance, the behaviour system

Figure 2. Summary of learning rules used in simulations. Graphs show how attack probability changes over learning events for each of five learning rules. From top down, each graph shows learning about an edible, a moderately defended and a well defended species (except E, which shows learning about an edible and a defended prey). A, General Learning Rule (Speed 1993a; Turner & Speed, 1996). Learning rate (a) varies positively from 0.5 to 1 with intensity of unconditioned stimulus (i.e. level of ‘niceness’ or ‘nastiness’ of prey). Stable asymptotic attack rate (k) varies continuously according to intensity of unconditioned stimulus (>0.5 for edible prey; <0.5 for defended prey). B, Bush Mosteller Learning Rule (in honour of Bush & Mosteller, 1955). Learning rate (a) is fixed at 0.5 for all prey and generates gradual learning. Stable asymptotic attack rate (k) varies continuously according to intensity of unconditioned stimulus (>0.5 for edible prey; <0.5 for defended prey). C, Owen & Owen Learning Rule (Owen & Owen, 1984). Learning rate (a) is fixed at 1. All learning is complete in a single trial. Stable asymptotic attack rate (k) varies continuously according to intensity of unconditioned stimulus (>0.5 for edible prey; <0.5 for defended prey). D, Sheppard & Turner Learning Rule (Turner et al., 1984). Learning rate (a) varies positively from 0 to 1 with level of ‘niceness’ or ‘nastiness’ of prey. For ‘neutral’ prey, a=0, and there is no learning, for very nice/nasty prey learning is complete in a single trial. Stable asymptotic attack rate (k) varies discontinuously. It is either 1 for edible prey or 0 for defended prey. E, Huheey Learning Rule (Huheey, 1976). Learning rate (a) is fixed at 1. All learning is complete in a single trial. Stable asymptotic attack rate (k) varies discontinuously. It is either 1 for edible prey or 0 for defended prey.

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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i. highly defended prey

Number of avoided prey

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Attack probability

Attack probability

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11

Time periods and number of avoided prey

0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

B

Attack probability Attack probability

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

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i. highly defended prey

Time periods

6

2

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ii. moderately defended prey

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T 1. Summary of the five learning rules. Extreme values are inclusive (i.e. the range 0 to 1 includes the values 0 and 1). Low values of k are associated with high levels of defence, and high values with edibility; low values of a are associated with neutral edibility, and high values with either high edibility or high levels of defence. The exact values run in our simulations may differ from the values in the originally published algorithms (see Turner & Speed, 1996 for details). Acronym

Name

Reference

k (learning asymptote) Any value between 0 and 1 Bush & Mosteller, 1955 Any value between 0 and 1 Turner et al., 1984 0 or 1 only

GR

Generalized rule

BM

OO

Bush and Mosteller rule Sheppard and Turner rule Owen and Owen rule

Owen & Owen, 1984

HH

Huheey rule

Huheey, 1964

ST

Values of parameters

Turner & Speed, 1996

Any value between 0 and 1 0 or 0.5 or 1 only

a (learning rate) Any value between 0.5 and 1 0.5 Any value between 0 and 1 1 only 1 only

proposed by Sheppard and Turner (Turner et al., 1984) is ST-ST; Huheey’s behaviour system (Huheey, 1964, 1976) is HH-HH; Owen & Owen’s (1984) is OO-OO; Speed’s (1993a) ‘Pavlovian Predator’ is GR-ST. A behaviour system which uses the

Figure 3. Summary of forgetting rules used in simulations. Each graph shows how a forgetting rule returns attack probabilities toward the naive state after learning has taken place. A, Sheppard & Turner Forgetting Rule (Turner et al., 1984). Forgetting occurs at the end of each time interval. Forgetting rate is low and fixed for all prey (φ=0.02). This graph shows the effects of forgetting from three contrasting starting points at time =1. B, Owen & Owen Forgetting Rule (Owen & Owen, 1984). Forgetting occurs at the end of each time interval. With the Owen & Owen predator, forgetting rate is inversely related to intensity of unconditioned stimulus. Thus mild experiences (e.g. middle curve) are forgotten about much more quickly than intense ones (e.g. top and bottom curves). C, Time Based Huheey Rule (cf. Huheey, 1976). A modification to Huheey’s original in that forgetting is time based and restores attack probabilities to naive values after a pre-determined interval. For very ‘nice’/‘nasty’ prey the duration of memory retention is greatest. The forgetting rate variable (φ) is zero until the end of the retention period when it changes immediately to one. The graph shows duration of retention after attacks on either (i) a highly defended species (dotted line) or (ii) a moderately defended species (solid line). D, Huheey Forgetting Rule (Huheey, 1976): Forgetting about the edibility of a prey happens after a pre-determined number of prey have been seen and avoided. It is therefore event based. For very ‘nice’/‘nasty’ prey the number of avoided prey is greatest. The forgetting rate variable (φ) is zero for the period of retention, but then changes immediately to 1. The graph shows duration of retention after attacks on either (i) a highly defended species (dotted line) or (ii) a moderately defended species (solid line). E, Stochastic Huheey Forgetting Rule (cf. Huheey, 1976). Forgetting is again event based. However, seeing and avoiding a prey triggers a small (2%) return of attack probability toward the naive value of 0.5. The forgetting rate variable (φ) is therefore zero unless a prey is seen and avoided; for that event its value is 0.02. F, Combined Forgetting Rule (cf. Huheey, 1976; Turner et al., 1984). Forgetting is both event and time based; both prey avoidance and the passage of a time interval trigger a small (2%) return of attack probability toward the naive value of 0.5. The forgetting rate variable (φ) is 0.02 for all uses of the algorithm.

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T 2. Summary of the seven forgetting rules. P0 the asymptote of forgetting, is 0.5 in all cases. A value of φ of 0 indicates no forgetting; for instantaeous forgetting, φ=1. See also Turner & Speed (1996), Table 2 Acronym

Name

Reference

Value of forgetting rate φ and rules for its application

Turner et al., 1984

0.02 in each time interval

OO

Sheppard and Turner rule Owen and Owen rule

Owen & Owen, 1984

HH

Huheey rule

Huheey, 1964

HT

Time based Huheey rule

Turner & Speed, 1996

HS

Gradual Huheey rule

Turner & Speed, 1996

CO

Combined rule

Turner & Speed, 1996

Any value between 0 and 0.05 in each time interval, according to strength of stimulus 0 until observation of a number (n) of identical prey, this number determined by unpleasantness or pleasantness of last experience; 1 thereafter 0 for time interval (n) determined by unpleasantness or pleasantness of last experience; 1 thereafter 0 unless identical prey seen and avoided, when it is 0.02 0.02; forgetting activated in each time interval and additionally if identical prey seen and avoided

OP

Observing prevents forgetting rule

Turner & Speed, 1996

ST

0.02 unless identical prey observed, when it is 0

Sheppard and Turner learning rule, OP forgetting rule and the fixed rate jog rule is ST-OP-FJ. The system used by Luedeman et al., (1981) corresponds broadly with our HHHT predator. Two earlier papers (Estabrook & Jespersen, 1974; Bobisud & Potratz, 1976) use idiosyncratic versions of HH-HH, involving the bizarre rule of subtracting from n the number of mimics previously encountered, but leaving n at its maximum if the number of mimics exceeds n. We have investigated the spectrum of mimicry for 29 combinations of memory and learning rules (Table 5). Learning rules GR, BM, ST and OO have each been paired with the HS, HH, HT ST, OO and CO forgetting rules. The HH learning rule has been paired with the HH and HT forgetting rules. In addition the ST learning rule has been combined to make the systems ST-OP, ST-OP-FJ and STOP-VJ. Simulation routines Learning and forgetting of the predator are simulated by the standard Monte Carlo technique of comparing the current attack probability with a pseudo-random number, and the application of equations 1 to 3 as sequential algorithms; eq. 1 is applied if an attack occurs; eq. 2 is applied after each time interval; eq. 3 is applied as described under ‘memory jogging rules’ above. A similar Monte Carlo routine controls the presentation of prey to the predator: prey appear randomly in proportion to their population densities; if the total density of all prey is less than 100%, then there are time intervals in which no prey appears. For details see Turner & Speed (1996). Four types of ‘virtual prey’ are presented to the predator. Model and Mimic share an identical appearance but can differ in edibility and density. Generally,

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T 3. Parameters for Model and MOC used in each behaviour system in simulations of the mimicry spectrum. ∗Forgetting does not define edibility. The forgetting rate φ is constant or is determined by formula from a or k (see Table 2 and Turner & Speed 1996). †Except when in combination with OP, OP-FJ and OP-VJ, when its value is 0.4. ‡Learning does not define edibility Rule GR (learning) BM (learning) ST (learning) OO (learning) HH-HH HH-HT

Learning asymptote

k learning rate a

0.2 0.2 0 0.2 0‡ 0‡

0.8 0.5 0.8† 1 1‡ 1‡

Forgetting parameter n ∗ ∗ ∗ ∗ 9 9

Model is the better defended of the two species. ‘Model’ and ‘Mimic’ represent a convenient and memorable terminology, but the terms are not intended to prejudge the mimetic relationship. Model and Mimic each have a corresponding control species (Speed, 1993a): MOC is identical to Model in level of defence and density, but has a unique appearance; MIC is similarly identical to the Mimic in levels of edibility and density but also has its own unique appearance (Turner et al., 1984, and Turner, 1984a, 1984b, 1987, termed these respectively Nasty and either Nice or Solo). MOC and MIC thus have the same respective characteristics as Model and Mimic except that they show no mimicry; they therefore control for the effects of mimetic resemblance between Model and Mimic. The simulation is thus performed with three attack probabilities: those for the two controls, and a probability for the Model and Mimic together. In the experiments on the mimicry spectrum, the four types of prey are each encountered in 20% of time intervals; no prey appears in the remaining 20%. The strength of defence of Model and MOC is held constant (Table 3 gives values) and the experimental variable is the acceptability of Mimic and MIC. This parameter varies from a point of high edibility through to a level of defence greater than that of Model. In the experiments on density-dependence, only Model and Mimic are presented to the predator; the experimental variable is the density of Mimic. Model appears in 20% of time intervals; Mimic appears in anything between none and 80% of intervals. Runs are conducted at selected Mimic edibilities. Our procedures for setting the values of the forgetting parameters in relation to the learning parameters, edibilities and other simulation conditions are as in Turner & Speed (1996). The only exception is in generating the avoid period in predators such as GR-HH or ST-HH. The formulae for determining the avoid number here is: n = INT {20(k − 0.5)} or n = INT {10a}

eq.4

where INT implies that only the integral part of the number is used. Each graph point is the result of 8 replications of runs of 8000 cycles after an ‘equilibration run’ of an arbitrary 1000 cycles; error bars are two standard errors of the mean.

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M. P. SPEED AND J. R. G. TURNER RESULTS

General mimetic relationships All predator behaviour systems generate a broadly similar ‘palatability spectrum’ (e.g. Fig. 4A) with a monotonic decline in the rate of attack on MIC as (from left to right) it becomes firstly less edible and then better protected on the right hand of the spectrum (cf. Morton Jones, 1932; Sargent, 1995). Gains and losses of Model and Mimic caused by mimicry can be gauged by comparison of attack rates of mimetic prey and their non-mimetic controls (Model vs. MOC, Mimic vs. MIC). In virtually all simulations Mimic gains protection from its mimicry, being attacked at lower rates than MIC (e.g. Fig. 4A). We categorize our results by the effect of Mimic on the protection gained or lost by Model. We distinguish four types of mimicry (Fig. 1, Table 4): Batesian and Muellerian mimics are as conventionally understood, edible or defended mimics which respectively harm and benefit their models. Quasi-Batesian mimics are defended mimics which nonetheless harm their models, and quasi-Muellerian mimics are conversely edible mimics which reduce predation on their models. In addition, at the infinitesimal changeover from one type of mimicry to another, there may be a point of neutral mimicry: this usually benefits the mimic but has no effect on the model. Figure 1 provides a digest of the spectrum results in these terms. The predators’ response to prey forms, particularly as they change density, is the crucial driving force of the evolutionary system. With positive effects of mimic density on predation rates (we term this positive density dependence) the mimic is advantaged by its initial rarity but loses its selective advantage as it becomes more common. New mimetic mutants therefore expand to a significant proportion of the prey population and then come to a balanced equilibrium with other forms; this tends to produce polymorphic mimicry (see e.g. Sheppard, 1958; Ford, 1964; Charlesworth & Charlesworth, 1975; Turner, 1980, 1987). Conversely mimics which have negative density-dependent effects on predation rates gain no advantage from initial rarity; a form gains fitness as it becomes commoner (nothing succeeds like success) and the population tends to remain or to become monomorphic. Mimics which have neutral density effects might tend to monomorphism or polymorphism, depending on the other evolutionary forces acting on the relevant genes. In some

Figure 4. Spectrum 1: The ‘Classical Spectrum’ (Batesian and Muellerian mimicries). Simulations of the mimicry spectra all show MIC, the Model-Mimic pair and MOC with the symbols shown here. The solid vertical line represents the position of neutral edibility, the dotted vertical line indicates the position of neutral mimicry. The arrow shows the point at which MOC and MIC are equally well defended (to the right of this point Mimic and MIC are more protected than MOC). Results are shown for (A) ST-ST and (B) HH-HT; Model and MOC remain constantly inedible. Mimic and MIC vary through the ‘palatability spectrum’ from being highly edible (far left) to having a high level of defence (far right). Neutral edibility is indicated by the solid vertical line. The point at which attacks on Model equal those on MOC is neutral mimicry. This is indicated by the vertical dotted line. The horizontal axis is a (A) or n (B—see Table 2). C, results of density-dependence simulations from the ‘Classical Spectrum’ (ST-ST). Mimic increases in density within lines and varies in edibility between lines (from top down, a=0.8, 0.0, 0.4, 0.8). For edible prey, k=1 and for defended prey, k = 0. An edible Mimic generates positive density-dependence, a defended Mimic generates negative densitydependence, and the neutral mimic generates density-independence.

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T 4. Types of mimicry described in this paper. ∗At least at low Mimic densities: see text. †At least at high densities

Model Mimic Effect on Model Direction of densitydependent predation Evolutionary prediction

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cases described below density dependent predation is not monotonic, and there is a more complex mapping of the ‘palatability spectrum’ and of density-dependence (see Appendix 2 for a logical demonstration). We categorize our results according to the general geography of mimetic relationships along the ‘palatability spectrum’. We describe the structure of the mimicry spectrum for each category, the relevant characteristics of density-dependent predation and finally the pertinent evolutionary predictions which follow (Table 5 and Fig. 1 provide a summary of these results).

Spectrum 1. The ‘classical’ spectrum with Batesian and Muellerian mimicries only Figure 4A, B shows the mimicry spectrum for the ST-ST and HH-HT predators respectively. These predators generate classical results. Mimic is always attacked less often than MIC, and thus always gains from mimicry. Model is attacked more than MOC on the (left hand) edible side of the spectrum but gains protection through mimicry in the (right) defended half. Mimicry is thus parasitic and Batesian when Mimic is edible, mutualistic and Muellerian when it is defended. A point of neutral mimicry is coincident with the point of neutral edibility in the centre of the spectrum (see solid/dotted lines). Only the ST–ST and the HH-HT predators generate this classical Batesian-Muellerian spectrum (Table 5). They are the only virtual predators which limit Model gain precisely to the defended half of the spectrum, and they have in common simple time-based memory systems. Density-dependent predation maps onto the mimicry spectrum simply and conventionally. An edible Mimic has a positive density effect on Model-Mimic predation rates (e.g. with ST-ST, Fig. 4C). Increases in the density of the Batesian mimic therefore dilute the protective quality of the aposematic signal. In the edible half of this spectrum selection will favour mimetic polymorphism (see Fig. 1). At neutral edibility there is no density effect on mimicry and Mimic makes no difference to Model predation, at any density. When Mimic is defended there is a negative effect of its density on predation rates (Fig. 4C). Increases in the Muellerian Mimic’s density thus enhance Model-Mimic protection. Hence throughout the defended half of the spectrum mimicry is predicted to be universally monomorphic (Fig. 1). These results replicate the findings of Turner et al. (1984) for the ST-ST system, and show that they can extend to a ‘switched’ equivalent (HH-HT).

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Spectrum 2. The ‘quasi-Muellerian spectrum’ with Batesian, quasi-Muellerian and Muellerian mimicry Three predator systems (ST-OP, ST-OP-FJ, ST-OP-VJ: Table 5) generate a mimicry spectrum (e.g. ST-OP, Fig. 5A) which is very similar to the classical result (in Fig. 4A). However, there is a narrow range of edible mimics which, despite their mild edibility, give some protection to their Models. This zone of quasi-Muellerian mimicry lies between the points of neutral edibility and neutral mimicry (i.e. between the solid and dotted vertical lines on the graph: cf. nomenclature in Speed, 1990, 1993a and MacDougall & Dawkins, 1998). Predators which generate this effect have the OP forgetting rule. The effect of density (Fig. 5B) is monotonic and negative on attack rates for defended Mimics. These prey are thus fully Muellerian and should all be monomorphic in their mimicry (Fig. 1). For prey with neutral edibility, the density-dependent effect on attack rate is also negative and monotonic. However for edible Mimics densitydependence can be a non-monotonic function and be positive up to a critical point, and then negative at higher densities. Hence with these predators, selection favours polymorphism in edible mimics except when these mimics are at a high density. Spectrum 3. The ‘Huheey spectrum’ with quasi-Batesian, but no true Muellerian mimicry Predators with HH or HS forgetting algorithms (Table 5, Fig. 6) generate Batesian mimicry in the edible region of the spectrum and quasi-Batesian mimicry for virtually all of the defended half of the spectrum (e.g. for ST-HS and HH-HH, Fig. 6A,B and Fig. 1). Thus Model loses protection (compared to MOC) from Mimic’s presence throughout. The exception lies at the point at which Model, Mimic and their controls are equally well defended and all four prey are attacked at the same rate (dotted vertical line, Fig. 6A,B). Here mimicry is not so much neutral as null. This result confirms the predictions from Huheey’s (1976) explicit deterministic solution (the HH-HH predator) which effectively forgets only when it sees but ignores prey. It shows also that the result is repeated with a stochastic form of this predator behaviour (e.g. HS forgetting). Density-dependence is positive and monotonic, whether Mimic is edible or defended (e.g. for ST-HS, Fig. 6C). A simple prediction can be made here that virtually all mimicry is potentially polymorphic (Fig. 1). The point at which Model and Mimic defences are equal is again the only exception. Here there is no effect at all of Mimic density on Model-Mimic attack rates (Fig. 6C), and no effect of mimicry on prey fitnesses. Hence mimicry here would be neutral with respect to selection and expected to decay as a useless adaptation under cumulative neutral mutation pressure. Spectrum 4. The ‘quasi-Batesian spectrum’ with Batesian, quasi-Batesian and Muellerian mimicry For a wide range of ‘virtual predators’ a zone of quasi-Batesian mimicry separates the zones of Muellerian and Batesian mimicries (Table 5; e.g. BM-ST, Fig. 7A and see Fig. 1). Mimics which are edible are always Batesian and those which are as

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well defended as their models, or nearly so, are truly Muellerian in that both Model and Mimic benefit from their mimicry. Muellerian mimicry (a zone, and not a singularity as MacDougall & Dawkins, 1998, state), is bounded to the left by a point of neutral mimicry. Between this point of neutral mimicry and the point of neutral edibility (between solid and the dotted vertical lines in Fig. 7A) the Mimic causes an increase in predation on its Model and is therefore quasi-Batesian (Fig. 7A). Fifteen behaviour systems divide the spectrum into three in this way: all combinations of the forgetting rules HT, ST, OO, and CO with the learning rules GR, BM, OO and ST (except for ST-ST: Table 5). Density simulations are shown for BM-ST and BM-OO (Fig. 7B,C). In the edible half of the spectrum, density-dependence is positive and monotonic for all predators (Fig. 7B). Thus all fifteen predators predict potential mimetic polymorphism in the edible half of the spectrum. However, in the defended half of the spectrum density effects are more complex. With BM-ST, density-dependence with defended prey is apparently monotonic (Fig. 7B). Using the GR-ST predator, Speed (1993a) also found monotonic density-dependence. However, with BM-OO, density-dependence is non-monotonic when Mimic has moderate defences (starred line in Fig. 7C). Initially attack rates rise slightly with Mimic density and then fall as Mimic becomes more common. This is consistent with the results of Owen & Owen (1984), who found a non-monotonic effect like that shown in Figure 7C. Speed (1999) has investigated the question of density dynamics in quasi-Batesian zones in more detail. A general conclusion which can be drawn is that monotonic density-dependence is generated if forgetting rates are constant (that is the probability of attack decays asymptotically, but at a constant rate independently of the power of the conditioning stimulus). Non-monotonic density-dependence is generated if the forgetting rate is variable, as in the OO forgetting rule in which decay is still asymptotic but is slower after experiences with better defended prey (Speed, 1999). This difference can be seen by plotting Owen & Owen’s explicit equation (Appendix 1): when φ varies with the strength of the stimulus (as in the OO-OO system—Fig. 8A) density-dependence is non-monotonic; when φ is equal for both Model and Mimic (as in ST forgetting, but still using the Owen & Owen equation—Fig. 8B) density-dependence remains monotonic. We might then expect all predators with OO forgetting to show the non-monotonic effect (see Speed, 1999). Despite these complications it can generally be concluded that all predators which generate a spectrum of Batesian, quasi-Batesian and Muellerian mimicry make evolutionary predictions in which mimetic polymorphism will be favoured in the edible region and some part of the defended region of the spectrum. If the OO forgetting rule is used and forgetting rates vary then this will generally be the case for quasi-Batesian mimics at the lower end of their density range.

DISCUSSION

Our findings (Fig. 1, Table 5) make a number of important points about mimicry. First, they show how sensitive predictions about mimicry are to relatively minor modifications in assumptions about learning and memory in predators. Second, the generation of the full classical Batesian-Muellerian mimicry spectrum is surprisingly difficult, turning up with only two of our 29 predator systems. To get a completely

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Figure 7. A, Spectrum 4: simulations of the mimicry spectrum which generate Batesian, quasi-Batesian and Muellerian mimicry (results from BM-ST). Horizontal axis is k. B & C, density-dependence with a predator that generates Batesian, quasi-Batesian and Muellerian mimicry. (B) BM-ST, (C) BM-OO, (top down, k = 0.8, 0.5, 0.3, 0.2).

Figure 6. Spectrum 3: simulations of the mimicry spectrum which generate the ‘Huheey Spectrum’ of Batesian and quasi-Batesian mimicry. (A) ST-HS, (B) HH-HH. Note that the qualitative effect is the same, even though HH-HT is entirely switched and ST-HS is a ‘cumulative’ model. The horizontal axis is a (in 4A) or n (in 4B—see Table 2). C, Density-dependence with a ‘Huheey’ type predator (STHS). Parameters as in Fig. 4C.

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T 5. Types of mimicry spectrum produced by combinations of learning and forgetting rules. B – Batesian mimicry; qB – quasi-Batesian mimicry; qM – quasi-Muellerian mimicry; M – Muellerian mimicry. Spectrum 1: ‘Classical Spectrum’ is B/M. Spectrum 2: ‘quasi-Muellerian Spectrum’ is B/qM/ M. Spectrum 3: ‘Huheey Spectrum’ is B/qB. Spectrum 4: ‘quasi-Batesian’ spectrum is B/qB/M. STOP-FJ and ST-OP-VJ are B/qM/M. See also Fig. 1. Rules in italics involve instantaneous learning or forgetting (‘switching’): those in bold type involve entirely event-dependent forgetting. ∗Densitydependence apparently monotonic. †Density-dependence non-monotonic at least for some quasi-Batesian mimics (see Speed 1999). ‡Density-dependence non-monotonic for some quasi-Muellerian mimics Learning rule GR BM ST OO HH

Forgetting rule HS B/qB∗ B/qB∗ B/qB∗ B/qB∗ —

HH B/qB∗ B/qB∗ B/qB∗ B/qB∗ B/qB∗

HT B/qB/M∗ B/qB/M∗ B/qB/M∗ B/q-B/M∗ B/M∗

ST B/qB/M∗ B/qB/M∗ B/M∗ B/qB/M∗ —

OO B/qB/M† B/qB/M† B/qB/M† B/qB/M† —

CO B/qB/M∗ B/qB/M∗ B/qB/M∗ B/qB/M∗ —

OP — — B/qM/M‡ — —

conventional spectrum (e.g. Turner et al., 1984) predators need to have extreme learning asymptotes for all prey and to have fixed-rate, time-based forgetting which is not subjected to ‘memory jogging’ effects. Once these assumptions are relaxed, then Batesian or Muellerian relationships extend in various ways beyond their conventional boundaries in the palatability spectrum. Third, the dynamics of densitydependence are not necessarily simple and monotonic as conventionally assumed. Variations in memory rules can generate rather more complex predictions. Our 29 predator behaviour systems cannot all, of course, be equally good representations of behaviour. We therefore evaluate rules for learning and forgetting, based on the findings of animal psychology and of laboratory experiments with frequencies of Batesian mimics. We then consider the evolutionary predictions made by each category of predator and evaluate their match to what we know about real populations. Evaluating the behaviour systems 1: psychology and behaviour Which aspects of behaviour are crucial? It is clear from Table 5 that it is the mode of forgetting that predominantly determines the type of mimicry spectrum. Forgetting that depends on the passage of time produces—with three exceptions—Spectrum 4, in which well defended prey are true Muellerian mimics and moderately defended prey are quasi-Batesian mimics (Fig. 1). Further with all these time-dependent rules, at least some prey are true Muellerian mimics. Conversely forgetting that depends purely on the passage of relevant events—in this case the observation of prey of a similar appearance— generates a ‘Huheey Spectrum’ (Spectrum 3) with quasi-Batesian instead of Muellerian mimicry (Fig. 1). If forgetting depends on a mixture of time and events (the CO rule) the spectrum contains both quasi-Batesian and Muellerian mimicries (Spectrum 4). The three exceptions all occur with time-dependent forgetting. First, if observation retards forgetting (the OP rule), or also in some circumstances reverses it (the OPFJ and OP-VJ rules) then moderately edible prey become quasi-Muellerian, having

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Figure 8. Density-dependent functions using Owen & Owen’s equation (from Owen & Owen 1984. See Appendix I). A, a forgetting is slower with the nastier Model (Model, kMO=0.2, φMO=0.3; Mimic kMI=0.3263157894738, φMI=0.35) giving a non-monotonic functions. B, forgetting rates are fixed and equal, (Model, kMO=0.2, φMO=0.3MO; Mimic kMI=0.3263157894738, φMI=0.3) giving a monotonic function. In both graphs XMO=1.6.

a strength in numbers typical of Muellerian mimicry (Spectrum 2: Fig. 1). The other exceptions occur when the stable asymptotic attack rate generated by learning is always extreme, instead of being continuously distributed according to the strength of the conditioning. Here continued conditioning eventually results either in total avoidance or in a one hundred percent decision to attack (ST and HH learning). In this case the spectrum can divide classically into traditional Batesian and Muellerian mimicry without quasi-Batesian mimicry (Spectrum 1: Fig. 1). The forgetting process exercises a further effect on density-dependent predation. If aversions and appetitive reactions decay over time at the same rate no matter how powerful the conditioning stimulus has been, density-dependence is monotonic. However it can be non-monotonic, for at least some quasi-Batesian mimics, if forgetting is

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slowed or accelerated according to the strength of the stimulus (OO forgetting). If forgetting is slowed or reversed by prey observations (as in ST-OP predators) then density-dependent predation can also be non-monotonic for edible prey. It is important to note a potential misunderstanding of the dynamics of forgetting which for example may have arisen in Huheey (1988): a constant rate of forgetting does not generate constant periods of avoidance, and better defended prey will still be avoided for longer with these behavioural models. A variable rate of forgetting somewhat magnifies this effect, but in practice it will be rather difficult to determine whether forgetting rates are variable simply by measuring avoidance periods. Evaluation using animal psychology The crucial questions which will determine whether mimicry is as conventionally supposed and then whether density-dependence will be simply monotonic, are therefore: (1) Is forgetting dependent on time or events? (2) Does observation without reinforcement counteract forgetting (‘remind’)? (3) Is forgetting a fixed exponential or does it vary with the intensity of the conditioning stimulus? (4) Are the asymptotes of learning always extreme or are they continuously distributed? Animal psychology can provide answers to some, but by no means all of these questions. We can draw from it only limited evaluations of predator behaviour. Question 1 can be answered in part. ‘Forgetting’ of Pavlovian memories is almost certainly dependent on the passage of time (see e.g. Spear, 1978; Spear et al., 1990, Bouton, 1993, 1994). This allows us to reject the predators with HH and HS forgetting, which cannot generate Muellerian mimicry, since these forget only as prey are observed but avoided (Fig. 3). However it is not clear how events such as prey observation affect time-based forgetting (question 2). On the one hand it could speed it up (as with the Huheey or combined CO predators), on the other it could slow it down or reverse it (as with the OP and the ‘memory jogging’ predators). The lack of decisive evidence (Guilford 1990) prevents proper appraisal of behaviour systems whose unconventional results rely on these properties of memory. ST-CO which generates quasi-Batesian mimicry and ST-OP-FJ (also ST-OP, and ST-OP-VJ) which generates quasiMuellerian mimicry are important examples. There is also apparently no decisive answer to question 3 (M.E. Bouton personal communication). We cannot therefore choose easily on empirical grounds between constant, ST forgetting and variable OO forgetting algorithms. The likely nature of density-dependent predation in the quasi-Batesian zone therefore remains ambiguous. Learning asymptotes do seem best modelled as continuous variables (question 4), at least as internal cognitive representations with control over levels of behaviour (Rescorla & Wagner, 1972; Wagner & Rescorla, 1972; Pearce & Hall, 1980; see Speed, 1993a, Macdougall & Dawkins, 1998 for discussions). We can therefore tentatively reject all systems which use only extreme values of learning asymptote; i.e. those which generated the classical Batesian-Muellerian mimicry spectrum (STST, HH-HT). This view of learning asymptotes also casts doubt on the predictions of our memory jogging predators which all rely on ST learning. However, memory

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jogging has not been extensively studied here and there remains a possibility that for example GR-OP will also generate quasi-Muellerian results. Animal psychology therefore enables us to reject predators who forget only after avoidance events (since forgetting is probably at least partly time-dependent), those which generate a classical Batesian and Muellerian spectrum, and our quasi-Muellerian predators (since learning is likely to be guided by continuous learning asymptotes). This leaves us favouring a spectrum of Batesian, quasi-Batesian and true Muellerian mimicry from predators which use the GR, BM and OO learning rules and time dependent forgetting (Fig. 1). However, the nature of density dynamics in the defended part of the spectrum remains equivocal. MacDougall & Dawkins (1998) have recently extended the GR learning rule (the ‘Pavlovian Predator’ of Speed, 1993a) by looking at the interesting question of recognition errors and attentional limitations. At present though we feel that their virtual predator needs further examination before its significance can be evaluated fully (Speed & Turner, in press). Evaluation using behaviour in reciprocal frequency experiments We have examined elsewhere (Turner & Speed, 1996) reciprocal frequency experiments with Batesian mimicry (Brower, 1960; Avery, 1985; Huheey, 1988). In these experiments models and mimics are presented to live predators in mixtures ranging from 0 to 100% of either, and the pattern of results is analysed using an equation from Huheey (1976). Such experiments are highly inconclusive about the heuristic validity of various behaviour systems (Turner & Speed, 1996). The feature of behaviour discriminated most readily by reciprocal frequency experiments is whether a behaviour system does or does not involve switching (Turner & Speed, 1996). However, our results reported here show that whether learning and forgetting are switched (single-trial, instantaneous) or gradual, has little or no influence on the type of mimicry spectrum produced. The fully quasi-Batesian spectrum of Huheey appears whether forgetting is switched (HH) or gradual (HS), so long as it is event-based. Similarly with learning rules, the OO system which ‘switches’ due to instantaneous learning gives the same qualitative results as the entirely gradual BM learning rule (Table 5). Likewise the ‘switched’ HH learning rule and its gradual analogue ST can both generate the classical Batesian/Muellerian spectrum. Moreover the outcome of reciprocal frequency experiments is dominated by the mode of learning (Turner & Speed, 1996). While the extent to which learning switches is indeed an important and interesting aspect of predator behaviour, it turns out now not to be crucial to the dynamics of the mimicry spectrum and of densitydependent predation on mimics. These dynamics are dominated by the mode of forgetting, primarily whether it is dependent on time or on the passage of relevant events, and whether it can be over-ridden by non-reinforced ‘reminders’. Reciprocal frequency experiments (see Huheey, 1988) do however help us to reject one memory rule. The OP forgetting rule (Turner & Speed, 1996) predicts results of a different order of magnitude to those seen with live, experimental predators and therefore lacks credibility. Evaluating the behaviour systems 2: conditions favouring polymorphism Virtually all simulated predators confer a positive effect of density on predation rates on all edible mimics and so make the conventional prediction that polymorphism

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is favoured in such Batesian mimics (the quasi-Muellerian predators are exceptional— below). With defended mimics the results are varied. The predators which generate entirely conventional results (Spectrum 1, Fig. 1), always generate negative densitydependent predation: monomorphism will be the rule for defended mimics. The predators which generate an entire spectrum with Batesian characteristics (Spectrum 3, ‘Huheey Spectrum’, Fig. 1) generate positive density-dependence throughout the spectrum. These systems make the unique prediction that in virtually all mimicry whether by defended or edible species polymorphism is favoured. The exception is the point at which mimic and model are equally well defended; here densitydependence is neutral and since mimicry is also selectively neutral, it would be degraded by the effects of mutation and random genetic drift. Predators which generate the fourth spectrum (Batesian/quasi-Batesian/Muellerian mimicries, 1) are expected to generate monomorphism in well defended mimics, and polymorphism in both edible and moderately defended (quasi-Batesian) mimics. In addition some of these predators generate non-monotonic relations in moderately well defended prey; in this case we might expect polymorphism with more complicated behaviour (oscillations, limit cycles or chaos) in the region of equilibrium, depending on the relative densities of the model and mimic species, with the mimic tending to polymorphism at low species densities and monomorphism at high species density (Owen & Owen, 1984). At very low densities we might expect most mimics, whether edible or defended, to tend towards polymorphism, as the quasi-Batesian zone of the spectrum will in this case probably extend across much of the defended half. Evidence from mimetic populations There seems to be ample evidence that polymorphism is largely confined to edible mimics. No apparently defended butterfly species for instance achieves the spectacular polymorphism of Pseudacraea eurytus, Papilio dardanus or Papilio memnon (e.g. Carpenter, 1949; Sheppard, 1961; Clarke et al., 1968). It is also the case that the majority of mimics that have their own defence are monomorphic. This is predominantly true for the mimicry rings of bees and wasps (Plowright & Owen, 1980). In a recent intensive study of the Ithomiine mimicry rings in Amazonian Ecuador, Beccaloni (1997) found that of 124 largely defended mimetic species (55 ithomiines, 34 other butterflies, 34 moths and one damselfly), 13 were polymorphic, generally with two forms in different mimicry rings; five of these are in groups traditionally supposed to be edible, and only one (Consul fabius) shows a further distinct sign of edibility (a cryptic underside; see for example Turner 1995, fig. 7). However there are a few apparently defended mimics which are polymorphic (e.g. Beccaloni, 1997). Outstanding among these have been various members of the South American butterfly genus Heliconius and the European moth Zygaena ephialtes (Turner, 1968, 1971a, 1977, 1987; Bullini et al., 1969; Brown & Benson, 1974; Sbordoni et al., 1979; Beccaloni, 1997). Huheey (1976) pointed out that his quasiBatesian system (HH-HH) could explain this polymorphism in apparently defended species. The problem is that his system explains too much: it predicts at least that polymorphism should be as widespread among defended mimics as among edible ones and this is not the case. The predominant monomorphism of defended mimics therefore suggests straight away that the systems with event-dependent forgetting, which predict almost ‘universal Batesian mimicry’ are not valid.

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This leaves a choice between three types of system, all of them with true Batesian mimicry at one end of the spectrum and true Muellerian mimicry at the other, but with different effects in the region of neutral edibility. The Classical Spectrum (Spectrum 1) divides Batesian from Muellerian mimicry exactly at the point of neutrality. Spectrum 2 has a zone of (monomorphic) quasi-Muellerian mimicry on the edible side of neutrality and Spectrum 3 has a (potentially polymorphic) zone of quasi-Batesian mimicry on the defended side of neutral edibility (see Fig. 1). It is extremely difficult to tell which of these situations is occurring in the real world, in part at least because our understanding of variations in prey acceptabilities is fragile (e.g. Ritland, 1991; Srygley & Kingsolver, 1998). There is little data on the relative acceptabilities of Batesian mimics, and therefore the prediction from the quasi-Muellerian spectrum (Spectrum 2) that polymorphism should be confined to the more profitable of edible mimics is at present untestable. There is however some suggestive evidence in favour of the quasi-Batesian/ Muellerian system, in the form of the apparently defended species (traditionally Muellerian mimics) which are polymorphic: the defended South and Central American butterfly Heliconius doris with three polymorphic forms (Turner, 1971a; Gilbert, 1984) was among the least well defended members of its genus when tested with birds (Brower et al., 1963). Other apparently noxious butterflies that are polymorphic include (i) some of the ‘tiger’ Heliconius species (Turner, 1968; Brown & Benson, 1974; Beccaloni, 1997); (ii) the African Danaus chrysippus and its apparent Muellerian co-mimic Acraea encedon and A. encedana (Pierre, 1973, 1974; Gordon, 1984; Smith et al., 1993, Owen et al., 1994); (iii) the European aposematic moth Zygaena ephialtes (polymorphic even in some areas which are not junction zones) (Bullini et al., 1969; Turner, 1971a; Sbordoni et al., 1979); and (iv) the apparently mildly inedible but polymorphic far eastern and Australian Hypolimnas bolina (Clarke & Sheppard, 1975; Marsh et al., 1977; Clarke et al., 1989). Brakefield (1985) has suggested that the polymorphism of the 2-spot ladybird Adalia bipunctata is mimetic, the red form copying the 7-spot Coccinella species and the melanic form the 4-spot Exochomus. The finding that A. bipunctata is only mildly undesirable to quail (Marples, 1990) suggests that it may be an unpalatable and potentially quasi-Batesian mimic (although see Marples et al., 1989 and Marples, 1993 for a more subtle analysis). The polymorphism of Heliconius melpomene and Heliconius erato, which Huheey (1976) cited in favour of his models, is almost certainly not explained in this way, however; it is amply demonstrated that it is produced by immigration and gene-flow in junction zones between their widespread and almost entirely monomorphic geographical races (Turner, 1971a, b; Mallet 1986; Mallet & Barton, 1989; Turner & Mallet, 1996; Mallet & Turner, 1997). The polymorphism of Danaus chrysippus has also recently been explained as the result of massive introgression between races (Smith et al., 1998). Dynamics of quasi-Batesian mimicry If as we have argued (see also Speed, 1993a, b, 1996) quasi-Batesian mimicry is likely to be a real phenomenon, the dynamics of quasi-Batesian mimics are of real interest. Clearly, a moderately defended species has the rather surprising 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).

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(2) It can act as a model for an edible Batesian mimic. (3) It can be a Muellerian co-mimic of a species as well defended as itself. (4) It can behave like a Batesian mimic of a species better defended than itself. A consideration of points (1) and (4) above suggests that when a moderately defended mimic greatly outnumbers its better defended ‘model’, the model produces virtually no reinforcement of the predator, and here the ‘mimic’ can in fact add protection. The direction of density-dependent predation on such a moderately defended species will be negative, as for a simple, independent aposematic species (also Joron & Mallet, 1998). In turn this indicates that all quasi-Batesian mimics are likely to generate non-monotonic density-dependent effects on predation rates (positive when rare, negative when very abundant). However, this has been demonstrated in our simulations only for systems with OO forgetting and this is probably because the simple virtual predators used here have invariant hunger levels and therefore are not completely sensitive to very high abundances of prey. Interesting questions arise from consideration of all four points about the dynamics of mimicry rings containing more than two species, for example a model, a quasiBatesian mimic and a true Batesian mimic, or about the density-dependence of quasi-Batesian mimics with changes in the density of their models.

CONCLUSIONS: HOW WELL DO WE UNDERSTAND THE MIMICRY SPECTRUM?

Our simulations make a number of important points which indicate limitations to existing mimicry theory and identify areas where data needs to be collected. We have shown, for instance that that the classical view of Batesian and Muellerian mimicry is not robust against variations in rules of learning and especially memory, and is indeed rather hard to generate without making rather restrictive assumptions. Psychological evaluations of published rules have helped us to reject some categories of behaviour. We find that, out of those predators simulated, those which generate a spectrum of Batesian, quasi-Batesian and Muellerian mimicry (Spectrum 4) are likely to be the most credible. However the effects of mimic densities on the attack rates of such predators remain uncertain. Reasonable cases can be made on psychological and behavioural grounds against predators which generate the ‘Classical’ Batesian/Muellerian spectrum (Spectrum 1), the ‘Huheey Spectrum’ (Spectrum 3 which generates no Muellerian mimicry) and those which can generate quasiMuellerian mimicry (Spectrum 4). Huheey (1976) originally suggested that previously unsuspected aspects of predator learning and forgetting could explain the fact that some apparently defended species are polymorphic. The suggestion was indeed fruitful: although we are inclined to reject his view that the mimicry spectrum lacks true Muellerian mimicry. Moderately defended mimics, we suggest, may at some times be Batesian-like and at others be truly Muellerian. Recent work by MacDougall & Dawkins (1998) reports interesting simulations incorporating generalized recognition errors by predators which develop this view further. We suggest however that these simulations need greater sensitivity testing, such as incorporating prey appearance or predator forgetting as variables. Reciprocal frequency experiments with live predators have been shown to help to determine whether predators use discontinuous ‘switching’ of their behaviours (Turner & Speed, 1996). We suspect that there has been an unstated assumption

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behind much current thinking that switching is significant in the mimicry spectrum. However the question of switching is shown here to make little contribution to our understanding of how predatory learning and memory generate mimetic relationships. Indeed the results are largely unaffected by whether the systems use switched or gradual rules. This has the encouraging implication that models which use explicitly soluble Markov processes (e.g. Luedeman et al., 1981), which necessarily rely on an assumption that behaviour switches, could give results which were still broadly applicable to predators that moderated their behaviour by cumulative learning and forgetting. A prediction that the ‘palatability spectrum’ is divided into Batesian, quasiBatesian and Muellerian mimicries accords well with an evaluation of patterns of polymorphism in real populations. This suggests that though mimetic polymorphism is indeed most common in edible species, it does extend in a quasi-Batesian manner, into some part of the defended region of the spectrum. This conclusion must be treated with some caution, however, since there is much yet to be learnt about predator psychology before mimetic relationships can be confidently described throughout the spectrum of prey acceptabilities.

ACKNOWLEDGEMENTS

Our thanks to Nichola Marples and Noe´l Holmgren for valued advice on the manuscript, to J. Wright of the Department of Psychology, University of Leeds for his guidance during this project, and to the Royal Society for an equipment grant to J.R.G. Turner. M.P. Speed has been supported by the Liverpool Hope University College Research Fund.

REFERENCES

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Speed MP. 1999. Robot predators in virtual ecologies: The importance of memory in mimicry. Animal Behaviour 57: 203–213. Speed MP, Turner JRG. in press. Robot predators, receiver psychology and doubts about Muellerian mimicry: Comments on MacDougall and Dawkins. Animal Behaviour. Srygley RB, Kingsolver JG. 1998. Red-wing blackbird reproductive behaviour and the palatability, flight performance, and morphology of temperate pierid butterflies (Colias, Pieris and Pontia). Biological Journal of the Linnean Society 64: 41–55. Turner JRG. 1968. Natural selection for and against a polymorphism which interacts with sex. Evolution 22: 481–495. Turner JRG. 1971a. Studies of Mu¨llerian mimicry and its evolution in burnet moths and heliconid butterflies. In: Creed ER, ed. Ecological genetics and evolution. Oxford: Blackwell Scientific Publications, 224–260. Turner JRG. 1971b. Two thousand generations of hybridisation in a Heliconius butterfly. Evolution 25: 471–482. Turner JRG. 1977. Butterfly mimicry: the genetical evolution of an adaptation. Evolutionary Biology 10: 163–206. Turner JRG. 1980. Oscillations of frequency in Batesian mimics, hawks and doves, and other simple frequency dependent polymorphisms. Heredity 45: 113–126. Turner JRG. 1984a. Mimicry: the palatability spectrum and its consequences. In: Vane-Wright RI, Ackery PR, eds. The biology of butterflies. (Symposia of the Royal Entomological Society of London; 11). London: Academic Press, 141–161. Turner JRG. 1984b. Darwin’s coffin and Doctor Pangloss – do adaptationist models explain mimicry? In: Shorrocks B, ed. Evolutionary ecology (British Ecological Society Symposium No 23), Oxford: Blackwell Scientific Publications, 313–361. Turner JRG. 1987. The evolutionary dynamics of Batesian and Muellerian mimicry: similarities and differences. Ecological Entomology 12: 81–95. Turner JRG. 1995. Mimicry as a model for coevolution. In: Arai R, Kato M, Doi Y, eds. Biodiversity and evolution. Tokyo: National Science Museum Foundation, 131–150. Turner JRG, Kearney EP, Exton LS. 1984. Mimicry and the Monte Carlo predator: the palatability spectrum and the origins of mimicry. Biological Journal of the Linnean Society 23: 247–268. Turner JRG, Mallet J. 1996. Did forest islands drive the diversity of warningly coloured butterflies? Biotic drift and the shifting balance. Philosophical Transactions of the Royal Society of London B351: 835–845. Turner JRG, Speed MP. 1996. Learning and memory in mimicry. I. Simulations of laboratory experiments. Philosophical Transactions of the Royal Society of London B351: 1157–1170. Wagner AR, Rescorla RA. 1972. Inhibition in Pavlovian conditioning: Application of a theory. In: Boakes RA, Halliday MS, eds. Inhibition and learning. London: Academic Press, 301–336. Wickler W. 1968. Mimicry in plants and animals. London: Wiedenfeld and Nicholson.

APPENDIX 1

The Owen-Owen equation The equation for defining the OO-OO predator from Owen & Owen (1984) is in the notation used here (also Turner & Speed 1996) P=1/{1+IIMO·[(1−kMO)·X/(X+φMO)]+IIMI·[(1−kMI)X/(X+φMI)]} where P is the mean attack probability on the Model-Mimic pair, IIMO is the relative frequency of Model and IIMI the relative frequency of Mimic, k is the learning asymptote and φ is the forgetting rate variable (the subscripts MO and MI denoting Model and Mimic respectively), and X is the total absolute abundance of prey in a population (this is assumed not to change over the course of a sampling season). Note that in Owen and Owen’s original equation 14, k represented forgetting rate, and r the asymptote of learning. There was also a constant K which always takes a value of 1 in Owen & Owen’s calculations. It has been omitted when used as a multiplier here.

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A y

+

Relative predation rate (zero when mimic is absent)

z – x

B

y

+

z – x

C +

y



Density of mimic

Figure 9. Elementary theory of relationships between density-dependence and harm or benefit to Model. The X axis measures Mimic’s density and the Y axis the difference made to the Model-Mimic predation rate by the presence of Mimic. Curves are hypothetical and illustrate monotonic and nonmonotonic density effects. x denotes a point at which it is known that the mimic benefits the model by decreasing the rate of predation at that density; likewise y denotes a point where it is known that a mimic causes an increase in the rate of predation. At z it is known that the mimic is neutral in its effect. (A) Provided density-dependence is monotonic, mimics consistently harm or benefit their models at all densities, and neutral mimics are neutral at all densities. It is also inevitable that a mimic which benefits its model always experiences negative density-dependence, and a mimic which harms its model always experiences positive density-dependence. (B) If density-dependence is not monotonic, mimics of any type may experience positive density-dependence at some densities and negative densitydependence at others. An apparently neutral mimic will in general be neutral only at certain nodal densities. (C) The case noted by Owen & Owen (1984) in which a mimic which harms its model at low densities may benefit it at high densities. Density-dependence clearly must not be monotonic.

APPENDIX 2

Elementary theory of relationships between density-dependence and harm or benefit to model. Figure 9 shows a number of hypothetical scenarios for density dependent predation. It is elementary to show (Fig. 9A, B) that, because a mimic at zero density can have no effect on the predation of its

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model, a mimic that benefits its model at some density must cause negative density-dependent predation in some part of the range of densities; and that conversely a mimic that harms its model must somewhere experience positive density-dependent predation. Further, if density-dependent predation rates are monotonic, then these results apply to the full range of density from zero to maximum (Fig. 9A). Thus provided density-dependence is monotonic, it follows that Batesian and quasi-Batesian mimics (as we have defined them) always experience positive density-dependent predation, and all Muellerian and quasi-Muellerian mimics experience negative density-dependent predation (Table 4). Thus the directions of density-dependent predation map one to one with the divisions of the mimicry spectrum, with a transition (neutral mimicry with no density-dependence) occurring at the point where Batesian (or quasi-Batesian) mimicry changes into Muellerian (or quasi-Muellerian) mimicry. In the conventional, classical mimicry spectrum this point of neutral mimicry coincides with the point of neutral edibility, but is displaced from it in other spectra (Fig. 1). However provided density-dependence is monotonic, the position of the point of neutral mimicry is invariant with density: there is some level of acceptability of the mimic at which mimicry is neutral at any possible density (Fig. 9A). On the other hand if density-dependence is not monotonic, this simple mapping breaks down. Again it is elementary to see that a mimic which harms its model may (at the simplest) cause positive density-dependent predation at low densities and negative density-dependent predation at high densities, and conversely for a mimic that benefits its model (Fig. 9B). In the extreme a mimic may harm its model at low densities, and then if the curve of predation dips far enough, benefit its model at high densities (Fig. 9C): Owen & Owen (1984) showed that with their behaviour system (OO-OO) this effect could occur for moderately defended mimics (Speed, 1999). In such cases the mapping of densitydependence onto the mimicry spectrum becomes more complicated, and the extent of the different zones of mimicry becomes density-dependent. For instance a spectrum might show a wide zone of quasi-Batesian mimicry when inspected at low density, but a much narrower zone at high Mimic density. In general the point of neutral mimicry within the spectrum will move according to the Mimic density at which the spectrum is plotted, and there will be no level of acceptability of the Mimic at which predation will be independent of Mimic density (Fig. 9B)

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