ANIMAL BEHAVIOUR, 1999, 57, 203–213 Article No. anbe.1998.0943, available online at http://www.idealibrary.com on

Robot predators in virtual ecologies: the importance of memory in mimicry studies MICHAEL P. SPEED

Environmental & Biological Studies, Liverpool Hope University College (Received 6 March 1998; initial acceptance 15 April 1998; final acceptance 15 July 1998; MS. number: 5811)

An important means of investigating gains and losses to prey caused by mimicry is through mathematical or computer constructs which represent and explore limited aspects of mimicry situations. Such studies use virtual predators which are usually simple automata, ‘robots’ that, through simple rules, vary virtual attack rates on virtual insect prey. In this paper I consider the effect of variations in predator memory and learning on mimicry dynamics. When there is mimicry between unequally noxious prey, the way that memory is modelled is shown to be crucial. If forgetting rates are fixed, an increase in the density of the least defended prey produces monotonic gains or losses in protection. However, if forgetting rate is inversely related in some way to degree of noxiousness of the prey then attack rates initially rise with the density of the least defended prey, reach a cusp and then fall. I show that the generation of this highly unconventional up–down result appears to be independent of variations in learning rate. This work shows how sensitive the predictions of virtual predators may be to relatively small changes in behavioural rules. 

Mimicry theory has, however, been dogged by ‘the problem of the palatability spectrum’. Variation in prey acceptability is well known both between species (e.g. Brower et al. 1963, 1968; Brower & Brower 1964; Marples et al. 1989; Sargent 1995) and within them (e.g. Brower et al. 1978; Ritland 1995). There have been lengthy debates about the significance of variations in prey acceptability, especially with respect to the distinguishing properties of Batesian and Mu ¨ llerian mimics. Arguments have focused on cases where levels of defence of mimetic prey are intermediate (see e.g. Huheey 1976, 1980, 1988; Benson 1977; Sheppard & Turner 1977; Turner 1987, 1995; Malcom 1991; Speed 1993a, b, 1996; Gavrilets & Hastings 1998; MacDougall & Dawkins 1998). The source of these arguments lies largely in the paucity of knowledge about aspects of animal psychology and the way that predators behave towards such prey in their natural habitats. One useful, although naturally limited, way to explore the basis of warning signals and mimetic relationships has been through theoretical studies by explicit solution (Huheey 1964, 1976; Brower et al. 1970; Pough et al. 1973; Leudeman et al. 1981; Hadeler et al. 1982; Owen & Owen 1984; Leimar et al. 1986) or by computer simulation (Turner et al. 1984; Speed 1993a; Turner & Speed 1996; MacDougall & Dawkins 1998). Such researchers construct ‘virtual ecologies’ inhabited by ‘robot predators’: simple predatory automata. These ‘virtual predators’ make decisions about attacks on imaginary

Few topics have provided such rich resources for evolutionary biologists as the study of warning signals and mimicry. Considerable light has been shed on difficult evolutionary problems through the study of warningly coloured insects (e.g. kin selected and other forms of altruism: Fisher 1930; Turner et al. 1984; Guilford 1988; the genetics of adaptation: Clarke & Sheppard 1971; Sheppard et al. 1985; arms races: Nur 1970; Turner 1987, 1995; evolutionary history: Turner 1988; Brower 1996; hybrid zones, shifting-balance and speciation processes: Mallet 1986; Mallet & Singer 1987; Jiggins et al. 1996). One reason for the enduring interest in mimicry lies in the opposed dynamics that discriminate Batesian from Mu ¨ llerian species. Batesian mimics are believed to parasitize their models’ defences, have a strength through rarity, a consequent tendency to polymorphism and mimetic supergenes to match. In contrast, Mu ¨ llerian comimics are assumed to be in mutualistic cooperation, have strength in numbers, strong tendencies to monomorphism and have little if any purposeful linkage between mimicry genes (e.g. Turner 1977, 1987). Batesian and Mu ¨ llerian mimicries therefore offer a natural laboratory for the study of often exquisite adaptations which have contrasting evolutionary trajectories. Correspondence: M. P. Speed, Environmental & Biological Studies, Liverpool Hope University College, Hope Park, Liverpool L16 9JD, U.K. (email: [email protected]). 0003–3472/99/010203+11 $30.00/0

1999 The Association for the Study of Animal Behaviour

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1999 The Association for the Study of Animal Behaviour

204 ANIMAL BEHAVIOUR, 57, 1

0.9

(a)

0.85 0.8 0.75 0.7 0.65 0.6

Proportion attacked

prey they encounter. They modify ‘virtual attack rates’ using simple algorithms to simulate learning and memory (e.g. Huheey 1964, 1976; Owen & Owen 1984; Turner et al. 1984; Speed 1993a). However, other variables such as hunger, response to toxins and discrimination errors are usually not considered (but see MacDougall & Dawkins 1998). Although these virtual predators and their simulated ecologies are highly simplified, they have proved useful representations for exploring causal features of real behavioural and ecological systems. Central to these studies, then, have been more or less informed speculations about the way that predators respond to prey of different nutritional and toxic qualities. In constructing such virtual predators there has been a tendency to allow variations in prey defences to be manifest as modifications to attack probabilities either by learning (Turner et al. 1984; Speed 1993a) or through variable forgetting rates (Huheey 1964, 1976; Estabrook & Jespersen 1974; Bobisud & Potraz 1976; Arnold 1978), but rarely in both. The work of Owen & Owen (1984) is, however, one of a few exceptions (see also Hadeler et al. 1982; Turner & Speed 1996). Owen & Owen (1984) theorized about mimetic relationships in the light of both learning and forgetting rates. Their findings, which are probably the most unconventional of any published to date in mimicry theory, are surprisingly underreported and underexplored. Traditional mimicry theory (e.g. Turner et al. 1984; Turner 1987), and even nontraditional versions (e.g. Speed 1993a), predict constant, monotonic effects of prey densities on attack rates (e.g. Fig. 1a). Owen & Owen’s virtual predator, however, does not always generate straightforward monotonic results. This is particularly the case when there is inequality in defence between mimics. Increases in the density of the less defended species can produce a single-cusped curve of the type shown in Fig. 1b. This is a startling prediction which suggests that a moderately defended mimic may cause a more noxious comimic to lose protection when rare, but eventually gain protection when the mimic is sufficiently common. In honour of Owen & Owen’s findings I call this general, nonmonotonic result the Owen–Owen Effect. Owen & Owen (1984), however, did not explore how this result follows from the construction of their virtual predator. Despite similarities in construction, my own virtual predator (the ‘Pavlovian Predator’: Speed 1990, 1993a) does not seem to produce this result. We thus have a potentially important insight into mimicry dynamics, but little understanding of which aspects of predator behaviour are important in generating the Owen–Owen Effect or how general it might be. I set out, then, to locate the behavioural cause of this unusual result and to consider whether it is robust against plausible variations in a behavioural program. I show that this curious dynamic results if forgetting rate varies such that forgetting is faster after experiences with less noxious prey. Moderately defended mimics can then dilute protection of their better defended partners by increasing forgetting rates, especially when rare. Later, when common, they can add net protection, by providing enough

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Figure 1. Each curve is a plot of the proportion of mimetic prey attacked versus density of a mimic species. In all figures, XMo refers to the number of Model in the prey population, λMo is the learning asymptote for Model, λMi the learning asymptote for Mimic, αfMo the forgetting rate of Model and αfMi the forgetting rate of Mimic. (a) Traditional monotonic density-dependent attack rates. Top line, Batesian mimic increases in density and predation rate increases, bottom line, a Mu¨llerian mimic increases in density and attack rate diminishes. Data generated using equation (1). XMo =1.6 (see Owen & Owen 1984), λMo =0.2, αfMo =0.3, Batesian: λMi =1.0, αfMi =0.0; Mu¨llerian λMi =0.2, αfMi =0.3. (b) Demonstration of the unconventional density-dependent effect, here called Owen–Owen Effect. Values for Model as above, for Mimic, λMi =0.3, αfMi =0.5. A represents attacks with Mimic virtually absent, B represents the cusp of the curve, C the density at which attack rates on Model and Mimic are equal to those on Model alone and D a point at which both Model and Mimic gain from their mimicry.

SPEED: MEMORY AND MIMICRY 205

encounters to offset predator forgetting. Since there are few data about the relationship of stimulus intensity and forgetting rates, we cannot at this stage be clear about the nature of mimicry when there are inequalities in chemical defence between mimics. SCENARIO AND GENERAL EQUATION

Terminology of Chemical Defence Terminology about prey defences is varied and is not a simple matter (Brower 1984). Common descriptors include at least: (un)palatable, (in)edible, (un)acceptable, distasteful, noxious, toxic, protected, poisonous and venomous. Commonly used terms such as ‘unpalatable’ can be ambiguous as they are used to describe aversive qualities that are purely gustatory (i.e. in pure bad taste) and/or potentially toxic. As a result I avoid the use of words such as ‘palatability’ and ‘distastefulness’. I consider that the sort of prey pertinent to the discussion are those that are chemically defended 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). Such prey will be termed defended or noxious. Much of the paper rests on the existence of a spectrum of levels of prey defence (and degrees of noxiousness) from cases of virtually no defence to the extreme of thorough toxicity. Whether innocuous but foul-tasting or foul-smelling prey can confer similar or persistent levels of protection as those that have toxic qualities is doubted (class II of chemical defence: Brower 1984). However, during a predator–prey arms race, some prey might generate tastes/smells that, although nontoxic, are so unpleasant as to make the prey effectively inedible. For the time being though, I exclude such prey from the following discussion.

Robot Predators in Virtual Ecologies Nearly all investigations in this area consider attack probability as the behavioural currency of importance. Species of prey vary in levels of defence and status as mimics. Owen & Owen (1984) use renewal theory to produce a simple model in which (1) a naive Owen & Owen predator attacks a novel prey with a probability of 1, (2) after an attack on a prey, attack probability moves immediately to a level ë, the asymptote of learning, which reflects the acceptability of the prey. Undefended, edible prey have a value of ë=1. For chemically defended prey ë<1, and with thoroughly noxious prey ë is around 0. (3) The predator forgets over a period by reverting attack rates to naive levels, by exponential decay. The rate of forgetting is inversely related to the level of noxiousness of a prey. Thus encounters with moderately defended species would be forgotten at a greater rate than encounters with a thoroughly noxious one. Mimicry is assumed to be exact and hence mimetic species share a single attack probability. In cases with mimicry between species, forgetting rate is set by experience with the species most recently sampled. In the ecology of the Owen & Owen virtual predator, hunger levels (and hence

naive attack rates) remain constant and prey populations are not more than marginally depleted by predation. The virtual predator of Speed (1993a), which generated broadly similar results, used a similar view of learning but followed Turner et al. (1984) in assuming a constant rate of forgetting.

Owen & Owen’s Equation Owen & Owen’s equation looked at the joint attack rate on a pair of identical species. In line with other authors (e.g. Turner et al. 1984) I use the name Model for the better defended of the pair and Mimic for the least defended (in the Owen & Owen original these were M1 and M2). ‘Model’ and ‘Mimic’ represent a convenient and memorable terminology, but the terms are not intended to prejudge the mimetic relationship. Owen & Owen’s equation (Owen & Owen 1984, page 234, equation 14) can be stated: P=1/{1+ÐMo((1
ëMo)X/(X+áf Mo))+ ÐMi((1
ëMi)X/(X+áfMi))} (1) where Mo and Mi refer, respectively, to the Model and Mimic species. These are visually identical but can vary in acceptability to predators; P is the mean attack probability on the Model–Mimic pair; ÐMo is the relative frequency of Model and ÐMi the relative frequency of Mimic; ëMo is the learning asymptote of Model and ëMi the learning asymptote of Mimic; áfMo is the forgetting rate variable for Model and áfmi is the forgetting rate variable for Mimic; and X (the sum of XMo and XMi) is the total absolute abundance of Models and Mimics in a population, which is assumed not to change over the course of a sampling season. Note that in Owen & Owen’s original equation, ë represented forgetting rate, and ó the asymptote of learning. I have modified this original terminology since it is at odds with that commonly used in animal psychology (see e.g. Rescorla & Wagner 1972). There was also a constant, K, in the original which always took a value of 1 in Owen & Owen’s calculations. I have omitted it when using it as a multiplier here. It is essential to note also that Owen & Owen did not attempt ‘a fully rigorous description of predation’ (Owen & Owen 1984, page 232). Instead their virtual predator was used to produce limited qualitative results. I carried out three main investigations based around the Owen & Owen predator, two using equation (1), the third using computer simulation. The first seeks to replicate and extend Owen & Owen’s results; the second to examine the role of memory in generating the Owen– Owen Effect; and the third to consider the significance of learning rates. REPLICATION AND EXTENSION OF OWEN & OWEN’S RESULTS

Replication of Owen & Owen’s Results I used equation (1) to generate the following results. When mimicry is Batesian (ëMi =1), a conventional

206 ANIMAL BEHAVIOUR, 57, 1

0.64

Low mimic Defence: moderate λ value

(a)

0.63

monotonic and positive shape results (Fig. 1a, top curve). When mimics are equally defended and mimetic, an increase in density of either one causes a monotonic decrease in proportion attacked (Fig. 1a, bottom curve). However, when there is an inequality in defence (ëMo = 0.2, ëMi =0.3; áfMo =0.3; áfMi =0.5; see Owen & Owen 1984, page 236) the up–down Owen–Owen Effect emerges as the less noxious Mimic species becomes common (Fig. 1b). Important points on the curve are labelled A–D for reference. Point B, the cusp of the curve, represents a critical density of Mimic, at which density dependence is momentarily stationary. To the left of B, density dependence is positive, to the right it is negative. Thus the qualitative results of Owen & Owen are replicated (see Owen & Owen 1984, page 236: note that Owen & Owen’s original graphs do not have values on their axes).

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The most unconventional and interesting part of Owen & Owen’s results is the region in which the better defended species loses by the presence of the less defended one (i.e. between A and C in Fig. 1b). Significant properties of this region for the evolution of mimicry are: (1) the range of Mimic densities over which Model loses and (2) the extent of Model loss, as defined by the height of the cusp (B). To examine how these components of the Owen–Owen Effect are affected by variations in psychological and ecological variables, I performed a limited set of explorations of ‘parameter space’. Here I explore the significance of three parameters on the shape of the Owen– Owen Effect: (1) the effect of variations in the values of Mimic defence as measured only by variations in the learning parameter, ëMi; (2) the effect of variations in the values of Mimic defence as measured only by variations in the forgetting parameter, áfmi; and (3) the effect of variations in density of the more noxious Model species, ÐMo. To examine the first two, all Model parameters are kept constant and Mimic varies only in density and ëMi or áMi as appropriate.

0.785

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B Low model density

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Size of Mimic population

Figure 2. Each curve is a plot of the proportion of the Model–Mimic pair attacked versus density of Mimic species. Conditions between curves vary. (a) Density and level of protection of Model remain constant, as in Fig. 1. Within curves, Mimic defence levels remain constant and density increases. Between curves, Mimic defence levels vary, by manipulation of λMi only. Protection of Mimic increases from top down: λMi =0.4, 0.3, 0.25, 0.21; αfMi =0.9 for all Mimic prey. (b) Density and level of protection of Model remain constant, as in Fig. 1. Within curves Mimic defence levels remain constant and density increases. Between curves, Mimic defence levels vary, by manipulation of αfMi only. Protection of Mimic increases from top down: αfMi =1, 0.5, 0.4, 0.3; λMi =0.3 for Mimic prey. (c) Level of protection of Model remains constant (same value as Fig. 1). Within curves Mimic defence levels remain constant and its density increases. Between curves, Model varies in density, being least common at the top. XMo (equation 1) top down: 0.1, 0.5, 1, 2. For Mimic, λMi =0.3, αfMi =0.8.

SPEED: MEMORY AND MIMICRY 207

Results

Summary Owen & Owen’s original results (Owen & Owen 1984) were replicated and extended. The dilution of protection is highest with large discrepancies in levels of noxiousness of mimetic prey (as measured by learning asymptote and forgetting rate) and when the better defended of a pair is relatively common. THE ROLE OF MEMORY IN THE OWEN–OWEN EFFECT I used Owen & Owen’s basic predator to look at the importance of memory in mimicry dynamics in two ways: in the first, forgetting rates are held constant for Model and Mimic, thereby introducing the constant forgetting rate of Turner et al.’s virtual predator (1984; Speed 1993a). In the second, learning asymptotes for Model and Mimic are equal (at zero). I used this to test whether the ‘classical’ results of Turner et al. (1984) are robust against inclusion of varied forgetting rates.

Constant Forgetting Rates When forgetting rates are invariant for all prey (á=0.3), the Owen–Owen Effect is not seen (Fig. 3). There appears to be a knife-edge of density neutrality, at which the existence of the less noxious mimic makes no difference to attack rates at any density in the range tested (middle line, Fig. 3). Rounding-up errors appear to cause ‘noise’ in the result, and a smooth horizontal

0.597484276729561 Positive density– dependent attack rate

Proportion attacked

In Fig. 2a, Mimic varies in noxious qualities only in terms of ë. Mimic becomes better defended (and hence ë decreases) from the top curve downward. The range of Mimic densities over which Model loses, and the extent of that loss, is largest with the least defended prey (top lines) and becomes smaller as Mimic becomes more noxious. In Fig. 2b, Mimic becomes better defended, in terms of a decreased forgetting rate only, as the curves are descended. Model loss from Mimic’s presence is again greatest with the least defended Mimics (top lines). Other factors being equal, then, the Owen–Owen Effect is greatest when the discrepancy between forgetting rates of mimetic prey is high. Note that when Mimic-inspired forgetting is very slow (and equal to that of Model) as with the bottom curve, the cusp, B, is not seen. In Fig. 2c, Model and Mimic defences are again unequal, but have constant values. Mimic increases in density within curves and Model becomes more common as the curves are descended. The region of Model loss and the height of the cusp both increase with Model density, such that greatest loss is seen in the bottom curve where Model is relatively common. If Model is sufficiently rare, as it is in the top curve, the positive density-dependent part of the curve is not seen with the values of Mimic density used.

'Neutral' density– dependent attack rate

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Figure 3. Each curve is a plot of the proportion of mimetic prey attacked versus density of a Mimic species. Here Model is constant in level of defence and density (values as in Fig. 1) and Mimic varies in acceptability between lines. Note that forgetting is constant for all prey (αfMi =αfMo =0.3), but λMi varies. Top line λMi =0.32631578947369, middle line λMi =0.326315789473685, bottom line λMi =0.32631578947368. All three lines appear to be corrupted by rounding-up errors. The middle line shows a neutral effect of Mimic density on attack rate.

line cannot be generated within the resolution of the standard PC (Dell, optiplex GL +575). In the conditions used, the qualitative nature of the predictions of the Owen & Owen predator is extremely sensitive to prey values. Variations in ëMi of only plus or minus 510
15 make attack rate respectively positively or negatively dependent on density (top and bottom lines in Fig. 3). Variations greater than this generate increasingly smooth curves.

Variable Forgetting Rates The ‘classical virtual predator’ of Turner et al. (1984) generated entirely conventional Batesian and Mu ¨ llerian mimicries with conventional monotonic density dynamics. This virtual predator varied rates of learning with different prey, such that learning rates with moderately defended prey were very slow and with highly noxious prey were high enough to facilitate single-trial learning. However, the Turner et al. (1984) virtual predator always tended to increase avoidance of any defended prey when sampled (i.e. it had an asymptotic attack rate for all defended prey of zero). The predator’s forgetting rates were invariant for all prey types. This virtual predator is important and influential (see Endler 1991). Although its learning algorithms have been critically evaluated

208 ANIMAL BEHAVIOUR, 57, 1

THE ROLE OF LEARNING IN THE OWEN–OWEN EFFECT

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Figure 4. Plots of the proportion of mimetic prey attacked versus density of a Mimic species. Again, Model is constant in level of defence and density and Mimic increases in density. Here though in terms of learning, both Model and Mimic are equally protected (λfMi =λfMo =0) but in terms of forgetting rate the two can vary in protection. Model forgetting, αfMo =0.2. For Mimic, from top down, αfMi =0.99, 0.8, 0.5, 0.2.

elsewhere (Speed 1993a), there has been no test to see whether the traditional predictions this predator makes are robust against variations in forgetting rates. In generating the graphs in Fig. 4, I have assumed that ë=0 for both Mimic and Model, but that the Mimic species, being the least noxious, may be forgotten about more quickly than the Model. This may be a reasonable, although not exact introduction of Owen & Owen’s forgetting rule into the predator of Turner et al. (1984). With fixed learning asymptotes, but greater forgetting rates for the less defended of a pair of mimics, the Owen–Owen Effect can be generated (Fig. 4). However, the conventional Mu ¨ llerian density dependence results when both learning asymptotes and forgetting rates are equal (bottom line, Fig. 4).

Summary The results in this section show that: (1) if Owen & Owen’s original predator is modified such that all forgetting rates are fixed, it no longer generates the Owen–Owen Effect; (2) the Owen–Owen Effect can be generated with variable forgetting, even if the learning asymptote is invariant between mimetic prey; (3) a simple version of Turner et al.’s (1984) classical Batesian– Mu ¨ llerian predator will not produce traditional monotonic Mu ¨ llerian mimicry if forgetting rates are variable with prey acceptability.

To examine whether learning rate is an important parameter I carried out a number of stochastic simulations (see Speed 1993a for details). I considered virtual predators that follow Owen & Owen’s (and Speed’s 1993a) use of ë, but that use learning and forgetting rates in different ways (see Table 1). I constructed three learning rate rules and four forgetting rate rules. Both fixed and variable rates of learning and forgetting are represented. Fixed rate rules are represented with high and low values. I used Owen & Owen’s original forgetting rate rule, in which the noxiousness of the last prey attacked sets the forgetting rate. A ‘sophisticated variant’ is also included, which takes its forgetting rate from an exponentially weighted moving average of previous experience (cf. Pearce et al. 1982). I used this to investigate whether variable forgetting rules other than Owen & Owen’s produce the Owen–Owen Effect. I used all possible permutations of learning and forgetting algorithms generating 12 different ‘virtual predators’. Each of these virtual predators meets six different sets of conditions of Model defence and density (details of simulation are given in the Appendix). For each virtual predator a Turbo-Pascal program generates points A and C in Fig. 1b. Point C lies at a Mimic density of 40% in all simulations (i.e. Mimics are seen in 40% of time intervals). The program then finds the highest point between A and C (the equivalent of B, if it exists) and finally point D. I used a two-way ANOVA with Tukey test for honest difference to analyse the arcsine square-root transformed results of simulations for each set of simulations with a virtual predator (with Model density and level of defence as independent variables). The Owen–Owen Effect is assumed to be generated only if, first, there is no significant difference between mean attack rates at A and C, and second, point B is significantly higher than both A and C. Points A and D are also compared to see if there is a difference between attack rates between extreme densities.

Results Table 1 summarizes the results. For each virtual predator with each prey condition, the table indicates whether the Owen–Owen Effect is generated and whether there is, overall, a difference between attack rates at extreme densities. A number of significant observations can be made here. (1) It is possible to replicate the Owen–Owen Effect with stochastic simulation, and with virtual predators with learning and/or forgetting algorithms different to Owen & Owen’s original. (2) The Owen–Owen Effect never turns up with fixed forgetting rate predators, but each virtual predator with variable forgetting produces it in at least two of the six prey conditions. This result is not affected by variations in the learning rate rule. (3) When it does occur the effect is most common with conditions of high Model defence. It

Forgetting rate

Low Moderate Moderate High High Low

Low Moderate Moderate High High

Low High Low High Low High

Low High Low High Low

D**

A*** A*** A*** A***

A***

A*** A*** A***

A***

High and fixed

High and fixed

High and fixed

High and fixed High

Low and fixed

High and fixed

Low High Low High Low

Low High Low High Low High

Low High Low High Low High

Low High Low High Low Sophisticated High variable

Variable

Forgetting rate

Level of model defence

***

Low Moderate Moderate High High

Low Moderate Moderate High High Low

Low *** Moderate *** Moderate *** High High Low

Low Moderate ** Moderate High High Low ***

Low

Model density

D*

D***

A*** A*** A*** A***

A***

A*** A*** A*** A*

A***

Forgetting rate

High

Level of model defence

Moderate Low and and fixed fixed

Moderate High and and fixed fixed

Low High Low High Low

Low High Low High Low High

Low High Low High Low High

Low High Low High Low Moderate Sophisticated High and fixed variable

Moderate Variable and fixed

Attack rate Owen– greatest at Owen A, D (*) or Learning Effect? no difference rate

***

Low Moderate Moderate High High

Low Moderate Moderate High High Low

Low Moderate Moderate High High Low

*** *** *** **

Low Moderate *** Moderate High High Low ***

Low

Model density

A*** A*** A*** A*** A** D*

A***

A*** A*** A*** A***

A***

Attack rate Owen– greatest at Owen A, D (*) or Effect? no difference

Asterisks in the Owen–Owen Effect columns indicate statistical confirmation of its presence. Asterisks in the ‘Attack-rate column’ indicate a significant difference between mean attack rate of points A and D. The letter before the asterisks indicates which is the higher of the two. Virtual predators are generated by stochastic simulation, programmed in Turbo-Pascal, as in Speed (1993a) and Turner & Speed (1996), with the exception that the naive attack rate is 1, instead of 0.5. For details of algorithms and values used, see Appendix. *P<0.05; **P<0.01; ***P<0.001.

Variable Low and fixed

Variable High and fixed

Low *** Moderate *** Moderate *** High High Low

***

Low High Low High Low High

Low

Low * Moderate *** Moderate High High Low ***

High

Model density

Attack rate Owen– greatest at Owen A, D (*) or Learning Effect? no difference rate

Low High Low High Low Variable Sophisticated High variable

Variable Variable

Learning rate

Level of model defence

Table 1. Results of tests for the Owen–Owen Effect using 12 computer-simulated virtual predators and six sets of prey conditions

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is also found much more often with rare and moderate than high Model densities. This is likely to be caused by point C being fixed at 40% Mimic density, which biases the results in this direction. (4) The sophisticated variable forget rule is superior to Owen & Owen ‘s original variable forget rule in generating the Owen–Owen Effect. (5) Fixed rate forgetting appears to produce density-independent attack rates for the parameters chosen (in only four cases of 24 is there a density effect in which mean attack rate at D>A; these are easily attributable to slightly high values for Mimic’s learning asymptote). GENERAL DISCUSSION Despite its absence in mimicry simulations of Turner et al. (1984) and Speed (1993a), an up–down densitydependent attack rate, here called the Owen–Owen Effect, can be replicated, using both Owen & Owen’s (1984) equations and also stochastic simulation. With Owen & Owen’s equation, learning asymptotes, forgetting rates and the density of the better defended mimic can all affect the shape of the curve, moving the position of its cusp. The effect seems robust against variations in assumptions about learning rate, but instead is very sensitive to assumptions about forgetting. If forgetting rates vary with prey defences then the Owen–Owen Effect can be produced. Significantly, the virtual predator used to support the traditional view of mimicry (Turner et al. 1984) will produce the Owen–Owen Effect if its forgetting algorithms are modified to include variable forgetting rates. In simulation the effect is enhanced if forgetting varies as a moving average of encountered values of the degree of defence of Mimic and Model.

Causes and Evolutionary Consequences of the Owen–Owen Effect When Mimic is absent the overall attack rate on Model is determined by (1) learning events, which lower attack probabilities, and (2) forgetting processes, which subsequently raise them. There is, roughly speaking then, an equilibrium attack rate which results from the opposed actions of learning and forgetting. A predator with fixed rate forgetting generates ‘traditional’ monotonic density functions by the effects of learning events on this equilibrium. Growth in the population of a less well-defended mimic can raise the equilibrium, and thus attack rates, if its learning asymptote is sufficiently high. On the other hand, if its defences are stronger it can lower overall attack rates by offsetting forgetting (Fig. 3; see Speed 1993a). However, the characteristic rise and fall of attack rates in the Owen–Owen Effect can only be explained through both the influence of learning and the influence of variable forgetting rates on attack probabilities. A rise in attack rates follows the introduction of the less welldefended Mimic (A to B, Fig. 1b). This happens because encounters with Mimic cause increases in forgetting rate which raise the position of the equilibrium. When more common, however, the same mimics can add net protection by providing sufficient learning encounters to offset

predator forgetting. Hence the cusped shape of the curve results because prey affect both forgetting and learning processes. The implications for the evolution of mimicry of the Owen–Owen Effect were covered, of course, by Owen & Owen themselves (Owen & Owen 1984); but in summary, it can be seen that quasi-Batesian mimicry (see Speed 1993a) is predicted up to the cusp of the curve (Fig. 1b) and this might be a cause of polymorphism in moderately defended mimics (see Speed 1993a; Turner 1995). Caution should be exercised though, as the results are essentially qualitative. The extent of any such dilution cannot at present be known. What can be said from the results is that the effect will be most important if the most noxious species is thoroughly nasty and/or relatively common and there is a large discrepancy in forgetting rates inspired by unequally defended mimetic species.

The Role of Virtual Predators in Mimicry Studies Virtual predator studies have usually taken the careful strategy of varying one or two components of predator behaviour and of prey ecology in order to isolate significant factors in the evolution of warning colour and mimicry. They have been especially useful in helping to formalize verbal arguments in mimicry studies (e.g. Batesian–Mu ¨ llerian differences across the ‘palatability’ spectrum: Turner et al. 1984; arms races and imperfect mimicry: Turner 1987). Of equal significance is their use in helping to crystallize key questions about predator behaviour, such as whether predators forget over time or with events (Huheey 1976, 1980, 1988; Turner 1987), whether asymptotes of learning are zero for all defended prey (e.g. Owen & Owen 1984; Turner et al. 1984; Speed 1993a; Turner 1995 for a review), whether recognition errors are important (MacDougall & Dawkins 1998) and here, whether forgetting rates vary with the intensity of an unconditioned stimulus. However, the results shown in this paper do indicate limitations to virtual predator studies. An important, traditional prediction of monotonic density dependence is shown not to be robust against quite minor modifications in assumptions about one aspect of predator psychology. Perhaps then we should treat with caution the results of these sorts of investigations until the significance of variations in a greater range of behavioural parameters has been explored. Such parameters might include: hunger levels, abundance of alternative prey, precision of mimicry, effects of prey defences on predator physiologies, variations in predator tolerances and variation in rates of detoxification. A major problem for mimicry theory remains the difficulty of obtaining good data about psychology and responses of captive or wild predators to their prey (but see Chai 1986 and Pinheiro 1996 for exceptions). Until such data are available, the construction of virtual predators (whether by verbal argument, mathematics or programming) will always include educated speculation. Mimicry theory thus has less sure foundations than is often acknowledged.

SPEED: MEMORY AND MIMICRY 211

Memory Dynamics and Mimicry Memory is an understudied component of the psychological landscape (Guilford & Dawkins 1991) and its importance in defining predator responses to perceived prey defences is usually ignored. Indeed some virtual predators do not even include forgetting as a parameter (see e.g. MacDougall & Dawkins 1998) which makes interpretation of their predictions difficult. Owen & Owen (1984) did not justify their use of the variable forgetting algorithm and data from animal psychology on this matter are equivocal at present (M. E. Bouton, personal communication). On the one hand there is evidence that memories for extinction are more labile than those for original aversive conditioning (Bouton 1993, 1994). However, this seems to be because extinction is a ‘second memory’ and not because the intensity of the unconditioned stimulus (US) in extinction is lower than in original conditioning (M. E. Bouton, personal communication). Roper & Redston (1987) have shown that variations in conditioned stimulus intensity affect memory for aversive feeding events in chicks. However, there is no systematic data in the literature that indicate the effect of varied US intensity on memory. This may be in part because forgetting rates are difficult to measure in captive animals. However, testing for the Owen– Owen Effect might provide a good alternative way of investigating effects of US intensity on forgetting rates. Whether forgetting is an actual deactivation of memories or merely opportunistic reversal of learnt behaviour, it is clearly an adapted phenotype (see Bouton 1994 for a discussion and Spear et al. 1990; Bouton 1993 for other reviews of recent memory work). Forgetting, broadly, allows animals to modify behaviour to take into account changes to their circumstances caused by the passage of time (e.g. Bouton 1994). Within an adaptive paradigm it seems reasonable to speculate that in general, forgetting rates might vary to match the ecological variability and importance of events in a manner suggested by Owen & Owen (see e.g. Arnold 1978; Macnamara & Houston 1987; Bouton 1994 for theoretical arguments). It is easy to imagine that predators whose forgetting rates are sensitive to the intensity and temporal arrangement of stimuli will be better adapted to their feeding ecology than those without such a function. At present, however, this remains an important question for animal psychology and mimicry studies. Acknowledgments I thank Mark Bouton and John Brinkman for very helpful advice; John Turner, Jim Mallet and two referees for useful and interesting comments on the manuscript; the Liverpool Hope Research Fund for financial support and the Year 3 Behavioural Ecology Class of 1997 at Liverpool Hope for stimulating and enjoyable discussions about problems in mimicry theory. References Arnold, S. J. 1978. The evolution of a special class of modifiable behaviours in relation to environmental pattern. American Naturalist, 112, 415–427.

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Appendix Stochastic simulation: for details see Speed (1993a) and Turner & Speed (1996). Simulations use 12 replications of 8000 time intervals.

Learning algorithms Changes to attack rate from learning are determined according to the equation, ÄP=ál(ë
P )

(A1)

ál is a learning rate parameter, ë is the learning asymptote and P is attack rate before encounter with a prey (other symbols as in equation 1 in the text).

Learning rules Variable ál changes after each attack and is calculated according to the equation: ál =0.5+(1
ë)/2

(A2)

which allows it to vary between 0.5 for palatable prey (ë=1) and 1 for truly noxious specimens (ë=0). High and fixed ál =1

(A3)

Thus learning is complete in a single trial for all prey (this is Owen & Owen’s learning rate rule). Moderate and fixed ál =0.5 Learning is thus gradual for all prey.

(A4)

SPEED: MEMORY AND MIMICRY 213

High and fixed

Forgetting algorithms Changes to attack rate caused by forgetting are determined according to the equation, ÄP=áf(1
P )

(A5)

áf =0.2

(A8)

thus forgetting is gradual and constant. Low and fixed áf =0.02

(A9)

áf is a forgetting rate parameter, and P is attack rate before encounter with a prey (other symbols as in equation 1 in the text).

thus forgetting is gradual and constant, but at a considerably lower rate.

Forgetting rules

Values for Model parameters

Variable áf changes after each attack according to the equation:

Parameters of Model used in the simulations are reported in Table A1. Density values refer to the three conditions of Model density used. Density is measured as the proportion of all prey in a virtual predator’s locality. All six possible sets of conditions of Model protection and density are used.

áf =ë

(A6)

where ë refers to the learning asymptote of the last prey attacked. Sophisticated variable áf changes after each attack according to the equation: áf =(áf 0.5)+(ë 0.5)

Table A1.

(A7)

where áf refers to the value of ál immediately prior to the attack and ë refers to learning asymptote of the prey just attacked. This algorithm generates an exponentially weighted moving average of ë values, which is then used as the forgetting rate parameter.

Model protection

λMo

Low High

0.3 0.05

Model density values

Low Moderate High

0.05 0.1 0.3