Multi-behavioral strategies in a predator�prey game: an evolutionaryalgorithm analysis

William A. Mitchell

W. A. Mitchell (wmitchell@indstate.edu), Dept of Biology, Indiana State Univ., Terre Haute, IN 47809, USA.

Behavioral games between predators and prey often involve two sub-games: ‘pre-encounter’ games affecting the rate ofencounter between predators and prey (e.g. predator�prey space games, Sih 2005), and ‘post-encounter’ games thatinfluence the outcome of encounters (e.g. waiting games at prey refugia, Hugie 2003, and games of vigilance, Brownet al. 1999). Most models, however, focus on only one or the other of these two sub-games.

I investigated a multi-behavioral game between predators and prey that integrated both pre-encounter and post-encounter behaviors. These behaviors included landscape-scale movements by predators and prey, a type of prey vigilancethat increases immediately after an encounter and then decays over time (‘ratcheting vigilance’), and predatormanagement of prey vigilance. I analyzed the game using a computer-based evolutionary algorithm. This algorithmembedded an individual-based model of ecological interactions within a dynamic adaptive process of mutation andselection. I investigated how evolutionarily stable strategies (ESS) varied with the predators’ learning ability, killingefficiency, density and rate of movement. I found that when predators learn prey location, random prey movement can bean ESS. Increased predator killing efficiency reduced prey movement, but only if the rate of predator movement was low.Predators countered ratcheting vigilance by delaying their follow-up attacks; however, this delay was reduced in thepresence of additional predators. The interdependence of pre-and post-encounter behaviors revealed by the evolutionaryalgorithm suggests an intricate co-evolution of multi-behavioral predator�prey behavioral strategies.

Predators and prey interact in behavioral evolutionarygames in which predators attempt to locate and kill preywhile prey attempt to avoid discovery and survive attack(Sih 1998, Brown et al. 1999, Kotler et al. 2002, Wolf andMangel 2007). Conceptually, a predator�prey game can besubdivided into two sub-games: a ‘pre-encounter’ game anda ‘post-encounter’ game. In the pre-encounter game,individuals attempt to manipulate their rate of encounterwith the other species. An example of the pre-encountergame is predator�prey habitat selection (Bouskila 2001,Alonzo 2002, Sih 2005, Roth and Lima 2007). In the post-encounter game, individuals attempt to manipulate theoutcome of an encounter; predators benefit from a killwhile prey obviously benefit from surviving the encounter.Examples of a post-encounter games include the ‘waitinggame’ between a predator and a refuging prey (Hugie 2003,2004), and the game in which prey vary vigilance based ontheir expectation of attack, while predators try to managesuch vigilance (Brown et al. 1999, Lima 2002). Mostmodels of predator�prey games focus on either the pre-encounter or post-encounter sub-game. But animals them-selves must deal with both, employing multi-behavioralstrategies that influence both the rate and outcome ofencounters. Here, I provide a novel, theoretical analysis of apredator�prey behavioral game that jointly considers suchmulti-behavioral strategies.

While it may simplify analyses to model pre- and post-encounter games independently, their behaviors may in factbe interdependent. For example, the optimal level of preyvigilance should depend on the encounter rate withpredators, which may in turn depend on pre-encounterbehaviors such as habitat selection (Liley and Creel 2008).Conversely, predators should select habitats on the basis ofprofitability (Quinn and Cresswell 2004), which maydepend on post-encounter behaviors such as prey vigilance.In another example, Lima (2002) suggested that predatorsmay manage prey behavior, such as vigilance, by spreadingout their attacks (encounters) across many prey on alandscape (Roth and Lima 2007). Given the interdepen-dence of pre- and post-encounter behaviors, it is surprisingthat they have not been considered jointly in previousmodels.

Mitchell and Lima (2002) investigated both pre- andpost-encounter behaviors in a simulation study designed tounderstand why some prey make frequent and randomlandscape-scale movements. The question of why preymove is an interesting one because movement costs energyand potentially increases predation risk (Norrdahl andKorpimaki 1998): movement may attract a visual predator’sattention, and a prey on the move may blunder into apredator. Mitchell and Lima’s individual based model(IBM) simulation showed that random prey movement

Oikos 118: 1073�1083, 2009

doi: 10.1111/j.1600-0706.2009.17204.x,

# 2009 The Authors. Journal compilation # 2009 Oikos

Subject Editor: Kenneth Schmidt. Accepted 15 January 2009

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may be part of a larger predator�prey ‘shell game’ in whichpredators can learn prey location while prey attempt to beunpredictable in time and space. These are pre-encounterbehaviors because they influence the rate of encountersbetween predators and prey. Mitchell and Lima’s (2002)IBM also considered the post-encounter behavior of preyvigilance. Specifically, the model considered ‘ratcheting’vigilance, defined as vigilance which could increase after anencounter with a predator, and then decay over time untilthe next encounter; ratcheting vigilance benefited preybecause they were able to increase vigilance when risk washigh and increase foraging when risk was low. Results of theIBM showed that prey using both strategies � randomizedmovement and ratcheting vigilance � survived better thanprey using only one anti-predator behavior.

While the IBM analysis of Mitchell and Lima (Mitchelland Lima 2002) suggests that prey movement and ratchet-ing vigilance behaviors could benefit prey, it could not byitself answer the important question of whether these multi-behavioral strategies of predators and prey could becoevolved evolutionarily stable strategies. Nor did theIBM provide the predator with a way to respond to or‘manage’ (sensu Lima 2002) the prey’s ratcheting vigilance.For example the predators’ best response to ratchetingvigilance may be to avoid rapid follow-up attacks on preythat ratchet their vigilance to higher levels.

Here, I provide a coevolutionary analysis of multiple,pre- and post-encounter behaviors influencing encounterrates and the outcomes of encounters between predatorsand prey. Due to the complexity of the behavioralinteractions, I used a computer-based evolutionary algo-rithm (Mitchell 1999, Eiben and Smith 2003, Ruxton andBeauchamp 2008). This algorithm embedded an indivi-dual-based model (IBM) within a dynamic adaptive processto solve for predator and prey multi-behavioral strategies.The goals of this study were to determine how thecoevolved, multi-behavioral strategies of predators andprey vary with parameters of the predator�prey interaction.The prey strategies I considered were between-patch move-ments and ratcheting vigilance, while the predator strategiesinvolved learning prey location, and scheduling attacks tomitigate the effect of ratcheting vigilance by prey. Iconsidered how these prey and predator behaviors variedwith the following attributes of predators: 1) ability to learnthe location of prey, 2) killing efficiency, 3) density, and 4)rate of movement.

The model

My model involved multiple, frequency-dependent beha-viors which varied probabilistically throughout an indivi-dual’s lifetime. The complexity of the model precludedstandard analytical techniques for finding evolutionarilystable strategies (Brown et al. 1999, Mitchell 2000). I chosetherefore to analyze the model using a computer-basedevolutionary algorithm (Eiben and Smith 2003). Myevolutionary algorithm (EA) described a dynamic adaptiveprocess that used an individual-based model (IBM) ofecological interactions to determine the fitness consequencesof behaviors. After each generation, surviving individualsreproduced subject to mutation of their strategies. Because of

the probabilistic nature of mutation and the outcome ofinteractions, I ran the IBM simultaneously on 400 land-scapes, containing a total of 12 000 prey and 400 to 1200predators to ensure that I obtained evolutionarily stablevalues. The evolutionary algorithm always yielded evolutio-narily stable values. I do not claim that the IBM and the EAfaithfully represent the details of actual predator�preyinteractions or biological evolution, respectively. I dohowever assume that the results provide meaningful,qualitative insights into coevolved predator�prey behavioralgames.

The individual-based model

The scenario modeled by the individual-based modelroughly corresponds to the ‘small-bird in winter’ paradigm,featuring Accipiter hawks and their avian prey (Lima 2002,Roth et al. 2006). In this paradigm, the fitness of the prey(small birds) is given by the probability of surviving theirnon-breeding season by avoiding both starvation andpredation, while the fitness of the predators (hawks)increases with the number of prey they consume. Typically,prey must manage a tradeoff between increased foraging(reduced starvation) and increased anti-predator behavior(reduced predation). Both prey and predators can makelandscape-scale movements, but predators typically move ata larger spatial scale than do prey and a single predator’sforaging area may encompass the non-overlapping foragingranges of multiple prey (Lima 2002).

All predator�prey behaviors and interactions took placeon a landscape containing 150 patches. The patches differedonly in foraging benefit available to prey, with ‘rich’ patchesproviding twice the foraging benefit of ‘poor’ patches. Allpatches were non-depleting in food, so prey experienced nodiminishing returns while foraging a patch. Rich patcheswere dispersed among poor patches such that a rich patchwas not neighbored by another rich patch in the same rowor column.

Prey

At the start of each generation, 12 000 prey were randomlydivided among 400 landscapes, resulting in 30 prey perlandscape. Each prey was assigned a randomly selectedhome range of three neighboring patches, one rich and twopoor. Home ranges of different prey were not permitted tooverlap. A prey moved among its three patches according toits unique three-element movement vector, each element ofwhich gave the probability of the prey moving to andoccupying a particular patch. I represent the prey’s move-ment vector as p�[p1, p2, p3], indexing the single richpatch in the home range by the subscript 1, so that amovement vector of [1, 0, 0] resulted in the prey spendingall of its time in the rich patch, while a movement vector of[1/3, 1/3, 1/3] resulted in the prey spending approximatelyequal time in all three home patches. Because the rich patchcontained twice the resources of the other two patches, aprey using the first movement vector could consume moreenergy over the course of a day.

Days were divided into ten periods, and prey could moveonce per period, while predators could move one, two or

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three times per period, depending on the experiment.Because predators and prey moved sequentially, rather thansimultaneously, each period was further divided into two ormore intervals demarcated by the times at which predatorsmoved. Within each interval prey allocated their timebetween foraging and vigilance. Prey vigilance increased ordecayed between intervals depending on whether or not apredator had been encountered in the previous timeinterval. This ‘ratcheting’ vigilance depended on threeparameters: vigBase was the base level of vigilance usedeven when a prey had not recently encountered a predator;vigMax was the maximum level of vigilance and was usedimmediately after an encounter with a predator; andvigMult was a multiplier that reduced vigilance if a predatorhad not been encountered in the previous time interval. Allthree vigilance parameters were constrained to lie in theinterval [0,1]. These three parameters modified vigilancelevel as follows. If a prey encountered a predator during atime interval, t, then the vigilance in the following timeinterval equaled the maximum vigilance.

vig(t�1)�vigMax (05vigMax51) (1a)

However, if the prey did not encounter a predator during t,then vigilance was reduced to either a proportion, vigMult,of the previous value, or vigBase, whichever was larger.

vig(t�1)�Max fvig(t)�vigMult; vigBaseg (1b)

Thus vigilance decayed to vigBase unless the prey encoun-tered another predator first.

Prey consumed energy at a rate that depended on thequality of patch in which it resided (rich or poor) and theproportion of time that it spent foraging as opposed tovigilant. Prey expended energy at a constant rate whetherforaging or vigilant. Thus, the change in energy state fromone time interval to the next was described by,

State(t�1) � State(t)�PatchValue(k)�(1�vig(t))�TI�metCost�TI (2)

where PatchValue(k) was the rate of energy intake by aforaging prey in patch k; this value equaled ten for richpatches and five for poor patches. The metabolic cost(metCost) equaled one for these simulations. TI was thelength of the time intervals, which depended on thefrequency of times the predator moved, and potentiallyattacked, each period. For example, if the predator movedonce per period, then TI equaled one half of a period,whereas if the predator moved twice each period, then TIequaled one third of a period. While I assume that anunsuccessful predator attack did not injure prey, Eq. 2indicates that even an unsuccessful predator attack maynegatively impact a prey by causing the prey to increase itsvigilance and thereby reduce its energy intake.

At the end of each day, each prey was subjected to aprobability of starvation that decreased exponentially withincreasing energy state.

Pr(starvation j State(10))�exp (�v�State(T)) (3)

The constant, v, represented the survival value of energy,which equaled 0.1 for all of the simulations I report here. Ialso made the conservative assumption that a prey beganeach day with an energy reserve of zero; that is to say, energyis only used for maintenance and does not carry over fromone day to the next. Thus prey could not increase (ordecrease) fat reserves across days. In order to survive to theend of one generation, a prey had to avoid both predationand starvation during 10 day/night cycles.

Predators

Depending on the experiment, from 400 to 1200 predatorswere randomly divided among the 400 landscapes, with oneto three predators per landscape. Unlike prey, predatorscould move over the entire 150 patch landscape. I made thesimplifying assumption that that the time required for apredator to move between patches and hunt within the newlyoccupied patch was independent of the virtual distanceseparating patches. A predator’s movement was determinedby its unique 150 element movement vector, in which eachelement gave the probability of moving to a particular patch.The predator calculated its movement vector just prior tomoving, based on its information about prey location.

Predators stored information about the location of preyin a separate vector of patch weights. At the start of eachgeneration, predators had no information about preylocation, so all patch weights were equal.

wi(t)�1=150 for all i in (0; 1; :: 150); and for t�0

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After hunting in a patch, the predator could decrease theweight of the patch if it contained no prey, or increase theweight if a prey were present. There were two cases in whicha patch could have no prey subsequent to the predatorhunting there: 1) the predator did not encounter a prey, or2) the predator encountered a prey, but then successfullyattacked and killed it. Both cases resulted in the predatorreducing patch weight according to a simple linear operator,

wi(t�1)�(1�l) wi(t) (5a)

where l was constrained to lie in the interval [0,1]. But ifthe predator encountered a prey in patch i and failed to killit, the patch weight could increase.

wi(t�1) � wi(t) (1�l)�l (5b)

I refer to l as the predator’s learning parameter, and itreflects the importance given by the predator to recentexperience (Beauchamp 2000). For these simulations, Iassumed that the predator only had the ability to learn preylocation at the spatial scale of the patch, not at the largerscale of the prey’s home range (or at multiple spatial scales;e.g. patch and prey home range).

The vector of patch weights reflected the predator’schanging assessment of prey locations. But it did notaccount for the effect of the prey’s ratcheting vigilance on

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the predator’s hunting success. For this, the predatorcalculated a ‘temporary’ patch weight for the patch mostrecently visited. The temporary patch weight reflected ashort-term modification of the patch’s weight based on thepredator’s immediate past experience. If the predator didnot succeed in killing a prey it has just attacked, thetemporary weight of that patch equaled the previous patchweight reduced by 1�d, where d was constrained to theinterval [0,1]. So, if a prey were present in patch i, thetemporary patch weight was,

vi�wi (1�d) (6a)

If there were no prey in the patch the temporary weightequaled the original patch weight.

vi�wi (6b)

I refer to d as the predator’s ‘delay’ parameter because itcould delay the return of a predator to a patch where it hadjust unsuccessfully attacked a prey. The delay parameteradjusted the temporary patch weight for a single move. Thistemporary patch weight was then used by the predator tocalculate its movement vector by normalizing its vector ofpatch weights, including the temporary patch weight, sothat the elements summed to one. The delay parametertakes advantage of the prey’s reduction in vigilance,allowing the predator to attack when the prey is morevulnerable.

A predator’s consumption of prey depended on theprobability of encountering prey, and the outcome of thatencounter, i.e. the probability that it killed an encounteredprey. I assumed that a predator attack would fail if the preywere vigilant during the attack, and succeed with aprobability equal to the predator’s killing efficiency para-meter (KEP) if the prey were not vigilant. Thus, theprobability that the predator killed an encountered preyequaled the proportion of time the prey was not vigilantmultiplied by the KEP,

Pr(Kill)�(1�Vig(t))�KEP (7)

Sequence of actions

Predators and prey moved sequentially rather than simulta-neously. This meant that there were two different ways thatpredators and prey could encounter one another. A predatorcould enter a patch that already contained a prey, or a preycould ‘blunder’ into a patch that already contained apredator. Thus sequential movement imposed a predationcost on prey that simultaneous movement would not. Aftereach opportunity for movement by either prey or predator,each predator scanned the patch in which it resided,attacking any prey contained therein, and killing the preywith a probability given Eq. 7. If the attack was successful,the predator’s state increased by one, and the predator

reduced the patch weight according to Eq. 5a. If the attackwas unsuccessful, the predator increased the patch weightaccording to Eq. 5b, and then used the delay parameter totemporarily discount the patch weight according to Eq. 6a.If the patch scanned by the predator contained no prey,then the predator decreased the patch weight according toEq. 5a. At the end of each ten day generation, survivingprey and predators reproduced, subject to mutation,according to the evolutionary algorithm as described inthe following section.

Evolutionary algorithm

A total of seven variables coevolved in the evolutionaryalgorithm. Two of these were variables determiningpredator behavior: predator learning (l) and delay (d).The other five variables determined prey behavior: the firsttwo elements of the movement vector (p1 and p2) � thethird element, p3, always equaled 1.0�p1�p2, because theprey always moved to one of the three patches � and thethree variables determining ratcheting vigilance (vigBase,vigMax and vigMult). In general, evolving parameters inevolutionary algorithms may be encoded as either binarystrings or continuous variables (Eiben and Smith 2003). Iused continuous coding because my biological variables ofinterest were inherently continuous (e.g. the probability ofprey moving to a patch, Table 1).

All individuals of a particular type (prey or predator)were initialized with identical values for their evolutionaryvariables. These variables changed values over the next 1000or more generations through a combination of selection andmutation. Selection of predators depended on their relativesuccess at capturing prey. At the end of each generation, Iranked predators from all landscapes by the number of preyconsumed and allowed the top 50% of the ranking tosurvive and reproduce. The prey that survived andreproduced were simply those that survived predation andstarvation. To avoid complications due to resource�con-sumer cycles (Abrams 1992), I allowed only enoughreproduction of randomly selected survivors to keep thenumber of predators and initial number of prey constantacross all generations.

Before individuals reproduced, their evolutionary vari-ables mutated with a probability of 0.2. While thismutation rate is much higher than a natural rate, it resultedin equilibrium values for all evolutionary variables. When avariable mutated, its original value was modified by addinga random number drawn from a normal distribution with amean of zero and a standard deviation of 0.1, subject toboundary constraints that the parameter must lie in theclosed interval, [0,1]. Mutating the movement vector of theprey was slightly more involved because the three prob-abilities summed to one, resulting in two degrees offreedom, and hence two parameters. If the mutated move-ment vector violated the boundary constraint that allparameters must lie in the closed interval, [0,1], the valueswere rejected and a new vector was generated until onesatisfied the constraint.

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Trials

Predator learning

To determine how the evolutionarily stable prey behaviorsdepend on predator’s ability to learn location, I ran threeseparate trials of the evolutionary algorithm with threedifferent types of predators, two of which did not learn preylocation, while the third did. The first type of non-learningpredator had no information on prey location at thebeginning of a generation, and was unable to acquireinformation through experience; these ‘uninformed’ pre-dators moved randomly among patches in the landscape. Imodeled uninformed predators by setting their learningparameter and mutation rate equal to zero. The second typeof non-learning predator was ‘omniscient’ in that it hadperfect information regarding the location of all prey, andtherefore had no need to learn prey location. The patchweight vector of omniscient predator contained zeros forpatches that harbored no prey, and 1/(number of survivingprey) for patches that contained prey; the patch weightsupdated after every prey move and prey capture to perfectlyreflect prey distribution prior to each move by the predator.Consequently, the omniscient predator always moved to apatch that contained a prey, chosen randomly fromavailable surviving prey. The ‘learning’ predator beganeach generation with no information on prey location,but was able to learn prey location according to Eq. 5 and 6.I modeled the learning predator by initializing its learningparameter, l, to zero, but then letting that parameter evolveby setting its mutation rate to 0.2.

For all trials considering the effect of predator learning, Iset predator killing efficiency parameter (KEP) equal to 0.1,predator density to one predator per landscape and the rateof predator movement to one move per period. The results Ireport are the averaged behaviors taken over all landscapesand all 12 000 prey and 400 predators.

Predator killing efficiency, predator density and rateof predator movement

I examined the effect of predator killing efficiency (KEP) oncoevolved predator and prey behaviors by running theevolutionary algorithm for six values of KEP (0.1, 0.2, 0.4,0.6, 0.8, 1.0). I also examined the effect of predator densityand rate of predator movement. I ran all predator densities

(1, 2 and 3 predators per landscape) with all values of KEP,holding the rate of predator movement to once per period. Iran all predator speeds (1, 2 and 3 moves per period) withall values of KEP, holding the predator density to one perlandscape.

Results

Effect of predator learning

Both types of non-learning predators (uninformed andomniscient) selected for relatively sedentary prey that spentmost of their time in their rich food patch (p1:0.9;Fig. 1a�b). In contrast, learning predators selected for morerandomly moving prey that divided their time more evenlyamong their three patches (p1:0.4; Fig. 1c). AlthoughFig. 1 shows only the value the probability of moving toand occupying the rich patch (p1), the values of p2 and p3

were each essentially half of 1 � p1; i.e. the equilibrium preymovement vector was approximately [p1, p2, p3]:[0.9,0.05, 0.05] for non-learning predators, and [p1, p2, p3]:[0.4, 0.3, 0.3] for learning predators. The predators’learning parameter coevolved with prey movement, increas-ing to 0.8 when prey were initially sedentary, but thendeclining to 0.6 as prey evolved random movement.

Prey evolved ratcheting vigilance in response to bothnon-learning and learning predators, as indicated by the factthat the equilibrium value of vigMax exceeded that ofvigBase in all cases (Fig. 2a�c). The equilibrium took longerto achieve in the case of the uninformed predator, while thelowest equilibrium values of vigBase and vigMax evolved inresponse to learning predators. The learning predator wasthe only one that could evolve a delay parameter (Fig. 2d).The low equilibrium value (:0.2) of the learning pre-dator’s delay parameter suggests that, for the parameters ofthis trial (KEP�0.1, predator density�1/landscape, rateof movement�1 move period�1) predators benefited littleby delaying their return to patches in which they had justencountered prey.

I should note two points about the results presentedhere. First, additional runs of the evolutionary algorithmindicated that the equilibrium values for all evolutionaryvariables did not depend on their initial values. Second, theaveraged predator values were ‘noisier’ than averaged preyvalues; i.e. they showed more variation around the trend.This is simply the result of predator values being averagedover fewer individuals than prey values.

Because the results of the evolutionary algorithmpresented here (and below) show a strong effect of KEPon prey movement, I decided to run a separate (non-evolutionary) simulation that compared the cost and benefitof movement at low and high KEP, where the cost isblundering into a predator, and the benefit is the reductionof mortality due to reduced follow-up attacks. As I was onlyinterested in the effect of KEP, I held other evolutionaryvariables constant (p1�0.4, vigBase�0.2, vigMax�0.85,vigMult�0.8, learning parameter�0.6, delay para-meter�0.2). The results of this simulation showed that,as KEP increased from 0.1 to 1.0, the proportion of killsdue to blundering (vs follow-up encounters) increased from

Table 1. Definitions of symbols

Symbol Definition

vig Proportion of time prey spends vigilantvigBase Base vigilance of preyvigMax Maximum vigilance after predator encountervigMult Vigilance multiplier in the absence of a predator

encounterpi Probability that prey occupies home range patch iv Survival value of energy statewi Predator’s patch weight for patch id Predator’s delay parametervi Predator’s temporary patch weight for patch il Predator’s learning parameterKEP Predator’s killing efficiency parameter (lethality)

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0.57 (90.025) to 0.71 (90.023). Thus, the predation costof movement effectively increases with predator lethality.

The effect predator density and killing efficiency

In trials investigating the effects of predator density andkilling efficiency, evolutionary variables equilibrated before900 generations elapsed. I therefore report the average ofeach evolutionary variable calculated across all individuals(12 000 prey or 400 to 1200 predators), over generations900 to 1000.

As predator killing efficiency (KEP) increased, preybecame increasingly sedentary in their rich patch(Fig. 3a). KEP also modified the effect of predator densityon prey movement. At intermediate values of KEP,increased predator density resulted in prey spending lesstime in their rich patch and moving more. But when KEPwas either low (0.1) or high (1.0), predator density had noeffect on prey movement (Fig. 3a).

Both KEP and predator density had small positive effectson base vigilance (Fig. 3b). Neither KEP nor predatordensity had a substantial effect on maximum vigilance orthe vigilance multiplier (Fig. 3c�d).

The predator’s learning parameter increased with KEP(Fig. 3e). KEP also modified the effect of predator densityon the predator learning parameter, mirroring the interac-tion of these two factors on prey movement: increasedpredator density resulted in a decrease in the predatorlearning parameter, but only at intermediate values of KEP(Fig. 3e); when KEP was low (0.1) or high (1.0), predatordensity had no effect on the learning parameter. Thepredator’s delay parameter increased with KEP, anddecreased with predator density. Furthermore, the effectof KEP on the delay parameter was diminished at higherpredator density (Fig. 3f).

The effect of rate of predator movement and killingefficiency

Predator rate of movement and killing efficiency interactedin their effect on prey movement. As described in theprevious section, prey increase p1 (become more sedentary)as KEP increases; but this effect occurred only whenpredators moved less frequently (once per period). Whenpredators moved more frequently (two or three times perperiod), KEP had no effect on prey movement; i.e. at higher

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Figure 1. Panel (a) through (c) show the effect of predator learning on the evolution of p1, which represents the prey’s probability ofmoving to and occupying a rich food patch. Each line represents the average of p1 calculated over 12 000 prey distributed across 400landscapes. Panel (a) and (b) illustrate that both types of non-learning predators selected for prey that spend most of their time sedentaryin their rich food patch. The trial with ‘uninformed predators’ illustrated in (a) was run for more generations to allow the vigilancevariables to achieve equilibrium (shown in Fig. 2). (c) learning predators selected for prey that moved almost equally among patches; themovement vectors of these prey were initialized so that they were sedentary in their rich patch ([p1, p2, p3]�[1.0, 0.0, 0.0]). (d) evolutionof the predator’s learning parameter, which increased quickly in response to initially sedentary prey, and then decreased somewhat as preyused random movement. The plotted line represents the average of the learning parameter calculated over all 400 predators. Plots ofpredator parameters generally show more ‘noise’ due to the smaller number of predators than prey in the simulation. For all of these trials,the predator killing efficiency equaled 0.1, predator density equaled one per landscape, and the rate of predator movement was once perperiod.

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rates of predator movement, the prey moved randomlyregardless of the predator’s killing efficiency (Fig. 4a).Higher rates of predator movement also resulted in highervalues of all vigilance parameters (Fig. 4b�d).

Increasing KEP resulted in increased values of thepredator learning parameter, as described in the previoussection. But just as predator speed eliminated the effect ofKEP on prey movement, it also reduced the effect of KEPon the predator learning parameter (Fig. 4e). Higher ratesof predator movement also resulted in a larger predatordelay parameter; that is to say, rapidly moving predatorswere less likely than slow moving predators to attack thesame prey in consecutive time intervals (Fig. 4f).

Discussion

My analysis suggests that predators and prey co-evolveevolutionarily stable, multi-behavioral strategies comprisingboth pre- and post-encounter behaviors. Pre-encounterbehaviors influence the rate of encounter, while post-encounter behaviors influence the outcome of encounters.In the pre-encounter game modeled here, predators attemptto increase the rate of encounter by learning prey location,while prey attempt to decrease the rate by becomingunpredictable in space. This game is tantamount to the

predator�prey ‘shell game’ analyzed by Mitchell and Lima(2002), which relied on an individual-based model (IBM)to demonstrate its plausibility. The evolutionary algorithmanalyzed here confirms that shell game behaviors are co-evolutionarily stable, and furthermore, co-evolve with post-encounter behaviors.

The evolution of prey movement in the pre-encountergame depended critically on whether predators learn preylocation. For the prey, the benefit of moving is that it is lesslikely to incur follow-up attacks once it has been discoveredby a predator. This benefit can outweigh the costs of lostforaging opportunities and increased likelihood of blunder-ing into a predator. But if predators cannot, or do not needto learn prey location, then prey receive no benefit fromrandom movement and should not pay the cost in a non-depleting world. Prey tended to move less when predatorswere more lethal (higher KEP). Moving prey pay the cost ofincreasing their rate of first encounters (blundering), toobtain the benefit of reducing their rate of re-encounters, atleast when predators learn prey location. But if the predatoris highly lethal, prey may be unlikely to survive even theirfirst encounter. In this case, the benefit of reducing re-encounters should be less than the cost of increasing firstencounters, and prey should reduce movement.

In the post-encounter game, prey can increase theirchance of surviving an encounter through the use of

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Figure 2. Panel (a) through (c) illustrate the effect of predator learning on the evolution of vigilance parameters. Each line represents anaverage of the variable calculated over 12 000 prey distributed among 400 landscapes. Ratcheting vigilance evolved in the presence of bothtypes of non-learning predators, as well as learning predators. During early generations, base vigilance (vigBase) increased more rapidlythan maximum vigilance (vigMax), resulting in constant levels of vigilance (�vigBase). Eventually vigMax overtook vigBase, resulting inratcheting vigilance. The scale for generations in (a) is twice that in the other figures because ratcheting vigilance took longer to evolvewhen predators were ‘uninformed’ non-learners. Vigilance parameters (and vigilance) were highest in the case of omniscient predators (b),and lowest in the case of learning predators (c). Panel (d) illustrates that the stationary value of the learning predators’ delay parameterremained low, indicating that for the parameter values of this trial predators showed a small tendency to delay their follow-up attacks ondiscovered prey. For all of these trials, the predator killing efficiency equaled 0.1, predator density equaled one per landscape, and the rateof predator movement was once per period.

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ratcheting vigilance which increases or decreases dependingon whether or not a predator has recently been encountered.Similar behavior has been observed in Starlings whichincrease their vigilance in response to recent encounterswith predators (Devereux et al. 2006). The predators in mymodel respond to ratcheting vigilance by delaying follow-upattacks on a recently attacked prey; the predator huntselsewhere, and then returns to attack after a previouslyencountered prey has reduced its vigilance. Such ‘preymanagement’ (sensu Lima 2002) by predators may be acommon feature of widely ranging predators that huntdispersed prey, such as raptors that hunt flocks of small

wintering birds distributed over a large landscape (Roth andLima 2007).

The effect of predator density on prey movement wasnot simple or additive; it depended on the predator’s killingefficiency. Increasing predator density should increase boththe cost and benefit of prey movement: there are moreopportunities to blunder into a predator, and moreopportunities to avoid re-encounters. At high predatorkilling efficiency, the cost of first encounters apparentlyexceeded the benefit of avoiding re-encounters, even aspredator density increased, and prey remained relativelysedentary. At low lethality, the prey were already using their

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Figure 3. Plots of the equilibrium values of four prey variables and two predator variables for a range of predator killing efficiency (KEP)and predator density (the number of predators per landscape). Each point is the average over the last 100 generations (generations 900 to1000) across all individuals (12 000 prey, and 400, 800 or 1200 predators, depending predator density). For all of these trials, the rate ofpredator movement was once per period. (a) illustrates the non-additivity of predator killing efficiency and density on the probability (p1)that the prey occupies its rich patch during a given period (b) base vigilance increases modestly with KEP, and predator density. Panel (c)and (d) show small responses in maximum vigilance and the vigilance multiplier. (e) the non-additivity of predator killing efficiency anddensity on the predator’s learning parameter. The pattern in this figure appears to mirror the response of the prey variable, p1. As KEPincreases, prey become more sedentary, resulting in predators acquiring a larger learning parameter. (f) as KEP increases, the predator ismore likely to delay the timing of follow-up attacks; predators are less likely to delay re-attacking discovered prey when they share thelandscape with other predators.

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patches almost equally, and the additional benefit of aslightly more even distribution across patches was appar-ently small, as the prey did not vary their movement patternas predator density increased. But at intermediate lethalityprey used only moderate movement in the presence of onepredator, and thus they had ‘room to move’. These preyresponded to increased predator density with increasedmovement.

The predators’ rate of movement had an even moredramatic effect on prey movement than did predatordensity, in general increasing prey movement. Indeed, athigher prey speeds, there was essentially no longer an effectof predator killing efficiency on prey movement. This resultcan be understood in terms of movement’s costs (blunder-ing into a predator) and benefits (reducing re-encounters).As predator speed increases, relatively more encountersresult from the predator re-entering a patch where it learned

prey location. Thus as speed increased, benefit of movementincreased more than the cost. This effect was strong. Itnegated the tendency of increased predator killing efficiencyto reduce prey movement. Even at the highest predatorkilling efficiency, prey movement was selected for whenpredators moved even twice the rate of prey.

In general, the predator’s learning parameter coevolvedwith prey movement. The predator benefited most fromlearning prey position when the prey are sedentary.Consequently, factors that increased the probability of theprey occupying its rich patch also increased the predator’slearning parameter. For example, the predator’s learningparameter increased with increasing predator lethality (inslow moving predators) and predator density at intermedi-ate lethality.

Predator density influenced the post-encounter game. Aspredator density increased, prey increased their base

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Figure 4. Plots of the stationary values of four prey variables and two predator variables for a range of predator killing efficiency (KEP)and predator speed (number of moves per period). Each point is the average over the last 100 generations (generations 900 to 1000) acrossall individuals (either 12 000 prey or 400 predators). For all of these trials, the number of predators per landscape equaled one. (a) KEPand predator speed have non-additive effects on the probability (p1) that the prey occupies its rich patch during a given period; whileincreased KEP increased p1 at low predator speed, it had no effect at higher predator speed. Panels (b, c and d) show that the preyvigilance parameters increased with predator speed. (e) predator learning parameter decreased with predator speed. (f) faster movingpredators showed a greater tendency than slow moving predators to delay follow-up attacks at known prey locations.

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vigilance, but not maximum vigilance or the multiplier.This can be understood by realizing that the base vigilanceis the prey’s defense against first encounters. Because highpredator density results in higher number of first encoun-ters, the prey established a higher level of vigilance thatprotects them against those first encounters. Predatordensity also influenced predators’ post-encounter behaviorby selecting for a reduction in the predator delay parameter.The predator delay parameter made the predator delay itsreturn to a patch where it had recently encountered a prey.The benefit of such a delay is that a prey with ratchetingvigilance would have reduced its vigilance, making the preymore vulnerable in the follow-up attack. But if the predatorshares prey with other predators, a delay in re-attacking in apatch where there is a known prey may result in thepredator losing that prey to another predator; i.e. higherpredator density discounts the future value of informationregarding prey location.

I argue that conceptualizing predator�prey interactions toinclude both pre- and post-encounter behaviors has generalapplicability. There are, however, differences between themodel described here and most other published models ofpredator�prey games. The pre-encounter, ‘shell game’described here differs from many other predator�preymodels of space use. Those models consider joint habitatselection in which the payoff of using a habitat depends onthe densities of predator and prey in that habitat (Hugie andDill 1994, Sih 1998, Cressman et al. 2004, Luttbeg and Sih2004). Such models may yield an ESS distribution of preyand predator between habitats (but see Schwinning andRosenzweig 1990, Cressman et al. 2004). At such an ESS,individual predators and prey may have no reason to switchhabitats. In the shell game model presented here, however, aprey that does not move invites repeated follow-up attacks:predators can learn patch quality through experience byupdating a vector of patch weights. Confronted with such apredator, prey may employ a tactic of random movement,and leave a patch even if it contains no other predator orprey.

The model presented here differs sharply from non-gametheory models anti-predator behavior that do not permitany response by predators (Lima 2002). In these non-gamemodels, prey associate a particular predation risk with aparticular patch and choose their habitat and/or vigilanceaccordingly. But game theory models have shown that thepredation risk in a particular patch varies with prey behaviorwhen predators can respond to maximize their own fitness(Hugie and Dill 1994, Brown et al. 1999, Bouskila 2001,Alonzo et al. 2003, Rosenheim 2004, Sih 2005). And whenpredators are responsive, prey should employ strategies tofrustrate the predator response.

The model described here ignores much of the complex-ity of actual predator�prey behavioral games. For example,a predator reduced the weight of an empty patch by thesame amount whether it failed to encounter a prey, or itencountered a prey which it then killed. But these two casesmay provide the predator with different informationregarding the chance that the patch will be occupied inthe future. A patch in which no prey is encountered at onetime period may be occupied at some later time if preymove among patches. But a patch in which a prey is killed isunlikely to be occupied in the future, unless prey disperse

among territories (which they did not in the currentmodel). A more sophisticated predator would distinguishbetween these two cases, and reduce the weight of an emptypatch more in the case of a kill. A model for such a predatorwould require at least two learning parameters. Anotherelaboration of the current model could permit the predatorto learn prey location on two spatial scales: the smaller scaleof patches as described in the present model, and a largerscale of prey home range size. Alternatively, detecting a preyin a patch could modify the weight of neighboring patchesas well as the patch in which the prey is detected. Addingsuch sophistication would increase the number of para-meters in the evolutionary algorithm, and increase thecomplexity and run time of the model. Future models ofmulti-behavioral predator�prey interactions will undoubt-edly benefit from the judicious addition of additionalparameters.

Another simplifying assumption of the present model isthat initial population sizes of predators and prey were heldconstant for all generations of the evolutionary algorithm.This assumption was made to reduce the algorithm’scomputational complexity, an important considerationgiven that the algorithm already models the complexity ofseven co-evolving behaviors. Furthermore, other models ofpredator�prey games that do not explicitly considerpopulation dynamics (Alonzo et al. 2003, Hugie 2003)have provided useful insights. Nevertheless, it is possiblethat population dynamics (e.g. cycles) could influence theco-evolution of multi-behavioral strategies, and future workshould evaluate the role of population dynamics in suchgames.

Predator�prey interactions involving both pre- and post-encounter games are probably common in nature.Unfortunately, such games are rarely considered in theore-tical or empirical work. The present work attempts toaddress this shortcoming. The resulting analysis yieldsevolutionarily stable strategies consisting of multiple, inter-dependent behaviors, that vary in non-additive and indirectways with parameters of the predator�prey system. Theseresults suggest that future work on predator�prey interac-tions could benefit by considering together two naturalconflicts between predators and prey: manipulating boththe rate and outcome of predator prey interactions.

Acknowledgements � I wish to thank S. L Lima, T. C. Roth andT. Steury for discussions about the ideas contained within thispaper. This research was supported by the National ScienceFoundation (IBN-0130758).

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