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Behavioral Ecology Vol. 14 No. 6: 807-817
© 2003 International Society for Behavioral Ecology
The waiting game: a "battle of waits" between predator and prey
Behavioural Ecology Research Group, Simon Fraser University, Burnaby, British Columbia, Canada
Address correspondence to D. M. Hugie. E-mail: don{at}hugie.net.
Received 14 May 2002; revised 16 December 2002; accepted 3 January 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Many prey respond to the presence of a predator by retreating into a shell or burrow, or by taking refuge in some other way that guarantees their safety but restricts further information from being obtained about the predator's continued presence. When this occurs, the individual predator and prey involved become opponents in a "waiting game." The prey must decide how long to wait for the predator to depart before re-emerging and potentially exposing itself to attack. The predator must decide how long to wait for the prey to re-emerge before departing in search of other foraging opportunities. I use a numerical approach to determine the evolutionarily stable waiting strategy of both players and examine the effects of various parameters on the ESS. The model predicts that each player's waiting distribution—the distribution of waiting times one would expect to observe for individuals in that role—will have a characteristic shape: the predator's distribution should resemble a negative exponential function, whereas the waiting time of the prey is predicted to be more variable and follow a positively skewed distribution. The model also predicts that very little overlap will occur between the players' waiting distributions, and that the predator will rarely outwait the prey. Empirical studies relating to the model and comparisons between the waiting game and the asymmetric war of attrition are discussed.
Key words: antipredator behavior, evolutionary game theory, foraging behavior, hiding, predator-prey, war of attrition.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Many prey respond to the presence of a predator by retreating into a shell or burrow, or by taking refuge in some other way that guarantees their safety but restricts further information from being obtained about the predator's continued presence. When such a prey takes refuge, the individual predator and prey involved both face a decision: the prey must decide how long to wait for the predator to depart before re-emerging and potentially exposing itself to attack; the predator must decide how long to wait for the prey to re-emerge before departing in search of other foraging opportunities. Apart from its obvious benefits, waiting also has costs for both the predator and the prey. For example, prey typically cannot forage while taking refuge, and predators are less likely to encounter other prey while they wait. However, in making their decisions, both the predator and the prey face more than a simple optimization problem because the costs and benefits associated with choosing any given waiting time will depend on the opponent's waiting decision. Indeed, neither the predator nor the prey will have a single "optimal" waiting time because such a pure strategy is vulnerable to an opponent that always chooses a slightly longer waiting time. Understanding the foraging behavior of predators and the antipredator behavior of prey in such situations requires a game-theoretic approach.
Examples of potential waiting scenarios are too numerous to review but include a wide range of taxa, both vertebrate and invertebrate (see also Dill and Fraser, 1997
). These include species that take refuge in the substrate, such as aquatic worms (e.g., Drewes and Fourtner, 1989), fiddler crabs (e.g., Frix et al., 1991), various fishes (e.g., gobies, garden eels), and burrowing mammals (e.g., MacHutchon and Harestad, 1990). Other species take refuge in a protective structure surrounding their body, such as polychaete tubeworms (e.g., Dill and Fraser, 1997), caddis fly larvae (e.g., Johansson and Englund, 1995), gastropod and bivalve mollusks, hermit crabs (e.g., Scarratt and Godin, 1992), barnacles (e.g., Dill and Gillett, 1991), turtles, and armadillos. As Dill and Fraser (1997) point out, functionally analogous behavior to refuge use occurs whenever an animal flees an area because a predator is detected and must then decide how long to continue its avoidance behavior without knowing whether or not the predator is still present (e.g., Koivula et al., 1995).
Despite the prevalence of waiting scenarios, there has only been one previous attempt to use game theory to understand the behavior of predators and prey in such situations: Johansson and Englund (1995) present a partial analysis of a game that does not yield evolutionarily stable strategies. In this article, I first model a "waiting game" contest between a solitary predator and a solitary prey. I then use a numerical approach to find each player's evolutionarily stable strategy (ESS) and to examine the effects of various parameters on the ESS. Unlike in Johansson and Englund's model, both players in the current model have an evolutionarily stable waiting strategy. Moreover, the model predicts that each player's waiting distribution—the distribution of waiting times one would expect to observe for individuals in that role—will have a characteristic shape over a wide range of parameter values. This prediction, and others, makes the model amenable to empirical testing despite its mathematical complexity.
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The Model |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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The game begins with a solitary predator in the immediate vicinity of a solitary prey that has just taken refuge. Although both players are assumed to be aware of the other's presence, it is irrelevant whether the prey initially detects the predator though a direct physical attack or by some remote sensory method. Let u and U be the waiting strategy of the prey and predator, respectively, each defined as a probability distribution over waiting time. Although time is a continuous quantity, a numerical solution demands that there be a finite number of waiting times from which each player can choose. I assume both players have the same
actions available to them, each associated with a waiting time 
i, where i = 1, 2, ... , . Waiting times are assumed to be increasing in i and correspond to the midpoints of equal divisions of time between 0 and time units. In this case, u = (p1, p2, ... , p) and U = (P1, P2, ... , P), where pi and Pi are the probabilities the prey and predator, respectively, choose action i. Note that lowercase and uppercase (Latin) letters are used for terms referring to the prey and predator, respectively, and Greek letters are used for all other terms.
The possible outcomes of the game are illustrated in Figure 1. At the beginning of the encounter, both the prey and the predator choose a waiting time according to their respective strategies. If the predator outwaits the prey, it has an opportunity to attack and captures the prey with probability 
. If the prey survives, it again takes refuge, and a new round begins with both players choosing a new waiting time. Several such rounds may occur before the encounter finally ends, because either the prey is captured or it outwaits the predator. To simplify the analysis, I assume that successive rounds are equivalent from the point of view of both players and, therefore, that the strategy used by either player does not change from one round to the next.
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It is important to explain what "outwaiting" means. The predator is said to outwait the prey if it has an opportunity to attack the prey when it re-emerges. Such opportunities will be determined by the actions of both players. For a prey choosing action i and a predator choosing action j, one can define a quantity
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is negative, the predator will always have an opportunity to attack the prey because it has chosen a longer waiting time. However, the predator will also have some chance of attacking the prey if it re-emerges just after the predator departs. It is unreasonable to assume that the opportunity to attack the re-emerging prey instantaneously drops to zero the moment the predator chooses to depart. This is particularly true if the predator leaves the area slowly, for example, because it is searching for other prey in the immediate vicinity. Thus, it is necessary to consider the possibility that the predator may outwait the prey even though it has technically departed at the time the prey re-emerges. Such "postdeparture" attacks may be distinguished from "predeparture" attacks that occur before the predator leaves. To incorporate this possibility into the game, I define 
(), the probability that the predator has an opportunity to attack the prey when it re-emerges:
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> 0 and v > 1. I assume the predator always has an opportunity to attack if the prey re-emerges before, or at the same time as, the predator departs (i.e., when 
0). If the prey re-emerges after the predator departs (i.e., when > 0), the probability that the predator will have an opportunity to attack initially decreases in an accelerating manner with increasing , then decelerates toward an asymptote of zero (Figure 2). The exact shape of () depends on the values of the parameters and v. In particular, is the value of for which () = 0.5, and v determines how rapidly () decreases at this point.
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To determine the ESS, it is necessary to calculate the expected payoff, in units of reproductive value, to the predator and prey for choosing a particular waiting time during a given round. The details of these calculations are described in Appendix A. In determining these payoffs, several useful quantities are calculated that summarize a typical interaction between a prey playing strategy u against a predator playing strategy U. These quantities are formally defined in Appendix A but are introduced here along with a general description of each player's payoff.
The expected payoff to both players depends on the probability the prey is captured by the predator and the total amount of time the player can expect to wait during the encounter. The probability that the predator will capture the prey during the encounter, 
(u, U), depends on the probability the predator will outwait the prey during a given round, 
(u, U), and the probability of prey capture during an attack opportunity (). Obviously, the two players have opposing interests in the prey being captured. However, the players also differ in the importance of prey capture to their payoff. Failing to avoid capture reduces the prey's fitness to zero, whereas failing to capture the prey only reduces the predator's payoff by the value of a single prey capture, V.
Unlike prey capture, the players' payoffs are similarly affected by the amount of time they wait. Waiting is assumed to be costly to both players for a number of reasons, including lost-opportunity costs. For example, prey typically cannot forage while taking refuge, and predators are less likely to encounter other prey while they wait. For simplicity, I assume that the loss in reproductive value of both players while waiting is a linear function of the total amount of time they wait during the encounter. The parameters a and A describe the cost of waiting—the rate at which the prey and predator, respectively, lose reproductive value while they wait. The expected time spent waiting during the encounter, 
(u, U) or 
(u, U), is the product of the expected number of rounds experienced during the encounter, 
(u, U), and the expected time spent waiting during each round, 
(u, U) or 
(u, U). As always, lowercase letters refer to the prey, uppercase to the predator. Note that in the case of the prey, the calculated waiting times are conditional on it surviving, because waiting costs are only relevant to the prey if it survives the encounter, and the prey is more likely to survive if it chooses longer waiting times.
Describing the ESS
Although determining the evolutionarily stable waiting strategy of each player is the immediate goal of the model, a more useful description of the ESS is provided by each player's waiting distribution—the expected distribution of waiting times a player would experience, or an observer would record, over the course of many encounters. In the case of the prey, the probability that an individual will wait time i is simply equal to the probability that it chooses that waiting time:
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i is
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i is equal to the probability that it will attack the prey or simply depart at that time, given that it has not already done so:
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As already mentioned, the expected amount of time a surviving prey and the predator will wait during a given round is (u, U) and (u, U), respectively. However, because the predator is not committed to its chosen waiting time, and because the prey is sometimes captured, the average waiting time actually chosen by each player must be calculated:
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Two additional quantities are useful to gain further insight into the dynamics of attacks at the ESS. The first is the proportion of attacks that occur after the predator has technically departed (see above):
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Finally, although not actually used to determine the ESS, it is useful to calculate the expected payoff to each player:
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Finding the ESS
It is important to keep in mind that although the game involves only a solitary predator and prey, both players are members of a population of individuals that could find themselves playing the game during their lifetime. In such populations, natural selection will favor individuals whose waiting strategy is the best, or optimal, response to the current strategy in the opponent population. However, because each role's best response depends on its opponent's strategy, the two populations will be locked in a coevolutionary struggle to outwit each other. Eventually, ESSs may emerge that not only are the best response to each other but also remain constant over time. Although the existence of such strategies is not guaranteed, their discovery would provide insight into the players' waiting behavior.
I searched for evolutionarily stable waiting strategies using the "best response with error" algorithm proposed by McNamara et al. (1997) and modified for the asymmetric case (see Appendix B). This method finds an ESS by iteration of the best response map with the added biological reality that players are assumed to make slight errors. The inclusion of error recognizes that natural selection is not a perfect optimizer for a variety of reasons ranging from genetic constraints to spatial and temporal variation in the environment. Error is introduced in a manner consistent with basic assumptions about how natural selection works: the greater the cost of an error to a player, the less likely it is to occur. During each iteration, the algorithm calculates each player's best response (with error) to its opponent's strategy in the previous iteration. A player's best response with error is a strategy that chooses the optimal waiting time(s) with the highest probability and suboptimal waiting times with a probability that rapidly decreases with increasing cost (see Appendix B). The amount of error introduced into the players' strategies is determined by the coefficient 
, where 0 < < 
. An ESS (with error) is found if both players settle on a strategy that remains constant from one iteration to the next.
The simplest interpretation of the best response with error algorithm is that it calculates the likely strategy in each player population resulting from natural selection during successive generations. There are obvious simplifications, however, including the assumption that populations are monomorphic and will evolve the best response (albeit with error) to the strategy in an opponent population in only one generation. The ability of a player to instantaneously respond to its opponent's strategy is a potential source of instability in the model because the response is to the opponent's strategy in the previous generation. This can cause the players' strategies to oscillate back and forth indefinitely. However, in real biological systems, genes that code for the proximate mechanisms necessary to realize a best response may not be immediately available and will take time to spread through the population. For this reason, damping is included in the model to slow down the change in each player's strategy from one iteration to the next (see Appendix B). Damping may be interpreted as determining what proportion of a population is replaced with mutants that follow the best response (with error) from one generation to the next (McNamara et al., 1997). The coefficient 
(where 0 < 1) determines the level of damping: the greater the value of , the greater the amount of damping in the model.
Computational details
The model was coded in ANSI C and executed on a SGI R4000 computer using the native IRIX 5.3 C++ compiler. Preliminary runs were initialized with random waiting strategies for both players. However, the model was found to converge slightly faster when initialized with both players following a pure strategy, with the predator choosing the minimum waiting time and the prey choosing the maximum waiting time.
Increasing the number of available waiting times, , was found to have no effect on the outcome of the model, apart from improving the resolution of the continuous time approximation, but dramatically increased the computational demands. Given the available computer resources, an acceptable compromise was found to be a value of of 750 time intervals. The maximum waiting time of both players was 1500 time units in all runs of the model, resulting in waiting time intervals of width 2.
Damping has a stabilizing influence on the model but does not influence the value of the ESS. Unfortunately, the number of iterations required for convergence increases rapidly with increased damping. Given the available computer resources, an acceptable compromise was found with a level of damping ( = 0.9995) that ensured convergence within 50,000 iterations.
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RESULTS |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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An ESS could always be found if the level of error in the model was sufficiently high. All ESSs had the same characteristics. Both players follow a mixed strategy—randomly selecting a waiting time from a distribution of possible waiting times. The predator's waiting distribution resembles a negative exponential function (Figure 3a) but with a "bump" below the prey's distribution (Figure 3b). The prey's waiting distribution is positively skewed—rapidly increasing after the predator is likely to have departed and then gradually decreasing with increased waiting time (Figure 3a,b).
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Perhaps the most striking feature of the ESS is the large difference in the waiting times of the two players (Table 1, Figure 3a,b). Most of the time the predator departs immediately. Less often, the predator waits a brief amount of time, departing before there is much chance of the prey re-emerging, and paying a small cost for its delay. Much less often, the predator waits long enough to have a significant chance of attacking the prey when it re-emerges. Although the accumulated cost of waiting is much higher for such events, it is partially offset by the potential benefit of capturing the prey. This asymmetry in the players' waiting times results in the predator rarely outwaiting the prey and encounters rarely lasting more than one round (Table 1). This observation is consistent over a wide parameter space and is a result of the large asymmetry between the players regarding the fitness consequences of an attack.
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The expected distribution of predeparture attacks closely follows the prey's waiting distribution but at a different scale (Figure 3, cf. b and c). This makes sense because the prey can only be attacked when it re-emerges. However, the average waiting time of predeparture attacks is always less than the average waiting time chosen by the prey (Table 1). Indeed, compared with the prey's waiting distribution, the distribution of predeparture attacks is slightly biased against attacks after long waiting times, although this is not obvious from Figure 3. This is because the predator is more likely to have departed when the prey re-emerges after waiting longer.
Closer inspection of the predator's waiting distribution reveals that its bump below the prey's distribution is owing to long waiting times chosen by the predator being censored by the re-emergence of the prey (Figure 3c). The fact that the predator is not committed to its chosen waiting time means that its expected waiting time during a given round is always less than the average waiting time it chooses (Table 1). Similarly, because the prey is more likely to be captured when it chooses a shorter waiting time, the expected amount of time the prey will wait, given that it survives a round, is always longer than the average waiting time it chooses. However, this difference is so small that it is not apparent from the values reported in Table 1.
Neither player's payoff for choosing a given waiting time is illustrated in Figure 3. One might expect that a player would receive exactly the same payoff regardless of which waiting time it selects from its mixed ESS. If this were not the case, and a particular waiting time yielded the highest payoff, the mixed strategy would not be evolutionarily stable because the population could be invaded by individuals who always played the optimal waiting time. However, this logic does not take into account the complexities of an ESS with error. The best response with error algorithm assumes that a player make errors, but that suboptimal waiting times are chosen with a probability that decreases rapidly with increasing cost. A player receives almost exactly the same payoff regardless of which waiting time it chooses from its mixed ESS (with error). Indeed, with the low level of error used in the model (except the run in which = 7 x 10-5) all waiting times chosen with a probability greater than 1 x 10-7 were associated with a payoff of at least 99.995% of the optimal waiting time. As expected, the most commonly chosen waiting time results in the highest payoff.
Although increasing the level of error has a stabilizing effect on the model, it also increases the expected waiting time of both players. Table 1 presents the results of the model with the minimum level of error necessary to achieve stability without damping ( = 7 x 10-5, = 0). Such a high level of error results in unrealistically long waiting times for the predator. This becomes apparent if one calculates the predator's "breakeven time"—the waiting time for which the expected benefit of an attack is exactly offset by the accumulated cost of waiting:
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= 0), stable solutions are only possible when the predator is so error-prone that it frequently experiences waiting times greater than E without having any significant chance of attacking the prey. By introducing sufficient damping into the model ( = 0.9995), the level of error can be reduced to a more biologically realistic level ( = 4 x 10-6) for which the predator rarely experiences waiting times greater than E, and such events are almost always associated with an attack (Figure 3, Table 1).
The influence of payoff parameters
The ESS is also affected by the parameters used to determine the payoff to each player. Investigation of these effects is simplified by arbitrarily assigning a value of 100 to f and F, the pre-encounter reproductive values of the prey and predator, respectively (see Appendix A). This does not imply that both players necessarily have the same absolute reproductive value, but simply provides a comparable scale for considering how both players evaluate the costs and benefits of waiting.
Payoff parameters may be classified according to whether they directly affect the model through the payoff to the predator, the prey, or both players. Two parameters have exclusive effects through the predator's payoff and also affect its break-even time, E. These are the predator's waiting cost, A, and the value of prey capture, V. I examined the effect of decreasing the value of A and increasing the value of V, such that E was increased from 100 to 150 time units in both cases (Table 1). Both manipulations effectively reduce the relative cost of waiting for the predator in so far as its expected net gain from attacking the prey at any given waiting time is increased. In both cases, decreasing the predator's relative waiting cost causes parallel movement in the waiting distributions of the two players, increasing the expected waiting time of both predator and prey (Table 1). This might be expected given that the predator can afford to wait longer in order to capture the prey and that the prey must respond by waiting longer to avoid being captured.
Perhaps surprisingly, the probabilities of the predator outwaiting and ultimately capturing the prey both decrease slightly. This observation seems counterintuitive. How could reducing the predator's relative waiting cost—and thereby conferring on it an apparent advantage—result in the predator having to wait longer in order to have a lower probability of capturing the prey? One explanation is that such a result does not necessarily mean the predator does worse: a reduction in the predator's relative waiting cost enables it to accept a lower probability of capturing the prey because the expected net gain from a capture is increased. Indeed, reducing the predator's absolute waiting cost, A, does increase its payoff, although the increase is too small to be reported in Table 1. However, the predator truly does worse when the value of prey capture, V, is increased.
Another possibility is that increasing the value of prey capture places the predator in an evolutionary dilemma. Initially, the predator might do better by waiting longer. However, once the prey responds in kind, the predator may end up doing worse but find it impossible to return to a shorter waiting strategy because doing so would result in an even lower payoff.
A final explanation is based on the way error is introduced into the model. The best response with error calculation does not guarantee that a player's payoff is reduced by the same amount owing to error in all situations, even when is held constant. This becomes apparent if one calculates the decrease in a player's payoff for following its best response with error rather than choosing its optimal course of action. Under different conditions this value—expressed as either a proportion or in absolute terms—varies slightly, even when is held constant. Thus, slight differences in the predator's payoff under different parameter values could partially be owing to differences in the level of error realized at the ESS.
The only parameter exclusively affecting the model through the prey's payoff is its waiting cost, a. I examined the effect of increasing and decreasing a by a factor of five relative to the predator's waiting cost, A. Unlike changes to exclusively predator parameters, changing the prey's waiting cost causes the waiting distribution of the two players to move in opposite directions (Table 1). As expected, when its waiting cost is increased, the prey trades off some of this additional cost by waiting less time and accepting a higher predation risk (Table 1). What is perhaps unexpected is the opposite motion of the predator's distribution. One might expect the predator to use such an opportunity to also wait less and thereby reduce its accumulated waiting cost. Instead, the predator waits longer, perhaps to take advantage of the improved opportunity to capture the prey. These observations are reversed when the prey's waiting cost is decreased (Table 1).
Although changes in the predator's relative waiting cost appear to affect the degree to which the predator "chases" the prey across waiting time, changes to the prey's waiting cost appear to determine the degree to which the players engage in attacks and, therefore, additional rounds of play. Not surprisingly, increased engagement results in a relatively large decrease in the prey's payoff (Table 1). What is unexpected is the slight decrease in the predator's payoff. Again, a possible explanation for this observation is variation in the level of error realized in the model under different parameter values.
Changes to the probability of prey capture during an attack opportunity, , and the details of the function (), affect the model through both player payoffs. Because partially determines the value of E, and therefore the relative cost of waiting for the predator, I examined the effects of changing in two ways. First, I followed the protocol used for the other parameters affecting E by increasing so that E was increased from 100 to 150 time units (Table 1). To control for the effects of changing the predator's relative waiting cost, I also increased to unity while keeping E constant by simultaneously decreasing V, the value of prey capture (Table 1). In this run of the model, the prey is of less value to the predator but is captured with certainty during an attack. As expected, such encounters never last more than one round because the prey either outwaits the predator or is killed (Table 1).
The qualitative results of both manipulations are the same (Table 1), suggesting the effect of on the ESS is independent of its effect on the predator's relative waiting cost. This view is also supported by the observation that changes in have qualitatively different effects on the ESS than do the previously discussed changes to parameters that determine E (Table 1). In particular, increasing causes the predator to reduce its waiting time while the prey's waiting time is increased. Such changes are analogous to those observed when the prey's waiting cost is decreased suggesting influences the ESS mainly through its effect on the prey's payoff. Interestingly, increasing the predator's ability to capture the prey during an opportunity to attack actually reduces the probability it will outwait and ultimately capture the prey. However, because the predator waits less time, and therefore has a lower accumulated waiting cost, its payoff is increased.
The final run of the model examines the effect of () on the ESS. This function is included in the model to recognize that the predator's opportunity to attack the prey when it re-emerges will not vanish instantaneously the moment the predator decides to depart but rather will decrease smoothly over time. This decrease may occur rapidly, as assumed in the default parameter values, or may decrease more slowly if, for example, the predator lingers in the immediate area for considerable time. I examined the effects of increasing the potential for postdeparture attacks by increasing the coefficient from 1 to 8 (see Figure 2). Not surprisingly, this results in an increased proportion of attacks occurring after the predator has technically departed, although such attacks remain relatively uncommon (Table 1). There is also a longer delay before the prey begin to re-emerge, although this is not revealed by average waiting times reported in Table 1 because the waiting distribution also becomes more peaked. These results indicate that () can affect the shape of the prey's waiting distribution, at least when the potential for postdeparture attacks is large. As expected, increasing the potential for postdeparture attacks results in a higher payoff to the predator and a lower payoff to the prey.
Clearly, the parameter space of the model is very large. For this reason, caution must be exercised when drawing general conclusions from a limited series of manipulations. I have attempted to draw conclusions that appear robust over a wide parameter space while avoiding those for which I have less confidence. For example, qualitative changes to the average waiting time chosen by the predator may not always result in corresponding changes to the waiting time it experiences because the latter depends on the prey's waiting strategy. Similarly, changes to the expected waiting time experienced by either player during a round need not affect its expected waiting time over the duration of the encounter in the same way because the latter also depends on the expected number of rounds that occur.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Waiting game contests are expected to be common in nature because prey often respond to predators by taking refuge in a way that restricts further information from being obtained about a predator's continued presence. Such antipredator behavior is extremely common, occurring in taxa ranging from aquatic invertebrates (e.g., Drewes and Fourtner, 1989
) to terrestrial mammals (e.g., MacHutchon and Harestad, 1990). The current model adds to our understanding of the foraging behaviour of predators and the antipredator behaviour prey in such situations. Not surprisingly, no single waiting time is optimal for either player because such a pure strategy is vulnerable to an opponent that always chooses a slightly longer waiting time. Instead, both players adopt a mixed waiting strategy—randomly selecting a waiting time from a distribution of possible waiting times. The most robust predictions of the model relate to the general shape and relative position of each player's waiting distribution and the distribution of attacks across waiting time (see Figure 3). In particular, the model predicts that (1) the predator's waiting distribution will resemble a negative exponential function (Figure 3a) but with a slight "bump" below the prey's distribution (Figure 3b); (2) the waiting time of the prey will be more variable and follow a positively skewed distribution (Figure 3a,b); (3) very little overlap will occur between the two players' waiting distributions and the predator will rarely outwait the prey (Figure 3a,b); and (4) the distribution of attacks across waiting time will resemble the prey's waiting distribution (Figure 3, cf. b and c) but with a slight bias against attacks at long waiting times. In addition, both the overlap between the players' waiting distributions and the likelihood of an attack are predicted to increase when the probability of prey capture during an attack opportunity is decreased or the prey's waiting cost is increased.
Testing the model
Empirical studies have tended to focus on the behavior of the prey in waiting scenarios using optimality logic to predict how individuals should trade-off predation risk and the lost-opportunity cost of waiting (e.g., Dill and Fraser, 1997; Dill and Gillett, 1991; Scarratt and Godin, 1992). Such studies have largely been successful in predicting changes in the prey's average waiting time in response to changes in its waiting cost: higher costs are associated with shorter waiting times. This is to be expected because predictions from an optimality approach and those of the waiting game are the same in this regard. However, empirical studies often report large amounts of positively skewed variation in the waiting time of prey (e.g., Dill and Fraser, 1997; Frix et al., 1991). Unlike an optimality approach, the current model predicts that such variation will occur as part of the prey's mixed strategy in a waiting game played against its predators.
Johansson and Englund (1995) studied a waiting game between bullheads (Cottus gobio; a predatory fish) and case-making caddis fly larvae (Halesus radiatus). After an unsuccessful attack by a bullhead, a larva withdrawals its head and legs into a case made of organic debris and remains motionless. The bullhead usually responds by orienting toward the larva and remaining motionless. Johansson and Englund's data reveal a reasonably good fit with the predictions of the waiting game. As expected, the waiting distribution of the bullheads resembles a negative exponential function with individuals rarely waiting longer than 200 seconds. Furthermore, the waiting distribution of the larvae rises sharply to a modal waiting time of 175 seconds before decreasing in a decelerating manner with increased waiting time. As predicted, the predator rarely outwaited the prey: only two of 133 larvae (1.5%) re-emerged before the bullhead departed.
Close inspection of Johansson and Englund's (1995; Figure 1) data reveals more overlap between the player's waiting distributions than might be expected from the current model. This is likely to be a common observation when data contain added variation owing to experimental error or differences in game parameters between observations. For example, bullheads may have varied in energy reserves and, therefore, in the value of prey capture. Differences did exist in both the intensity and duration of the initial attack by the bullheads, and both were positively correlated with their subsequent waiting time. Moreover, the larvae appeared able to assess the motivation of the bullheads because they waited longer when attacked with greater intensity by a simulated predator and because the bullheads outwaited fewer larvae than expected from the overlap in the observed waiting distributions (two instead of nine of 133 larvae). This suggests that Johansson and Englund's data contain added variation due to differences in game parameters between observations. In fact, nearly half of the variation in the waiting time of bullheads was owing to intra-individual variation. Such noise in the data will tend to increase the apparent overlap between the players' waiting distributions, obscure the "bump" in the predator's distribution, and cause the prey's distribution to appear less skewed.
Theoretical issues
Johansson and Englund (1995) attempted to model the game between bullheads and caddis fly larvae but failed to find ESSs. They did, however, identify similarities between this game and the asymmetric war of attrition (Hammerstein and Parker, 1982), a model originally developed to describe animal conflicts that are settled by means of aggressive displays.
Like the waiting game, the asymmetric war of attrition is a contest between two opponents in different roles (e.g., "owner" and "intruder"), and the winner is the individual that persists the longest. However, three important features of the asymmetric war of attrition make it unsuitable for modeling the waiting scenario between a predator and a prey. First, regardless of its role, the winner in the war of attrition is not required to pay the full cost of its chosen persistence level once its opponent concedes defeat. By contrast, an important feature of the waiting game is that the prey has no way of knowing when the predator departs and must, therefore, always pay the full cost of its chosen waiting time. Second, unlike the asymmetric war of attrition, the waiting game does not always end after just one round: both players will find themselves facing another round of waiting whenever the prey survives an attack and again takes refuge. Third, the characteristics of the ESS in the asymmetric war of attrition critically depend on the assumption that the players occasionally make errors and believe they are in the same role. This seems reasonable for opponents deciding who has the most to gain from winning, or pays the least for persistence, during an aggressive interaction. However, it seems unlikely that a prey would ever choose its waiting time believing that it was the predator in a waiting scenario or visa versa!
The stability criterion used in the current game is the "ESS with error" proposed by McNamara et al. (1997). This approach recognizes that organisms will sometimes make errors and include suboptimal actions in their strategy but that natural selection will make such errors less likely for actions with higher costs. This is important because such errors will affect both the equilibrium strategies and the stability of real biological games. Indeed, error has an important stabilizing effect on the current game and is particularly important in shaping the predator's waiting distribution. This becomes apparent if one considers the robustness of the predator's waiting distribution to changes in payoff parameters but its sensitivity to the level of error in the model. In fact, most of the time the predator is predicted to not wait long enough to have any realistic chance of outwaiting the prey. This may seem disconcerting to some. However, as the model demonstrates, brief waiting times owing to error, but with little cost to the predator, are expected to occur even with low error levels. This does not mean the predator is not interacting strategically with the prey. The high level of damping required to stabilize the game under realistic error levels demonstrates the existence of such interactions. However, the dire consequence of an attack for the prey ensures there is little overlap between the players' waiting distributions. The importance of such interactions to the prey can be demonstrated by preventing the predator from attacking the prey in the model. In this case, the prey's waiting distribution dramatically shifts to the left and completely overlaps the predator's distribution.
As with many biological games, there are several variations on the basic waiting scenario presented here. One variation is the scenario in which the prey takes refuge to avoid injury to some part of its body, rather than death. Examples include invertebrates, such as barnacles (see Dill and Gillett, 1991) or polychaete worms (see Dill and Fraser, 1997), that withdraw into a protective structure to avoid loosing a portion of a feeding or respiratory appendage. This version of the game may not differ much from the current one because sublethal injury is mathematically similar, although not equivalent, to the situation in the current game in which there is a low probability of capture when the predator has an opportunity to attack. In such systems, one would expect greater overlap in the player's waiting distributions with the predator more frequently outwaiting the prey.
As with many game-theoretic models, the current one assumes that both players have perfect knowledge regarding the parameters that affect their own payoff and the payoff to their opponent. However, it is unlikely that a player will always know the details of its opponent's internal state or the state of external factors that affect either player's payoff. Such complications may not change the general predictions of the model. For example, variation in the cost of waiting among prey may affect the waiting tendency of different individuals but is not expected to cause them to adopt a pure strategy: an individual suffering a high waiting cost may be more likely to choose a short waiting time but is not expected to do so with certainty because this would leave it vulnerable to predators that detect the regularity in its waiting behavior. Furthermore, even if individual differences in waiting strategy do exist, this does not mean that the overall strategy of individuals in that role, described at a population level, will necessarily deviate from the general predictions of the basic model. Further investigations are required to determine the degree to which state dependence and imperfect knowledge affect the results of the model.
The current model also assumes that the prey is unable to gain any information about the predator's presence while taking refuge. However, in some of the examples of waiting scenarios described in the Introduction, the prey may be able to gain such information while waiting. Sih (1992) considered how long prey should remain in a refuge under such circumstances assuming that no strategic interactions occur between the predator and prey. A more complete, but complicated, model would incorporate both approaches. Clearly, the availability of such information will affect the outcome of the game. In the extreme case in which the prey has perfect knowledge about the predator's presence, it would simply wait until the predator departs before re-emerging. The goal of the current model is to capture the essence of the waiting game with the minimum complexity. As such, it provides a reasonable starting point for future theoretical and empirical studies.
Natural selection is not expected to produce predators that are simple-minded executioners in search of prey that are content to simply wait to be discovered and eaten. For this reason, understanding the foraging behavior of predators and the antipredator behavior of prey requires consideration of the strategic nature of predator-prey interactions. The waiting game provides an example of how such an approach can lead to a clearer understanding of commonly observed behavior by both predators and their prey.
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APPENDIX A |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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The players' payoffs
To determine the ESS, it is necessary to calculate the expected payoff, in units of reproductive value, to the prey and predator for choosing action i during the first round of the encounter and adopting its respective strategy, u or U, during any subsequent rounds that occur. The expected payoff to both players depends on the probability the prey is captured by the predator and the total amount of time the player can expect to wait during the encounter. I begin by calculating the probability that a prey choosing action i during a given round will outwait a predator playing strategy U:
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To calculate the total amount of time that a prey can expect to wait given that it survives the encounter, I first calculate the expected number of rounds which will occur during the encounter:
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For simplicity, I assume the loss in reproductive value of both players from waiting is a linear function of the total amount of time they wait during the encounter. In this case, the expected payoff to the prey for choosing action i is
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I next calculate the expected payoff to the predator for choosing action i. I begin by calculating the probability a predator choosing action i during a given round will outwait a prey playing strategy u:
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(u, U). Thus, the probability a predator choosing action i during a given round will ultimately capture the prey during the encounter is
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To calculate the total amount of time a predator can expect to wait during the encounter, I first calculate the amount of time a predator choosing action i can expect to wait during a given round:
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(u, U), and therefore, the expected amount of time a U-strategist will spend waiting during the encounter is
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Finally, the expected payoff to the predator for choosing action i is
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APPENDIX B |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Determining the ESS
The evolutionarily stable waiting strategy of each player was determined following the method proposed by McNamara et al. (1997)
, modified for the asymmetric case. This method finds an ESS by iteration of the best response map with the added biological reality that players are assumed to make errors. During each iteration, one first calculates the "canonical cost" (sensu McNamara and Houston, 1986) to each player for choosing each action given its opponent's current strategy:
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(U) and 
(u) are the reproductive values of the prey and predator, respectively, given an optimal choice of action:
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of normalizing these costs so they represent a proportional loss of reproductive value relative to choosing the optimal action.
Next, errors in decision making are introduced by assigning a weight to each action according to its canonical cost. I use the following function, described by McNamara et al. (1997), to assign a weight to each action for both players:
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(where 0 < < ), is the same for both players.
Each action is then assigned a new probability according to its relative weight:
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(U) and B(u), respectively, is defined as the strategy that chooses action i with probability 
i(U) and 
i(u), respectively (see McNamara et al., 1997). Under this strategy, the optimal action is chosen with the highest probability, where suboptimal actions are chosen with some positive probability that rapidly decreases with increasing canonical cost.
The procedure for calculating the best response with error becomes problematic in the current game because the predator is not committed to pay the remaining cost of its chosen waiting time once the prey emerges. This means that extreme waiting times have more or less the same canonical cost for the predator because they all result in an attack opportunity after a similar, and much shorter, waiting time. As a result, the predator's best response with error becomes sensitive to the maximum waiting time allowed in the model () because extreme waiting times carry significant weight in the calculation of B(u). The problem lies in the way the best response with error algorithm considers all actions made available to the predator even though there is an infinite number of extreme waiting times that result in approximately the same payoff. The only immediate solution to this problem is to avoid choosing a value of that is unrealistically long.
Once each player's best response with error is calculated for the current iteration 
, its strategy in the next iteration is determined by
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< 1. The value of determines the level of damping in the model. Note that I reversed the scale of relative to the method of McNamara et al. (1997) so that the greater the value of , the greater the level of damping in the model.
Under the above scheme, the strategy combination (
, 
) is defined as the ESS with error if b() = and B() = and is found by iteratively applying the above calculations until the waiting strategy of each player converges (see McNamara et al., 1997).
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ACKNOWLEDGEMENTS |
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I am grateful to Larry Dill, David Lank, Michael Mesterton-Gibbons, Bernie Roitberg, and Ron Ydenberg for providing valuable comments on the manuscript. I also thank Evan Cooch for his assistance in implementing the computer code and Jaclynne Campbell for sketching the hare and fox in Figure 1. This work was supported by a Natural Sciences and Engineering Research Council Canada grant to L. M. Dill (A6869).
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References |
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TOPABSTRACTINTRODUCTIONThe ModelRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Dill LM, Fraser AHG, 1997. The worm re-turns: the hiding behavior of a tube-dwelling marine polychaete, Serpula vermicularis. Behav Ecol 8:186-193.
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Frix MS, Hostetler ME, Bildstein KL, 1991. Intra- and interspecies differences in responses of Atlantic sand (Uca pugilator) and Atlantic marsh (U. pugnax) fiddler crabs to simulated avian predators. J Crust Biol 114:523-529.[CrossRef]
Hammerstein P, Parker GA, 1982. The asymmetric war of attrition. J Theor Biol 96:647-682.[CrossRef]
Johansson A, Englund G, 1995. A predator-prey game between bullheads and case-making caddis larvae. Anim Behav 50:785-792.[CrossRef]
Koivula K, Rytkönen S, Orell M, 1995. Hunger-dependency of hiding behaviour after a predator attack in dominant and subordinate willow tits. Ardea 83:397-404.
MacHutchon AG, Harestad AS, 1990. Vigilance behaviour and use of rocks by Columbian ground squirrels. Can J Zool 68:1428-1432.
McNamara JM, Houston AI, 1986. The common currency for behavioral decisions. Am Nat 127:358-378.[CrossRef]
McNamara JM, Webb JN, Collins EJ, Székely T, Houston AI, 1997. A general technique for computing evolutionarily stable strategies based on errors in decision-making. J Theor Biol 189:211-225.[CrossRef][ISI][Medline]
Scarratt AM, Godin J-GJ, 1992. Foraging and antipredator decisions in the hermit crab Pagurus acadianus (Benedict). J Exp Mar Biol Ecol 156:225-238.[CrossRef]
Sih A, 1992. Prey uncertainty and the balancing of antipredator and feeding needs. Am Nat 139:1052-1069.[CrossRef]
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