Foraging in a group is a common pattern of animal behavior,

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1 Behavioral Ecology Vol. 11 No. 4: An evolutionarily stable strategy for aggressiveness in feeding groups Etienne Sirot Université de Bretagne Sud, 12 avenue Saint-Symphorien, Vannes, France Animals searching for food in groups display highly variable degrees of aggressiveness. In this paper I present an individualbased game theoretical model of how gregarious animals should adjust their level of aggressiveness to their environmental conditions. In accordance with behavioral observations, the predicted level of aggressiveness increases progressively with decreasing food availability and increasing animal density. The proposed model also predicts a positive influence of food energy value and handling time on the level of aggressiveness within the group. In addition, the model provides information about the influence of aggressive behavior on individual foraging success, interference, and population dynamics. Adaptive behavioral rules for aggressiveness in consumers are predicted to respond to both competitors and food density in a way that contributes to stabilization of the dynamics of population systems. Key words: aggression, evolutionarily stable strategy, game theory, intraspecific interference, optimal foraging. [Behav Ecol 11: (2000)] Foraging in a group is a common pattern of animal behavior, which results from the aggregated distribution of food in the environment (Goss-Custard et al., 1984) or from particular advantages, such as improved hunting success (Creel and Creel, 1995) or increased protection from predators (Caraco et al., 1980). Group foraging has important costs, however, caused by competition within the group. First, animals within a group have to share the resource (Creel and Creel, 1995; Free et al., 1977). Second, animals within a group frequently interact aggressively, which consumes both time and energy (Ens et al., 1990). The present study investigated the second phenomenon, referred to as interference competition (Goss-Custard, 1980), from an adaptive point of view. As energy is the primary reason for which animals feed, optimal foraging models commonly use the rate of energy gain to evaluate benefits of foraging strategies (Stephens and Krebs, 1986). It is clear that the time and energy budget of an animal searching for food with competitors depends on its tendency to accept confrontation (Goss-Custard et al., 1998). Too much aggressiveness leads to an important loss of time and energy. On the other hand, animals displaying no or very little aggressiveness might lose most of the food items they encounter to the advantage of their competitors and therefore have a poor feeding rate. Recent studies (Goss-Custard et al., 1998; Stillman et al., 1997) suggest that animals feeding in groups become aggressive only if aggressiveness pays in terms of energy gain. As stressed by Stillman et al. (1997), explanations of interference patterns based on a description of individual behavior (Beddington, 1975; Moody and Houston, 1995; Ruxton et al., 1992) neglect the adaptive component of this behavior. Two recent theoretical studies considered this aspect. The first one relied on a detailed simulation model on interference in the European oystercatcher Haematopus ostralegus (Stillman et al., 1997). The second one studied the behavior of kleptoparasites (Broom and Ruxton, 1998). In the present paper, I make allowance for the fact that individuals may be more or less prone to resort to kleptoparasitism, but also more or less prone to defend their own prey against kleptoparasites (Norris and Johnstone, 1998). Address correspondence to E. Sirot. sirot@univ-ubs.fr. Received 21 December 1998; revised 3 September 1999; accepted 3 October International Society for Behavioral Ecology In the wild, the level of aggression between animals foraging in groups is influenced by food availability (Dolman, 1995; Kotrschal et al., 1993; Smith and Metcalfe, 1997; Stokes, 1962) and density of animals in the group (Fenderson and Carpenter, 1971; Goss-Custard, 1980; Jones, 1983; Vines, 1980). I studied, on a theoretical basis, the influence of these two factors on the level of aggressiveness in groups of animals. Both time and energy costs of aggressiveness are taken into account in the model. Aggression is taken in a broad sense and includes all levels of disturbance in food searching that can be imposed by one animal on another (Goss-Custard, 1980). These levels of aggressiveness may vary considerably, even within a species, from a simple display of threat to a real attack (Brown, 1964). However, as most of these fights do not lead to injuries, the model does not take into account direct mortality caused by injuries [see Houston and McNamara (1991) for a study on fatal fighting]. Nor does the model consider the decrease of feeding rate due to food depletion (see Free et al., 1977). In order to take into account both time and energy losses caused by conflicts, the model uses the rate of energy gain to assess the pay-off of the different possible strategies (Stephens and Krebs, 1986). In addition, the game theoretical approach permits consideration of the influence of others behavior on the fitness of each individual, which is necessary when studying interactions between individuals (Maynard Smith, 1982). As a starting point, the hawk-dove model (Maynard Smith and Price, 1973) provides a general basis for the representation of conflictual encounters. The model Description of foraging behavior The model considers a group of conspecific animals searching for food simultaneously. Food items (or prey) are randomly distributed over a given patch of land. If an animal locates a food item while not in the vicinity of a competitor, it eats this food unreservedly. If this animal locates a feeding animal, or if it is spotted while feeding, it may choose to be aggressive and be ready to fight for the food or to withdraw at once if its opponent is aggressive. The aim of the model is to predict the proportion of time each animal should be aggressive and the proportion of time it should avoid confrontation, taking into account ecological conditions (food availability and density of competitors). I assume that conflicts never involve more than two animals, and, following Maynard Smith and Price

2 352 Behavioral Ecology Vol. 11 No. 4 (1973), I refer to the aggressive strategy as the hawk strategy and to the nonaggressive, timorous one as the dove strategy. When an animal playing dove meets an animal playing hawk, it flees and leaves the food to the latter. When two hawks compete for a food item, they stand and eventually fight for a certain time, the probability of gain being 0.5 for each of them. When two doves meet, the confrontation is settled quickly and peacefully, and the probability of gain is once again 0.5. Since the outcome of confrontation does not depend on who is the attacker and who is the defender, the strategy of the individual is represented by a unique variable, P, which is the probability that it will be a hawk. Representation of the outcome of animal conflicts uses the following parameters: V: energy gain for the consumption of one food item. T: average duration of a confrontation between two animals playing hawk. C: energy loss per time unit of confrontation between two hawks. With these parameters, the average payoff in energy is V/2 CT for a hawk encountering a hawk and V for a hawk encountering a dove. It is 0 for a dove encountering a hawk and V/2 for a dove encountering a dove. New parameters represent the environment in the model: : encounter rate with food items or prey. reflects food abundance. h: handling time for one food item or prey. D: density of foraging animals in the group. 1 : probability of being in the vicinity of a competitor while spotting or consuming a food item. This means that the animal will have to face an opponent for the possession of this food, be it the one that comes upon a feeding animal or the one that is approached while feeding. 2 : probability that the prey encountered is already the object of a fight between two animals playing hawk. In this case, we will assume that the animal will not take part in the dispute and will go on searching for other food items. Appendix A shows how one can calculate 1 and 2. In what follows, E(P, P ) is the payoff (i.e., the average rate of energy gain) for an individual playing strategy P (i.e., being a hawk with probability P, and a dove with probability Q 1 P), in a population in which the animals are hawks with probability P, and doves with probability Q 1 P. Each individual encounters prey per unit of searching time; 1 of them will also be coveted by a competitor, which will induce an encounter between the two animals, while 2 of them are already the object of a fight, and the studied animal will withdraw. The remaining (1 1 2 ) prey will be eaten unreservedly. We will now focus on the 1 encounters opposing the animal with one and only one competitor. The outcome of these encounters will depend on the respective strategies P and P of the two opponents. A proportion PP of these encounters will lead to a confrontation with an unpredictable winner, a proportion PQ will lead to a prompt victory of the studied animal, a proportion QP will lead to a rapid withdrawal from its part, and a proportion QQ will lead to a quick and peaceful settlement because both animals avoid intensification of events. When the two animals meet, one of them has already invested on the average h/2 time units handling the food in activities such as extracting seeds or prey from soil or burrows or breaking shells. Each opponent has the respective probability 0.5 of being the defender or the attacker. Therefore, on average, every individual involved in a contest has already invested h/4 time units for the food item beforehand. With these specifications, an animal playing hawk encountering another hawk spends, on average, T h/4 additional units of time in the encounter and the consumption of the food because it has a 0.5 probability of winning. It will lose no time in combat and spend on average h/2 units of time handling the food if it meets a dove. An animal playing dove will lose no additional time if it meets a hawk because it will flee. It will spend an average h/4 more time units if it meets another dove because it has a 0.5 probability of winning and consuming the food. I assume that the average rate of energy gain is equal to the ratio of the average energy gain for one period of searching time and the total time spent searching, consuming and fighting for this food [see Turelli et al. (1982) for a justification of this approach]. We therefore obtain: E(P, P ) { (1 )V 1 2 [PP (V/2 CT) PQ V QQ V/2]} 1 {1 (1 )h 1 2 [PP (h/2 T) PQ 3h/4 QP h/4 1 QQ h/2]}. (1) The evolutionarily stable level of aggressiveness The evolutionarily stable level of aggressiveness, P*, is reached if no mutant can invade a population in which all animals play strategy P*, which means that no other strategy can beat strategy P* in this population. P*, the evolutionarily stable strategy, or ESS (Maynard Smith, 1982), is characterized by the following mathematical property: E(P, P *) E(P *, P *) for P P *, or E(P, P *) E(P *, P *) and E(P, P) E(P *, P), for P P *. (2) Appendix B shows how one can calculate the value of P*. It shows that for any given set of parameters (, D, V, h, C, T), at least one ESS exists. The equations do not permit an analytical study (see Appendix A), but repeated calculations, performed over wide ranges for all parameters always yield a unique ESS, so we may reasonably assume that for all conditions a single ESS P* exists. Because natural populations are assumed to be at the evolutive equilibrium, P* is a prediction for real behavioral patterns. Foraging success and interference The individual feeding rate in a population playing the ESS is calculated by setting P P P* and Q Q 1 P* in Equation 1: Feeding rate E(P*, P*). Following Hassell and Varley s (1969) example, experimentalists study intraspecific interference by plotting individual feeding rate (or parasitization rate; Hassell and Rogers, 1972) as a function of animal density on logarithmic axes (Sutherland and Koene, 1982). To compare the predictions of the model with the results of other studies on interference, I plotted log(feeding rate) as a function of log(animal density) for different levels of food availability in the environment. In the present model, the effect of aggressiveness on feeding rate may stem from two assumptions: first, the assumption that the animals are aggressive and use up some time and energy in disputes over food, and second, the assumption that they do so adaptively and change their level of aggressiveness according to environmental parameters. To separate the effects of these two factors, I also plotted log(feeding rate) as a function of log(animal density) for a population with a fixed level, P, of aggressiveness. This last curve takes into account the effect of aggression but not that of adaptative flexibility in behavior.

3 Sirot An ESS for aggressiveness 353 RESULTS The evolutionarily stable level of aggressiveness This section studies the influence of parameters, D, V, h, C, and T on the predicted level of aggressiveness, P*. Repeated calculations for different values of the parameters provided qualitative but unambiguous predictions, and Figure 1 shows representative examples of the data. In each graph of this figure, P* is plotted as a function of encounter rate with food,, and density of foraging animals, D. In addition, the values of parameters V, h, C, and T are changed between graphs to show their influence on P*. The first conclusion that can be drawn from Figure 1a is that both food availability and density of foraging animals influence the predicted level of aggressiveness. For a fixed level of food availability, animals become more aggressive as their density increases, but the increase is smooth, and in most situations the animals should be partially aggressive (i.e., 0 P* 1). Appendix B shows that the pure dove strategy is never an ESS, and the numerical results show that P* does not tend to 0 when D tends to 0 for a fixed level of food availability. Therefore, the animals should always be at least partially aggressive, even if their own density is low. On the other hand, the ESS can be the pure hawk strategy if density of competitors is high and food availability is low. The model also predicts that the level of aggressiveness should rise as food abundance decreases. For a fixed animal density, P* always converges to 1 as tends to 0. The particular case where 0 was not presented because it has no biological meaning, but values of P* obtained for low values of illustrate the tendency. Although the pure dove strategy is never an ESS, the numerical results show that, for a fixed animal density, the ESS P* always converges to 0 as tends to infinity. In this case, the average searching time to find food tends to 0, and the optimal strategy is to always avoid conflict and go on searching for undisputed food. Increasing food energy value, V, while keeping other parameters constant always increases the value of P*. Therefore, the model predicts that animals searching for food items of high energy value should be more aggressive toward one another. This result stems from the game-theoretical approach of the model. As V increases, the payoff for animals playing hawk increases when they encounter a dove, and the hawk behavior becomes more advantageous. Increasing the value of h while keeping other parameters constant also increases the value of P*, so animals are expected to be more aggressive when the prey are, for example, more difficult to break open. The reason is that increasing handling time, h, while keeping fighting time, T, constant decreases the relative time cost of aggressive behavior compared to other activities. Again, the hawk attitude becomes more advantageous. In addition, longer handling times increase encounter rates between animals. In Figure 1b, parameters V and h are increased in the same proportions, compared to Figure 1a, and food profitability, V/h, is the same in both cases. V and h are higher in the second case, and the animals are more aggressive. Hence, the model predicts that food profitability, whose influence on foraging behavior is well known (Stephens and Krebs, 1986), cannot solely determine the level of aggressiveness in a feeding group. If two kinds of prey have the same profitability, the one with the highest energy value will arouse more aggressiveness in the group. In Figure 1c, the energy expenditure per time unit of agonistic encounter, C, is increased, while the average duration of the conflict T is reduced, compared to Figure 1a. Fights become shorter and more violent, with the same total energy cost, CT. As a result, animals are more aggressive for all possible levels of food availability and density in the group. The Figure 1 Influence of food availability and density of animals on the level of aggressiveness. The evolutionarily stable level of aggressiveness (P*) is represented as a function of encounter rate with food items ( ) and density of foraging animals (D). Parameter values: (a) a 0.1, V 10, h 1, C 1, T 2. (b) Increased food energy value and handling time; a 0.1, V 40, h 4, C 1, T 2. (c) Increased energy cost and reduced time cost of fighting; a 0.1, V 10, h 1, C 4, T 0.5.

4 354 Behavioral Ecology Vol. 11 No. 4 Figure 2 Effect of the evolutionarily stable level of aggressiveness on the feeding rate of animals.the individual feeding rate in groups of animals playing the evolutionarily stable strategy is represented, on a logarithmic axis, as a function of encounter rate with food ( ) and density of animals in the group (D, logarithmic axis). Parameter values: a 0.1, V 10, h 1, C 1, T 2. explanation is that confrontations now cost less time, which increases the average rate of energy gain for aggressive animals. Figure 3 Effect of different behavioral rules for aggressiveness on individual feeding rate. The individual feeding rate is plotted as a function of animal density, on logarithmic axes, for a population of pure dove strategists (dotted line: P 0), for a population displaying a constant level of aggressiveness (broken line: P 0.223), and for a population playing the evolutionarily stable strategy (solid line: P P*). Parameter values: 2, a 0.1, V 10, h 1, C 1, T 2. Foraging success and interference Figure 2 shows the influence of food availability in the environment on the interference curve log(feeding rate) f [log(animal density)]. For a fixed level of food abundance, the interference curve is curvilinear, suggesting that it may seem linear only if animal density describes a small interval. The interference coefficient, m, which measures the strength of interference, is defined as the absolute value of the slope of the linear regression between log(animal density) and log(feeding rate) (Hassell and Varley, 1969). Figure 2 predicts that the value of m will depend on the range of animal densities over which it is calculated. It will be higher if it is calculated over a range of high animal densities. Because the actual relationship is not linear, calculating m over the whole range of possible densities is pointless. However, m may be calculated for every possible animal density as the absolute value of the slope of the interference curve for that particular point. In addition, the model predicts that, for a fixed density of animals in the group, the interference coefficient will depend on the level of food availability. It will always be lower if food is abundant because the interference curve is less steep. Figure 3 represents the interference curve for a fixed level of food availability and for different kinds of populations. The members of the first population are never aggressive (they play the fixed strategy, P 0). Members of the second population are partially aggressive, with a constant level of aggressiveness (P 0.223; this value corresponds to the ESS when animal density, D, is set to 1, its lowest value). Members of the third population play the ESS, P*. Their aggressiveness changes with their density. The three curves of the figure have a concave shape, so interference is predicted to increase with animal density in the three cases. The curves, however, cannot be superposed. For a given animal density, the slope of the curve is always steeper for the second population than for the first one, and steeper for the third population than for the second one. In the first population, animals are never aggressive, but their feeding rate decreases when their density is high because they share more and more food as the frequency of encounters with other animals increases. The decrease in feeding rate is accentuated in the second population, in which animals display a fixed level of aggressiveness. Their feeding rate decreases not only as a result of energy loss in confrontations, but also because, for one period of searching time, they spend more time in these confrontations. In the third population, animals are more aggressive as their density increases, and the decrease in their feeding rate is even more accentuated. This is so because energy and time losses increase more rapidly when the encounter rate with other animals increases. DISCUSSION Using a game theoretical model, this study investigated in what conditions gregarious animals should be aggressive with one another while searching for food. Both time and energy costs of aggression are incorporated in the model because they were observed in experimental studies (Ens et al., 1990; Goss-Custard, 1980; Stillman et al., 1997). The model shows that these two types of costs should influence individual behavior. It also predicts that aggressiveness should strongly vary with ecological conditions. The model predicts a progressive increase in the level of aggressiveness as density of the group increases, which implies that, for a broad set of ecological conditions, the animals should be partially aggressive. Experimental studies suggest that the level of aggressiveness of animals is flexible (Goss- Custard et al., 1998; Vines, 1980) and progressively increases with density in the group (Burger et al., 1979; Goss-Custard, 1977, 1980; Smith and Metcalfe, 1997; Vines, 1980). The original hawk dove model (Maynard Smith and Price, 1973) predicted partial preferences for aggressiveness. The present model shows how these partial preferences may vary with ecological conditions. Previous studies highlighted two other sources of behavioral variability in populations playing the hawk dove game: physiological differences among individuals (Houston and McNamara, 1988) and errors in decision-making (McNamara et al., 1997). The present model also studies the variations in aggressiveness linked with variations in food abundance. In accordance with recent field observations (Cresswell, 1998; Dolman, 1995; Kotrschal et al., 1993; Triplet et al., 1999), the model predicts more intense interference competition at low prey density. Several authors (Norris and Johnstone, 1998; Sutherland and

5 Sirot An ESS for aggressiveness 355 Koene, 1982; Vines, 1980) suggested that the influences of food availability and animal density on individual feeding rate may be confused in the field because high densities of animals are often found in areas where food is abundant (Goss-Custard et al., 1984). The present study confirms the need to assess both resource availability and density in the group while studying interference. It predicts that experimental studies in which the gradient of animal density reflects a gradient of food abundance might underestimate the interference coefficient. In a recent study, Dolman (1995) traced interference curves for flocks of wintering snow buntings Plectrophenax nivalis foraging on patches with high or low density of seeds and showed a clear influence of food availability on the interference coefficient. He found no significative effect of group density on food intake rate on rich patches and a curvilinear interference curve for poor patches, which suggests that interference increases considerably with animal density in this kind of environment. These curves are similar in shape to the interference curves predicted by the present model (see Figure 2), so this model may provide a behavioral explanation for this particular pattern. In any case, the conclusion that the interference coefficient should be higher when food density is low is in agreement with the conclusion of Dolman s study on snow buntings. For a fixed density of food, the interference curves obtained with the model are qualitatively equivalent to those measured in experimental studies (Dolman, 1995; Stillman et al., 1996) and with those predicted by other models that use an individual-based approach to describe interference (see Moody and Houston, 1995). Interference coefficient is predicted to increase with animal density, from negligible to important values, which dramatically reduce individual foraging success. However, the adaptive patterns of aggression described here increase the value of the interference coefficient, compared to a model that neglects possible agonistic encounters between the animals or compared to a model that assumes a fixed level of aggressiveness (see Figure 3). The present model suggests that a part of the interference measured in the field should be attributed not only to aggressive behavior but also to its adaptive component. Through its effect on individual fitness, interference competition has a stabilizing effect on population dynamics (Free et al., 1977; Hassell and Varley, 1969), and aggressiveness is known to regulate density in animal populations (Ayer and Whitsett, 1980). The present model, which predicts a positive correlation between the animal density and the interference coefficient, reinforces this conclusion. The stabilizing effect is, in fact, twofold because the model also predicts that animals should be more aggressive if food is scarce. Hence, if the population of prey is sparse, animals feeding on them will lose much time in agonistic encounters, their feeding rate will be low, and the prey will be somehow protected. On the other hand, if prey are abundant, the predators will not be very aggressive toward one another. Their feeding rate will be high and the prey population will be depleted at a high rate. Hence, the strategies of aggression predicted by the model are bound to contribute to stabilization of the dynamics of predator prey systems through their effect on both predator and prey populations. APPENDIX A Determination of encounter rates The probability that the animal will have to face an opponent before consuming a given food item is 1, whether the animal is the first or the second one to reach this food. The probability that an encountered food item is already disputed by two animals playing hawk is 2. Both 1 and 2 depend on ecological parameters and level of aggressiveness, P in the population. I consider the time budget of an animal in this population. The proportions of time spent searching, feeding, and fighting by this animal are derived from Equation 1. They are, respectively: 1 1/{1 (1 1 )h 2 [P 2 1 (h/2 T) P Q 3h/4 Q P h/4 Q 2h/2]}, 2 [ (1 1 )h (P 1 h/2 P Q 3h/4 Q P h/4 Q h/2)] {1 (1 1 )h 2 [P 2 1 (h/2 T) P Q 3h/4 Q P h/4 Q 2h/2]}, and ( P T) {1 (1 1 )h 2 2 [P 1 (h/2 T) P Q 3h/4 Q P h/4 Q 2h/2]} The number of fights taking place simultaneously, and per surface unit, is D 3 /2. Similarly, the density of food items being handled by animals is D 2. The area of discovery for a searching animal is a (i.e., the area it visits per unit of searching time; Hassell and Varley, 1969). Because this animal encounters prey per time unit, food density in the environment is /a. The proportion of food items that are the object of a fight is 2. We get 3D/2 2 (A1) /a The sum of the probability that the prey is already being handled while encountered and the probability that it is not handled, nor disputed, when encountered, but sighted by a searching animal before the animal has finished to handle it is 1. We get 2D 2D a 1Dh (1 e ) (A2) /a /a Equations A1 and A2 allow one to numerically compute probabilities 1 and 2 for a population in which all animals play strategy P. APPENDIX B Determination of the evolutionarily stable level of aggressiveness, P* The first step is to determine the optimal strategy that may be played in a population in which all animals play strategy P. Therefore, we derivate E(P, P ) with respect to P. This operation yields E F(P ) (P, P ), P [G(P, P )] 2 where F(P ) is a function of P only, while G(P, P ) is a function of both P and P. This implies that the sign of ( E/ P)(P, P ) depends on P only. Depending on the value of P, three cases must be considered:

6 356 Behavioral Ecology Vol. 11 No. 4 E (P, P ) 0. a. P The best strategy that may be played within the population is P 1. P is an ESS if and only if P 1. E b. (P, P ) 0. P The best strategy that may be played within the population is P 0. P is an ESS if and only if P 0. E c. (P, P ) 0. P All possible strategies have the same payoff in the population. We see easily that ( E/ P)(P, P ) 0 for P 0, which means that a pure dove population will be invaded by any aggressive mutant, and P 0 is never an ESS. Finally, two cases must be considered: a. ( E/ P)(P, P ) 0 for all P values. There is one and only one ESS: P* 1. b. ( E/ P)(P, P ) 0 for P 1. As ( E/ P)(P, P ) is a continuous function of P, ifp is fixed, there is at least one value of P in the interval [0,1] such that ( E/ P)(P, P ) 0. We call the first one P 1.Ina population that plays P 1, all strategies have the same payoff, and any mutant strategy may subsist. It will not, however, be able to invade the population. Indeed, any beginning of invasion by a mutant strategy would drive the average level of aggressiveness in the population away from P 1, in a way that would decrease the fitness of the mutant compared to that of the animals which play strategy P 1. So the representation of the mutant in the population would decrease until it is negligible. Therefore P 1 is an ESS. If the equation ( E/ P)(P, P ) 0 has three or more solutions between 0 and 1, then the system admits more than one ESS. It cannot be proven analytically that this never happens because the equations have to be solved numerically. However, this situation never occured in the computations, though wide ranges of parameter values were explored. So we will assume that in any case a unique ESS exists. I thank Gerard Driessen, Vlastimil Křivan, and two anonymous reviewers for helpful comments on earlier versions of the manuscript and Sarah Le Faouder for improving the English. 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