Behavioral Ecology Vol. 14 No. 1: 2-9
© 2003 International Society for Behavioral Ecology
Resource defense in a group-foraging context
aDépartement des Sciences biologiques, Université du Québec à Montréal, Case postale 8888, succursale Centre-Ville, Montréal, QC H3C 3P8, Canada bDepartment of Biology, Concordia University, 1455 ouest, bd. De Maisonneuve, Montréal, QC H3G 1M8, Canada
Address correspondence to L.-A. Giraldeau. giraldeau.luc-alain{at}uqam.ca.
Received 9 November 2001; revised 14 March 2002; accepted 8 April 2002.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONREFERENCES |
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When foraging in groups, animals frequently use either scramble or contest tactics to obtain food at clumps found by others. The question of which competitive tactic should be used has been addressed from two different perspectives: a simple optimality approach and a game theoretic approach. Surprisingly, both approaches make strikingly different predictions about how per-capita frequency of aggression within groups should change as a function of food abundance and competitor density. Resource defense theory typically predicts dome-shaped relationships between the per-capita frequency of aggression and both food abundance and competitor density, whereas game theoretic models predict an increase in aggression with competitor density and a decline in aggression with increased food abundance. We developed a game theoretic model to explore whether the predictions of resource defense theory and the game theoretic approach can be reconciled. Our model assumes that players have different competitive abilities and can adopt roles of either finder or joiner that affect the quantity of food that can be gained from a food clump. In accordance with earlier game theoretic models, we predict an increase in aggression with competitor density when animals compete by pair-wise contests. However, when food clumps can be challenged by more than one competitor, both the costs and benefits of defending increase with competitor density, which results in a dome-shaped relationship between the two variables. Our model predicts that aggression should always decrease as the density of food clumps increases.
Key words: aggression, competitor asymmetry, evolutionarily stable strategy model, finder's advantage, foraging groups, resource defense.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONREFERENCES |
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When animals forage in groups, they often eat food found by others (Giraldeau and Beauchamp, 1999
). This social foraging behavior can lead to intraspecific variation in competitive tactics that can vary from aggressive encounters to apparently peaceful scrambles for the food (Giraldeau and Caraco, 2000). Gregarious populations of Barbados ground doves (Zenaida aurita), for instance, switch from aggressive food defense with escalated fighting to scramble tactics depending on the size and temporal predictability of the food patches they exploit (Goldberg et al., 2001). Similar behavioral plasticity in aggressive responses has been noted in a number of animal taxa including fish (Grant et al., 2002), birds (Zahavi, 1971), and mammals (Monaghan and Metcalfe, 1985).
The factors that govern the use of competitive tactics, including aggression, have been modeled from at least two different perspectives: a simple optimality approach used to predict the economic defendability of resources (Brown, 1964; Emlen and Oring, 1977; Grant, 1993) and a game theoretic approach aimed at predicting the occurrence of escalated fighting within populations (Maynard Smith, 1976; Maynard Smith and Price, 1973; Sirot, 2000). Simple optimality models have focused primarily on the resource characteristics that promote resource defense (i.e., their economic defendability; Brown, 1964). For example, optimality models predict that economic defendability increases as a resource becomes more clumped in space but less clumped in time (Grant, 1993).
The hawk-dove game, a classic example of the game theoretic approach, pits two strategies, one that escalates (hawk) and one that does not (dove), against each other and evaluates the evolutionarily stable strategies (ESSs) under different sets of conditions (Maynard Smith and Price, 1973). When both opponents play dove, they equally share the resources without any costs for sharing, but when meeting a hawk, the dove leaves all the resources to the hawk without a fight. However, when both opponents play hawk, they engage in an escalated fight until one of them decides to retreat. Although the aggressive hawk strategy can be an ESS in some cases, the model predicts that the nonaggressive dove can only exist as part of a mixed ESS. The frequency of hawk in the mixed ESS, and hence the proportion of encounters resulting in escalated fights, depends on the relationship between the value of winning and the cost of losing an aggressive encounter.
Both approaches make strikingly different predictions about how per-capita frequency of aggression within groups should change as a function of resource abundance and competitor density. Resource defense theory typically predicts that an individual's aggressiveness, often measured as the proportion of encounters with competitors that result in aggression (Robb and Grant, 1998), should increase and then decrease with an increase of either food abundance or competitor density. When food is scarce or competitors abundant, the energetic value of the food does not exceed the energetic costs of aggression. As the number of animals competing for a food clump decreases, as a result of an increased food abundance or a decreased number of foragers within the group, the net benefit of fighting increases, and so should the frequency of aggressive interactions. However, when food is abundant or competitors scarce, nonaggressive individuals can acquire the same amount of food as aggressive individuals without paying the cost of aggression (Grant et al., 2000, 2002). So resource defense theory predicts that the total number of aggressive acts, the per-capita number of aggressive acts, and the intensity of aggressive encounters should follow a dome-shaped relationship with respect to either food or competitor abundance. The hawk-dove game, in contrast, predicts that the frequency of aggressive individuals (hawks) in the population should increase with competitor density but decrease with increased food abundance (Sirot, 2000). When the frequency of hawks increases in the population, peaceful sharing becomes rarer, as do interactions with doves. Conversely, increasing the proportion of aggressive individuals increases the probability that two hawks meet, and hence increases the proportion of encounters resulting in escalated fights. So the hawk-dove game predicts that the average intensity of fighting, the frequency of fighting, and the per-capita frequency of fighting should increase as competitor density increases but decrease with increased food abundance. Experimental evidence for dome-shaped responses of aggression exists with respect to competitor density (Goldberg et al., 2001), food abundance (Carpenter and MacMillen, 1976; Grant et al., 2002), and competitor-to-resource ratio, which integrates competitor number and food abundance (Grant et al., 2000). In contrast, however, many observations of birds foraging in shared patches support the linear predictions of the hawk-dove game (see Sirot, 2000).
In this study we extended the game suggested by Sirot (2000) to reconcile the contrasting predictions of resource defense theory and the game theoretic approach. Sirot's game involves symmetric players competing in pairs for single prey items in which one player has already invested some handling without any energy gain. However, most instances that involve competition within groups involve asymmetric players. In addition, when animals forage in groups, they often exploit clumps of food that can be shared (Giraldeau and Caraco, 2000). When group foragers exploit clumps of food, the finder often obtains a fraction before it is joined, and possibly displaced, by other group members (Giraldeau et al., 1994). Finally, when animals forage in groups, a large number of individuals often converge at the clump to partake of the discovery, forcing the finder to defend the resource against a large number of intruders (Chapman and Kramer, 1996). Game theoretic models are often highly sensitive to such assumptions. Therefore we believe it is important to extend Sirot's model (2000) to consider the potential consequences of these modifications. In this study we modified the game suggested by Sirot (2000) by allowing clumps of food that can be shared, asymmetric players that have different probabilities of winning confrontations, and two roles, finder and joiner, that affect the quantity of food that can be gained from a clump. Finders obtain a finder's advantage, a portion of a discovered clump it eats before the arrival of joiners (Beauchamp and Giraldeau, 1996; Giraldeau et al., 1990; Vickery et al., 1991). In addition, we considered both situations where individuals compete by pair-wise contests and situations allowing for the presence of more than two competitors on the clump. We asked how these changes toward greater realism affect the relationship between per-capita aggression, competitor density, and food abundance.
A two-forager hawk-dove game with energetic costs of fighting
We first considered a group in which resources are contested by pairs of foragers. The animals can compete either by using an escalated aggressive hawk behavior or a nonfighting dove behavior. When two doves meet, they share the resource and hence adopt a scramble competitive tactic. When an individual playing dove meets one playing hawk, the hawk threatens the dove, who immediately retreats, leaving behind all of the remaining resource for the hawk; an aggressive displacement has occurred. However, when two hawks meet, they engage in an escalated fight until one of them, based on the asymmetry in fighting abilities, decides to retreat. As in Sirot (2000), the losing hawk is not injured. Like Sirot (2000) we suppose that individuals do not simultaneously arrive at the contested resource; one is the finder and the other the joiner. However, unlike Sirot (2000), we assume that this role asymmetry affects the payoffs available to the players because only the finder can have access to the finder's advantage—that is to a portion of the total resource it can consume exclusively before the joiner arrives. Because each individual can play either hawk or dove, there are four potential conditional strategies:
- (H,H) Both animals play hawk.
- (H,D) The finder plays hawk, the joiner plays dove.
- (D,H) The finder plays dove, the joiner plays hawk.
- (D,D) Both animals play dove.
- (H,D) The finder plays hawk, the joiner plays dove.
We assume that the replacement value in terms of fitness units of losing a resource is given by the resource's energetic content (but see Parker, 1984). The resource corresponds to a clump of food that contains F indivisible items. The animal that discovers a food clump obtains a finder's advantage, a portion a (0 
a F) of the total resource that it can use exclusively before the joiner arrives (Giraldeau et al., 1990; Mangel, 1990; Ranta et al., 1996; Vickery et al., 1991). Animals are assumed to differ in their fighting abilities. Consequently, when both play hawk, the asymmetry in fighting ability influences which individual retreats first. We consider that each animal is characterized by its competitive ability relative to the average competitive ability of the other group members. So we express the asymmetry between two contestants, one that finds and the one that joins, by the parameter x (0 < x < 1), which is the probability that the finder wins the remaining (F - a) items against an average individual from the population and (1 - x) is its probability of losing the contest against an average individual. A value of x > 0.5 indicates that the finder has a higher competitive ability than its average opponent, whereas a value < 0.5 implies that it has a lower competitive ability than the average population member. A pair of contestants is increasingly asymmetric in terms of fighting ability as x moves away from 0.5 in either direction. It is assumed in this model that individuals have perfect information about their competitive ability relative to that of their opponent and their role (finder or joiner). Escalated fighting is energetically costly for both the winner and the loser; C represents this energetic cost.
There are four possible payoffs for both the finder and the joiner. Let EF(H,H) be the payoff to the finder when both animals play hawk, EF(H,D) be the payoff to the finder when it plays hawk and the joiner plays dove, and so on. Similarly, let EJ(H,H) be the payoff to the joiner when both animals play hawk, EJ(H,D) be the payoff to the joiner when the finder plays hawk and it plays dove, and so on. Figure 1 shows the payoff matrix of this game.
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To determine under which conditions each strategy is an ESS, we compare the payoff of the joiner when it plays hawk and dove for a given behavior of the finder and then deduce from the choice of the joiner the finder's best decision given a range of asymmetries between players. We assume that the finder and the joiner decide sequentially whether to compete aggressively because the finder arrives on the clump before the joiner and hence has more information about the value of the resource for which they compete when it is joined. As a consequence, the time required for the joiner to gain sufficient information about the benefits associated with each alternative and decide to play hawk or dove is likely longer than for the finder.
Analysis
The joiner's best strategy
If the finder plays hawk, the joiner should also play hawk when EJ(H,H) > EJ(H,D). Resolving this inequality, the joiner should always play hawk against a hawk-playing finder when x < (F - a - C)/(F - a). This is most likely to be the case when player asymmetry is large and the finder's likelihood of winning is small, or when the clump is large such that the finder's advantage is relatively small or when the costs of fighting are negligible. If the finder plays dove, then the joiner should play hawk when EJ(D,H) > EJ(D,D), which is always true except in the trivial case where a = F, which is of no interest because there is no food left for joiners anyway.
The finder's best strategy
From above we know that the joiner will always play hawk when x < (F - a - C)/(F - a). When this happens the finder's best strategy is to play hawk whenever EF(H,H) > EF(D,H)—that is, when the contestants' asymmetry increases and the player's probability of winning is x > C/(F - a).
In the situation where x > (F - a - C)/(F - a), the joiner's best strategy will depend on the finder's strategy. The joiner should play hawk when the finder plays dove but should play dove when the finder plays hawk. Knowing this, it follows that the finder's best strategy is to play hawk when EF(H,D) > EF(D,H), which is always true because it requires F > a.
Predictions
We can define three ESSs areas in the a - x plane (Figure 2):
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(H,H) and hence frequent escalated fighting is an ESS if C/(F - a) < x < (F - a - C)/(F - a),
(D,H) and hence frequent aggressive displacements is an ESS if x < (F - a - C)/(F - a) and x < C/(F - a), and so frequent aggression of moderate intensity is expected in this kind of group,
(H,D) and hence unchallenged finding is an ESS if x > (F - a - C)/(F - a), and so no aggression but no sharing is expected in this kind of group.
The model predicts that one of the two individuals will always be willing to escalate and play hawk, so (D,D), the peaceful sharing of the food clump by the finder and its joiner, is never an ESS in this game. Which solution qualifies as an ESS depends on the size of the finder's advantage, the extent of expected asymmetry between players, and whether the finder likely wins or loses. When opponents are closely matched and the expected asymmetry is low (i.e., when x = 0.5) there are only two ESS solutions: (H,H) for small finder's advantages up to a < (F - 2C) and for finder's shares beyond this threshold (H,D) is the ESS. So, for near symmetric populations, escalated fights should be observed for small finder's advantages and energetic costs of the fight. For large finder's advantage or large costs of fights, it does not pay to challenge the finder; the dove saves the energetic cost of the fight that would at best provide it with only small remaining amounts of food. Thus, for any value of a, F, and C, the finder should always escalate when provoked, but the joiner should respond by escalation only when the finder's advantage (a) is small and less than (F - 2C).
Increasing the competitor asymmetry reduces the area in which escalated interactions occur (Figure 2). For small finder's advantages, the finder should play dove when its x is small and it is likely to lose, while joiners should play hawk. As the asymmetry between players decreases, we enter a zone of (H,H). As asymmetry increases again, but the finder now expects to win more often, it should play hawk more such that the ESS is (H,D). At a larger finder's advantage, the increase in asymmetry simply switches from (D,H) when the finder expects to lose to (H,D) when it expects to win without any (H,H) transition when players are nearly symmetrical. At very large values of the finder's advantage or very large energetic costs of escalation (a > F - 2C) the ESS is (H,D), regardless of the degree of asymmetry. In other words, it never pays an individual to attempt to escalate against a finder when the finder's share is so large that the remaining food available to the joiner would not cover the energetic costs of the fight. So the model predicts that escalated fighting will be common when the finder's advantage is moderate and the players are closely matched in fighting abilities.
The finder's advantage is usually assumed to depend on the time required for joiners to reach the discovered food clumps. Increasing the density of players likely reduces the intercompetitor distance and should reduce the finder's advantage. If that assumption is correct, then for intermediate levels of competitor asymmetry, increasing population density moves the population from an ESS of (H,D) of unchallenged finders and no aggression to (D,H) with aggressive displacements with little escalated fighting to all-out escalation (H,H) over every food clump discovery. Hence, the intensity of aggression is expected to increase with population density.
Increasing food abundance can be achieved in two ways: increasing the total number of food items in a clump or increasing clump density. Increasing the number of items per clump will have little effect on the absolute size of the finder's advantage but will reduce its relative importance. Increasing F, therefore, for intermediate levels of asymmetry among players moves the population in the same direction as an increase in population density: little aggression when F is small, and as F increases, a step increase in aggression when the (H,H) threshold is crossed. Increasing clump density, in contrast, will have no effect on the finder's advantage and hence on the level of aggression. To summarize, this game predicts an increase in aggressiveness with increased competitor density and clump richness, but no effect of clump density on the use of competitive tactics.
Up to now we have been interested in determining the conditions under which a pair of animals should escalate fighting when one discovers a clump of food. We have not considered the consequences of each individual's decision on its future rate of food intake. However, if fights are energetically costly, they are also likely to involve a time cost. Consequently, the benefits of playing hawk may depend both on the value of the resource, the costs of fighting, and the foragers' opportunities of discovering other food clumps.
A two-forager hawk-dove game with temporal costs
The game is as above, except that we now express the payoffs of both strategies as the rate of energy intake, E/T. The encounter rate with food clumps is 
, and the mean duration of a contest is t. Figure 3 shows the payoff matrix of this game for a symmetric population.
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Analysis
When the finder plays hawk, the joiner should also play hawk when EJ(H,H) > EJ(H,D)—that is, when a < F - 2C. When the finder plays dove, the joiner should still play hawk if EJ(D,H) > EJ(D,D), which is true when a < F. If a > F - 2C, the joiner will play hawk if the finder plays dove but dove if the finder plays hawk. Then the finder should play hawk when EF(H,D) > EF(D,H)—that is, when a < F. This gives rise to three ESS areas in the a -
plane (Figure 4):
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(H,H) is an ESS if a < F - 2C and
< (F - a - 2C)/2at.
(D,H) is an ESS if a < F - 2C and > (F - a - 2C)/2at.
(H,D) is an ESS if a < F - 2C.
Predictions
Once again, (D,D), the sharing of resources by the finder and joiner, is never an ESS. When the competitors are symmetric, we now predict that pure hawk will be an ESS for a much narrower range of conditions than when we considered only energetic costs (Figure 4). This reduced area of pure hawk is because now the temporal costs are such that in some cases the finder has no interest in behaving aggressively when the time spent fighting is likely longer than the time needed to find another clump. For this reason, increasing both the duration of contests (t) and the encounter rate with food clumps () further reduces the area in which (H,H) is an ESS.
In this game increasing the finder's advantage will influence the ESS only for low clump encounter rates, moving from pure hawk and hence escalated fighting to mixed strategies, first aggressive displacements at intermediate values of the finder's advantage to unchallenged finding at very large finder's advantage. So in this model too, we predict that increased competitor density will lead to an increase in aggressiveness within groups inasmuch as this will lead to a smaller finder's advantage. Increasing clump richness will have a similar effect. However, now increasing clump density should lead to higher clump encounter rates and so for a given finder's advantage we predict that aggressiveness should decline with clump density.
Up to now we have considered only pairs of interacting foragers. In many instances, however, a large number of individuals converge at the clump to partake of the discovery, forcing the finder to defend the resource against a large number of intruders. To explore the effect of the number of intruders on the finder's decision whether to defend we model a situation where all group members that detect a food finder attempt to join, arrive simultaneously at the clump, forcing the finder either to chase all joiners out of the clump and it gains exclusive access to the food or to cease defending and the resource is equally shared between the foragers.
Two-forager game with temporal costs and n challengers
Now we modify the assumption of dyadic encounters at food clumps and allow the finder to be challenged simultaneously by G - 1 joiners (G 
2). Each individual concurrently searches for its own food while searching for opportunities to join resources uncovered by others, an information-sharing system (sensu Giraldeau and Beauchamp, 1999). All assumptions are as above except that now we express the finder's advantage as a function of group size such that a = [(F + 
) - G] for a > amin otherwise a = amin. As group size increases, we assume that the density of foragers increases, so that the mean interindividual distance between group members declines. Hence, the time taken to detect and reach a companion's discovery also declines, so the finder's advantage decreases linearly as group size increases, until it reaches a threshold value amin. The parameter amin corresponds to the minimal amount of total resource the finder can exploit exclusively before the joiners arrive, while indicates the number of food items by which the finder's advantage decreases when group size increases by 1.
When the joiners arrive, the finder can play one of two strategies: hawk, escalated fighting in an attempt to monopolize the clump or dove, a non-aggressive sharing of the clump with the joiners. We assume that the joiners do not compete aggressively, but escalate when provoked by the finder. So if the finder plays dove it shares the remaining (F - a) food items with the (G - 1) joiners. If the finder plays hawk, its probability of winning now depends on both its competitive ability x relative to the average competitive ability of the joiners and the number of joiners that compete for the food. As previously, we assume that during each confrontation with an opponent playing hawk the finder has an average probability x of winning (0 < x < 1). Therefore, if the G - 1 joiners all play hawk, the finder has a probability xG-1 of winning and (1 - xG-1) of losing. When the finder loses, it leaves the clump, and the victorious joiners share the remaining food. Consequently, a finder expects to experience a mean of n = (1 - xG-1)/(1 - x) escalated fights. Because each escalated fight lasts for time t with no energetic cost, we express the payoffs of both strategies as a rate of energy intake, E/T.
To test for evolutionary stability, we calculate the rate of energy intake for a finder playing either dove or hawk. When the finder plays dove, the joiners do not compete aggressively, which implies that the remaining food items are equally shared between the G foragers without any cost. So a finder's rate of energy intake when it plays dove is WD with:
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Analysis
Once again, the model predicts that pure dove can never be an ESS. Indeed, the rate of energy intake of a finder playing hawk (Equation 2b) is greater than that of a finder playing dove (Equation 1) in a population of joiners that play dove. This indicates that the finder should always aggressively defend the clump, forcing the joiners to escalate in order to prevent the finder from gaining exclusive access to the food. Pure hawk is an ESS only when [a + xG-1(F - a)]/(1 + nt) > [a + (F - a)G—that is, when the rate of energy intake of a finder playing hawk in a population of joiners that play hawk (Equation 2a) is greater than that of a finder playing dove in a population of joiners that play dove (Equation 1).
When the finder's advantage (a) increases, the finder's rate of energy intake increases whether it plays hawk in a population of joiners playing hawk or dove in a population of joiners playing dove, although at a faster rate if it does not attempt to monopolize the whole resource (Figure 5). As a consequence, the likelihood of hawk being an ESS decreases as the finder's advantage increases. Inasmuch as increasing the density of competitors reduces the finder's advantage, hawk should exist as a pure ESS more frequently in large group sizes or at high forager densities. However, as group size increases, the rate of energy intake of a finder playing hawk in a population of joiners playing hawk (Equation 2a) decreases because both its probability of winning (xG-1) decreases while the number of contests (n) increases. Consequently, from this game we should expect that the likelihood of hawk being an ESS is maximal at intermediate competitor densities. When both strategies can invade each other, they can coexist within the population. If p is the proportion of joiners playing hawk and 1 - p the proportion of joiners playing dove, we can express the rate of energy intake of a finder playing hawk, WH(p) according to p, as
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The finder should cease competing aggressively when its expected gain if it plays hawk becomes smaller than its expected gain if it plays dove. Thus the stable proportion of joiners playing hawk p* can be found by setting WD =
{a + [(F - a)G]} equal to WH(p) =
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At equilibrium,
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Predictions
The model predicts that the ESS proportion of joiners playing hawk, p*, and hence the frequency of aggression should be maximal for intermediate group sizes or competitor densities (Figure 6). When the finder can consume a large portion of the clump before the joiners arrive, the value of the remaining resource is very low, and the finder has no interest to defend. Increasing group size first leads to a decrease of the finder's advantage so that the value of the resource for which animals compete increases with group size, which is expected to increase the frequency of aggression. However, as soon as the increase in group size no longer affects the finder's advantage, the proportion of individuals playing hawk is predicted to decrease. In this case the value of the remaining resource for which animals compete is fixed at (F - amin), while the finder has a reduced probability of winning and an increased nonforaging time lost to contests when group size increases. This means that the costs of defending increase with increasing group size, which in turn decreases the value for the finder of behaving aggressively.
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Increasing the encounter rate with food clumps decreases the proportion of individuals playing hawk (Figure 7). This can be explained by the fact that when the probability of discovering new clumps of food is high, the time needed to find another clump is then shorter than the time spent in fights. For the same reason, increasing the mean duration of contests (t) leads to a decrease in the frequency of aggression.
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The frequency of aggression is predicted to increase as the number of food items per clump (F) increases, and as the finder's advantage (a) decreases. Indeed, (F - a) - (F - a)/G represents the additional gain a finder that defends can obtain compared to a finder that does not defend, which means that the benefits of defending may increase when (F - a) increases. In addition, increasing clump size (F) does not affect the mean number of contests when the finder's share (a/F) remains constant. However, because a is unlikely to be affected by the size of F, the finder's share (a/F) likely declines with increasing clump size, causing aggression to increase in frequency.
Finally, we predict that the frequency of aggression should increase with an increase in the finder's competitive ability (x). When the finder has a low competitive ability, its probability of winning is low compared to the cost of fighting, in terms of lost of opportunities to discover new clumps of food. However, the influence of competitive ability on the proportion of individuals playing hawk should be much more important when food density is low, when the finder's share is weak, and when contests last a short time.
Discussion
Competitor density
Like Sirot's (2000) model, our two-person games predict that the frequency of aggression should increase with population density, but for different reasons. Sirot's model is based on two major assumptions: that resources cannot be shared between contestants and that food discoveries are not necessarily challenged by another group member. This implies that when the density of competitors is low, a high proportion of food items are consumed without interactions. However, as competitor density increases, the encounter rate with conspecifics increases, and hence the proportion of unchallenged prey items decreases, which leads to an increase in the frequency of aggression. In contrast, we assumed that clumps of food can be shared and that the finder of the food clump can consume a certain portion of the total resource before the joiner arrives. Increasing competitor density decreases the mean interindividual distance between group members, which increases the remaining resource for which joiners can compete. Hence, the benefits of defending increase with competitor density, and so should aggressiveness and the observed frequency of aggression. However, when food discoveries can be challenged by more than one competitor, we predict a dome-shaped relationship between the frequency of aggression and the density of competitors. This arises because the time the finder has to spend fighting to chase the other competitors out of the clump increases as the number of joiners increases, while at the same time the probability that it will succeed in defending the resource declines. Consequently, increasing competitor density increases not only the benefits but also the costs of defending, and hence we predict that the frequency of aggression should peak at an intermediate population density. Resource defense theory makes a similar prediction and has been supported by several empirical studies (see Table 1).
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Food abundance
In accordance with previous game theoretic models (Broom and Ruxton, 1998
; Sirot, 2000) we predict that aggressive interactions should occur more frequently when food clumps are scarce and rich. Numerous empirical studies have demonstrated that aggression increases as resources become more clumped in space (Goldberg et al., 2001; Grant and Guha, 1993; Monaghan and Metcalfe, 1985). Because increasing the degree of spatial clumping is likely to reduce the density of food clumps by increasing their richness, these results give some support to our predictions but do not allow us to distinguish the effect of clump density from clump richness. Moreover, Grant et al. (2000) showed that increasing the ratio of competitors to the number of resources causes aggression in Japanese medaka (Oryzias latipes) first to increase and then decrease. This result is consistent with the prediction of resource defense theory and appears to contradict a key prediction of our model (see Table 1). The contradiction, however, may only be apparent. In our analysis we assumed that clump discovery is rare and occurs sequentially. However, in reality the probability of simultaneous discoveries may increase as clump density increases. Given that aggressiveness should be maximal for intermediate numbers of competitors on a clump, it follows that, if animals divide equally among the discoveries, the number of animals competing for each food discovery may decrease as clump density increases, leading to an increase followed by a decrease in the frequency of aggression.
Although there is no direct empirical evidence supporting this prediction, the results of Robb and Grant's (1998) study in Japanese medaka fish can be interpreted as an argument for a dome-shaped relationship between the number of simultaneous discoveries and the level of aggression. In their experiment they manipulated both the spatial and temporal clumping of resources. When the encounter rate with food clumps was low, (i.e., when resources were temporally dispersed), increasing the probability of simultaneous discoveries by decreasing the level of spatial clumping resulted in an increase in total aggression. However, in temporally clumped trials, chase rate was higher in the spatially clumped than in the spatially dispersed trials. Given that the probability of simultaneous discoveries may also depend on group size, experiments on aggression should control for this parameter in the future when testing for the effects of food abundance or competitor density (see Robb and Grant, 1998).
Competitor asymmetry
Our analysis predicts that when animals compete by pair-wise contests, the frequency of aggression should decrease as the expected asymmetries between contestants in both their fighting ability and their expected gain increase. Because escalated fights involve both costs and benefits, they should occur only when the expected gain for each opponent outweighs the cost of fighting. When the asymmetry in fighting ability is large, it never pays the individual with the lowest competitive ability to attempt to escalate because its expected gain is then smaller than the energetic cost of fighting. Similarly, when the finder can consume a large portion of the total resource before the joiner arrives, the remaining food available becomes insufficient to cover the cost of fighting, forcing the joiner to retreat, whatever its relative competitive ability. Because our model allows for the presence of more than two competitors on food clumps, it gives a slightly different prediction: Higher levels of aggressiveness are expected in groups exhibiting a large asymmetry between the finder and the other group members. This prediction, however, stems from the model's assumption of no competition between joiners. Several empirical studies report that the frequency of aggression increases as the asymmetry in body size or dominance status between opponents decreases (Brunkow and Collins, 1998; Evans and Shehadi-Moacdieh, 1988; Furuichi, 1983; Saito, 1996; Turner, 1994), which supports our prediction. In contrast, escalated fights are reported to be relatively uncommon (Enquist et al., 1990), so our model may overestimate the frequency at which escalated interactions should occur. The overestimation could be related to our assumption that both the duration and the energetic cost of escalated fights are fixed, while encounter duration may increase when the asymmetry in fighting ability decreases (e.g., Enquist et al., 1990). Moreover, we assumed in this model that animals have perfect knowledge about their fighting ability relative to that of their opponent and decide whether or not to compete aggressively accordingly. When they have incomplete information, it would be very costly and risky to engage in an escalated fight, and then we would expect aggressions to occur less frequently. Finally, our model assumes that when the finder is challenged by more than one competitor, the available resource does not decline. A more realistic assumption of the model would allow the joiners to exploit the food clump while the finder is engaged in a fight. Reduction of resource levels during fights would probably reduce the level of aggressiveness and hence the frequency of aggression.
Conclusion
In this study, we have attempted to reconcile predictions of resource defense theory and the game theoretic approach. To do so we developed a game theoretic model that allows asymmetric players with roles of either finder or joiner and allows for the presence of more than two competitors on food clumps. Unlike previous game theoretic models that predict that a certain proportion of encounters should always result in escalated fights, our analysis predicts that in a wide range of conditions, one of the competitors should always retreat without a fight, leaving all the remaining resource to its opponent. The difference between the two approaches is related to our assumption that the quantity of food that can be gained from a clump depends on whether the animal is the finder or a joiner, which in turn affects its expected gain and hence its decision whether to compete aggressively. As numerous empirical examples indicate that the costs and benefits associated with fighting differ between defenders and intruders and that animals adjust their behavior patterns accordingly (see Chapman and Kramer, 1996), it seems crucial to consider this role asymmetry to predict the frequency of aggression, not only when groups exploit divisible clumps of food but also when they compete for single prey items (see Sirot, 2000). When individuals compete for indivisible items there may be no finder's advantage, but the finder will have invested both time and energy to capture and possibly handle the prey before it is joined. This role asymmetry should affect the individual's net expected gain and hence its defense tactic (e.g., Erlandsson, 1988). Like previous hawk-dove games (Maynard Smith and Price, 1973; Sirot, 2000), however, the current model predicts that dove is never an ESS. The question remains, therefore, why a number of group-foraging animals share food clumps without any overt aggression. It is possible that such nonaggressive use of food clumps can arise from iterated play among familiar players (Giraldeau and Caraco, 2000). Future models of resource defense within groups should explore this possibility.
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ACKNOWLEDGEMENTS |
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We thank the UQÀM behavioral ecology discussion group for comments on an earlier draft and especially Bill Vickery for mathematical advice. This research was financially supported by Natural Sciences and Engineering Research Council of Canada research grants to L.-A.G. and J.W.A.G. F.D. was financially supported by a postdoctoral fellowship from Fondation Fyssen (France).
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TOPABSTRACTINTRODUCTIONREFERENCES |
|---|
Beauchamp G, Giraldeau L-A, 1996. Group foraging revisited: information-sharing or producer-scrounger game? Am Nat 148:738-743.[CrossRef]
Broom M, Ruxton GD, 1998. Evolutionarily stable stealing: game theory applied to kleptoparasitism. Behav Ecol 9:397-403.
Brown JL, 1964. The evolution of diversity in avian territorial systems. Wilson Bull 76:160-169.
Brunkow PE, Collins JP, 1998. Group size structure affects patterns of aggression in larval salamanders. Behav Ecol 9:508-514.
Carpenter FL, MacMillen RE, 1976. Threshold model of feeding territoriality and test with a Hawaiian honeycreeper. Science 194:639-642.
Chapman MR, Kramer DL, 1996. Guarded resources: the effect of intruder number on the tactics and success of defenders and intruders. Anim Behav 52:83-94.[CrossRef]
Emlen ST, Oring LW, 1977. Ecology, sexual selection, and the evolution of mating systems. Science 197:215-223.
Enquist M, Leimar O, Ljungberg T, Mallner Y, Segerdahl N, 1990. A test of the sequential assessment game: fighting in the cichlid fish Nannacara anomala. Anim Behav 40:1-14.
Erlandsson A, 1988. Food-sharing versus monopolising prey: a form of kleptoparasitism in Velia caparai (Heteroptera). Oikos 53:203-206.[CrossRef]
Evans DL, Shehadi-Moacdieh M, 1998. Body size and prior residency in staged encounters between female prawns, Palaemon elegans Rathke (Decapoda: Palamonidae). Anim Behav 36:452-455.[CrossRef]
Furuichi T, 1983. Interindividual distance and influence of dominance on feeding in a natural Japanese macaque troop. Primates 24:445-455.[CrossRef]
Giraldeau L-A, Beauchamp G, 1999. Food exploitation: searching for the optimal joining policy. Trends Ecol Evol 14:102-106.[CrossRef][Medline]
Giraldeau L-A, Caraco T, 2000. Social foraging theory. Princeton, New Jersey: Princeton University Press.
Giraldeau L-A, Hogan JA, Clinchy MJ, 1990. The payoffs to producing and scrounging: what happens when patches are divisible? Ethology 85:132-146.
Giraldeau L-A, Soos C, Beauchamp G, 1994. A test of the producer-scrounger foraging game in captive flocks of spice finches, Lonchura punctulata. Behav Ecol Sociobiol 34:251-256.[CrossRef][Web of Science]
Goldberg JL, Grant JWA, Lefebvre L, 2001. Effects of the temporal predictability and spatial clumping of food on the intensity of competitive aggression in the Zenaida dove. Behav Ecol 12:490-495.
Grant JWA, 1993. Whether or not to defend? The influence of resource distribution. Mar Behav Physiol 23:137-153.
Grant JWA, Gaboury CL, Levitt HL, 2000. Competitor-to-resource ratio, a general formulation of operational sex ratio, as a predictor of competitive aggression in Japanese medaka (Pisces: Oryziidae). Behav Ecol 11:670-675.
Grant JWA, Girard IL, Breau C, Weir LK, 2002. Influence of food abundance on competitive aggression in juvenile convict cichlids. Anim Behav 63:323-330.[CrossRef]
Grant JWA, Guha RT, 1993. Spatial clumping of food increases its monopolization and defense by convict cichlids, Cichlasoma nigrofasciatum. Behav Ecol 4:293-296.
Mangel M, 1990. Resource divisibility, predation and group formation. Anim Behav 39:1163-1172.[CrossRef]
Maynard Smith J, 1976. Evolution and the theory of games. Am Sci 64:41-45.[Web of Science][Medline]
Maynard Smith J, Price GR, 1973. The logic of animal conflict. Nature 246:15-18.[CrossRef]
Monaghan P, Metcalfe NB, 1985. Group foraging in wild brown hares: effects of resource distribution and social status. Anim Behav 33:993-999.[CrossRef]
Parker GA, 1984. The producer/scrounger model and its relevance to sexuality. In: Producers and scroungers: strategies of exploitation and parasitism (Barnard CJ, ed). New York: Chapman & Hall 127-153.
Ranta E, Peuhkuri N, Laurila A, Rita H, Metcalfe NB, 1996. Producers, scroungers and foraging group structure. Anim Behav 51:171-175.[CrossRef]
Robb SE, Grant JWA, 1998. Interactions between spatial and temporal clumping of food affect the intensity of aggression in Japanese medaka. Anim Behav 56:29-34.[CrossRef][Web of Science][Medline]
Saito C, 1996. Dominance and feeding success in female Japanese macaques, Macaca fuscata: effects of food patch size and inter-patch distance. Anim Behav 51:967-980.[CrossRef]
Sirot E, 2000. An evolutionarily stable strategy for aggressiveness in feeding groups. Behav Ecol 11:351-356.
Turner GF, 1994. The fighting tactics of male mouthbrooding cichlids: the effects of size and residency. Anim Behav 47:655-662.[CrossRef]
Vickery WL, Giraldeau L-A, Templeton JJ, Kramer DL, Chapman CA, 1991. Producers, scroungers, and group foraging. Am Nat 137:847-863.[CrossRef][Web of Science]
Zahavi A, 1971. The social behaviour of the white wagtail Motacilla alba alba wintering in Israel. Ibis 113:203-211.[Web of Science]
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