Behavioral Ecology Advance Access originally published online on August 28, 2007
Behavioral Ecology 2007 18(6):1040-1044; doi:10.1093/beheco/arm073
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Dilution games: use of protective cover can cause a reduction in vigilance for prey in groups
a Faculty of Veterinary Medicine, University of Montréal, PO Box 5000, St-Hyacinthe, Québec, Canada J2S 7C6 b Institute of Biomedical and Life Sciences, University of Glasgow, Glasgow G12 8QQ, United Kingdom
Address correspondence to G. Beauchamp. E-mail: guy.beauchamp{at}umontreal.ca.
Received 20 March 2007; revised 4 June 2007; accepted 16 July 2007.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHE MODELRESULTS AND DISCUSSIONGENERAL DISCUSSIONREFERENCES |
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Antipredatory vigilance usually decreases in groups. The generally accepted "collective detection" explanation implies that because there are more eyes to scan the surroundings for predators, individuals in a group can lower their personal investment in vigilance without increasing their predation risk. The role of other factors, such as numerical risk dilution caused by the mere presence of companions, has been neglected. In a model, we explore a dilution game when foragers in groups have access to protective cover. We show that foragers can take advantage of risk dilution and that this leads to changes in vigilance with group size without the need to invoke collective detection. We identify a cost to maintaining high levels of vigilance as less vigilant foragers gather food faster and so depart the group sooner (to reach cover) leaving more vulnerable stragglers behind. In groups, there is a scramble to reach safe sites that can induce a reduction in vigilance levels. Such a mechanism operates less forcefully in large groups because individuals in these groups are less vulnerable to the departure of an individual. We also demonstrate that individuals should adopt lower levels of vigilance, to reach safe sites sooner, when predator evasion is compromised or when the rate of food intake is high. The model provides new insights into the mechanisms underlying changes in vigilance with group size in animals.
Key words: antipredatory vigilance, collective detection, ESS model, group size, protective cover, risk dilution.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS AND DISCUSSIONGENERAL DISCUSSIONREFERENCES |
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The decrease in antipredatory vigilance with group size ranks as one of the most widely documented patterns in animal behavior (Krause and Ruxton 2002
). Ironically, the seminal models of vigilance commonly cited in textbooks did not predict that vigilance should necessarily decrease with group size. Rather, models predicted that because more eyes are available in a group to scan the surroundings for predators, individuals could reduce vigilance without increasing individual risk. However, these models left open the question as to why individuals should reduce vigilance levels in larger groups rather than keep vigilance levels steady and enjoy the benefits of reduced risk of predation (Pulliam 1973; Dehn 1990). Implicitly, these models assumed that individual behavior is modified so as to keep the risk of surviving an attack constant, but the motivation for this assumption was not made clear. Subsequent elaboration of the basic model of vigilance based on game theory implied that a reduction in individual vigilance levels can pay off in a group because foragers can obtain more food, thus increasing fitness, while at the same time benefiting from the vigilance of their companions (Pulliam et al. 1982; McNamara and Houston 1992). A further benefit of low vigilance levels in larger groups is the ability to monopolize a greater share of limited resources when exploitation speed is at a premium (Beauchamp and Ruxton 2003). Such factors are expected to lead to a reduction in vigilance with group size provided that group members that detect an imminent attack convey this information to companions.
The possibility that antipredatory mechanisms other than collective detection can produce a reduction in vigilance in groups has attracted little attention (Lima 1990). This is unfortunate because a substantial body of empirical work suggests that the levels of within-group communication about predation risk implicitly assumed by collective detection models are not always observed in nature (Lima 1995a; Lima 1995b; Hilton et al. 1999; Kaby and Lind 2003; Quinn and Cresswell 2005). Besides collective detection, another antipredatory mechanism may be risk dilution, which takes place because predators often attack only one prey per group leading to a decrease in individual predation risk for group foragers. Here, we explore a dilution game between group foragers that have access to protective cover and which is based on balancing minimization of time spent exposed to predators with minimizing the chances of being killed in attacks that do occur.
In many species of fish (Dill 1990), birds (Caraco et al. 1980), and mammals (Blumstein et al. 2003; Creel and Winnie 2005), for instance, individuals allocate their time between foraging in areas exposed to potential predators and resting or hiding in areas where they are protected from predators. For example, house sparrows (Passer domesticus), a small granivorous bird species, emerge from the vegetation cover to forage in open areas exposed to predators such as cats and hawks (Barnard 1980; Lima 1987a). A primary concern in such a species should be to reduce exposure time to predators while at the same time accumulating enough food. Individuals in a group foraging away from cover could all adopt a high level of vigilance to reduce individual risk. However, we surmise that a forager in this highly vigilant group could benefit from reducing unilaterally its vigilance levels because it would obtain food more quickly than its companions and be able to retreat to cover more quickly, thus reducing exposure time. The best response by other companions should therefore be a similar reduction in vigilance because the cost of failing to do so is a sharp increase in vulnerability toward the end of the foraging bout as more and more companions leave the group for cover leaving more vigilant and so slower eating foragers behind. In the model, we examine whether the dilution game can produce a decrease in vigilance rate in groups in the absence of collective detection in habitats with varying levels of predation risk, when animals accumulate resources at different rates or detect predators with different ease.
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THE MODEL |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS AND DISCUSSIONGENERAL DISCUSSIONREFERENCES |
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We consider a group of N individuals. Each individual begins the day in the foraging area where they can collect food but are vulnerable to predation. The maximum food limit (R) and the maximum rate at which an individual can gather food F set the upper limit to time spent foraging. The food limit may be related to satiation or to the achievement of a fixed requirement. We assume that antipredatory vigilance and food gathering are entirely incompatible. Thus, if an individual devotes a proportion
of its time to vigilance, then to obtain the maximum food limit it remains in the foraging area for time T = R/F(1 – ).
Thus, increased investment in antipredatory vigilance increases the length of time for which the individual must face the threat of predatory attacks. However, increased vigilance also increases the likelihood of surviving an attack. Specifically, we assume that the predator attacks the foraging area at rate A. We assume that during some time interval of length t, a fixed number of foraging individuals n remain in the foraging area. We are interested in the survival of a focal individual that is vigilant for a fraction of the time. During each attack, the predator selects a prey individual at random. If the focal individual is not selected, then it is certain to survive. If it is the one targeted, then we assume that the attack is of one of 2 types. One type, which occurs with probability P, can be combated by vigilance. We assume that the probability of not surviving such an attack is (1 – )2 (Houston et al. 1993, see also Bednekoff and Lima 2004). The other type of attack, which happens with probability (1 – P), is such that vigilance provides no defense, and the prey is always captured no matter its vigilance rate. Thus, the probability of not surviving each attack on the group is simply
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Our aim is to find the evolutionarily stable level of vigilance (*). To do this, we assume that there are N – 1 "field" individuals that all have vigilance level f and one "mutant" that has vigilance m. An individual's strategy is defined fully by the vigilance level it adopts. The evolutionarily stable strategy (ESS) is simply the value of f, such that no value of m leads to higher total probability of survival.
In order to define total survival, we must first calculate the times in the feeding area for the field and the mutant individuals. We assume that both these are capped by a maximum value (Tmax) that could be interpreted as the day length. Thus,
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- m < f, and so the mutant leaves the feeding area first. In this case, the survival of the mutant is given by
and the survival of the field individuals by
where, for convenience, we have defined the following terms:
- m > f, and so field individuals leave the feeding area first. In this case, the survival of the field individuals is given by:
and the survival of the mutant by:
We calculate the evolutionary fitness of both phenotypes as the probability of surviving multiplied by the amount of food collected. Thus, the evolutionary fitness of the mutant and field individuals are given by:
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In theory, * could be found by algebraic manipulation; however, the equations are sufficiently complex that a closed form solution cannot be obtained and the final transcendental equation that must be solved for * is too complex to be instructive. Thus, we find * by numerical methods as used by Pulliam et al. (1982). Specifically, we proceed by setting the vigilance of field group members very high and finding the best response by the mutant forager, which is initially to adopt a lower level of vigilance. If the fitness of the mutant exceeds that of the field group members, we lower the vigilance of the field group members and reiterate the comparison between mutant and field group members. The first value of vigilance where the mutant can no longer increase its fitness by adopting a lower level of vigilance is defined as the ESS. We later explore the possibility of alternate ESS values found by starting at the low vigilance condition.
By default, we set Tmax = 600, the maximum food limit at 200, and F at 0.5. This means that an individual feeding maximally (i.e., i = 0) can obtain the food limit in only 66% of the time available. At the other extreme, an individual can have a i value of 33% and still obtain the food limit in the time permitted. Hence, vigilance levels above 33% indicate that foragers sacrifice total food intake for increased probability of surviving an attack. Conversely, a vigilance level below 33% indicates a strategy involving eating quickly enough that the food limit (R) is reached before Tmax, and the foragers can reduce their time in the potentially dangerous foraging area. Such individuals are exposed to fewer attacks but have an increased probability of being killed in any attacks that they do experience. We set A at 0.02 attacks per unit time so that there are 12 attacks on average during the maximum time available. We set P = 0.8. Thus, vigilance provides no survival benefit in 20% of attacks. Variables used in the model and their values are provided in Table 1.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS AND DISCUSSIONGENERAL DISCUSSIONREFERENCES |
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We examine the effect of each parameter sequentially holding the remaining parameters at their default values. With respect to predation risk, when predation rate is high, vigilance decreases with group size (see the diamonds and squares in Figure 1), but individuals have greater than 33% vigilance and thus fail to obtain the food limit despite remaining in the foraging area for the maximum time allowed (Tmax). Notice that it would never be attractive for individuals to leave before Tmax if they had not yet consumed R food units. Prior to consuming R food units, food intake increases linearly with time spent in the foraging area, whereas probability of surviving a given time interval decreases slower than linearly with the length of the interval. Because fitness is defined to be probability of survival multiplied by food gathered, it is never advantageous to leave voluntarily before obtaining the food limit.
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For high attack rates, individuals always eat sufficiently slowly that they remain under predation risk for the maximum time (Tmax). Increasing group size increases the probability of surviving each attack substantially (simply by dilution), and so it becomes advantageous for individuals to enhance their fitness by focussing more on food gathering than on vigilance. Thus, for high predation risk, vigilance declines, and so feeding rate increases with increasing group size (see the diamonds and squares in Figure 1).
However, when the predation rate is lower, individuals can leave the foraging patch earlier than Tmax to reach cover and still obtain the food limit. This can be seen in the triangles of Figure 1 where vigilance, especially in the small groups, drops to values below the threshold vigilance that allows individuals to obtain the food limit before the Tmax. We find that individuals leave earlier in smaller groups than in larger groups as vigilance is lower in smaller groups. Our interpretation is that when predation risk is lower, the risk of being targeted by an attack is low. Therefore, lowering vigilance only increases vulnerability marginally whereas providing a large increase in feeding rate and a large decrease in exposure time. The departure of a mutant forager also has a large impact when group size is small. In a group of 2, for instance, the vulnerability of the remaining forager increases by 100% when the companion leaves the group (this explains why the single individual is much more vigilant than individuals in small groups for the triangle situation in Figure 1). In effect, the mutant takes the group hostage and forces the remaining foragers to leave early as well. However, the departure of one companion is not expected to have such a forceful impact in a larger group where dilution effects can remain strong even when the group loses a forager. In this case, larger groups can adopt a higher level of vigilance but still opt to leave earlier than the time limit.
The same scenario occurs when feeding rate is modified. When the feeding rate is low, individuals have little option but to remain in the foraging patch until the time limit (see the diamonds and squares in Figure 2). When the rate is higher, individuals can leave before the time limit (see the triangles in Figure 2), and the effect is again stronger in small groups than in larger ones. Here, the option of lowering vigilance and leaving earlier is attractive because individuals can obtain food sufficiently quickly to compensate for greater vulnerability to any given attack.
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When the probability of avoiding detected attacks is high, individuals remain in the foraging patch until the limit (squares and diamonds in Figure 3). When this probability decreases (triangles in Figure 3), individuals can opt to leave the patch earlier, and again the effect is stronger in smaller groups. This occurs because when the value of scanning to avoiding detected attacks is low, it pays to decrease exposure time and increase food intake rate.
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In the above 3 cases, vigilance decreases sharply when group size reaches a certain value and then increases afterward. The exact value depends on the choice of parameter values. However, a different choice of parameter values has no effect on the qualitative trends that we reported. The abrupt changes in vigilance with group size that we noted in the above 3 cases reflect in part the discrete nature of group size. We surmise that conditions that allow foragers to reach cover before the time limit can only be achieved once a given group size has been reached creating this threshold effect in vigilance.
We find that there can be 2 ESS values of vigilance associated with each group size. For the default parameter values, for instance, vigilance levels can be evolutionarily stable even when very low (triangles in Figure 4). The lowest levels of stable vigilance can be found by reversing the algorithm for finding the ESS and starting with low vigilance levels for the field group members. Therefore, we conclude that although the scope for a reduction in vigilance is strongest in smaller groups, low levels of vigilance can still occur in large groups in the dilution game. However, our expectation is that the very low vigilance ESS (triangles in Figure 4) will rarely be encountered in the real world. Most groups do not arrive simultaneously in the foraging area but rather start with a single individual arriving first. That single individual would be expected to be highly vigilant, and this high-vigilance starting point is likely to lead to the higher vigilance ESS (squares in Figure 4) after other individuals have arrived. In addition, if the detector of an attack could escape with a higher probability, then it would not pay to maintain vigilance levels really low (McNamara and Houston 1992).
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GENERAL DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS AND DISCUSSIONGENERAL DISCUSSIONREFERENCES |
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We examined the relationship between vigilance levels and group size when foragers exploit resources in areas where they are exposed to predators but can hide in protective cover once sufficient food has been obtained. Our key insight is that foragers can benefit from the presence of companions through risk dilution, and we need not assume that collective detection is necessary to lower vigilance in groups. In this context, there is a cost to maintaining high levels of vigilance as less vigilant foragers gradually leave the group, leaving slower eating stragglers behind. In groups, there is therefore a scramble to reach safe sites, and this scramble can induce a reduction in vigilance levels. Such a mechanism operates more forcefully in small groups because individuals in these groups are more vulnerable to being targeted in any given attack, and the defection of a companion increases vulnerability to a greater extent compared with individuals in larger groups.
We have shown that when predation risk is low, food intake rate is high, or the probability of avoiding detected attacks is compromised, vigilance is predicted to first decrease and then to increase as group size increases. We have shown that in the larger groups, it is possible to obtain low levels of vigilance given that low vigilance is also an ESS in the dilution game. However, as explained above, we consider this low vigilance ESS unlikely to occur in nature. Vigilance in most studies is usually lower in large groups, which is not in line with the above prediction that vigilance can increase with group size in larger groups under some conditions. Therefore, other factors must be invoked to account for the common finding that vigilance is lower in larger groups. Factors not considered here may lead to a smoother decrease of vigilance with group size. For instance, scramble competition for food is predicted to reduce vigilance levels in large groups to allow foragers a greater relative share of limited resources (Beauchamp and Ruxton 2003). Collective detection of predatory threats could also allow a reduction in vigilance in larger groups. We stress that in the dilution game modelled here, vigilance can decrease with group size without invoking the collective detection mechanism but that in larger groups this effect is more marginal.
Earlier models of vigilance have examined vigilance in foraging groups where foragers have no option but to remain in the same group (Pulliam 1973; Pulliam et al. 1982; Lima 1987b; McNamara and Houston 1992). In one model in particular, the game theory approach of pitting a single forager with a higher or lower vigilance value than the rest of the group produces incoherent consequences for the group in this situation (Pulliam et al. 1982). This appears to arise because the fitness function for the field group members assumes that group size remains the same even though focal individuals with lower vigilance have already left. Our model relaxes these restrictions and provides a coherent framework, both spatially and temporally, to explore the consequences of individual behavior on vigilance in a group.
The model presented here predicts that one key factor in the decrease in vigilance with group size is the availability of protective cover. Therefore, the decrease in vigilance with group size should be reduced or even disappear when individuals forage in protected areas where there is little predation risk and therefore little need to reach cover to decrease predation risk. Empirical evidence suggests that the group-size effect on vigilance is indeed less pronounced in more protected areas (Barnard 1980; Lima et al. 1999). Only the explanation of the group-size effect based on scramble competition would predict a decline in vigilance with group size when individuals forage in protected areas for limited resources (Beauchamp and Ruxton 2003).
The model also makes additional predictions. When the probability of avoiding detected attacks is compromised, animals that have access to protective cover should reduce vigilance levels to gain access to the refuge more quickly. This is also predicted to occur when the rate of food intake increases. In both cases, the opposite would have been predicted to occur when no refuge is available. Without a refuge, an increase in vigilance when avoiding detected attacks is limited or when the rate of food intake is high can increase safety. However, when protective cover is available, safety may be best achieved by leaving the risky foraging area as soon as possible. These predictions are empirically testable: predator detection ability can be easily manipulated in the field, by partially blocking the field of view of prey for instance (Bednekoff and Lima 2005; Fernandez-Juricic et al. 2005), whereas the rate of food intake may be manipulated by changing food density.
The model assumes that foragers cease foraging voluntarily only after accumulating the food limit and that foraging occurs in a single bout. This is appropriate for our particular description of fitness. However, for other fitness functions, the food limit could be treated as a decision variable so that individuals could vary both vigilance levels and the amount of time spent foraging in a bout implying that foraging could occur in many bouts throughout the day. A potential advantage of such a strategy is that foragers could adjust total foraging time in response to changes in the relative risks of having insufficient reserves to last the night and being preyed on while gathering food. The consequences of flexibility in foraging time on changes in vigilance in groups of different sizes could be explored more fully in future work. However, the contribution of this paper is to suggest the existence of a dilution game between group foragers when protective cover is available. The next stage must be to evaluate the relative importance of dilution and collective detection in the natural world when protective cover is available.
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ACKNOWLEDGEMENTS |
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We thank A. Houde, T. Stankowich, 4 anonymous reviewers, and, particularly, P. Bednekoff for constructive comments on earlier versions of this work.
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REFERENCES |
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