Behavioral Ecology

This Article Abstract FREE Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Search for citing articles in: ISI Web of Science (10) Request Permissions Google Scholar Articles by Mesterton-Gibbons, M. Articles by Dugatkin, L. A. Search for Related Content PubMed Articles by Mesterton-Gibbons, M. Articles by Dugatkin, L. A. Social Bookmarking          What's this?

Behavioral Ecology Vol. 10 No. 4: 377-390
© 1999 International Society for Behavioral Ecology

On the evolution of delayed recruitment to food bonanzas

Michael Mesterton-Gibbons^a and Lee Alan Dugatkin^b

^a Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA ^b Department of Biology, University of Louisville, Louisville, KY 40292, USA

Address correspondence to M. Mesterton-Gibbons. E-mail: mmestert{at}mailer.fsu.edu

Received 18 June 1998; revised 4 December 1998; accepted 17 December 1998.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Whereas food sharing by immediate recruitment to food bonanzas is relatively common, especially among birds, delayed recruitment from overnight roosts is comparatively rare, although it has been studied extensively in the common raven (Corvus corax). Two hypotheses have been advanced to explain the evolution of delayed recruitment. Under the status-enhancement hypothesis, delayed recruiting is favored because the recruiter's social status increases with the number of followers it leads to a food source. The posse hypothesis also focuses on the number of individuals recruited to a site, but in this case aggregation is favored because larger groups are more likely to usurp a carcass defended by a pair of territorial adult ravens. We used a game-theoretic model to explore the logic of immediate versus delayed recruitment in the light of these hypotheses. In particular, we identified three critical values of the probability of immediate recruitment: that below which delayed recruitment is a cooperative strategy, that below which delayed recruitment is an evolutionarily stable strategy, and that below which a mutant strategy of delayed recruitment will invade a population of immediate recruiters to reach fixation. The model demonstrates that either status enhancement or the posse effect may alone suffice for the evolution of delayed recruitment to food bonanzas via mutualistic information sharing at communal roosts.

Key words: communal roosts, cooperation, Corvus corax, delayed recruitment, evolutionary game theory, food sharing, information centers, mutualism, ravens, social status.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Food sharing has been well documented in many species of birds and mammals (see Dugatkin, 1997; Packer and Ruttan, 1988, for reviews). Often food sharing involves individuals finding a food source and then immediately recruiting others to the site of the newly discovered food. For example, Elgar (1986) found that house sparrows (Passer domesticus) emit a "chirrup call" (Summers-Smith, 1963) that attracts conspecifics to a newly found food source, and male chickens appear to emit food calls to increase their chances of attracting a female (Collias and Joos, 1953; Evans and Marler, 1994; Marler et al., 1986a,b). Various other cases of food-sharing signals have been documented in birds. Many of them are associated with communal breeders and "information centers" (Ward and Zahavi, 1973), as in the cliff swallow (Hirundo pyrrhonota) (see e.g., Brown, 1986; Brown et al., 1991).

In addition to empirical work like that on the sparrow and other species, this type of recruitment to newly discovered food items has been the focus of at least two game-theoretic models (Caraco and Brown, 1986; Newman and Caraco, 1989). These models suggest that the phenomenon is largely an example of "by-product mutualism" (Brown, 1987), in which cooperation is an incidental consequence of otherwise selfish behavior, although in certain circumstances "reciprocal altruism" (Axelrod and Hamilton, 1981), in which individuals keep score of pairwise interactions, may play a role as well. The mutualism arises because failing to call increases the risk of predation, whether on the foragers or on their fledglings, and so immediate recruitment is favored.

There is, however, another kind of recruiting behavior—delayed recruitment—that has yet to be explored theoretically, although it has been the focus of some experimental work (see Rabenold, 1987, and references below). In delayed recruitment, an individual who finds a food source waits for a considerable period of time before using the resource itself or recruiting others to it. In ravens (Corvus corax), for example, individuals often return to their communal roost after spotting a large carcass but wait until the following morning before leading other birds to the bonanza (Heinrich, 1989). Here, we present a game-theoretic model that explores conditions favoring delayed versus immediate recruitment of individuals to food sites.

The model we develop is general in nature, but it takes as its empirical base the well-studied food-sharing and recruitment system of ravens (Heinrich, 1988a, b, 1989; Heinrich and Marzluff, 1991; Marzluff and Heinrich, 1992). For the purposes of our model, the juvenile raven system is ideal for a variety of reasons: (1) juvenile ravens are vagrants and aggregate into roosts. Both immediate and delayed recruitment exist (Heinrich, 1988a, b, 1989, 1994a). (2) Communal roosts are thought to function as "information centers" (Loman and Tamm, 1980; Heinrich 1994a; Heinrich et al., 1994; Marzluff et al., 1996). (3) Yelling, the mechanism by which some recruiters attract other ravens to a prey item, has been well studied (Heinrich and Marzluff, 1991; also see Heinrich et al., 1993, for more on a related issue). (4) Change of social status as a result of recruitment has been examined in a number of different contexts (Heinrich 1994b; Heinrich and Marzluff, 1991). Social status appears to be important in ravens not only for access to food (Harriman and Berger, 1990; Heinrich and Marzluff, 1991), but also with respect to making males more attractive to mates (Gwinner, 1964). (5) The mechanism by which ravens recognize food items has been examined both from an ecological and a cognitive perspective (Heinrich, 1995a, b; Heinrich et al., 1995). (6) The means by which recruiters and followers are able to overcome territorial adults defending a food source are well known (Marzluff and Heinrich, 1992). (7) Foraging often occurs at large, discrete, and rare carcasses (Heinrich, 1989).

The raven system also holds greater promise for studying recruitment as a possible case of cooperative behavior. In this context, it is usually difficult to distinguish among mutualism, reciprocity, and two other categories of cooperative behavior—kin-selected and group-selected behavior (Dugatkin, 1997; Mesterton-Gibbons and Dugatkin, 1992). In the case of ravens, however, it is reasonable to rule out three of the four categories a priori. DNA fingerprinting studies of relatedness within and between groups strongly suggest that kinship does not play a significant role in raven foraging groups (Parker et al., 1994), and raven communal nests show high turnover rates (Heinrich, 1988a, b; Heinrich et al., 1994), which seem likely to preclude both the deme structure needed for group selection and the repeated interactions necessary for reciprocity. So one can focus on mutualism to study cooperative elements of raven recruitment.

In the light of his extensive empirical work, Heinrich (1989) has formulated two hypotheses concerning factors that favor delayed recruitment in ravens (which are not mutually exclusive). We refer to them here as the "status-enhancement" and "posse" hypotheses. Under the status-enhancement hypothesis, delayed recruiting is favored because the recruiter's social status increases with the number of followers it leads to a food source. Delayed recruitment might allow more individuals to be informed of who has discovered a bonanza, and hence significantly improve the recruiter's status; on the other hand, the effect is diluted when several foragers independently discover the same bonanza. The posse hypothesis also focuses on the number of individuals recruited to a site, but in this case aggregation is favored because larger groups are more likely to usurp a carcass defended by a pair of territorial adult ravens (Marzluff and Heinrich, 1992). Note that a carcass is inaccessible to ravens unless opened by a heterospecific (e.g., a coyote or wolf), because their large (7.9-9.5 cm) bills, although well suited to shearing meat from bones and pecking into crevices among them, are too weak to penetrate the hide of large mammals (Heinrich, 1988a).

Both the posse and status-enhancement hypotheses are consistent with the notion that larger groups allow ravens to overcome their fear of most new items they encounter. Thus, although their "neophobia" is an interesting phenomenon in its own right (Heinrich, 1995b; Heinrich et al., 1995), it cannot shed light on the distinction between status enhancement and the posse effect as factors favoring the evolution of delayed recruitment.

Here we present a formal game-theoretic model of immediate versus delayed recruitment. We then use this model to explore the question of when delayed recruitment (DR), via information sharing at a communal roost or other information center, can be expected to evolve from immediate recruitment (IR) in a population whose food source is rare and ephemeral, but bountiful. The model is based on highly idealized assumptions and therefore has numerous limitations, but we postpone a discussion of these limitations to the end of the paper, so that we can focus on the assumptions in developing the model. A list of symbols appears in Appendix C.

	Mathematical model

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Consider a population of unrelated overwintering juveniles that forage independently by day, but roost together at night in groups of size N + 1. They search during a period of L daylight hours from early morning until dusk. Food bonanzas are defined to be opened carcasses that are rare and ephemeral, but bountiful. For the sake of simplicity, but consistent with this definition, we assume that an opened carcass lasts only a day, but is ample enough to satiate any juvenile that exploits it. Moreover, we assume that the rate of renewal is precisely one per day; thus the object of search is to find the opened carcass, wherever in a huge area it may have randomly appeared. We scale expected future reproductive success so that feeding at a bonanza increases it by 1 unit of fitness. Also, we assume that juveniles feed solely at bonanzas, and we ignore the effect of predation. Thus any additional fitness increment is due to a rise in social status. We assume that the fitness of a discoverer increases by units for every animal it recruits, and we will refer to this assumption as the status-enhancement effect.

Each day, an isolated individual locates the bonanza with probability b, and hence with probability 1 — b finds no bonanza. We assume that 0 < b < 1. The time until a focal individual locates the bonanza by itself is a random variable Z, continuously distributed between 0 and with cumulative distribution function (cdf) G and probability density function (pdf) g, i.e.,

(1)

where a prime denotes differentiation. Because Z has a continuous distribution, two individuals cannot discover the bonanza simultaneously, although any number of individuals may discover the bonanza at some time during the day.

Individuals cease to search either at the end of the day or when they become privy to the bonanza. But the focal individual need not discover the bonanza for itself. Let W denote the time until one of the N nonfocal individuals locates the bonanza. When W is also continuously distributed between 0 and , but with a different cdf and pdf, say H and h, respectively. Because 1 — Prob(W w) is the probability that none of N independent foragers locates the bonanza by time w, we have

(2)

and

(3)

If Z < W < L, then the focal individual ceases to search at time Z. If Z > W, however, then its stopping time depends on its strategy.

For the sake of simplicity and tractability, each individual in the population is assumed to be either an immediate recruiter (strategy IR) or a delayed recruiter (strategy DR). Both types respond to recruitment by others: this aspect of behavior is assumed not to be under selection. If an IR strategist discovers the bonanza, it immediately recruits all individuals within range. It then remains at the carcass with its gang of recruits until all return to the roost at dusk (i.e., at time L). If another individual subsequently discovers the carcass itself, independently of recruitment, then it neither rises in status nor joins the gang; however, it is able to satiate itself if the gang acquires access to the food. (If necessary, we can think of such late discoverers as lurking in the vicinity of the carcass until the gang has departed and gorging themselves rapidly before likewise returning to the roost.)

Each juvenile will have access to the food if it is not defended, or if it is defended but the gang contains enough recruits to repulse the resident adults. Let denote the probability that the bonanza is defended by a resident pair, and let (I) denote the probability that a gang of I juveniles (including its leader) will repulse the residents. For the sake of simplicity and tractability, we assume that a lone animal has no chance of repulsing the residents, but that the probability of a gang's success thereafter increases in direct proportion to the number of recruits, with maximum probability (when the entire roost of juveniles is at the bonanza). That is, we assume a linear relationship of the form

(4)

where > 0. Note that Equation 4 implies (1) = 0 and (N + 1) = . With this assumption, the probability of access for a gang of size I is A(I) = 1 - + (I), and Equation 4 implies

(5)

We will refer to the assumption embodied in Equation 4 as the posse effect; it holds whenever bonanzas are defended with positive probability but is absent when bonanzas are undefended. In other words, there is a posse effect if and only if > 0.

As already stated, we assume that in an IR population, at most one animal per day, the first discoverer, can rise in status. In a DR population, on the other hand, several animals may rise in status by discovering the carcass independently; however, the concomitant increase of fitness is shared equally among them. A DR strategist that discovers the bonanza delays recruitment to the following day at dawn. If it is the sole discoverer of the bonanza, then the other N juveniles follow it at dawn from the roost to the site of the carcass; but if several animals discover the bonanza, then each is equally likely to lead. An animal that fails to discover the bonanza will fail to rise in status, but its fitness will increase by 1 unit if some DR strategist discovered the bonanza and the flock gains access to the food. Thus, conditional upon access, the increase of expected future reproductive success to a DR strategist in a population of DR strategists is

(6)

where E denotes expected value and D is the number of other discoverers. It is shown in Appendix A that the above expression reduces to b(1+) (1 - (1 - b)^N+1) - b. Thus, on multiplying Equation 6 by the probability A(N + 1) of access, we find that the increase of expected future reproductive success to a DR strategist in a population of DR strategists is R defined by the matrix in Table 1.

View this table: [in this window] [in a new window]   Table 1 The reward matrix

 

Now suppose that the population contains a mutant IR strategist (in addition to N DR strategists). If this focal individual finds a bonanza at time Z, then it will immediately recruit the other individuals—all of them DR strategists—within its range of attraction. Let F(Z) denote the number of other individuals still foraging at time Z, and let r denote the recruitment probability for each. In ravens, immediate recruitment appears to be predominantly vocal (Heinrich, 1988b; Heinrich and Marzluff, 1991), although visual cues may play a lesser role (Heinrich et al., 1993; Marzluff et al., 1996). Thus, because individuals are assumed to search at random over a large area, we can interpret r as the ratio of the area of the animal's call range to the total search area for the roost. Then the expected number of recruits is rF(Z). The focal individual's fitness will increase by 1 unit for access to the food plus rF(Z) units for the status of a discoverer; and the size of the gang will be rF(Z) + 1, so that the probability of access will be A(rF(Z) + 1), where A is defined by Equation 5. Even if the IR strategist fails to discover the bonanza, however, i.e., if Z > L, its fitness will increase by 1 if one of the DR strategists discovers it (i.e., if W < L) and the group as a whole gains access the following dawn. But a payoff of 1 with conditional probability 1 - (1 - ) is equivalent to a payoff of 1 - (1 - ) with conditional probability 1. Thus the focal IR strategist's payoff against the N DR strategists is

(7)

and so the increase of expected future reproductive success to an IR strategist in a population of DR strategists is

(8)

which reduces to the expression in Table 1 after a calculation described in Appendix A.

As remarked above, if the focal IR strategist is not a mutant but belongs instead to a population of IR strategists, then we must allow for the possibility that it ceases to search before the end of the day, not because it has discovered the bonanza itself, but because another IR strategist has discovered the bonanza and the focal individual is near enough to be recruited. In that case, the focal individual's status will not rise, but its fitness will still increase by 1 unit if (immediate) recruitment yields a large enough gang for access to the carcass. Figure 1 shows the joint sample space of the random variables Z and W, which are independent. If the focal individual is first to discover the carcass, then its fitness increases by 1 plus times the expected number of recruits, or 1 + rN. If it is not the first discoverer of the carcass (implying Z > W), then its fitness will increase by 1 if either it is recruited or subsequently it discovers the carcass for itself. In the triangle shaded in Figure 1, the focal individual is recruited with conditional probability r and with conditional probability 1 - r is not recruited, but subsequently finds the bonanza itself. In either case, its reward is 1. In the open rectangle to the right of this triangle, however, the focal individual enjoys a reward of 1 only if it is recruited. But a payoff of 1 with conditional probability r is equivalent to an expected payoff of r with conditional probability 1. Thus the focal individual's payoff, conditional upon access to the carcass, is

(9)

and so its increase in expected future reproductive success, conditional on access to the carcass, is

(10)

Whenever the reward is positive, regardless of whether the focal individual is the discoverer, a recruit, or neither, the expected size of the gang of juveniles is rN + 1. So the probability of access is A(rN + 1) = 1 - (1 - r), and the (unconditional) increase of expected future reproductive success to an IR strategist in an IR population is

(11)

which reduces to the expression in Table 1 after a calculation described in Appendix A.

View larger version (30K): [in this window] [in a new window]   Figure 1 Joint sample space of Z and W.

 

To complete the reward matrix in Table 1, now suppose that the population consists of N IR strategists and a mutant DR strategist. This focal individual obtains an immediate reward only if it responds to an IR strategist's call; if it finds the bonanza itself, then its reward is delayed to the following dawn. Thus we must allow for the fact that if the focal DR strategist is first discoverer, its expected number of recruits is no longer rN, but rather the number of IR strategists that were not made privy to the bonanza the previous day (all others being satiated, by assumption). An IR strategist is not made privy to the bonanza if its fails to find it, and it is not the case that both another IR strategist finds it and the first animal is recruited. On using Equation 1, the probability of this event is

(12)

The corresponding expected number of recruits is qN. Thus, although the probability of access remains 1 - (1 - r) when Z > W, for Z < W it changes to A(Nq + 1) = 1 - (1 - q). So, by analogy with Equations 9-11, the increase of expected future reproductive success to a DR strategist in an IR population is

(13)

which reduces to the expression in Table 1 after a calculation described in Appendix A.

Depending on the signs of R - T and S - P, either IR or DR is potentially an evolutionarily stable strategy, or ESS (sensu Maynard Smith, 1982) of the game with reward matrix defined by Table 1. If R - T and S - P are both positive, then DR is the only ESS; if both are negative, then IR is the only ESS; if R - T is positive but S - P is negative, then both strategies are evolutionarily stable; and if R - T is negative but S - P is positive, then neither strategy is an ESS, but each can infiltrate the other, so that a mixture of both will be evolutionarily stable within the population. For further discussion of mathematical details, see Mesterton-Gibbons (1992).

	The evolutionarily stable strategy

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Following Mesterton-Gibbons and Dugatkin (1992), we will refer to the strategy that yields the higher reward to the population as the cooperative strategy. Thus, IR is the cooperative strategy if P > R, and DR is the cooperative strategy if R > P. In the latter case, our game is an (N + 1)-player analogue of the "cooperator's dilemma" we described in our earlier paper, the games being equivalent for N = 1. But here we are concerned with larger group sizes. Note, therefore, that the additional constraint 2R > S + T (Mesterton-Gibbons and Dugaktin, 1992; Equation 1b), which applies only in the case where N = 1, is irrelevant. If, in addition, T > R and P > S (so that T > R > P > S), then our game is an (N + 1)-player analogue of the well-known "prisoner's dilemma" (Axelrod, 1984; Mesterton-Gibbons, 1992), in which the strategic dominance of strategy IR prevents the population from achieving the higher reward associated with the cooperative strategy DR.

The posse effect in isolation
In general, the identity of the cooperative strategy will depend on the relative magnitude of r, but an exception occurs when there is no status-enhancement effect, (i.e., when = 0). This special case is of considerable interest because we would like to explore whether the posse effect alone suffices for the evolution of delayed recruitment. Setting = 0 in Table 1 we find that

(14)

which is invariably positive because b,, r, and are all probabilities between 0 and 1. Thus DR is is always the cooperative strategy in the absence of a status-enhancement effect. In addition, with = 0 we have

(15)

and

(16)

where we define a critical recruitment probability r₁ by

(17)

By inspection, R - T is invariably positive, but the sign of S - P depends on r. Thus, in the absence of a status-enhancement effect, DR is always an ESS. Moreover, if r < r₁, then DR is the only ESS. If, on the other hand, r > r₁, then both IR and DR are evolutionarily stable, but if IR is the primitive state, then the noncooperative strategy will prevail. We can now answer the question of interest. If the immediate recruitment probability, r, exceeds r₁, then the posse effect alone does not suffice for the evolution of cooperation. If, however, r lies below this critical value, then the cooperative strategy DR is a dominant strategy and will invade an IR population. In sum, the posse effect alone suffices for the evolution of delayed from immediate recruitment if the recruitment probability, r, is sufficiently small.

The status-enhancement effect in isolation
It is also of considerable interest to explore whether the status-enhancement effect alone suffices for the evolution of delayed recruitment. Setting = 0 in Table 1 we find that

(18)

(19)

and

(20)

where r₁ is defined by Equation 17, and we define two other critical recruitment probabilities by

(21)

and

(22)

For N2 these inequalities satisfy

(23)

and the ESSs are as follows:

(24)

For N 3, on the other hand, although

(25)

we now have r₀ > r₂ > r₁ only if the probability of discovery b is lower than a critical value b_c (defined by r₁ = r₂), which decreases with N; for b_c < b < 1, we have r₀ > r₁ > r₂ instead. The ESSs are now as follows:

(26)

In the region where neither DR nor IR is an ESS, a mutant IR strategist can infiltrate a DR population and a mutant DR strategist can infiltrate an IR population, but neither mutation can reach fixation. The population will therefore consist of a polymorphic mixture of IR and DR strategists.

In practice, the size of a roost is rarely in single digits. For example, at the New Vineyard roosting site in the forests of western Maine, which Marzluff et al. (1996) observed over 24 nightfalls between 15 November and 20 December 1988, the size of a roost was between 12 and 46 on 18 occasions, with 3 instances of 9 and 1 each of 5 and 80 (and 1 instance in which all of 77 arriving birds apparently relocated after dark). In every case, the size of an actual roost was at least 5, suggesting N 4. Thus also N 3, and the relevant conditions are Equation 26, which we illustrate in Figure 2 for = 0.5 and various values of N. Note that the region where both IR and DR are an ESS appears to be of little consequence unless the size of the roost is in single digits, which was scarcely observed.

View larger version (23K): [in this window] [in a new window]   Figure 2 Evolutionarily stable strategy (ESS) regimes in the parameter space 0 b, r 1 for = 0 and = 0.5. The thick solid curve is r = r0, above which immediate recruitment (IR) is the cooperative strategy, below which delayed recruitment (DR) is the cooperative strategy. The thin solid curve is r = r2, below which DR is an ESS, above which DR can be infiltrated by IR. The dashed curve is r = r1, above which IR is an ESS, below which IR can be infiltrated by DR. IR or DR in corners of graphs denotes a cooperative ESS, and P denotes a polymorphism. In general, r1 and r2 are independent of but decrease with respect to N and b, whereas r0 decreases with respect to all three parameters (and approaches 1 - b in the limit as , N ); however, the dependence of r0 on N is weak.

 

We can now answer the question of whether the status-enhancement effect alone suffices for the evolution of delayed recruitment. The answer is yes, and for at least moderately large roosts the conditions for DR to infiltrate the primitive state are essentially the same; but the conditions for DR to invade and become a monomorphic ESS are potentially much more stringent.

The combined effect
In the more general case where both and are positive, we denote by ₀, ₁, ₂, respectively, the critical recruitment probabilities below which DR is a cooperative strategy, below which DR can infiltrate IR, and below which DR is an ESS. Note that, in terms of these new parameters, the result that DR is invariably a cooperative ESS when = 0 becomes ₀ = 1 = ₂ for = 0.

On inspecting Table 1, we readily find that

(27)

where r₁ is defined by Equation 17. In other words, the critical recruitment probability, below which DR can infiltrate an IR population, depends only on N and b. In general, however, both ₀ and ₂ depend on , , and , as well as on N and b. Because the analytical expressions are cumbersome, they are presented in Appendix B. What they reveal is that both ₀ and ₂ increase with respect to or but decrease with respect to or N (although the decrease of ₀ with N is weak), and that ₀ still decreases with respect to b but that ₂ no longer decreases with respect to b unless is sufficiently large. Furthermore, it is shown in Appendix B that

(28)

Hence, if IR is cooperative (r > ₀), then IR is invariably also an ESS (r > ₁). We illustrate the possibilities in (Figures 3,4,5,6).

View larger version (22K): [in this window] [in a new window]   Figure 3 ESS regimes in the parameter space 0 b, r 1 for = 0.25, = 0.5 and N = 10. The thick solid curve is r = 0, above which immediate recruitment (IR) is the cooperative strategy, below which delayed recruitment (DR) is the cooperative strategy. The thin solid curve is r = 2, below which DR is an ESS, above which DR can be infiltrated by IR. The dashed curve is r = 1, above which IR is an ESS, below which IR can be infiltrated by DR. IR or DR in corners of graphs denotes a cooperative ESS, P denotes a polymorphism.

 

View larger version (22K): [in this window] [in a new window]   Figure 4 ESS regimes in the parameter space 0 b, r 1 for = 0.5, = 0.5 and N = 10. As in Figure 3, the thick solid curve is r = 0, the thin solid curve is r = 2 and the dashed curve is r = 1. IR, immediate recruitment; DR, delayed recruitment.

 

View larger version (23K): [in this window] [in a new window]   Figure 5 ESS regimes in the parameter space 0 b, r 1 for = 0.5, = 0.25 and N = 10. As in Figure 3, the thick solid curve is r = 0, the thin solid curve is r = 2 and the dashed curve is r = 1. IR, immediate recruitment, DR, delayed recruitment.

 

View larger version (21K): [in this window] [in a new window]   Figure 6 ESS regimes in the parameter space 0 b, r 1 for = 1 and = 0.5. As in Figure 3, the thick solid curve is r = 0, the thin solid curve is r = 2 and the dashed curve is r = 1. Note that all three critical recruitment probabilities are independent of when = 1 (see Appendix B). IR, immediate recruitment, DR, delayed recruitment.

 

	DISCUSSION

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
We have explored the question of when delayed recruitment (DR), via information sharing at a roost or other information center, can be expected to evolve from immediate recruitment (IR) in a population whose food source is rare and ephemeral, but bountiful. We have modeled the dilemma of whether to delay recruitment in terms of six ecological parameters, namely, N + 1, the size of the roost; b, the probability per day that an isolated forager locates a food bonanza through random search; , the marginal value in terms of expected future reproductive sucess of a unit of increased social status, scaled with respect to the food value of a bonanza; , the probability that the bonanza is defended; , the maximum probability of access to a defended bonanza (i.e., the access probability for the entire roost); and r, the probability of immediate recruitment, which we interpret as the ratio of the area of an animal's call range to the total search area for the roost. All of these parameters are fixed, in particular b, , and N. In the real world, of course, both the probability of a discovery and its effect on social status may vary among individuals; the size of a roost may vary with time; animals may not forage randomly (e.g., ravens may travel directly above a highway, apparently in search of road kills); and they may not search alone. For example, of the 87 ravens that Heinrich (1989) observed over four winters flying over the countryside away from all known food sources, only 69% were singles (29% were in pairs and 2% were in larger groups). Moreover, even under natural conditions (as opposed to the experimental conditions created by Heinrich and his co-workers), a rare bonanza may last for several weeks—as opposed to 24 h, which our model assumes. Thus the model is highly idealized. In the terms of Levins (1966), we sacrifice realism to generality and precision: our work is purely analytical.

Nevertheless, the point of a game-theoretic model in behavioral ecology is not to replicate the game of nature, but rather to abstract the essence of a strategic interaction so that its logic can be rigorously explored (Mesterton-Gibbons and Adams, 1998). The simplicity of such a model is its strength, and a variety of such models, each by itself inadequate in a different way, can yield a "robust theorem which is relatively free of the details of the model," its truth being "the intersection of independent lies" (Levins, 1966: 423).

Among inadequacies of the model not already mentioned are the following: (1) We have assumed a linear relationship of the form (I) = (I - 1)/N between the number I of recruits in a gang and its probability of repulsing a resident pair, whereas a nonlinear relationship that included the effect of saturation would be more realistic. For example, Heinrich's (1988b) observations of ravens suggest (I) = 1 for I9. (2) We have also assumed a linear relationship between the number of conspecifics that a discoverer recruits and its rise in status, whereas a nonlinear relationship would again be more realistic. For example, there could be a critical number of recruits beyond which status would rise dramatically and lead to breeding as a yearling rather than as a 2 year old. (3) We have assumed that the only available strategies are those of obligate IR strategist or obligate DR strategist, whereas it would be more realistic to allow more flexible use of the two behaviors (perhaps depending partly on the timing of discovery). (4) We have attributed a rise in status only to animals who discover a bonanza independently and only to the first such discoverer in a population of IR strategists; whereas in nature, being privy to a bonanza may yield fitness gains through status enhancement regardless of how that information is acquired, but especially where it is acquired from accompanying the discoverer. As acknowledged already, however, the more fundamental inadequacy in this regard is our assumption that animals forage independently.

On the other hand, there are several ways in which our assumptions appear to yield quite good approximations. First, although a bonanza may sometimes persist for several days, at other times a heavy snowfall or a mammalian scavenger may eliminate the resource rapidly, long before it could all be eaten (Marzluff et al., 1996), which is consistent with our assumption that a bonanza suffices to satiate any animal who is privy to it. Second, we have ignored predation, but ravens are relatively immune to being predated once past the nestling stage (Heinrich, 1988b). Third, our implicit assumption that the composition of a gang will vary from day to day is consistent with observations that roost composition in ravens is highly fluid (Heinrich, 1988b; Parker et al., 1994; Marzluff et al., 1996). Finally, we have ignored variation in energetic costs, but this assumption would have strategic consequences in the context of our model only if such costs differed significantly between IR and DR strategists. The costs are the same for all searching animals, regardless of whether they are IR or DR strategists, however, and at night all animals share a common roost. Thus any difference in costs must be due to the greater likelihood that a focal individual in an IR population will return to the roost early by virtue of being recruited and hence engage in some other daytime activity. But a difference in costs between an IR population and a DR population cannot affect the evolutionary stability of either; at worst, it affects whether IR or DR is the cooperative strategy, and even this effect is ignorable if the costs of other daytime activities are comparable to those of searching (or if the costs of either are negligible compared to those of surviving the night).

In general, either IR or DR may be the cooperative strategy (i.e., the strategy that yields a higher reward to the population), and either IR or DR may be an ESS, in the sense of Maynard Smith (1982). Broadly speaking, IR is cooperative only when b and r are both relatively high, but if IR is cooperative, then it is invariably an ESS. More precisely, IR is a cooperative ESS when r exceeds a critical value ₀, which increases with respect of and , but decreases with respect to , N, and b (and always exceeds 1 — b). As described earlier, however, mutualistic food sharing via immediate recruitment has been adequately explained by previous work (e.g., Newman and Caraco, 1989), and so we do not discuss it further. We concern ourselves instead with the case where a relatively low value of either b or r yields a higher population reward from pooling knowledge at an information center.

In other words, for the remainder of the section, we assume that r < ₀, so that the cooperative strategy is DR. Two other critical (immediate) recruitment probabilities now determine whether DR is an ESS, and if so, whether it would evolve in a population of immediate recruiters. DR is an ESS when r is less than a critical value, ₂, which increases with respect to and , but decreases with respect to and N. DR will invade an IR population to reach fixation when r is less than a second critical value, ₁, which is independent of , , and , but decreases with respect to b and N (and is always less than 1 — b). Thus, the condition for DR to evolve from an IR population is that

(29)

(which is invariably less than ₀).

In terms of our model, Heinrich (1989) has conjectured that delayed recruitment evolved in ravens via either a statusenhancement effect or a posse effect. We have confirmed that either alone may suffice. That is, a posse effect may suffice in the absence of a status-enhancement effect ( > 0, = 0), and a status-enhancement effect may suffice in the absence of a posse effect ( > 0, = 0). In either case, delayed recruitment evolves mutualistically, and where mutualism suffices, reciprocity is unnecessary. In terms of Mesterton-Gibbons and Dugatkin (1997), who stressed the significance of disparate ecological time scales in balancing the evolutionary ledger, when a behavior that is potentially altruistic on a short time scale serves the actor's self-interest on an intermediate time scale, it is unnecessary to invoke reciprocal interactions over a much longer time scale to rationalize the behavior. Here, the intermediate time scale is 24 h, the long time scale is the duration of a winter, and the short time scale is part of a day, during which the behavior DR is potentially altruistic because there could be many other juveniles in the vicinity when a DR strategist refrains from calling, in which case there is a short-term loss of fitness in terms of social status.

In fact, Heinrich has argued against reciprocity in ravens, and it is interesting to explore this additional conjecture in the light of our model. We would expect DR to be sustained by reciprocity only in a region of parameter space where DR is cooperative but the players are locked in a prisoner's dilemma (i.e., where T> R> P> S). Is there such a region? From Figures 2,3,4,5,6, but especially Figure 3, the answer is yes, but only if is sufficiently large, and the larger the value of , the bigger the size of the region. Indeed, it is shown in Appendix B that DR fails to be an ESS, and hence reciprocity over many days is necessary to sustain cooperation, only if exceeds the critical value

(30)

which increases with respect to , , and b, but decreases with respect to N. Thus our model predicts that reciprocity should be important in the evolution of delayed recruitment only if the status-enhancement effect is sufficiently strong (in the sense that at least exceeds _c), and so if Heinrich is right that reciprocity is absent in ravens, then our model suggests that any status-enhancement effect must be comparatively weak.

It is tempting to infer that the principal factor for the evolution of delayed recruitment in ravens is probably a posse effect, but any such inference must be tentative because some of our assumptions may be unrealistic in ways that bias the model's predictions away from a major role for the status-enhancement effect in the evolution of delayed recruitment and thus exaggerate the importance of the posse effect. Two assumptions are particularly vulnerable to this criticism: the assumption that carcasses last only a day (whereas in nature they may last for several weeks), and the assumption that a gang's probability, , of repulsing a resident pair increases linearly with the number I of recruits (whereas empirical evidence suggests that = 1 for I 9). On the other hand, the assumption that an IR strategist fails to rise in status when it discovers a bonanza previously located by another IR strategist may introduce little or no bias in this direction (because a DR strategist does rise in status when it discovers a bonanza previously located by another DR strategist). The assumption that animals feed solely at bonanzas may introduce an opposite bias because the presence of other food sources could reduce potential for fitness gains through status enhancement. Thus we cannot draw any firm conclusions about the relative importance of the posse and status-enhancement effects in ravens.

Regardless of whether status enhancement is important, however, Equation 29 implies that DR will evolve in an IR population only if r<₁ or, on using Equations 17 and 27, if

(31)

In principle, we can estimate r as

(32)

In practice, data for both numerator and denominator are hard to find. Nevertheless, Marzluff et al. (1996) suggest that the distance from which an individual can be attracted is at least 10 km, and it is known that ravens can forage to a distance of at least 90 km per day (Heinrich 1988b). It therefore seems reasonable to suppose that the numerator and denominator of Equation 32 can be approximated by (10²) and (90²), respectively, yielding r = (1/9)² = 1/81 0.01, so that Equation 31 is satisfied if (1 — b)^N — 80b + 79 > 0. For N 4, as the data strongly suggest, this inequality reduces to b < 0.9875. Alternatively, and more conservatively, one could use Marzluff et al.'s (1996) map to estimate that their study area is km 1830 km, and in turn use this number as an estimate of total foraging area, which increases the estimate of r from about 0.01 to 100 / 1830 0.17. Then, for N 4, Equation 31 reduces to b < 0.793. Either way, the constraint on b seems likely to be satisfied because carcasses are extremely rare and ephemeral (Heinrich, 1988b). Thus the data tentatively suggest that conditions would favor a DR mutant in an IR population.

When first hypothesizing that raven roosts are information centers, Heinrich (1988b: 153) expressed concern over "how vagrant (and presumably unrelated) ravens can evolve a system of deliberate information sharing without being swamped by cheaters." Cheating matters only if information sharing is kin selected or reciprocative, however (Mesterton-Gibbons and Dugatkin, 1992, 1997; Dugatkin, 1977); if information sharing is mutualistic instead, then cheating ceases to be an issue. Marzluff et al. (1996) effectively recognized this point in disputing the importance of stable group membership to the operation of an information center. But although their argument is reinforced by their empirical results, it remains purely verbal. In contrast, our mathematical model shows quite rigorously that the logic of delayed recruitment qua mutualism is sound. Thus our model strengthens Heinrich (1988b, 1994a) and Marzluff et al.'s (1996) contention that raven roosts are information centers.

	APPENDIX A

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
The reward matrix
To calculate the reward matrix, we make use of some combinatorial results associated with the binomial distribution. Specifically, for 0 < p < 1, we have

(33)

(34)

(35)

(36)

where denotes the number of ways of choosing k objects from N. We will calculate each element of the reward matrix in turn.

First, we calculate R. From Equation 5 we have A(N + 1) = 1 - (1 - ), while Equation 33 and Equation 36 with p = b in Equation 6 yield

(37)

The product of the two expressions is R in Table 1.

Next we calculate T. The focal individual is now an IR strategist. To calculate the expected value in Equation 8, we must average over the distributions of both F(Z) and Z. Because F(Z) = k when N — k others have ceased to search, and because, in a population of DR strategists, an animal has ceased to search at time Z with probability G(Z), we have

(38)

Thus, if we temporarily fix the value of Z and use _T(Z) to denote the average of Equation 8 over the distribution of F(Z), for Z < L we find that

(39)

on using Equations 33-35 with p = 1 - G(Z). Note that, because G(L) = b and G'(z) = g(z), by Equation 1, the substitution x = 1 - G(z) implies

(40)

Using this result for i = 0, 1, 2 and noting that, because Z and W are independent random variables, they are distributed over the infinite quadrant in Figure 1 with joint probability density g(z) h(w) = G'(z)H'(w) per unit area, we now find that Equation 8 becomes

(41)

which readily reduces to the expression for T in Table 1.

Next we calculate P. First we note that

(42)

(43)

and

(44)

On using Equation 3 and the substitution x = 1 - G(w), we find that

(45)

because G(0) = 0, and G(L) = b. Substitution from Equation 45 into Equations 42-44 and then into Equations 10 and 11 now readily yields the expression for P in Table 1.

Finally, we calculate S. From Equations 42 and 45 we have Prob [Z < min (L, W)] = [1 - (1 - b)^N+1]/(N + 1). Substituting into Equations 10 and 13, we find that

(46)

from which the expression for S in Table 1 now readily follows.

	APPENDIX B

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
The critical recruitment probabilities
Here we calculate the critical recruitment probabilities, ₀ and ₂, in the case where both and are positive. If we think of R - T as a function of r, say , then R - P = (r) = B₀ - B₁r - B₂r², where B₀, B₁ and B₂ defined by

(47)

are all positive. Thus the graph of is that of an inverted parabola, with (0) > 0, '(r) = -B₁ - 2B₂r < 0, and

(48)

Because 0 < b < 1, this expression is negative for any N 1. So R - P is positive when r is less than the positive root of the quadratic equation (r) = 0, that is, when 0 < r < ₀, where

(49)

In general, this expression depends on , , , b, and N; however, when = 1 it depends only on , b, and N (because B₀, B₁ and B₂ are all directly proportional to , which therefore scales out of the quadratic equation for ₀). Furthermore, for r = 1 - b we have

(50)

which is positive, because 0 < 1 - (Nb + 1) (1 - b)^N < 1 for 0 < b < 1, N 1. Thus ₀ > 1 - b, confirming Equation 28.

Similarly, if we think of R - T as a function of r, say , then R - T = (r) = C₀ - C₁r - C₂r², where

(51)

and

(52)

where _? is defined by Equation 30. Because (0) = C₀ > 0 and '(r) = -C₁ -2C₂r < 0, R - T must be positive when (1) > 0, or < _?. If, on the other hand, > _? so that (1) < 0, then R - T is positive when r is less than the positive root of the quadratic equation (r) = 0. Collating these results, we find that R - T is positive when 0 < r < ₂, where

(53)

Again, this expression depends only on , b, and N when = 1.

	APPENDIX C

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 

View this table: [in this window] [in a new window]   List of symbols used

 

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
This research was supported by National Science Foundation awards DMS-9626609 and DMS-9626637 to the authors. We are grateful to Bernd Heinrich for an invaluable discussion and to an anonymous reviewer for detailed and constructive criticism of the original manuscript.

	REFERENCES

TOP ABSTRACT INTRODUCTION Mathematical model The evolutionarily stable... DISCUSSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Axelrod R, 1984. The evolution of cooperation. New York: Basic Books.

Axelrod R, Hamilton WD, 1981. The evolution of cooperation. Science 211:1390-1396.[Abstract/Free Full Text]

Brown C, Brown MB, Shaffer ML, 1991. Food-sharing signals among socially foraging cliff swallows. Anim Behav 42:551-564.

Brown CR, 1986. Cliff swallow colonies as information centers. Science 234:83-85.[Abstract/Free Full Text]

Brown JL, 1987. Helping and communal breeding in birds. Princeton, New Jersey: Princeton University Press.

Caraco T, Brown JL, 1986. A game between communal breeders: when is food-sharing stable? J Theor Biol 118:379-393.

Collias NE, Joos M, 1953. The spectrographic analysis of sound signals in domestic fowl. Behaviour 5:175-188.

Dugatkin LA, 1997. Cooperation among animals: an evolutionary perspective. New York: Oxford University Press.

Elgar M, 1986. House sparrows establish foraging flocks by giving chirrup calls if the resource is divisible. Anim Behav 34:169-174.

Evans CS, Marler P, 1994. Food calling and audience effects in male chickens, Gallus gallus: their relationships to food availability, courtship and social facilitation. Anim Behav 47:1159-1170.

Gwinner, E, 1964. Untersuchungen uber das Ausdrake-und Sozialverhalten des Kolkraben (Corvus corax corax L.). Z Tierpsychol 21:657-748.

Harriman AE, Berger R, 1990. Physical correlates of drinking priority among juvenile common ravens (Corvus corax).Percept Mot Skills Res Exch 71:839-847.

Heinrich B, 1988a. Food sharing in the raven, Corvus corax. In: The ecology of social behavior (Slobodchikoff CN, ed). San Diego, California: Academic Press;285 -311.

Heinrich B, 1988b. Winter foraging at carcasses by three sympatric corvids, with emphasis on recruitment by the raven, Corvus corax. Behav Ecol Sociobiol 23:141-156.[Web of Science]

Heinrich B, 1989. Ravens in winter. New York: Simon and Schuster.

Heinrich B, 1994a. Does the common raven get (and show) the meat? Auk 111:764-769.

Heinrich B, 1994b. Dominance and weight changes in the common raven, Corvus corax. Anim Behav 48:1463-1465.

Heinrich B, 1995a. An experimental investigation of insight in common ravens, Auk 112:994-1003.[Web of Science]

Heinrich B, 1995b. Neophilia and exploration in juvenile common ravens, Corvus corax. Anim Behav 50:695-704.

Heinrich B, Delia K, Knight T, Schaumberg K, 1994. Dispersal and association among common ravens. Condor 96:545-551.

Heinrich B, Marzluff JM, 1991. Do common ravens yell because they want to attract others? Behav Ecol Sociobiol 28:13-21.

Heinrich B, Marzluff J, Adams W, 1995. Fear and food recognition in naive common ravens. Auk 112:499-503.

Heinrich B, Marzluff J, Marzluff C, 1993. Common ravens are attracted by appeasement calls of food discoverers when attacked.Auk 110:247-254.

Levins R, 1966. The strategy of model building in population biology. Am Sci 54:421-431.

Loman J, Tamm S, 1980. Do roosts serve as `information centres' for crows and ravens? Am Nat 115:285-289.[Web of Science]

Marler P, Dufty A, Pickert R, 1986a. Vocal communication in the domestic chicken: I. Does a sender communicate information about the quality of food referent to a receiver? Anim Behav 34:188-193.

Marler P, Dufty A, Pickert R, 1986b. Vocal communication in the domestic chicken: II. Is a sender sensitive to the presence and absence of a receiver? Anim Behav 34:194-198.

Marzluff JM, Heinrich B, 1992. Foraging by common ravens in the presence and absence of territory holders: an experimental analysis of social foraging. Anim Behav 42:755-770.

Marzluff JM, Heinrich B, Marzluff C, 1996. Raven roosts are mobile information centres. Anim Behav 51:89-103.

Maynard Smith J, 1982. Evolution and the Theory of Games. Cambridge: Cambridge University Press.

Mesterton-Gibbons M, 1992. An introduction to game-theoretic modelling. Redwood City, California: Addison-Wesley.

Mesterton-Gibbons M, Adams ES, 1998. Animal contests as evolutionary games. Am Sci 86:334-341.

Mesterton-Gibbons M, Dugatkin LA, 1992. Cooperation among unrelated individuals: evolutionary factors. Q Rev Biol 67:267-281.

Mesterton-Gibbons M, Dugatkin LA, 1997. Cooperation and the prisoner's dilemma: toward testable models of mutualism versus reciprocity. Anim Behav 54:551-557.[Web of Science][Medline]

Newman J, Caraco T, 1989. Cooperative and noncooperative bases of food-calling. J Theor Biol 141:197-209.

Packer C, Ruttan L, 1988. The evolution of cooperative hunting. Am Nat 132:159-194.[Web of Science]

Parker PG, Waite TA, Heinrich B, Marzluff J, 1994. Do common ravens share ephemeral food resources with kin? DNA fingerprinting evidence. Anim Behav 48:1085-1093.

Rabenold PP, 1987. Recruitment to food in black vultures: evidence for following from communal roosts. Anim Behav 35:1775-1785.

Summers-Smith JD, 1963. The house sparrow. London: Collins.

Ward P, Zahavi A, 1973. The importance of certain assemblages of birds as `information centres' for finding food.Ibis 115:517-534.

CiteULike    Connotea    Del.icio.us    What's this?

This article has been cited by other articles:

				K. Wagner, J. A. Reggia, J. Uriagereka, and G. S. Wilkinson Progress in the Simulation of Emergent Communication and Language Adaptive Behavior, March 1, 2003; 11(1): 37 - 69. [Abstract] [PDF]

				J. R. Stevens and D. W. Stephens Food sharing: a model of manipulation by harassment Behav. Ecol., May 1, 2002; 13(3): 393 - 400. [Abstract] [Full Text] [PDF]

				S. R. X. Dall Can information sharing explain recruitment to food from communal roosts? Behav. Ecol., January 1, 2002; 13(1): 42 - 51. [Abstract] [Full Text] [PDF]

				Z. Barta and L.-A. Giraldeau Breeding colonies as information centers: a reappraisal of information-based hypotheses using the producer--scrounger game Behav. Ecol., March 1, 2001; 12(2): 121 - 127. [Abstract] [Full Text] [PDF]

This Article Abstract FREE Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Search for citing articles in: ISI Web of Science (10) Request Permissions Google Scholar Articles by Mesterton-Gibbons, M. Articles by Dugatkin, L. A. Search for Related Content PubMed Articles by Mesterton-Gibbons, M. Articles by Dugatkin, L. A. Social Bookmarking          What's this?

Online ISSN 1465-7279 - Print ISSN 1045-2249

Copyright © 2009 Oxford University Press

Oxford Journals Oxford University Press

• Site Map
• Privacy Policy
• Frequently Asked Questions

Other Oxford University Press sites: Oxford University Press Oxford Journals China Oxford Journals Japan American National Biography Booksellers' Information Service Children's Fiction and Poetry Children's Reference Corporate & Special Sales Dictionaries Dictionary of National Biography Digital Reference English Language Teaching Higher Education Textbooks Humanities International Education Unit Law Medicine Music Online Products Oxford English Dictionary Reference Rights and Permissions Science School Books Social Sciences Very Short Introductions World's Classics