Foraging under competition: Evolutionarily stable patch-leaving strategies with random arrival times. 1. Scramble competition

Our objective is to determine the evolutionarily stable strategy [14] that is supposed to drive the behavior of foragers competing for a common patchily distributed resource [16]. Compared to [18], the innovation lies in the fact that random arrival times are allowed. In this first part, we investigate scramble competition: the game still yields simple Charnov-like strategies [4]. Thus we attempt to compute the optimal long-term mean rate gamma* [11] at which resources should be gathered to achieve the maximum expected fitness: the assumed symmetry among foragers allows us to express gamma* as a solution of an implicit equation, independent of the probability distribution of arrival times. A digression on a simple model of group foraging shows that gamma(N)* can be simply computed via the classical graph associated to the marginal value theorem-N is the size of the group. An analytical solution allows us to characterize the decline in efficiency due to group foraging, as opposed to foraging alone: this loss can be relatively low, even in a "bad world," provided that the handling time is relatively long. Back to the original problem, we then assume that the arrivals on the patch follow a Poisson process. Thus we find an explicit expression of gamma* that makes it possible to perform a numerical computation: Charnov's predictions still hold under scramble competition. Finally, we show that the distribution of foragers among patches is not homogeneous but biased in favor of bad patches. This result is in agreement with common observation and theoretical knowledge [1] about the concept of ideal free distribution [12,22].