A simpler, more general method of finding the optimal foraging strategy for Bayesian birds

Abstract

Oaten's (1977)stochastic model for optimal foraging in patches has been solved for a number of particular cases. A few cases, such as Poisson prey distribution and either systematic or random search, are easy to solve. In other cases, such as binomial prey distribution and random search, the form of the optimal strategy may be found using a theorem of McNamara, although more work is required to find which particular rule of the proper form is actually best. More generally (but not completely generally), optimal strategies may be found using dynamic programming. This requires that the number of prey found up to a particular time is a sufficient statistic for the number of prey remaining in a patch. This requirement cannot be dispensed with, but other simplifying assumptions that were used in the past are not necessary. In particular, it is not necessary, even for the sake of convenience, to assume that prey distribution has a form convenient for Bayesian analysis, such as a beta mixture of binomials or a gamma mixture of Poissons. Any prey distribution may be used if whatever prey are in a patch are located at random, and if search either is systematic for discrete time or for continuous time, or is random for continuous time. In earlier work, some pains had to be taken to find the rate of finding prey achieved by a given candidate strategy, but this is not necessary if expected gains and expected times are calculated routinely for each potential stopping point during dynamic programming. A new, simple method of finding optimal strategies is illustrated for discrete time and systematic search. This paper is based on a talk given at the Fifth Hans Kristiansson Symposium held in Lund, Sweden in August, 2003. The subject of the symposium was Bayesian foraging.