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The optimal-patch-use problem in predation theory is investigated by use of a stochastic discrete model to match experimental situations when deterministic continuous models are inappropriate. We first consider three elementary strategies, differing in when to leave the patch in which the predator has been foraging; namely, (1) a fixed time has passed, (2) a fixed number of prey has been captured, and (3) the interval between two successive catches has exceeded a fixed time. Each of these fixed quantities has same value that optimizes the strategy concerned, given certain conditions. The optimized strategies are compared to determine the most efficient. It is shown that the ranking of the strategies depends critically on the type of prey distribution between patches, other things being equal; e.g., strategy 3, the best when the variance of prey distribution is sufficiently high, tends to be the worst when the variance is minimal. The above strategies do not use full information for estimating the number of prey remaining in the patch; for example, every time interval between two successive catches might be used by a more sophisticated predator. The statistical decision theory reveals, however, that the number of prey already taken (n) and the total time spent foraging (t) in the patch are the minimal sufficient statistics for estimating the number of unexploited prey under the random-search assumption. In other words, under the assumption of random search the estimator r is a function of n and t only at most, and detailed knowledge of the distribution of the time intervals is immaterial. Thus, the sophisticated predator should leave the patch when the estimator r(n,t) falls below a certain value that optimizes the strategy adopted. The estimator r depends on the prey distribution: If prey are distributed contagiously, estimated value of the remaining prey jumps up each time a capture is made, but it steps down if prey are distributed regularly. However, r may be a function of n or t alone, depending on the type of the between-patch distribution of prey. In particular, r is a function of n only for a completely regular distribution, and the best strategy is reduced to elementary strategy 2. If the distribution is random (Poisson), r is a function of t only. Then the best strategy becomes simply elementary strategy 1. It is pointed out that whether the predator under observation actually behaves optimally in accordance with the distribution of prey should be revealed by plotting the number of captures against the length of period for which the predator stayed in each patch.
Iwasa et al. (Fri,) studied this question.