Switching, Functional Response, and Stability in Predator-Prey Systems

Abstract

We consider a model for the functional response of a predator feeding on two species of prey. We assume that the predator searches randomly, at constant speed, for randomly distributed prey, but that whether it attacks a contacted prey will depend on the species of the prey, the species of the last meal, and the time since the last meal. We establish a formula for the expected time between meals on species 1, and take the functional response on species 1, the number of species 1 eaten per unit time by a single predator, to be the reciprocal of this. The general formulae are somewhat complicated, but if it is assumed that the probability of attack is independent of time, it is possible to obtain a fairly simple expression for functional response. This functional response is not necessarily stabilizing; it will be if there is a sufficiently strong (relative to the handling times) tendency for the predator to be more likely to attack a contacted prey if it is the species that was eaten last. (More detail is given at the end of the section Switching and Stability.) We then consider relative attack rates (still assuming that preference is independent of time since the last meal) first by taking the ratio of the two functional responses (for species 1 and species 2) and next by regarding the sequence of meals as a Markov chain. The answers are the same: the relative attack rate is a function only of relative density, and switching occurs provided only that the product of the probabilities of attacking the same species as last eaten is greater than the product of the probabilities of attacking the other species. It is possible to have switching and yet not have a stabilizing functional response.