Ideal free dispersal in a predator-prey system

Roughly speaking, a population is said to have an ideal free distribution on a spatial region if all of its members can and do locate themselves in a way that optimizes their fitness, allowing for the effects of crowding. Dispersal strategies that can lead to ideal free distributions of populations that use them have been shown to exist and to be evolutionarily stable in a number of models for a single population. Those models include reaction-diffusion-advection equations and the analogous models using discrete diffusion or nonlocal dispersal described by integrodifferential equations. Furthermore, in the case of reaction-diffusion-advection models and their nonlocal analogues, for environments that are static in time there are strategies that allow populations to achieve an ideal free distribution by using only local information about environmental quality and/or gradients. In this paper, we extend some of these ideas and results to certain Lotka-Volterra type predator-prey systems. In the case of single-species models it is often possible to do the analysis via methods based on monotonicity, but in the predator-prey context those fail so we use methods based on a Lyapunov functional.