Ideal Free Distribution in a Multiple Predator-prey System

We study the mechanism and effects of ideal free distributions (IFDs) on an ecological community consisting of n prey and m predator species, for any positive integers n and m, by considering the corresponding diffusive Lotka-Volterra system with time-periodic coefficients. We define a notion of joint IFD in a timeperiodic environment, and give necessary and sufficient conditions for it to be achieved by a subcollection of prey and predator species with suitable dispersal strategies. Next, we show, via construction of a Lyapunov function, that such dispersal strategies are evolutionarily stable, in the sense that if a subcollection of prey and predator species adopts an ideal free dispersal strategy, then the total community must converge to an IFD for large time; if a unique combination of prey-predator species adopts an ideal free strategy, then it can drive all other species to extinction. Conversely, if a combination of prey-predator species adopts a non-ideal free dispersal strategy, then it can be invaded by some suitable mutant strategies. Our results provide insight into the evolution of spatial distribution of ecological communities with predator-prey interactions.