Predator–Prey Dynamics and Ideal Free Distribution in a Heterogeneous Environment

Abstract

The concept of an ideal free distribution (IFD) is extended to a predator–prey system in a heterogeneous environment. We consider reaction–diffusion–advection equations which describe the evolution of spatial distributions of predators and prey under directed migration. Modification of local interaction terms is introduced, if some coefficients depend on resource. Depending on coefficients of local interaction, the different scenarios of predator distribution are possible. We pick out three cases: proportionality to prey (and respectively to resource), indifferent distribution and inversely proportional to the prey. These scenarios apply in the case of nonzero diffusion and taxis under additional conditions on diffusion and migration rates. We examine migration functions for which there are explicit stationary solutions with nonzero densities of both species. To analyze solutions with violation of the IFD conditions, we apply asymptotic expansions and a numerical approach with staggered grids. The results for a two-dimensional domain with no-flux boundary conditions are presented.