Effects of dispersal for a predator-prey model in a heterogeneous environment

Abstract

In this paper, we study the stationary problem of a predator-prey cross-diffusion system with a protection zone for the prey. We first apply the bifurcation theory to establish the existence of positive stationary solutions. Furthermore, as the cross-diffusion coefficient goes to infinity, the limiting behavior of positive stationary solutions is discussed. These results implies that the large cross-diffusion has beneficial effects on the coexistence of two species. Finally, we analyze the limiting behavior of positive stationary solutions as the intrinsic growth rate of the predator species goes to infinity.

Dynamics and pattern formation in diffusive predator-prey models with predator-taxis

“…The relatively simple models which describe the behaviors of one predator and one prey have been extensively studied. Some problems of stage structures were proposed since there are always two stages in the growing process of the majority of species, such as immature and mature stages [6,8,26,27,32,33]. A reaction diffusion model with stage structure for the predator was proposed in [8],…”