Effects of cross-diffusion for a prey-predator system in a heterogeneous environment

This article concerns the stationary problem of a cross-diffusion prey-predator system with a protection zone for the prey. We first give the necessary condition and sufficient condition for the existence of coexistence states of the two species, by applying the bifurcation theory. Furthermore, the asymptotic behavior of coexistence states is established as the cross-diffusion coefficient of the prey tends to infinity. We also analyze the corresponding limiting system.