Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone

Abstract

In this work, we study the change of behavior of positive solutions in a Leslie predator-prey model when a simple protection zone and cross-diffusion for the prey are introduced. We analyze the effects of cross-diffusion and protection zone on the bifurcation continuum of positive solutions. The asymptotic behavior of positive solutions is also discussed as the cross-diffusion and the birth rate of the predator tend to infinity, respectively. Finally, for small birth rates of two species and large cross-diffusion for the prey, the detailed structure and stability of positive solutions are established. Our results indicate that the environmental heterogeneity, together with large cross-diffusion, can produce much more complicated stationary patterns, moreover, our research here reveals significant difference from those studied in Du et al. (J Differ Equ 246:3932-3956, 2009), Oeda (J Differ Equ 250:3988-4009, 2011) and Wang and Li (Nonlinear Anal Real World Appl 14:224-245, 2013).