Global stability and stationary pattern of a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response

This paper is concerned with a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response subject to the homogeneous Neumann boundary condition. Firstly, by upper and lower solutions method, we prove the global asymptotic stability of the unique positive constant steady state solution. Secondly, introducing the cross diffusion, we obtain the existence of non-constant positive solutions. The results demonstrate that under certain conditions, even though the unique positive constant steady state is globally asymptotically stable for the model with self-diffusion, the non-constant positive steady states can exist due to the emergency of cross-diffusion, that is to say, cross-diffusion can create stationary pattern. Finally, using the bifurcation theory and treating cross diffusion as a bifurcation parameter, we obtain the existence of positive non-constant solutions.