Investigating the effect of three different types of diffusion on the stability of a Leslie-Gower type predator-prey system

In this paper, we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response. We mainly study the effect of three different types of diffusion on the stability of this system. The main results are as follows: (1) in the absence of prey diffusion, diffusion-driven instability can occur; (2) in the absence of predator diffusion, diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable; (3) in the presence of both prey diffusion and predator diffusion, the system can occur diffusion-driven instability and Turing patterns. At the same time, we also get the existence conditions of the Hopf bifurcation and the Turing-Hopf bifurcation, along with the normal form for the Turing-Hopf bifurcation. In addition, we conduct numerical simulations for all three cases to support the results of our theoretical analysis.