Permanence, stability, and coexistence of a diffusive predator-prey model with modified Leslie-Gower and B-D functional response

This paper investigates a diffusive predator-prey system with modified Leslie-Gower and B-D (Beddington-DeAngelis) schemes. Firstly, we discuss stability analysis of the equilibrium for a corresponding ODE system. Secondly, we prove that the system is permanent by the comparison argument of parabolic equations. Thirdly, sufficient conditions for the global asymptotic stability of the unique positive equilibrium of the system are proved by using the method of Lyapunov function. Finally, by using the maximum principle, Poincare inequality, and Leray-Schauder degree theory, we establish the existence and nonexistence of nonconstant positive steady states of this reaction-diffusion system, which indicates the effect of large diffusivity.