Dynamics of a diffusive predator–prey model with modified Leslie–Gower and Holling-type III schemes

The diffusive predator–prey system with modified Leslie–Gower and Holling-type III schemes is considered here. Firstly, stability analysis of the equilibrium for a reduced ODE system is discussed. Secondly, we obtain that the system is permanent. Thirdly, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using the method of Lyapunov function. Finally, we establish the existence and nonexistence of nonconstant positive steady states of this reaction–diffusion system, which indicates the effect of large diffusivity.