Global asymptotical stability and persistent property for a diffusive predator–prey system with modified Leslie–Gower functional response

A diffusive predator–prey system with modified Holling–Tanner functional response and no-flux boundary condition is considered in this work. A sufficient condition which ensures persistence of the system is obtained. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using a comparison method. It is shown that our result supplements and complements one of the main results of Shi et al. [H.B. Shi, W.T. Li, G. Lin, Positive steady states of a diffusive predator–prey system with modified Holling–Tanner functional response, Nonlinear Analysis: Real World Applications 11 (2010) 3711–3721].