Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response

This paper is concerned with a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. The uniform persistence of the solutions semiflows, the existence of global attractors, local and global asymptotic stability of the positive constant steady state of the reaction–diffusion model are discussed by using comparison principle, the linearization method and the Lyapunov functional method, respectively. The global asymptotic stability of the positive constant steady state shows that the prey and predator will be spatially homogeneously distributed as time converges to infinities.