Qualitative analysis on a diffusive and ratio-dependent predator–prey model

Recently, ratio-dependent predator–prey models have been favoured by many biologists as they better describe predator–prey interactions where predation involves a researching, sharing or competing process. The present paper concerns a predator–prey model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann boundary condition and its main objective is to provide further qualitative analysis of the dynamics of the system and its corresponding steady-state problem. The global stability of the unique positive constant steady state is discussed, and the non-existence and existence results for positive non-constant steady states (stationary patterns) when one of the diffusion rates is large or small are also established. The existence result of positive non-constant steady states suggests that this predator–prey system enjoys a richer dynamics compared to the one with the linearly prey-dependent functional response. In particular, our study reveals the critical role of the predator diffusion in capturing the stationary patterns. The mathematical techniques in the paper can find applications in some other predator–prey models.