Stability and instability analysis for a ratio-dependent predator–prey system with diffusion effect

In this paper, we consider a ratio-dependent predator–prey system with diffusion. And we mainly discuss the following problems: (1) stability and Hopf bifurcation analysis of the positive equilibrium for the reduced ODE system; (2) Diffusion-driven instability of the equilibrium solution; (3) Hopf bifurcations for the corresponding diffusion system with homogeneous Neumann boundary conditions. In order to verify our theoretical results, some numerical simulations are also included, respectively.