Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type III functional response

The spatial, temporal, and spatiotemporal dynamics of a reaction-diffusion predator-prey system with ratio-dependent Holling type III functional response, under homogeneous Neumann boundary conditions, are studied in this paper. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented. For the reaction-diffusion model, firstly the parameter regions for the stability or instability of the unique constant steady state are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next, it is proved that the model exhibits Hopf bifurcation, which produces temporal inhomogeneous patterns. Finally, the existence and nonexistence of nonconstant steady- state solutions are established by bifurcation method and energy method, respectively. Numerical simulations are presented to verify and illustrate the theoretical results.