Diffusion-Induced Instability of the Periodic Solutions in a Reaction-Diffusion Predator-Prey Model with Dormancy of Predators

Abstract

A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and stability of the equilibrium solutions and derive precise conditions on system parameters so that the system can undergo Hopf bifurcations around the positive equilibrium solution. Moreover, the properties of Hopf bifurcation are studied in detail. For the reaction-diffusion system, we are able to derive conditions on the diffusion coefficients so that the spatially homogeneous Hopf bifurcating periodic solutions can undergo diffusion-triggered instability. To support our theoretical analysis, we also include several numerical results.