Dynamic behaviors of a predator–prey model with weak additive Allee effect on prey

Abstract

This paper deals with a reaction–diffusion system modeling predator–prey interaction with additive Allee effect. We first analyze nonnegative constant equilibrium solutions and study the stability of the unique positive solution. Then we investigate the steady state bifurcation and Hopf bifurcation from the unique positive constant solution, respectively. The main tools used here include the bifurcation theory, the space decomposition and the implicit function theorem. Moreover, we describe the global structure from a simple eigenvalue. Finally, for the case d1=0, we prove the existence of far-from-equilibrium steady states with jump discontinuities. Moreover, by applying the singular perturbation method, we give a proof of the existence of large amplitude solutions when d1 is sufficiently small.