Stability and Hopf Bifurcation of a Reaction–Diffusion System with Weak Allee Effect

Abstract

A delayed three-component reaction–diffusion system with weak Allee effect and Dirichlet boundary condition is considered. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained via the implicit function theorem. Moreover, taking delay as the bifurcation parameter, the Hopf bifurcation near the spatially nonhomogeneous steady-state solution is proved to occur at a critical value. Especially, the direction of Hopf bifurcation is forward and the bifurcated periodic solutions are unstable. Finally, the general results are applied to four types of three-species population models with weak Allee effect in growth.