Stability and Hopf bifurcation for a three-component reaction–diffusion population model with delay effect

A delayed three-component reaction–diffusion population model with Dirichlet boundary condition is investigated. The existence and stability of the positive spatially nonhomogeneous steady state solution are obtained via the implicit function theorem. Moreover, taking delay ττ as the bifurcation parameter, Hopf bifurcation near the steady state solution is proved to occur at the critical value τ0τ0. The direction of Hopf bifurcation is forward. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the stability of bifurcating periodic solutions occurring through Hopf bifurcations is investigated. It is demonstrated that the bifurcating periodic solution occurring at τ0τ0 is orbitally asymptotically stable. Finally, the general results are applied to four types of three species population models. Numerical simulations are presented to illustrate our theoretical results.