Properties of Hopf bifurcation to a reaction-diffusion population model with nonlocal delayed effect

This paper is concerned with a reaction-diffusion population model with nonlocal delayed effect and zero-Dirichlet boundary condition. Under the condition when the delayed feedback control is dominant, the normal form for spatially nonhomogeneous Hopf bifurcation from the sufficiently small positive equilibrium is computed by means of the normal form method and the center manifold theorem for partial functional differential equations. It is revealed that Hopf bifurcations appearing at the small positive equilibrium are forward and all bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. To verify the validity of the obtained theoretical results, numerical simulations are also provided.