Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model

A delayed reaction-diffusion model of the Fisher type with a single discrete delay and zero-Dirichlet boundary conditions on a general bounded open spatial domain with a smooth boundary is considered. The stability of a spatially heterogeneous positive steady state solution and the existence of Hopf bifurcation about this positive steady state solution are investigated. In particular, by using the normal form theory and the centre 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 the first bifurcation point is orbitally asymptotically stable while those occurring at the other bifurcation points are unstable.