Stability and Bifurcation of a Delayed Reaction-Diffusion Model with Robin Boundary Condition in Heterogeneous Environment

In this paper, we investigate a reaction-diffusion model with delay and Robin boundary condition in heterogeneous environment. The existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions and periodic solutions are studied by employing the Lyapunov-Schmidt reduction method. Moreover, the Hopf bifurcation direction is derived. It is observed that Robin boundary condition plays a crucial role in the Hopf bifurcation. More precisely, when the boundary effect is stronger than the interaction of the species within the region, there is no Hopf bifurcation no matter how the time delay tau changes. Finally, we illustrate our general theoretical results by an application to the Nicholson's blowflies model.