Hopf Bifurcation and Stability of a Competition Diffusion System with Distributed Delay

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In this paper we investigate the effects of time delay and diffusion rate on the stability of the positive steady state for a competition diffusion system with distributed delay. We obtain the condition of instability of the positive uniform steady state. Regarding the time delay as bifurcation parameter, we reduce the original system on the center manifold and get the stability of the Hopf bifurcation periodic solutions. Finally, regarding the diffusion rate as a bifurcation parameter, we show that the onedimensional competition diffusion system with infinite delay and Dirichlet boundary condition exhibits the spatiotemporal structures near the steady state of the system.