Competitive exclusion and coexistence of a delayed reaction–diffusion system modeling two predators competing for one prey

In this work, we study a time delayed reaction–diffusion system with homogeneous Neumann boundary conditions. This system describes two predators competing for the same prey. By the method of upper and lower solutions, we obtain sufficient conditions for the competitive exclusion principle to hold and sufficient conditions of the global asymptotic stability of positive constant solution. By taking time delay as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant solution are proved to occur for a sequence of critical values of the delay parameter. It is shown that there are three coexistence forms for the three species: steady states, spatial homogeneous and inhomogeneous periodic oscillations.