Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion

This paper deals with a delayed three-species Lotka–Volterra food chain model with diffusion effects and homogeneous Neumann boundary conditions. By taking the sum of delays as the bifurcation parameter, spatially homogeneous and nonhomogeneous Hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay parameter. In addition, sufficient conditions for global asymptotic stability of the positive constant steady state are derived by using an iteration technique. Furthermore, in order to determine the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions, the formulas are given by using the normal form theory and the center manifold reduction for PFDEs. Finally, to verify our theoretical predictions, some numerical simulations are also included.