Stability and Hopf bifurcation for a delayed prey–predator system with diffusion effects

This paper is concerned with a delayed Lotka–Volterra prey–predator system with diffusion effects and Neumann boundary conditions. The main purpose is to investigate the stability of spatially homogeneous positive equilibrium and give the explicit formulae determining the direction and stability of Hopf bifurcation. By linearizing the system at positive equilibrium and analyzing the associated characteristic equation, the stability of positive equilibrium and the existence of Hopf bifurcation are demonstrated. By means of the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), the direction and stability of periodic solutions occurring through Hopf bifurcation are determined. Finally, in order to verify our theoretical results, some numerical simulations are also included.