HOPF BIFURCATION ANALYSIS FOR A DELAYED LESLIE-GOWER PREDATOR-PREY SYSTEM WITH DIFFUSION EFFECTS

A delayed predator-prey diffusion system with homogeneous Neumann boundary condition is considered. In order to study the impact of the time delay on the stability of the model, the delay tau is taken as the bifurcation parameter, the results show that when the time delay across some critical values, the Hopf bifurcations may occur. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution have been established. The effect of the diffusion on the bifurcated periodic solution is also considered. A numerical example is given to support the main result.