The Dynamic Analysis For The Predator-Prey System With Four Delays

A predator-prey system with four time delays is considered. This system is a reasonable generalization of the corresponding one studied in the recent literature. By constructing suitable Lyapunov functionals, both local and global stabilities of the positive equilibrium are given under its existence hypothesis. It is shown that the positive equilibrium will loss its stabilities and the Hopf bifurcation then occurs as the delay surpasses the critical values. Then based on the center manifold theorem and the Poincarénormal form theory, the direction of the bifurcation, the stability and the period variance of the period solution produced by the Hopf bifurcation are studied. Finally, it is proved that the period solutions appeared during the Hopf bifurcation will persist globally by using the global extension method.