Hopf Bifurcation Theory In The Population Dynamics

This paper is divided into two parts. In the first part, the Leslie predator-prey system with two delays is considered.We establish the existence of local Hopf bifurcation.The direction and stability of the bifurcating periodic solutions are determined by applying the center manifold theorem and normal form theory.Due to the global Hopf bifurcation theorem by Wu, the conditions to guarantee the global existence of periodic solutions are given.Numerical simulations are presented to support the theoretical results found.In Part 2, a delayed predator-prey system with stage structure is investigated. Sufficient conditions for the system to have multiple periodic solutions are obtained when the delay is sufficiently large by applying Bendixson's criterion. The results obtained here are development and generalization of the results in [63].