Volterra System With Time Delay Hopf Bifurcation Analysis

In this paper, by using the bifurcation method of ordinary differential equations, Hopf bifurcation for two classes of delayed volterra predator-prey models are studied. By analyzing the characteristic equation at the positive equilibrium, and the sufficient conditions of the system occurring Hopf bifurcations are given. In addition, by using the normal form method and center manifold theorem, the nature of bifurcation is analyzed,such as, bifurcation direction, the stability of bifurcation periodic solutions, and period. Four chapters are included in this paper, and the tree of this paper is as follows:Chapter one is the introduction. Firstly, the background and research value are briefly discussed. Secondly, the main content of this paper and problem are presented.In the chapter two, a delayed volterra predator-prey system is considered. By applying the theorem of Hopf bifurcation and regarding the delayτas the bifurcation parameter, the stability of the positive equilibrium is discussed. It is found that Hopf bifurcations occur when the delayτpasses through a sequence of critical values. In addition, by using the normal form method and center manifold theorem, we obtain the formula to determine the bifurcation direction, the stability of bifurcation period ic solutions, and period. Finally, numerical example is included to illustrate the theoretical result. Related conclusions have been improved and generalized by our results.In the chapter three, a predator-prey system with two delays is investigated. By applying the theorem of Hopf bifurcation skillfully and choosing the delaysτ1, andτ2 as the bifurcation parameters in turn, and analyzing the characteristic equation of the linearized system of original system at the positive equilibrium, the stability of the positive equilibrium is researched, and the sufficient conditions of the positive equilibrium occurring Hopf bifurcations are given. That is, there is a small amplitude of the periodic solution near the equilibrium point, and its biological significance showed that the two groups to coexist in the form of periodic oscillations. The methods used in this chapter is relatively new. Also, the results established can be considered as the complement of the known ones. The conclusion and prospect is shown in chapter four. By comparing two classes of predator-prey models from several ways,work on this thesis is summarized, and pointed out some problems needing for further research.