Study On Stability And Branch Of Several Predator - Prey Models

We investigate the stability, optimal harvesting problem and Hopf bifurcation for several kinds of predator-prey models by using the Liapunov stability theory, Pontrya-gin’s maximum principle and center manifold theorem. Some new results are obtained. The thesis is divided into four sections.Chapter 1 Preference, we briefly introduce the studying background and the main work of this paper.Chapter 2 The stability and the optimal harvesting problem of the model with two competing preys and one predator in which the species are infected by some toxicants are discussed. The model is as follows The optimal harvesting policy is presented by Pontryagin’s maximum principle.Chapter 3 We consider a delayed prey-predator system with stage structure for predator which is the following Firstly, when Υ= 0, we study the property for the positive equilibrium, obtain the sufficient condition of global stability for the positive equilibrium by constructing the Liapunov function and present the optimal harvesting policy. Secondly, the existence of Hopf bifurcation is examined at the coexistence equilibria when Τ≠0. Moreover, we use center manifold theorem to analyze the property of the Hopf bifurcation.Chapter 4 The model with Beddington-DeAngelis functional response and two delays is investigated. By choosing the two delays as the bifurcation parameters, we get the sufficient conditions for the stability and the existence of Hopf bifurcation. Further-more, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution.