Hopf Bifurcation Analysis And Control For Two Classes Of Predator-Prey Models With Time Delay

Predator-prey model plays an important role in population dynamics, the dynamical property of this model is widely concerned and deeply studied by many scholars in recent years. Based on the fact that time delay will influence dynamical behavior that can be considered to reflect the laws of nature more accurately. The thesis mainly considered the problem of Hopf bifurcation and bifurcation control for two classes of predator-prey models with time delay.Firstly, the research background, significance and status of predator-prey model are reviewed. Based on that, the research content of this paper and corresponding preliminary knowledge are introduced.Secondly, a predator-prey model with two time delays, ratio-dependent and stage structure for the predator is investigated. By choosing the two time delays as the bifurcation parameter and utilizing stability theory for delay differential equation, the local stability is analyzed; utilizing Hopf bifurcation theorem, the existence of Hopf bifurcation is established; furthermore, based on the normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution, and the correctness of the theory is verified by numerical simulation.Thirdly, a delayed predator-prey model with disease in the predator and nonlinear incidence rate is investigated. The sufficient conditions for the local stability and the existence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter. A new hybrid strategy is proposed to control the Hopf bifurcation, in which state feedback and parameter perturbation are used to delay the onset of an inherent bifurcation. Meanwhile, the bifurcation direction and the stability of the periodic solution of the controlled system are discussed, and the hybrid control can effectively control the Hopf bifurcation by numerical simulation.Finally, we summarize the work of this thesis, and the future research direction is prospected.