Stability And Hopf Bifurcation For Several Predator-Prey Systems

In recent years, the study of Hopf bifurcation problems has been one of important subjects in dynamical systems and has been applied extensively in many fields such as Mechanics, Physics, Chemistry, Biology, Ecology, Control, Numerical calculations, En-gineering technology and Economics and Social sciences and so on. The predator-prey system is the fundamental structure in the population dynamics and studying these sys-tems is important to understand the real world. Based on those facts, we consider stability and Hopf bifurcation of two kinds of predator-prey systems with multiple delays.In Chapter 2, we study the stability and Hopf bifurcation of a delayed two predator-one prey system with Hollingâ…¡functional response. Firstly, by applying Hopf bifurcation theory, the conditions of stability of the positive equilibrium and existence of the periodic solution with two same delays and inequality are respectively obtained, when the gestation of two predators and mature time of the second predator are considered at the same time. Secondly, an explicit algorithm determining the direction of Hopf bifurcation and the stability of bifurcated periodic solutions is given by the normal form method and the center manifold theorem. Furthermore, the global Hopf bifurcation theorem is used to get the condition of the global existence of Hopf bifurcation. Meanwhile, some numerical simulations are carried out to illustrate our analytic results.In Chapter 3, the stability and Hopf bifurcation of the above system are investigated with feedback delay to the prey and the second predator, respectively. Some conditions of stability of equilibria and the existence of local Hopf bifurcation are obtained. Further-more, by using the normal formal method and the center manifold theorem, we give the properties of periodic solution bifurcated from the positive equilibrium. Some numerical simulations are also given to verify our theoretical results in the end.In Chapter 4, the stability and Hopf bifurcation in a predator-prey system with stage-structure and harvesting is presented and studied, whose coefficients are delay-dependent. The switch phenomena occurs on the positive equilibrium with the delay increasing. That is, the positive equilibrium switches from stability to instability again to stability when the delay crosses over a sequence of critical values. Some numerical simulations are in-cluded. Simulations also show that changing the harvesting may also destroy the stability of system, or even causes the predator to die.