Study On The Stability And Bifurcation Analysis For Delayed Predator-prey Models

The predator-prey models are very important in the population dynamics, which gain considerable attention in ecological and biological fields. The stability of solutions and a variety of phenomena produced by the oscillation of the solutions have increasingly become one of the significant research topics. Delay is studied, because the trends of the development and change of things in each moment not only depend on the current state, but also depend on the history of the state. In general, delay will have a great impact on the topological structure of solutions of differential equations. It also can change the stability of equilibrium point and cause bifurcation and chaos phenomena. Therefore, it is necessary to study the effect of time delay for the predator-prey dynamics.In this thesis, based on Lyapunov stability theorem, the center manifold theorem, the norm form, and numerical simulations methods, the stability and bifurcation of delayed predator-prey model are systematically studied. The main work is summarized as follows.First, we study the dynamics of prey-dependent predator-prey model. A Lyapunov function with LaSalle’s invariance principle is constructed to prove the global stability of the equilibrium indicating the extinction of top-predator. The conditions including the stability and Hopf bifurcation of coexisting equilibrium are given, the properties of Hopf bifurcation are analyzed by using the center manifold theorem and the norm form method. The conclusion that bifurcation periodic solution is a global existence is reached by using the numerical simulation.Secondly, we investigate the phenomena of Hopf bifurcation with delayed ratio-dependent predator-prey models. The prey-dependent response functions are only related to the diet which appeared "paradox of enrichment", so a more suitable function should be based on both prey and predator, which is the so-called ratio-dependent functional response. The time-delayed3D Gause-type predator-prey model with ratio-dependent functional response is considered and τ is seen as the bifurcation parameter. The pos-itive equilibrium stability and Hopf bifurcation point of the existence of the model are studied. The results show that when the delay passes through certain critical values, the phenomenon of Hopf bifurcation will be produced.Thirdly, we consider the phenomena of Hopf-Fold bifurcation with delayed ratio- dependent Gause-type predator-prey models. The model specification type is given by the center manifold method, the dynamical properties of the model are analyzed from high codimension bifurcation and a biological explanation is given. The critical value of the branch near complete set is given in theory. Numerical simulations of periodic solutions, periodic solutions and population explosion, are consistent with the results of the theoretical analysis. The research reveals the extreme changes in regulating population through a range of mechanisms, and plays an important role in maintaining the balance of ecosystem.Finally, we prove the global stability of delayed ratio-dependent Gause-type predator-prey models. The nature of boundary solution and persistence of the system are analyzed by using the comparison theorem. The conclusion is that if the intermediate predator in a low consumption ability caused their extinction, then the prey species will survive, and at the same time the top predator will become extinct. Then through the vari-able transformation method, an equivalent system is obtained. A Lyapunov function with LaSalle’s invariance principle is constructed to get the conclusion that the model of the coexistence equilibrium is globally asymptotically stable. Through numerical simulation, a fact is found that when the delay τ is large enough, the coexistence equilibrium gradu-ally loses its stability, and a stable periodic solution emerges. This reveals that delay has an important effect on the nature of system dynamics.