The Bifurcation Of Two Kinds Of Ecological Model And Numerical Simulation

In ecology, it is well known that there are two kinds of mathematical models in the theory of population dynamical models:the continuous-time models and the discrete-time models described by differential and difference equations respectively. Delay and outside interference has a larger impact on the density changes of biological populations. The asymptotically stability of Host-Parasitoid model with the distributed and discrete delay and of a discrete-time prey-predator model with food-limited and distributed is investigated, including the existence and asymptotic stability of solutions, the existence and direction of Hopf bifurcation periodic solution, the existence and stabi-lity of Flip bifurcation and Neimark-Sacker periodic solution.Biological population growth is constrained by external factors and time delay. A class Host-Parasitoid Model of discrete-time and continuous-time delay, Holling functional response and survival rate is studied, including the unconditional stability of equilibrium, the existence of Hopf Bifurcation, the direction of bifurcation at the critical value, and the stability of bifurcation solution. With time lag as a parameter, sufficient conditions of the asymptotically stability of the model and of the existence of bifurcation periodic solution are derived by using the theory of characteristic value. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are discussed by using the normal form theory and the center manifold theorem, Finally, some numerical examples supporting our theoretical predictions are also given.The equilibrium of continuous-time system of a single parameter branch have the Hopf bifurcation and Fold (or cut) bifurcation in the study of dynamical systems; but the equilibrium of discrete-time system of single-parameter branch referred to as Flip bifurcation which consist of folding bifurcation and flip (double period) bifurcation, and bifurcation of Neimark-Sacker (or torus). In chapter3, the discrete-time predator-prey model with food restrictions and disturbs is studied For this model, the existence and stability of two equilibrium are analyzed. It is shown that the model undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R+2by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits with period13,15,28,cascades of period-doubling bifurcation in orbits with period2,4,8,quasi-periodic orbits and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model.