Three Species Food Chain System Hopf Bifurcation And Chaos

The dynamic of prey-predator including the stability, the existence of peri-odic solutions, even chaotic occurrence is most important questions in predator-prey theory. It has received a great deal of attention of many mathematicians and biologists. It has become a very important part in mathematics and ecol-ogy. It has a great value in both theory and application. The bifurcation theory was applied in the biological systems more and more frequently, especially the Hopf bifurcation theory, which is very important to ascertain whether a certain biological system with delay has periodic orbit. Recently, the study bifurcation analysis and control of chaos for a ratio-dependent three species food chain is progressive development constantly. In this paper, basing on the three species food chain model (Hasting-Powell model), the two new three species food chain model is investigate by increasing time delay and developing ratio dependence through modification of Holling type II functional responses.In Chapter 2, we study the dynamical properties of a three species food chain model with time delay. we take the delay as the bifurcation parameter and find that when the delay passes through the critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. By applying the normal form theory and center manifold argument, we obtain the explicit for-mulas determining the stability of bifurcating periodic solutions, the direction and the period of Hopf bifurcations.A numerical simulation is given to support our results.In Chapter 3, a ratio-dependent three species food chain system with time delay is investigated. By the Hopf bifurcation theory, we choose the delay as bifurcation parameter and find that when the delay passes through the critical value, the stability of positive equilibrium changes and bifurcates a family of periodic orbits. Similarly, we use Hassard method and the center manifold theory. We can further obtain the formula of the direction and stability of periodic orbits. Finally, a numerical simulation is given to support our results. Chaos phenomena appears with the delay increasing.