Hopf Bifurcation In Biological Systems And Chaos

In the recent several decades, 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. In Chapter 2, a partial dependent prey-predator system with delay is discussed through Hopf theory. Take the delay as the bifurcation parameter, the results show that, when the delay pass through the critical value, the stability of equilibrium change, and a family of periodic orbits bifurcating from the equilibrium. Using Hassard method and the center manifold theory, the formula of estimating the direction and stability of periodic orbits are also obtained.Recently, the study on the chaos control is progressive development constantly, some methods to control chaos were proposed one after another, thereinto that the method of time-delayed feedback control is easier to manipulate, and the biological systems affected by the time delay aren't avoid in the reality. Hence, it's reasonable to control the chaos in biological systems using delay. In Chapter 3, a hybrid ratio-dependent tree species food chain model is discussed by the theory of functional dif-ferential equation. By taking delay as bifurcation parameter, the results showing that when the time pass through the critical value, the stability of the positive equilibrium changing from unstable to stable, hence chaos vanishes and a family of periodic orbits bifurcating from the positive equilibrium. Using Hassard method and the center man-ifold theory, the formula of estimating the direction and stability of periodic orbits are also obtained. And chaos phenomena appears again with the delay increasing. Finally, numerical simulations are also included.