Hopf Bifurcation Of Several Classes Of Biomathematics Models

As a branch of ecology, population ecology has been developed very extensively and systematically by utilizing mathematical theory. In recent years, the prey-predator system has received a great deal of attention by mathematicians and biologists for its widely applications. On the other hand, the dynamics of neural networks have been investigated with the special attention given by Professor Hopfield of California Institute of Technology, who introduced the continuous neural networks model firstly. In this paper, the Hopf bifurcation problems of several classes of biomathematics models are investigated. By choosing the delay as a bifurcation parameter, it is shown that Hopf bifurcation occurs as the delay passes through a sequence of critical values. Furthermore, a formula for determining the direction of Hopf bifurcation and stability of the bifurcating periodic solutions is given by using the normal form method and center manifold theory. The paper consists five sections.As an introduction, in Section One, the background and the current research status of biomathematics models are given. The motivations of this work are briefly addressed.In Section Two, some elementary tools are listed.In Section Three, we study a class of delayed prey-predator model with Beddington- DeAngelis functional response which is based on the known model, the conditions for the stability of the positive equilibrium point is obtained; furthermore, the delay bound and the conditions of the existence of Hopf bifurcation are obtained. Then, a delayed three-level food chain model with Beddington-DeAngelis functional response is investigated; the stability of the four nonnegative equilibria are analyzed and the conditions of the existence of Hopf bifurcation are obtained by using the same method. A formula for determining the direction of Hopf bifurcation and stability of the bifurcating periodic solutions is given by using Hassard's method. The results can be considered as the generization of the results obtained by Lu and Chen.In Section Four, a tri-neuron model with distributed delays is discussed. The conditions for the existence of Hopf bifurcation are obtained, and a formula for determining the direction of Hopf bifurcation and stability of the bifurcating periodic solutions is given by using the normal form method and center manifold theory. We generize the results obtained by Liao and Zhao.The conclusion is shown in Section Five, and some problems are concluded.