Stability And Hopf Bifurcation Of A Kind Of Ecological System

Mathematics bionomics is the most complete and systematic branch of the mathematical biology. In recent years, population ecology model has been received great concern due to its extensive applications. With the rapid development of ecological mathematics, its researches of the dynamic properties concerning the predator-prey systems have become the focus of attention. It is well known that pests are the foe to crops, and great losses are caused to crops due to pests each year. To control the pests, it is necessary to understand the growth and development as well as the behavior of them. On the other hand, study on the dynamics of ecological system with delay had attracted great attention of many scholars in the field of system biology. The dynamic characteristics of kinds of ecological system are investigated by mean of stability theory and bifurcation theory of dynamical system, center manifold theorem and normal form theory. Details as follow:1. The purposes and present development of the system investigated about predator–prey,Hopf bifurcation theory and time delay system as well as expounds the concept and definition of stability, bifurcation, flip bifurcation theory, Hopf bifurcation theory, center manifold theorem, Hurwitz criterion and Lyapunov coefficient method. Briefly summarized and illustrates the conditions of flip bifurcation and Hopf bifurcation occurs, and the stability of the equilibrium. Finally, this article made a brief introduction for the main work.2. We studied the dynamical behaviors of a discrete predator–prey system with Holling type ? functional response. More precisely, we investigated the local stability of equilibria,flip bifurcation and Hopf bifurcation of the model by using the center manifold theorem and the bifurcation theory. And analyzed the dynamic characteristics of the system in two-dimensional parameter-spaces, one can observed the system exist a “cluster”phenomenon when the parameters pass through Hopf bifurcation curve. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model. The results show that we can more clearly and directly observe the chaotic phenomenon, period-adding and Hopf bifurcation from two-dimensional parameter-spaces and the optimal parameters matching interval can also be found easily.3. A stage-structured population model with birth pulse is considered, and the dynamics,the existence and stability of Hopf bifurcation and the effects of different parameters on the dynamic behavior of this model are also discussed by using the Lyapunov coefficient method and the bifurcation theory. We have analyzed dynamic characteristics of the system in two-dimensional parameter-spaces. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model. Through the nonlinear dynamicsin two-dimensional parameter-spaces, we can observe the chaotic phenomenon, period-adding and intermittent chaos phenomenon.4. We studied the dynamic behavior of a delayed SIR epidemic model system with nonlinear infection rate. Though the analysis of the distribution of characteristic roots of equation for the linearization of the system at positive equilibrium points, we discuss the stability of positive equilibrium points and conditions causing local Hopf bifurcation by using the Hurwitz criterion. Finally, some numerical simulations are presented to verify the obtained theoretical results. The results show that Hopf bifurcations occur at the positive equilibrium as the delay ? crosses some critical values.