Study On The Prey-Predator System With Nonlinear Functional Response Function

The prey predator model plays a significant role in the population dynamics models. The most famous ecological model based on the two population of the predator and the prey is dissected by two mathematical ecology scholars of V.Volterra and A.J.Lotka. In population ecology, in order to achieve a reasonable explanation and prediction of some ecological phenomena, we establish a mathematical model which can accurately describe the ecological system, and analyze some ecological phenomena by means of mathematical calculation and theory. Thus we can provide an effective solution to the problem of ecology. By 1965 Holling had proposed three types of functional response functions based on the different species. In view of the functional response function, the model of the prey predator model with functional response is highly concerned. Generally speaking, nonlinear functional response function has relatively obvious effects on the structure of the system, which can change the stability of the equilibrium point and cause the branch. So, it has important theoretical value and practical significance to research the dynamic model of the prey predator with nonlinear functional response function. Through the comprehensive application of Jacobian matrix comparison principle, stability theory and bifurcation theory, this paper mainly study the dynamical properties of the predator prey model with Holling type functional response function. Then the stability of the two kinds of equilibrium points of the target model is analyzed and proved, and the bifurcation of the model is discussed. The full text structure is arranged as follows:In the first chapter, we firstly summarize the origin, development and research significance of the prey predator dynamics model. And the development and trend of the feed system is introduced.In the second chapter, the model of the prey predator dynamics with nonlinear functional response function of Holling II type is introduced. We emphasis on the nature of the two kinds of equilibrium points, and deduce the possibility of the Hopf bifurcation at the equilibrium point.In the third chapter, we investigate the dynamic model of the prey predator model with Holling II type nonlinear functional response function. Then, the dynamic characteristics of the two kinds of equilibrium points are obtained by using the comparison principle, stability method and bifurcation theory. And the sufficient conditions of the Hopf bifurcation are gained. Finally, the existence of the saddle node bifurcation and the trans critical bifurcation is proved with the help of the Sotomayor’ s theorem.Finally, the conclusions and problems of the prospect about the research content are addressed.