Hopf Bifurcation And Stability Analysis For A Delayed Predator-prey System With Stage Structure And Functional Response

So far, the stability, Hopf bifurcation and so on for the stage-structuredpredator-prey system with time delay have gotten a lot of results. To examinethese dynamics model not only has a wide range of theoretical biology, butalso has important practical application value. This paper will discuss the localstability of the nonnegative equilibrium and the existence of Hopf bifurcationof a delayed predator-prey system with stage-structure and functional response.Formulae are derived to determine the direction of bifurcations and the stabilityof bifurcating periodic solutions by using normal form theory and the centermanifold theorem. The main contents in this paper can be summarized as follows:In the first chapter, we ?rst introduce the biology background of the predator-prey system. Then we introduce the biological signi?cance of time delay, stage-structured and functional response. Next we introduce the result of correlationmodels. Finally, we will give out this article's research content.In the second chapter, we show the theory of characteristic equation ofnonlinear functional di?erential equation and Hopf bifurcation theorem.In the third chapter, we discuss the nonnegative equilibrium of a delayedpredator-prey system with stage-structure and functional response. The localstability of the nonnegative equilibrium. By choosing the time delay as a pa-rameter and analyzing the characteristic equation of the linearized system, weinvestigate the local stability of the nonnegative equilibrium. In addition, we getsufficient conditions for the existence of Hopf bifurcations. In the second sectionof this chapter, we derive formulae to determine the direction of bifurcations andthe stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. In the third section, numerical simulations arecarried out to illustrate the theoretical results.