Stability And Bifurcation Analysis For A Predator-Prey Model With Two Delays

Population Ecology, with its core being the study of Population Dynamics,is an important part of Ecology. With the development of science and technol-ogy, mathematical knowledge shows increasingly broad application prospects inthe study of Ecology and we can use mathematical methods to study ecologicalphenomena, in order to interpret them and even control them. When establishingmathematical models that describe ecological phenomena, delay can be consid-ered to refect the laws of nature more accurately. The existence of delays tendsto evolve more complex dynamical properties in the system, thus bring more com-plexity and difculties to our study. Nevertheless, delay is inevitable in ecologicalphenomena, the study of which is very important to understand the real world.In this thesis, our major work is to discuss a three species predator-prey modelwith two delays. We will discuss stability and Hopf bifurcation of equilibria byemploying stability theory of functional diferential equations, bifurcation theory,center manifold reduction and normal form approach. This thesis is organized asfollows:In Chapter1, the background and the motivation for the study of predator-prey model and dynamic system with delays are presented.In Chapter2, we list the fundamental theories which will be used in our anal-ysis, i.e. the theories of stability and Hopf bifurcation, center manifold theorem,normal form theory.In Chapter3, we discuss stability of positive equilibria and prove the exis-tence of Hopf bifurcation under some conditions. Linear stability is investigatedby analyzing the associated transcendental equation,and then some sufcient con-ditions are derived to ensure that all the characteristic roots have negative realparts.In Chapter4, the direction of Hopf bifurcation, the stability and periodic ofbifurcation periodic solutions on the center manifold are determined. Based on thecenter manifold reduction and the normal form method, we provide an algorithmfor determining the bifurcation direction and stability of the bifurcation periodicsolutions.In Chapter5, numerical simulations are given to illustrate our theoreticalresults.Finally, we summarize the qualitative properties of a predator-prey model with two delays and predict the prospects of its future research.