Stability And Bifurcation Of Two Predator-prey Models

The interaction between species is an important topic in the field of population ecology.In recent years, the interaction species with diseases or harvest rate have attracted muchattention due to their significant nature in both theory and application.In this paper, by means of qualitative theory and bifurcation theory of differentialequations, we study two types of the predator-prey models. We study three kinds ofbifurcations of positive equilibriums in the first model. For the second model, we get thesufficient and necessary conditions for local asymptotic stability of the nonnegativeequilibrium.The first chapter introduces the significance of studying eco-epidemiological models,and then the main works of the thesis are introduced. The second chapter describes somebasic knowledge in the qualitative and bifurcation theories of differential equations. In thethird chapter, we study a predator-prey model with constant harvest rate. We analysis thenature of the equilibrium and discuss the existence of the equilibrium, we study the Hopfbifurcation, Bogdanov-Takens bifurcation and Fold-Hopf bifurcation, and we obtain stabilitycoefficient of the Hopf bifurcation curve and the bifurcation curves near theBogdanov-Takens, also give the bifurcation diagram of the Fold-Hopf bifurcation. In ChapterIV, we focus on a four-dimensional SIS ecological model with standard incidence and diseasespreading only in predator or prey. By means of qualitative and stability theory of ordinarydifferential equation, we study boundedness of solutions and the nature of the nonnegativeequilibrium, further give the conditions for local asymptotic stability of the boundaryequilibrium and the Hopf bifurcation of the positive equilibrium.