Dynamical Analysis Of Two Eco-Epidemiological Models

In this paper,by using the differential equations qualitative theory and the knowledge of bifurcation theory,two kinds of eco-epidemiological model are studied.One is that the disease was only spread in the predator with ecological nonlinear incidence rate of the epidemic model.The other is eco epidemic model of constant rate harvesting prey.For the two models,we discussed the boundedness of the solutions and then we give the exit conditions for the boundary balance points and the positive equilibrium points.And next we discuss the stability conditions for the balance points,At last,the bifurcation conditions of the systems are derived.This paper is divided into four chapters: The first chapter mainly introduces the significance and the research status of the eco-epidemiological model,as well as the main research work and the purpose of this paper.The second chapter describes the basic concepts and research methods.In the third chapter,we study the eco-epidemiological model with nonlinear incidence rate and the disease only spread in the predator,for which the nonlinear incidence rate is20 b I S.In this chapter,we analyze the boundedness of the solutions and the existence and stability of boundary equilibrium and positive equilibrium by using the differential equations qualitative theory and the knowledge of bifurcation theory.Then we analyze the conditions of Bogdanov-Takens bifurcation of positive equilibrium,so we get the corresponding saddle node bifurcation curves,Hopf bifurcation curves and homoclinic bifurcation curves.At last,as for the Hopf bifurcation of the positive equilibrium,we analyze the direction of bifurcation and the existence of stable limit cycles.In Chapter IV,we study the eco-epidemiological model with constant harvesting rate in the prey,and we are devoted to studying the local asymptotic stability of positive equilibrium of the model,and obtain the positive equilibrium of the model and its existence conditions.Finally,we analyze the degenerate Hopf bifurcation near a higher-order equilibrium by Liapunov-Schmidt method.