Bifurcation Of A Predator-prey System With A Nonlinear Incidence Rate

This paper considers predator-prey systems with disease in the prey, Assume that the predator eats only the infected prey, and the incidence rate is nonlinear. As we know, there are many references on predator-prey models and epidemiological models, but little attention has been paid so far to merge these two important areas of research. In this paper, by means of qualitative theory and bifurcation theory of differential equations, we study the eco-epidemiological models of this type. We study the dynamics of the models in terms of local analysis of equilibria and bifurcation analysis of a boundary equilibrium and a positive equilibrium. Our results show a much wider range of dynamical behaviors than do those with predator-prey systems or epidemiological models due to higher dimension of the systems. This artical is divided into four chapters.The first chapter introduces the significance of studying bifurcation and eco-epidemiological models and arrangements for this article.The second chapter describes the basic concepts and methods of this paper.In Chapter III, we study an eco-epidemiological model with nonlinear incidence rate. The main purpose of this chapter is to present local analysis and bifurcation analysis of the model, we analyse the boundedness of solutions and local stability of equilibria, by using bifurcation methods and techniques, we study Bogdanov-Takens bifurcation near a boundary equilibrium, and obtain a saddle-node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The Hopf and generalized Hopf bifurcation near the positive equilibrium is analyzed, the existence of one or two limit cycles is also discussed. Numerical simulation results are given to support the theoretical predictions.In Chapter IV, we analysis an eco-epidemiological model with Logistic growth in prey. we analyse the existence and local stability of equilibria, by using bifurcation methods and techniques, we study Bogdanov-Takens bifurcation near a boundary equilibrium including a saddle-node bifurcation, a Hopf bifurcation and a homoclinic bifurcation. The Hopf bifucation near the positive equilibrium is analyzed by using the projection method for center manifold computation. In some case,the conditions for the existence of a limit cycle are obtained.