Application Of Bogdanov-Takens Bifurcation With Epidemiological Model

This paper considers predator-prey system with a nonlinear incidence rate and disease in the prey, The prey system has an invarible input rate.There are some references about predator-prey models and epidemiological models,but little attention has been paid to merge these two important areas of research.In the paper,by means of bifurcation theory and qualitative theory of ordinary differential equations,we study the eco-epidemiological models of this type.We study the dynamics of the model on local stability of equilibria and Hopf bifurcation of two-dimensional systems'positive equilibrium and Hopf bifurcation three-dimensional systems'positive equilibrium , we also discuss Bogdanov-Takens bifurcation of two dimensional systems. we have some complicated dynamical behaviors than do those with epidemiological models because of higher dimension of this systems.Numerical simulation results are given to verify the theoretical results.This article is divided into four parts.The first part describes the significance for studying bifurcation of the eco-epidemiological model and arrangements for this paper.The second part introduces some basic concepts and methods of this article.In third part ,we study an eco-epidemiological model with nonlinear incidence rate and disease in the prey, The prey system has an invarible input rate.The main purpose of this part is to study lacal stability of equilibria and Hopf bifurcation of the three-dimensional systems,we also give some stability conditions.The three-dimensional Hopf bifurcation near the positive equilibrium is analyzed by using the projection method for center manifold computation. Numerical simulation results are given to verify the theoretical results.The fourth part, we still study the same systems,but the systems is the twe-dimemsional.We obtain the condition of Hopf bifurcation in the twe-dimensional system,we receive the limit cycles and discuss limit cycles'stability,we also give the corresponding numerical simulation.We obtain the condition and give the proof with appearance of Bogdanov-Takens bifurcation.We study Bogdanov-Takens bifurcation including a saddle-node bifurcation,a Hopf bifurcation and a homoclinic bifurcation.