Bifurcation Theory In A Class Of Tumor-immune Competition Systems With Delay

Recently, human beings are facing a serious malignant tumor problem, so more and more scholars focus on researches in this field. At the same time, some experts and scholars have been establishing tumor immune models and studying the related dynamical behaviors. This paper aims at studying the model proposed by Kuznetsov and Taylor in1994. Inspired by Mayer etc., we introduce the influence of time delay in Kuznetsov’s model, and study the dynamical behaviors of this model.This paper is divided into four chapters. In the first chapter, we introduce the background of tumor-immune and the model that we will research. In the second chapter, we study a tumor model with delay, and show the properties of the Hopf-periodic solutions of this system, including the direction of Hopf bifurcation, the periodic and stability of Hopf bifurcation periodic solutions. Especially, the existence of global Hopf bifurcation is given in this section. In the third chapter, we consider a more general tumor model with delay:first, we analyze the linear stability of this model and the existence of the Hopf bifurcations, then, we analyze the bifurcation of the model, including the codimension one and codimension two bifurcations. In the last chapter, we consider the reaction-diffusion tumor-immune model with delay, and we discuss the influence of diffusion with the help of the basic theory of partial functional differential equations.