The Dynamic Behavior Of A Class Of An Immunogenic Tumor Model

Recently, human beings have been facing a serious malignant tumor problem, so more and more scholars focus on research in this field. At the same time, some experts and scholars have been establishing tumor immune models and studying the related dynamic behavior. This paper aims at studying the model proposed by Kuznetsov and Taylor in 1994. Inspired by Mayer etc., we introduce the influence of time delay in Kuznetsov’s model, and study the dynamic behavior of this model.In the first section, we introduce the background of the research and some known results of the tumor-Immune model. In section two, we give the conditions for the existence of the tumor-free equilibrium. We also prove that the tumor-free equilibrium is globally asymp-totically stable. Also, we give conditions for the existence of a tumor-present equilibrium. The tumor-present equilibrium is not unique. In the last section, time delay is considered. Local bifurcation theory is applied to explore the rich dynamic behavior of the model. The normal form of the model is derived. It is found that Hopf bifurcation occurs when Υ crosses some critical value. Using the theory of normal form and center manifold, general formulas for the existence, direction, period and stability of hopf bifurcation are given. A numerical example and simulations are given to illustrate the bifurcation analysis and the obtained results.