Stability And Hopf Bifurcation Of A Tumor Immune Models With Time Delay And Diffusion

In this paper,the stability and Hopf bifurcation of the tumor immune model with time delay and diffusion are studied.The main work includes three aspects.First,the stability and Hopf bifurcation of the local tumor immune model with time delay are investigated.The stability analysis of the equilibria of the model and the conditions for generating Hopf bifurcation are given.The stability of the bifurcating periodic solution and the direction of Hopf bifurcation are discussed by using the normalization theory and the central manifold theorem.Then,the stability and Hopf bifurcation of the tumor immune model with self-diffusion are studied.It is shown that self-diffusion does not cause Turing instability.Finally,the stability and Hopf bifurcation of the tumor immune model with time delay and diffusion are discussed.By analyzing the characteristic equation of the equilibria,the asymptotic stability of the equilibria and the condition that the model has Hopf bifurcation are obtained.The results show that the interaction of time delay and diffusion can cause Turing instability.