Stability and bifurcation analysis of a tumor-immune system with two delays and diffusion

A tumor-immune system with diffusion and delays is proposed in this paper. First, we investigate the impact of delay on the stability of nonnegative equilibrium for the model with a single delay, and the system undergoes Hopf bifurcation when delay passes through some critical values. We obtain the normal form of Hopf bifurcation by applying the multiple time scales method for determining the stability and direction of bifurcating periodic solutions. Then, we study the tumor-immune model with two delays, and show the conditions under which the nontrivial equilibria are locally asymptotically stable. Thus, we can restrain the diffusion of tumor cells by controlling the time delay associated with the time of tumor cell proliferation and the time of immune cells recognizing tumor cells. Finally, numerical simulations are presented to illustrate our analytic results.