Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay of immune system where the immune cell has a probability p in killing tumor cells. We find increasing time delay destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability p on Hopf bifurcation values is also discussed.