Hopf Bifurcation For A Tumor Model With Immune Response On Tumor Surface And Time Delay

Medical research has shown that the interactions between tumor and immune cells are closely related to the shape of the tumor.For the tumor without blood vessels,the immune response rate depends on tumor’s surface area.Besides,researchers note that the immune system cannot attack tumor cells as soon as they appear,and it takes time to activate and differentiate immune cells.Therefore,a two-dimensional tumorimmune model with surface immune effect and delay is proposed in this paper.Firstly,it is proved that zero equilibrium is always unstable.In order to remove the influence of fractional power,a reversible transformation is introduced.The existence and uniqueness of positive equilibrium are proved.The linearized systems at boundary equilibrium and positive equilibrium are calculated and the roots of the corresponding characteristic equations are analyzed,respectively.It is found that the boundary equilibrium is always unstable and the stability of positive equilibrium switches with the delay.Moreover,a sufficient condition such that system undergoes Hopf bifurcation at positive equilibrium is determined.By using center manifold theorem and normal form method,the reduced equation restricted onto the center manifold is derived and the properties of local Hopf bifurcation are obtained.Then we prove that the connected component connecting each center is unbounded by using the equivariant degree theory.It is also demonstrated that the periodic solutions of system are uniformly bounded and there is no the nonconstant periodic solution with ? period.Thus,the global Hopf bifurcation result is established.Finally,we choose parameters based on biological background and carry out simulations.We simulate the joint effect of the killing rate to tumor of CTL immune cells and the growth rate of tumor on the occurrence of Hopf bifurcation;a stable region on the r-k plane is obtained.The phenomena that the volume of tumor and the number of immune cells are asymptotically stable and oscillate periodically are exhibited,respectively.We further present the variations of amplitude for periodic solutions as the delay increases.This explains the existence of global Hopf bifurcation.Finally,how the recruitment rate,death rate,killing rate to tumor of immune cells and the growth rate of tumor affect the first bifurcation value is simulated.Additionally,the impact of the first bifurcation value on survival time is shown,which indicates that decreasing immune delay will enlarge survival time.