Complicated dynamics of tumor-immune system interaction model with distributed time delay

In this paper, we propose a distributed delay model to investigate the dynamics of the interactions between tumor and immune system. And we choose a special form of delay kernel which combines two delay kernels: a monotonic delay kernel representing a fading memory and a nonmonotonic delay kernel describing a peaking memory. Then, we discuss the effect of such delay kernel on system dynamics. The results show that the introduction of nonmonotonic delay kernel does not change the stability of tumor-free equilibrium, but it can induce stability switches of tumor-presence equilibrium and cause a rich pattern of dynamical behaviors including stabilization. Moreover, our numerical simulation results reveal that the nonmonotonic delay kernel has more complicated effects on the stability compared with the monotonic delay kernel.