Dynamic Analysis Of Two Kinds Of Tumor-immune System With Delay And Diffusion

Tumor is one of the diseases that endangers human health.Tumor disease has become a difficult problem for medical workers because of its difficult treatment.Understanding the mechanism of tumor-immune system is the premise of tumor treatment.Delay affects the dynamic character of tumor-immune system.Two kinds of tumor-immune models with delay are established in order to study the effect of delay on the interaction between tumor cells and immune system.The results have certain practical significance for medical personnel understanding the pathogenesis of tumor.In the first model,the tumor-immune system with delay and diffusion is established.The delay of immune cells recognizing tumor cells is selected as parameter and the stability of equilibrium is analyzed by using bifurcation theory.The conditions for the system generating stable equilibrium and undergoing Hopf bifurcation are established.The normal form of Hopf bifurcation reduced on the central manifold is derived by using the multiple time scales method,and the method to judge the stability of periodic solutions is given.Finally,the above theoretical analysis is verified by numerical simulations.The results show the human body has higher immunity,and the immune system can control the deterioration of tumor cells when the delay of immune cells recognizing tumor cells is small.In the second model,considering both the delay of immune cells identifying tumor cells and the delay of tumor cells growth,the tumor-immune system with two delays and diffusion is established.The stability of equilibrium and the existence of Hopf bifurcation are derived and the normal form of Hopf bifurcation reduced on the central manifold is obtained by multiple time scales method.The theoretical analysis is verified by numerical simulations.The results show that the proliferation of tumor cells can be inhibited by controlling the delay of tumor cells growth and the delay of immune cells identifying tumor cells.