Hopf Bifurcation Analysis Of The Interaction Model Of Immune System-cancer Cell With Distributed Delay

Time lag is ubiquitous in nature and human society,which has a critical effect on the dynamic behavior of the dynamical system.In this dissertation,the dynamic system of immune system-cancer cell interaction with distributed time lag is considered.In the process of immune response,it usually takes a certain time for the immune system to response after being stimulated by antigens.However,in fact,the response time will not be fixed at a certain constant value,but will float around a certain value.Therefore,the delay is distributed near an average value,and the distributed delay model will be more accurate.Under the support of NSFC(Nos.11372282 and 11972327),in this dissertation mainly Hopf bifurcation of immune system-cancer cell interaction model with distributed response time delay under two different kernel functions i.e.Dirac-Delta and Gamma is considered,and the influence of anti immune activity of tumor on the stability of the system is also investigated.The improved frequency domain method is a general method.The parameter spaces of Hopf bifurcation under different kernel functions are obtained,which is no longer confined to some special kernel functions.In the calculation process,at last only the kernel functions need to be substituted.Firstly,the model is transformed into a linear system with nonlinear feedback,that is,the characteristic equations corresponding to the phase are calculated and analyzed by the improved frequency-domain method.Then the Hopf bifurcation of the system is analyzed.The necessary conditions for the existence of Hopf bifurcation and the algebraic equation for calculating Hopf bifurcation point are discussed.Then the direction of Hopf bifurcation and the stability of periodic solution of Hopf bifurcation near the equilibrium of the model are analyzed by Nyquist stability criterion and Graphic Hopf bifurcation theorem,and the first-order approximation frequency domain diagrams of eigenvalue trajectory and amplitude trajectory are plotted on the complex plane.It is verified by the nonlinear dynamics simulation software Win PP that the numerical simulation results are consistent with those of the theoretical analysis in this dissertation,which further implies the validity and correctness of the theoretical analysis.The Hopf bifurcation observed in this dissertation indicates the existence of a stable oscillatory periodic solution,which corresponds to the equilibrium period of immune editing.Choosing appropriate parameters to make it a stable equilibrium point,corresponding to the clearance period of immune editing,which will benefit to the control and treatment of the cancer.The innovation of this dissertation is the improved frequency domain method is employed to study the more realistic immune system-cancer cell interaction model with distributed response delay and its Hopf bifurcation is investigated.