考虑到免疫效应细胞的激活及作用环节具有不可避免的时滞因素，肿瘤免疫系统可以用泛函微分方程来刻画。时滞对系统的影响是本质的，会导致系统出现分支现象。本项目将对具有时滞肿瘤免疫系统进行分支问题研究，刻画其发生突变的一些临界现象，预测肿瘤在免疫细胞作用下的变化规律，为免疫治疗提供理论依据。具体地，根据肿瘤生长方式与肿瘤免疫疗法特点，引入时滞及脉冲等影响因素，建立时滞肿瘤免疫系统的数学模型；利用泛函微分方程的Razumikhin型定理、中心流形与规范型理论，以免疫时滞与活化率为参数，研究系统的结构稳定性及可能出现的各种分支情况，如Hopf分支，Hopf-zero分支等，揭示肿瘤免疫系统的动力学行为；分析肿瘤细胞在免疫细胞作用下生长变化的临界情况，找出控制肿瘤生长的关键因素，提出肿瘤免疫的控制方案，为肿瘤免疫治疗提供理论支持。

Considering the inevitable delay factors in the activation and action of immune response cells, the tumor-immune system can be characterized by functional differential equations. The effect of time delay on the system is essential, which leads to the appearance of a bifurcation of the system. This project investigates the bifurcation problems of tumor-immune systems with time delay, depicting some critical phenomena of its mutation, predicting the change rule of tumor under the action of immune cells, and providing a theoretical basis for immunotherapy. Specifically, based on the characteristics of tumor growth and tumor immunotherapy, through introducing time delay factors and impulsive effects, we establish the mathematical model of the delayed tumor-immune system. Using the Razumikhin type theorem, the center manifold and the normal form theory in functional differential equations, by taking the time delay of immune and activation coefficients as parameters, we study the structural stability of the system and the various bifurcations of the system such as Hopf bifurcation, Hopf-zero bifurcation and other bifurcation phenomena. The dynamic behaviors of immune cells and tumor cells are revealed. The obtained phenomena of bifurcation analysis are used to predict the critical condition of growth and metabolism of tumor cells and immune cells, and to deduce the key factors that control tumor growth. This project will present tumor-immune control scheme to provide theoretical support for tumor immunotherapy.