| Tumor immunotherapy is an important branch in the fields of bio-medicine. In the early eighties, bio-mathematicians has paid much attention to tumor immune mechanism, and numerous experiments and research are conducted in this area,which show the mechanisms on how the immune system including innate and adaptive immune protect organism from tumor cells. With the development of mathematical biology, Kuznetsov proposes a well-known tumor immune system. In recent twenty years, the outcomes of the system, such as stability and bifurcations,play an important role in cancer treatment.There is usually a time delay for the effector cells take effect on tumor cells in human body, and this process can be described by delay systems. The bifurcation analysis is the mainly adopted method to study the dynamical behavior of systems with parameters. The bifurcation is a phenomenon that the dynamical behavior of a system changes qualitatively as the parameter varies in the vicinity of a critical value. Local Hopf bifurcation is common for the systems described by differential equations, in which a fixed point loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane, and a small amplitude limit cycle will be generated.A class of tumor immune system with time delay is studied in the paper,mainly focus on the stability of equilibrium and Hopf bifurcations. First, the tumor immune system with time delay feedback is introduced, and the existence of equilibria and their stability are shown. Second, the conditions for the occurrence of Hopf bifurcation are provided. The quantities which determine the diction of bifurcation and the stability of bifurcated periodic solutions are also presented.Finally, numerical simulations are carried out using MATLAB. The results reveals that the number of tumor and immune cells in an organism would behave oscillations over time due to their competition. |
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