Aims and Objective: In order to understand the dynamic mechanisms of tumor growth
and make a contribution to develop anti-cancer treatment strategies, a mathematical model for tumor
growth with two-time delays is proposed in this article.
Materials and Methods: First, the relationships among host cells, tumor cells and effector cells, and
the biological meaning of two-time delays are explained. Moreover, the system stability is discussed
by analyzing the characteristic equation of the model. In addition, the existence and properties of
oscillatory dynamic are also researched by using normative theory and central manifold method.
Finally, the numerical simulations are performed to further illustrate and support the theoretical
Results: Both two-time delays in the model can affect the dynamics of tumor growth. Meanwhile,
the system can experience a Hopf bifurcation when the delay crosses a series of critical values.
Further, a clear formula is deduced to determine the Hopf bifurcation and the direction of stability of
the periodic solution. Finally, these results are verified by using numerical simulation.
Conclusion: The results demonstrated that the time from identifying tumor cells to making the
appropriate response for the immune system and the time needed for competition between host cells
and tumor cells for natural resources and living space is significant for tumor growth. These findings
in this paper may help us better understand the behaviors of tumors and develop better anti-cancer