A Delayed Dynamical Model Describing the Effect of Tumor-Induced Vascular Endothelial Growth Factor on Angiogenesis: Backward Bifurcations and Global Dynamics

Dynamic models have been successfully used for the prediction of long-term evolutionary behavior of the growth of tumor cells. In this paper, based on previous research, time delay in the process of the growth of tumor cells is further considered and a class of dynamic models with two-time delays and general functional response function is proposed. This model describes the effect of tumor-induced vascular endothelial growth factor (VEGF) on angiogenesis. First, the model is reduced to a two-dimensional one by means of the quasi-steady-state approximation method. This model has forward and backward bifurcations of the equilibria. Then, local stability of all equilibria, global stability of the tumor cell-free equilibrium and the vascular endothelial cell-free equilibrium, and the persistence of this model are studied in detail. It is found that time delay of tumor-induced vascular endothelial cells proliferation does not change local stability of the equilibria, while time delay associated with the growth of tumor cells can lead to the existence of Hopf bifurcation periodic solutions.