Analysis of the free boundary problem of vascular tumor growth with periodic nutrient supply and time delay terms

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient $\sigma (r,t)$ through blood vessels, and the tumor attracts blood vessels at a rate proportional to $\alpha (t)$. So, the boundary value condition

holds on the boundary, where the function $\psi (t)$ is the concentration of nutrient externally supplied to the tumor. Considering that the nutrients provided by the outside world are often periodic, the research in this paper assumes that $\psi (t)$ is a periodic function. Sufficient conditions for the global stability of zero steady state are presented. Under certain conditions, we prove that there exists at least one periodic solution to the model. The results are illustrated by computer simulations.