Time delays in a double-layered radial tumor model with different living cells

Abstract

This paper deals with the free boundary problem for a double-layered tumor filled with quiescent cells and proliferating cells, where time delay $\tau >0$ in cell proliferation is taken into account. These two types of living cells exhibit different metabolic responses and consume nutrients $\sigma$ at different rates ${\lambda }_1$ and ${\lambda }_2$ ( ${\lambda }_1⩽{\lambda }_2$). Time delay happens between the time at which a cell commences mitosis and the time at which the daughter cells are produced. The problem is reduced to a delay differential equation on the tumor radius $R\left(t\right)$ over time, and the difficulty arises from the jump discontinuity of the consumption rate function. We give rigorous analysis on this new model and study the dynamical behavior of the global solutions for any initial $\phi \left(t\right)$.