phases, by its age and its maturity. The maturity describes the development of the cell. It is the concentration of what composes a cell, such as proteins or other elements one can measure experimentally. The maturity is supposed to be a continuous variable ranging from $m=0$ to $m=1$ in the two phases.

Cells enter the proliferating phase with age $a=0$. We suppose that they can be lost with a rate $\gamma$. Every proliferating cell with maturity $m$ divides, at mitosis, with a rate $\kappa=\kappa(m,a)$. We assume that the mapping $m\mapsto \kappa(m,a)$ is continuous, the mapping $a\mapsto\kappa(m,a)$ is piecewise continuous, and that there exist $\underline{\tau}\geq0$ such that $$\kappa(m,a)=0 \qquad \textrm{ if } a\leq\underline{\tau}, \ m\in[0,1],$$ and $0<\overline{\tau}<\infty$ such that $$\int_0^{\overline{\tau}}\kappa(m,a)=+\infty \qquad \textrm{ for } m\in[0,1].$$ This latter assumption describes the fact that a proliferating cell which has not died must divide before it reaches the age $a=\overline{\tau}$. Therefore, the age of a proliferating cell ranges from $a=0$ to $a=\overline{\tau}$.

Notice that $\underline{\tau}$, which represents the minimal time a cell has to spend in the proliferating phase before it divides, can vanish. When $\underline{\tau}>0$, cells cannot divide immediately after their entrance in the proliferating phase. This is probably the most realistic situation 