the age variable ranges from $a=0$ to $a=+\infty$. We suppose that the resting cells can die at a rate $\delta$, which corresponds to cellular differentiation, and that the reintroduction rate $\beta$, from the resting phase into the proliferating phase, depends on the maturity and the total density of cells in the resting phase.

We denote by $p(t,m,a)$ and $n(t,m,a)$ respectively the population density in the proliferating and the resting phase, at time $t\geq 0$, with maturity $m$ and age $a$. We define the total densities of proliferating and resting cells at time $t$ and a maturity level $m$, respectively, by $$P(t,m)=\int_{0}^{\overline{\tau}}p(t,m,a)da,$$ and $$N(t,m)=\int_{0}^{+\infty}n(t,m,a)da.$$

Each phase is described by a partial differential equation. Hereafter, $p=p(t,m,a)$ and $n=n(t,m,a)$. The two conservation equations are $$\label{p(t,m,a)}
\frac{\partial p}{\partial t}+\frac{\partial p}{\partial a}+\frac{\partial (V(m)p)}{\partial m} =
-\left[\gamma(m)+\kappa(m,a)\right]p,$$ $$\label{n(t,m,a)}
\frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}+\frac{\partial (V(m)n)}{\partial m} =
-\left[\delta(m)+\beta(m,N(t,m))\right]n.$$

The functions $\gamma$ and $\delta$ are supposed to be positive and continuous on the interval $[0,1]$.

For the moment, $\beta$ is supposed to be a continuous and positive function. In fact, as we are studying a cellular population, we could consider only non-negative solutions ; hence, to give a correct definition of $\beta$, we will suppose that the mapping 