since fast proliferations are usually observed in pathological cases.

All cells are supposed to age with unitary velocity, that means $$\frac{da}{dt}=1,$$ and to mature with a velocity $V(m)$, which is assumed to be the same in the two phases and satisfies $V(0)=0$, $V(m) > 0$ for $m \in (0,1]$, $V\in C^{1}[0,1]$ and $$\int_{0}^{m} \frac{d\theta}{V(\theta)} = + \infty, \quad \textrm{ for } m \in (0,1].$$ Since $\int_{m_{1}}^{m_{2}} \frac{d\theta}{V(\theta)}$ is the time required for a cell with maturity $m_{1}$ to reach a maturity $m_{2}$, this last condition on $V$ implies that a cell with maturity zero is unable to mature. For example, this condition is satisfied if $V(m)\underset{m\to 0}{\sim}\alpha_{1}m^{p}$, with $\alpha_{1}>0$ and $p\geq 1$.

After the division, each daughter cell enters immediately the resting phase with age $a=0$. If the maturity of the mother cell at the cytokinesis point was $m$, then we suppose that the two daughter cells have maturity $g(m)$ at birth. It is natural to assume that g is an increasing and continuous function from $[0,1]$ into itself. We also suppose that $g\in C^{1}[0,1)$ and $g(m)<m$ for $m \in (0,1)$. Furthermore, we suppose that $g^{-1}(m)=1$ if $m>g(1)$.

A cell can stay its entire life in the resting phase, so 