$x\mapsto \beta(m,x)$ is constant for $x\leq 0$, with $\beta(m,x)=\beta(m,0)$. For example, Mackey [@mackey1978] considered the function $\beta$ defined by $$\label{betalipschitz}
\beta(m,x)= \left\{ \begin{array}{ll}
\displaystyle\frac{a(m)}{x^{n}+b(m)}, & \textrm{ if } x\geq0, \\
\displaystyle\frac{a(m)}{b(m)}, & \textrm{ if } x<0,
\end{array} \right.$$ for all $m\in[0,1]$, where $a$ and $b$ are two positive and continuous functions on $[0,1]$ and $n\geq 1$.

System (\[p(t,m,a)\])-(\[n(t,m,a)\]) is completed by boundary conditions modelling the flux between the resting phase and the proliferating phase.

The first condition, $$\label{bcp(t,m,a)}
p(t,m,0)=\beta (m,N(t,m)) N(t,m),$$ represents the efflux of cells leaving the resting phase to the proliferating phase.

The second condition, $$\label{bcn(t,m,a)}
n(t,m,0)=2 (g^{-1})^{\prime}(m)\int_{\underline{\tau}}^{\overline{\tau}}
\kappa(g^{-1}(m),a)p(t,g^{-1}(m),a)da,$$ describes the fact that, firstly, cells with maturity $m$ entering the resting phase come from proliferating mother cells with a maturity $g^{-1}(m)$ and, secondly, that age of a dividing mother cell is between $\underline{\tau}$ and $\overline{\tau}$ and this cell divides with a rate $\kappa$.

We also consider the maturity flow $\pi_{s} : [0,1] \to [0,1]$, defined for $s \leq 0$, satisfying the ordinary differential equation $$\left\{ \begin{array}{rcll}
\displaystyle\frac{d\pi_{s}(m)}{ds} & = & V(\pi_{s}(m)),& s \leq 0, \\
\pi_{0}(m)       & = & m, &
\end{array} \right.$$ for all $m\in[0,1]$, which represents the evolution of cells maturity to reach a maturity $m$ 