-N_{n-1}(s,\pi_{-(t-s)}(m))| e^{-I(t-s)} ds, \\
            & \leq & \varepsilon L^{n+1}\theta^{n}\int_{\overline{\tau}}^{t}e^{-I(t-s)} e^{-\rho(s-\overline{\tau})} ds, \\
            & \leq & \varepsilon L^{n+1}\theta^{n}e^{\rho\overline{\tau}}e^{-It}\int_{\overline{\tau}}^{t}e^{(I-\rho)s} ds.\end{aligned}$$ Hence, finally, we obtain $$\begin{aligned}
|N_{n+1}(t,m)-N_{n}(t,m)| & \leq & \varepsilon L^{n+1}\theta^{n}e^{\rho\overline{\tau}}e^{-It}\Big[ 1+2\tilde{\zeta}e^{\rho\overline{\tau}} \Big]\int_{\overline{\tau}}^{t}e^{(I-\rho)s}ds, \\
            & \leq & \varepsilon (L\theta)^{n+1} e^{-\rho(t-\overline{\tau})}.\end{aligned}$$ This means that (\[hyprecurrence2\]) is true for $n\geq 1$.\
Then, we have, for all $(t,m)\in[\overline{\tau},T]\times[0,1]$, $$\begin{aligned}
|N_{n}(t,m)| & \leq & \sum_{i=1}^{n} |N_{i}(t,m)-N_{i-1}(t,m)| + |N_{0}(t,m)|, \\
    & \leq & \sum_{i=1}^{n} \varepsilon (L\theta)^{i} e^{-\rho(t-\overline{\tau})} +  \varepsilon e^{-\rho(t-\overline{\tau})}, \\
    & \leq & \varepsilon e^{-\rho(t-\overline{\tau})}\Big(1+L\theta+ \cdots + (L\theta)^{n} \Big).\end{aligned}$$ Since $L\theta < 1$, we conclude that, for all $t\in[\overline{\tau},T]$ and $m\in[0,1]$, $$|N^{\varphi}(t,m)|\leq \frac{\varepsilon}{1-L\theta}e^{-\rho(t-\overline{\tau})}.$$ This result is true for any $T>\overline{\tau}$. So, it is true for $t\geq\overline{\tau}$, and the proposition is proved.

Global stability of the population of primitive cells {#stabilitydde}
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We are interested, in this section, in the behaviour of the population of immature cells. That is, the cell population with maturity $m=0$.

Let $N^{\varphi}$ be a solution of Problem (\[N(t,m)\])-(\[icmf\]) for an initial condition $\varphi\in\mathcal{D}(A)$. We know that $$\lim_{m\to 0} V(m)\frac{\partial}{\partial m}N^{\varphi}(t,m) = 0.$$ Then, the function $x$ defined, for all $ t\geq 0$, by $$x(t)=N^{\varphi}(t,0),$$ is a solution of the following delay differential equation $$\label{dde}
x'(t) = -(I_{0}+\beta_{0}(x(t)))x(t) + 2\int_{\underline{\tau}}^{\overline{\tau}} Z(a)\beta_{0}(x(t-a))x(t-a)da,$$ for $t\geq 