Then, the result of Proposition \[stabgloblip\] represents a normal behaviour of the cellular population.

If the function $x\in\mathbb{R}\mapsto\beta(m,x)$ is differentiable and decreasing on $(0,+\infty)$ for all $m\in[0,1]$, then, under the assumption $$(2\tilde{\zeta}+1) \sup_{m\in[0,1]}\beta(m,0) <I,$$ the trivial solution $N\equiv 0$ of Problem (\[mildformulation\]) is globally exponentially stable.

If the function $x\mapsto\beta(m,x)$ is differentiable and decreasing on $(0,+\infty)$ for all $m\in[0,1]$, we obtain that $$\frac{\partial}{\partial x}(x\beta(m,x)) \leq \beta(m,x) \leq \beta(m,0),$$ for $x\in(0,+\infty)$ and $m\in[0,1]$. Then, the function $x\mapsto x\beta(m,x)$ is Lipschitz continuous on $\mathbb{R}$ and the proposition \[stabgloblip\] concludes the proof.

For example, if $\beta$ is defined by (\[betalipschitz\]) and if, in addition, $$(2\tilde{\zeta}+1)\sup_{m\in[0,1]}\bigg(\frac{a(m)}{b(m)}\bigg) < I,$$ then the trivial solution $N\equiv 0$ of Problem (\[mildformulation\]) is globally exponentially stable.

Our next objective is to prove the global stability of solutions under more general conditions, less restrictive than the condition (\[condstab\]) given in Proposition \[stabgloblip\].

We need the following assumption : there exists $C\geq0$ such that $$\label{integralefinie}
\int_{m}^{g^{-1}(m)} \frac{d\theta}{V(\theta)}\leq C,$$ for all $m\in(0,1]$.

The biological interpretation of this condition is easy. That is, the time $\int_{m}^{g^{-1}(m)} \frac{d\theta}{V(\theta)}$ required for a daughter cell to reach the maturity $g^{-1}(m)$ of its mother is bounded. For example, if $V(m)\underset{m\to 0}{\sim}\alpha_{1}m$, with $\alpha_{1}>0$, then (\[integralefinie\]) is equivalent 