$x^{\psi}\geq 0$ on $[\overline{\tau},\overline{\tau}+\underline{\tau}]$.\
Using the same arguments and the method of steps, we obtain the result for any $t\geq\overline{\tau}$.

As we are studying a cellular population, it is more interesting to consider non-negative initial conditions. Then we introduce the set $$C^{+}=\big\{ \psi\in C[0,\overline{\tau}] \ ; \ \psi(t)\geq 0, \ \textrm{ for } \ t\in[0,\overline{\tau}] \big\}.$$ The global stability of the population of primitive cells is given by the following result.

\[ddestability\] Assume that $I_{0} > (2z-1)\beta_{0}(0)$ where $z=\int_{\underline{\tau}}^{\overline{\tau}}Z(a)da$. Then the trivial solution $x\equiv 0$ of Equation (\[dde\]) is globally stable on the set $C^{+}$, that is, for all $\psi\in C^{+}$, $$\lim_{t\to +\infty} x^{\psi}(t)=0.$$

We prove the global stability of the trivial solution of (\[dde\]) by using a Liapunov functional.\
We define the functions $\lambda$ and $\Lambda$ by $\lambda(x)=x\beta_{0}(x)$ and $\Lambda(x)=\displaystyle\int_{0}^{x}\lambda(s)ds$, for all $x\geq0$. Let $H:C^{+} \to \mathbb{R}$ be the mapping defined, for all $\psi\in C^{+}$, by $$H(\psi)=\Lambda(\psi(\overline{\tau}))+\int_{\underline{\tau}}^{\overline{\tau}}Z(a) \bigg( \int_{\overline{\tau}-a}^{\overline{\tau}}\lambda^{2}(\psi(\theta))d\theta \bigg) da.$$ We show that $H$ is a Liapunov functional relative to Equation (\[dde\]). We have $$\begin{aligned}
\overset{\bullet}{H}(\psi) & = & \lim_{t \to 0} \frac{1}{t}(H(x_{t}^{\psi})-H(\psi)), \nonumber \\
    & = & \overset{\bullet}{\psi}(\overline{\tau})\lambda(\psi(\overline{\tau}))+\int_{\underline{\tau}}^{\overline{\tau}}Z(a) (\lambda^{2}(\psi(\overline{\tau}))-\lambda^{2}(\psi(\overline{\tau}-a)))da.\nonumber\end{aligned}$$ Since $$\overset{\bullet}{\psi}(\overline{\tau}) = -(I_{0}+\beta_{0} (\psi(\overline{\tau})))\psi(\overline{\tau}) + 2\int_{\underline{\tau}}^{\overline{\tau}} Z(a)\lambda(\psi(\overline{\tau}-a))da,$$ then $$\begin{aligned}
\overset{\bullet}{H}(\psi) & = & -[I_{0}+\beta_{0} (\psi(\overline{\tau}))]\beta_{0} (\psi(\overline{\tau}))\psi^{2}(\overline{\tau}) \nonumber 