\\
        &   & +\int_{\underline{\tau}}^{\overline{\tau}}Z(a) \big[\lambda^{2}(\psi(\overline{\tau}))+2\lambda(\psi(\overline{\tau}))\lambda(\psi(\overline{\tau}-a))-\lambda^{2}(\psi(\overline{\tau}-a))\big]da, \nonumber \\
        & = & -[I_{0}+\beta_{0} (\psi(\overline{\tau}))]\beta_{0} (\psi(\overline{\tau}))\psi^{2}(\overline{\tau}) + 2\int_{\underline{\tau}}^{\overline{\tau}}Z(a) \lambda^{2}(\psi(\overline{\tau}))da \nonumber \\
        &   & -\int_{\underline{\tau}}^{\overline{\tau}}Z(a) \big[\lambda(\psi(\overline{\tau}))-\lambda(\psi(\overline{\tau}-a))\big]^{2}da.  \nonumber\end{aligned}$$ Hence, $$\overset{\bullet}{H}(\psi) \leq  -[I_{0}-(2z-1)\beta_{0} (\psi(\overline{\tau}))]\beta_{0} (\psi(\overline{\tau}))\psi^{2}(\overline{\tau}).$$ Let $u$ be the function defined, for $x \geq 0$, by $$u(x)=r(x)\beta_{0}(x)x^{2},$$ where, for all $x\geq 0$, $$r(x)=I_{0}-(2z-1)\beta_{0}(x).$$ $r$ satisfies $r(0)>0$ and $\lim_{x \to \infty} r(x)= I_{0}\geq 0$. Since $\beta_{0}$ is decreasing on $\mathbb{R}^{+}$, $r$ is a monotonous function. Therefore, $r$ is a positive function on $\mathbb{R}^{+}$.\
Hence $u$ is non-negative on $[0,+\infty)$ and $u(x)=0$ if and only if $x=0$. Consequently, every solution of Problem (\[dde\])-(\[icdde\]) with $\psi\in C^{+}$ tends to zero as $t$ tends to $+\infty$.

Combining this result with the result of local stability shown in Section \[sectionlocalstability\], we can obtain global stability for a class of solutions of Equation (\[N(t,m)\]) or its integrated version (\[mildformulation\]).

Global stability {#globalstability}
================

In this section, we give a general criterion for global stability of Equation (\[mildformulation\]). We first recall a definition.

Let $\Omega$ be a subset of $C([0,\overline{\tau}]\times[0,1])$. A solution $\overline{N}$ of Problem (\[mildformulation\])-(\[icmf\]), with initial condition $\overline{\varphi}$, is said to be [*globally stable on the set $\Omega$*]{} if, for all $\varphi\in\Omega$ and all $m\in[0,1]$, $$\lim_{t\to +\infty} |N^{\varphi}(t,m)-\overline{N}(t,m)|=0.$$ This solution $\overline{N}$ 