and $t \geq 0$.

Hence, an integrated formulation (variation of constant formula) of Equation (\[N(t,m)\]) is given by $$\label{mildformulation}
N(t,m) = \varphi(\overline{\tau},\pi_{-(t-\overline{\tau})}(m))K(t-\overline{\tau},m) +G(N)(t,m)-J(N)(t,m),$$ if $t\geq \overline{\tau}$, and $$\label{icmf}
N(t,m) = \varphi(t,m),\ \  \textrm{ if } t\in[0,\overline{\tau}],$$ for an initial condition $\varphi\in C([0,\overline{\tau}]\times[0,1])$, the space of continuous functions on $[0,\overline{\tau}]\times[0,1]$, endowed with the supremum norm $||.||$, where $$\begin{aligned}
G(N)(t,m) = 2\!\!\int_{\overline{\tau}}^{t}\!\bigg(\!\int_{\underline{\tau}}^{\overline{\tau}}\!\! \zeta (a,\pi_{-(t-s)}(m))N(s-a,\Delta(a,\pi_{-(t-s)}(m)))\times  \\
\beta (\Delta(a,\pi_{-(t-s)}(m)),N(s-a,\Delta(a,\pi_{-(t-s)}(m)))  da \bigg) K(t-s,m) ds,\end{aligned}$$ where $$\zeta(s,m)=(g^{-1})^{\prime}(m)k(g^{-1}(m),s)\xi(s,g^{-1}(m)),$$ and $$J(N)(t,m)=\int_{\overline{\tau}}^{t}K(t-s,m)\beta(\pi_{-(t-s)}(m),N(s,\pi_{-(t-s)}(m)))N(s,\pi_{-(t-s)}(m))ds,$$ for $t \geq \overline{\tau}$ and $m \in [0,1]$.

One can also obtain, using the same technique, an integrated form of Equation (\[P(t,m)\]).

Under classical Lipschitz conditions on the nonlinear part $x\mapsto x\beta(m,x)$, we can easily show, using an iterative method and the method of steps, that, for each initial data $\varphi\in
C([0,\overline{\tau}]\times[0,1])$, the problem (\[mildformulation\])-(\[icmf\]) has a unique continuous solution $N^{\varphi}$, which is nonnegative.

Local stability {#sectionlocalstability}
===============

We are interested, in this section, in the local stability of the trivial solution $N\equiv0$ of Equation (\[N(t,m)\]) or its integrated version (\[mildformulation\]). First, we recall a useful definition.

Let $\Omega$ be a subset of $C([0,\overline{\tau}]\times[0,1])$. We say that a solution $\overline{N}$ of Problem (\[mildformulation\])-(\[icmf\]), with an initial condition $\overline{\varphi}$, is [*locally stable on the set $\Omega$*]{} if, for 