at time $0$ from a time $s \leq 0$. $\pi_{s}$ satisfies $\pi_{s}(0)=0$ and $\pi_{s}(m) \in (0,1]$ for $s\leq 0$ and $m \in (0,1]$.

One can remark that, for all $s\leq 0$ and $m\in[0,1]$, $$\pi_{s}(m)=h^{-1}(h(m)e^{s}),$$ where the continuous function $h :[0,1] \to [0,1]$ is defined by $$h(m)=\left\{ \begin{array}{ll}
exp\bigg(-\displaystyle\int_{m}^{1} \frac{d\theta}{V(\theta)} \bigg),&\textrm{for } m \in (0,1],\\
0,& \textrm{for } m=0.
\end{array} \right.$$

Integrating the equations (\[p(t,m,a)\]) and (\[n(t,m,a)\]) over the age, using the boundary conditions (\[bcp(t,m,a)\]) and (\[bcn(t,m,a)\]) and the method of characteristics, we obtain that $N(t,m)$ and $P(t,m)$ satisfy, for $t\geq\overline{\tau}$ and $m\in[0,1]$, $$\label{N(t,m)}
\begin{array}{c}
\displaystyle \frac{\partial }{\partial t}N(t,m)+\frac{\partial }{\partial m} (V(m)N(t,m)) =
-(\delta (m)+\beta (m,N(t,m)))N(t,m)
+2(g^{-1})^{\prime}(m)\times \vspace{1ex}\\
\displaystyle\displaystyle\int_{\underline{\tau}}^{\overline{\tau}} k(g^{-1}(m),a)
\xi(a,g^{-1}(m))\beta(\Delta(a,m),N(t-a,\Delta(a,m)))N(t-a,\Delta(a,m))da,
\end{array}$$ $$\label{P(t,m)}
\begin{array}{c}
\displaystyle \frac{\partial }{\partial t} P(t,m)+\frac{\partial }{\partial m} (V(m)P(t,m)) = -\gamma (m)P(t,m)+\beta (m,N(t,m))N(t,m)\\
\displaystyle -\int_{\underline{\tau}}^{\overline{\tau}}k(m,a)\xi(a,m)
\beta(\pi_{-a}(m),N(t-a,\pi_{-a}(m)))N(t-a,\pi_{-a}(m))da,
\end{array}$$ where $$\xi (t,m)=\exp\bigg\{-\int_{0}^{t} \gamma (\pi _{-s}(m))+V'(\pi _{-s}(m))ds\bigg\},$$ and $$k(m,a)=\kappa(m,a)\exp\left(-\int_0^a\kappa(\pi_{s-a}(m),s)ds\right).$$

The mapping $\Delta : [0,+\infty)\times[0,1] \to \mathbb{R}^{+}$ is defined by $$\Delta(s,m) = \pi_{-s}(g^{-1}(m)).$$ $\Delta$ is non-increasing with respect to $s$ and non-decreasing with respect to $m$. Furthermore, for $m \in (0,1]$, $$\label{delta}
\Delta(s,m) < m \quad \textrm{ if and only if } \quad s > \int_{m}^{g^{-1}(m)} \frac{d\theta}{V(\theta)}.$$

We can notice that the solutions of Equation (\[P(t,m)\]) depend on the solutions of Equation 