(\[N(t,m)\]) whereas the converse is not true. In fact, if we know the behaviour of the solutions of Equation (\[N(t,m)\]), which describes the process of replication, then we can deduce the behaviour of the solutions of Equation (\[P(t,m)\]). Hence, we will focus our study on Equation (\[N(t,m)\]), and particularly on the stability of its trivial solution.

Let $C[0,1]$ be the space of continuous functions on the interval $[0,1]$, endowed with the supremum norm. We give, by using, for example, the semigroup theory, an integrated version of Equation (\[N(t,m)\]). First, we recall the expression of the semigroup generated by the linear operator $$A:\mathcal{D}(A) \subset C[0,1] \to C[0,1],$$ defined by $$\mathcal{D}(A) = \Big\{u \in C[0,1] \ ; \ u \in C^{1}(0,1], \ \lim_{m \to 0} V(m)u'(m) =0\Big\},$$ and $$Au(m)=\left\{\begin{array}{ll}
    -(\delta (m) +V'(m))u(m)-V(m)u'(m), & \textrm{if} \ m \in (0,1], \\
    -(\delta (0) +V'(0))u(0), & \textrm{if} \ m =0.
    \end{array} \right.$$

We can easily show (one can refer to [@webb1996] for a similar operator) that the operator $A$ is the infinitesimal generator of a $C_{0}$-semi-groupe $(S(t))_{t \geq 0}$ defined for $\psi \in
C[0,1]$, $m \in [0,1]$ and $t \geq 0$, by $$(S(t)\psi)(m)=\psi(\pi_{-t}(m))K(t,m),$$ where $$K(t,m)=exp\bigg\{-\int_{0}^{t} \delta (\pi_{-s}(m)) +V'(\pi_{-s}(m)) ds \bigg\},$$ for all $m \in [0,1]$ 