usually depends on $\tau$ and the condition $\Delta(m)<m$, used by Mackey and Rudnicki in the example $V(m)=rm$, $r>0$, and $g(m)=\alpha m$, $0<\alpha\leq 1$, which yields to $\Delta(m)=\alpha^{-1}e^{-r\tau}m$, is not true in the general case, for all $\tau>0$.

In this work, we consider a general situation, when the age at cytokinesis is distributed with a kernel supported on an interval $[\underline{\tau},\overline{\tau}]$ with $0\leq\underline{\tau}<\overline{\tau}<+\infty$. This yields to the boundary condition (\[bcn(t,m,a)\]).

We obtain a first order partial differential equation with a distributed time delay and a retardation of the maturation variable, which depends also on the time delay. In our model, the nonlinear part depends on the maturity variable (see Equations (\[N(t,m)\]) and (\[P(t,m)\])).

The population behaviour of the model obtained is analysed in the case $$\label{equationintro}
\lim_{t\to +\infty} N(t,0)=0.$$ In the interpretation of this model, $N(t,0)$ corresponds to the population of the primitive cells type, and Condition (\[equationintro\]) corresponds, for a long time, to an insufficient supply of the primitive cells.

We prove that Condition (\[equationintro\]) guarantees, under local stability (resp. local exponential stability) of the partial differential equation, global stability (resp. global exponential stability) of the system. These results are obtained under the assumption that the cells have enough time, 