in the proliferating phase ($\underline{\tau} > \tau_{0}$), to sufficiently increase their maturity.

It is also proved that, if this minimum time of replication $\underline{\tau}$ is too large, then the destruction of the population of stem cells (cells with small maturity) affects the entire system, and the population extincts for a finite time (see Corollary \[extinction\]).

The paper is organised as follows. In Section \[presentationmodel\], we present the model. We obtain a system of two partial differential equations with distributed time delay and a nonlocal dependence in the maturity variable. We give an integrated formulation of our model by using the classical variation of constant formula. In Section \[sectionlocalstability\], we study the local stability of the trivial solution. In Section \[stabilitydde\], we investigate the global stability of the delay differential equation satisfied by the population of the primitive cells type. We use a Liapunov functional to prove the global stability of the trivial solution of this equation. In Section \[globalstability\], we provide a criterion for global stability of the trivial solution in our model. This criterion is based on Condition (\[equationintro\]) and local stability of the trivial solution.

Presentation of the model {#presentationmodel}
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Each cell is characterised, in the two 