proliferating phase (see [@burns]). The cells in the resting phase can not divide. They mature and, provided they do not die, they eventually enter the proliferating phase. In the proliferating phase, if they do not die by apoptosis, the cells are committed to divide and give birth, at the point of cytokinesis, to two daughter cells. The two daughter cells enter immediately the resting phase and complete the cycle. In the resting phase, a cell can remains indefinitely.

The model in [@mackey1993] has been analysed by Mackey and Rey in 1995 [@mackey1995_1; @mackey1995_2], Crabb et al. in 1996 [@crabb1996_1; @crabb1996_2], Dyson, Villella-bressan and Webb in 1996 [@webb1996] and Adimy and Pujo-Menjouet in 2001 [@adimypujo; @adimypujo2]. In these studies, the model in [@mackey1993] was simplified by assuming that all cells divide exactly at the same age. In the most general situation in a cellular population, it is believed that the time required for a cell to divide is not identical between cells (see the works of Bradford et al. [@bradford] about experiments on mice). To our knowledge, this hypothesis has been given for the first time in [@mackey1993] for particular cases, and only numerical studies have been investigated.

In [@webb2000] and 