[@webb2000_2], Dyson et al. considered a time-age-maturity structured equation in which the age for a cell to divide is not identical between cells. They presented the basic theory of existence and uniqueness and properties of the solution operator. However, in their model, the division is represented by the following boundary condition $$\label{bcwebb}
p(t,m,0)=\int_{0}^{+\infty} \beta(a)p(t,m,a)da,$$ which considers only one phase (the proliferating one), and the intermediary flux between the two phases is not represented in this model.

In [@mackey1994] (1994) and [@mackey1999] (1999), Mackey and Rudnicki studied the behaviour of solutions of the model considered in [@mackey1993], but only in the case when all cells divide at the same age. They obtained a first order partial differential equation with discrete time delay $\tau$ and a nonlocal dependence in the maturity variable $\Delta(m)$ due to cell replication. That is, $$\label{eqmackey}
\frac{\partial u}{\partial t}(t,m)+V(m)\frac{\partial u}{\partial m}(t,m) = f(u(t,m),u(t-\tau,\Delta(m))),$$ with $\Delta:[0,1]\to[0,1]$ a continuous function satisfying $\Delta(0)=0$ and $\Delta(m)<m$ for $m\in(0,1]$.

They gave in [@mackey1999], which is more general than [@mackey1994], a criterion for global stability in such equations. However, these authors considered only the special case when the term $f(u,v)$ in (\[eqmackey\]) does not depend on the maturity variable. Moreover, the nonlocal function $\Delta(m)$ 