J. Math. Biol. (1999) 38: 195—219 A new criterion for the global stability of simultaneous cell replication and maturation processes Michael C. Mackey 1 , Ryszard Rudnicki 2  Departments of Physiology, Physics and Mathematics, and Center for Nonlinear Dynamics, McGill University, 3655 Drummond Street, Montreal, Canada H3G 1Y6. e-mail: mackey@cnd.mcgill.ca  Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8, 40-013, Katowice, Poland Received: 26 August 1996 / Revised version: 22 March 1997 Abstract. We analyze a population model of cells that are capable of simultaneous and independent proliferation and maturation. This model is described by a first order partial differential equation with a time delay and a retardation of the maturation variable, both due to cell replication. We provide a general criterion for global stability in such equations. Key words: Cell cycle — Hematology — Time-age-maturation 1 Introduction Time—age—maturation models for age structured biological popula- tions have arisen in many contexts, the first of which was the modeling of human demographics as described in Keyfitz (1968), Pollard (1973), and Henry (1976). The comprehensive book of Metz and Diekmann (1986) can be consulted for an excellent survey of many of the more recent applications outside the demographic area, as well as an exposition of how such models may be formulated from the relevant biology. One of the areas in which such ‘‘time—age’’ or ‘‘time—maturation’’ models have been used with great success is that of cell replication and maturation, and these applications date from almost 40 years ago [Von Foerster (1959), Trucco (1965a, b; 1966), Oldfield (1966), Nooney (1967), Rubinow (1968, 1975)]. Recently Mackey and Rudnicki (1994)