Global Stability of a Partial Differential Equation with Distributed Delay due to

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Global Stability of a Partial Differential Equation with Distributed Delay due to Cellular Replication? Mostafa Adimy† and Fabien Crauste‡ Year 2002 Laboratoire de Mathematiques Appliquees Universite de Pau et des Pays de l'Adour Avenue de l'universite, 64000 Pau, France Abstract In this paper, we investigate a nonlinear partial differential equation, arising from a model of cellular proliferation. This model describes the production of blood cells in the bone marrow. It is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay. Our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population. 1 Introduction and motivation This paper analyses a general model of the blood production system based on a model proposed by Mackey and Rey in 1993 [12]. The initial form of this model is a time-age- maturity structured system and it describes the dynamics of proliferative stem cells and precursors in the bone marrow. It consists in a population of cells which are capable of both proliferation and maturation. In this model, the period of life of each cell is divided into a resting phase and a pro- liferating phase (see [4]). 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.

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Global Stability of a Partial DierentialEquation with Distributed Delay due toCellular Replication∗Mostafa Adimy†and Fabien Crauste‡Year 2002

LaboratoiredeMathematiquesAppliqueesUniversitedePauetdesPaysdel’AdourAvenuedel’universite,64000Pau,France

AbstractIn this paper, we investigate a nonlinear partial dierential equation, arising froma model of cellular proliferation. This model describes the production of blood cells inthe bone marrow. It is represented by a partial dierential equation with a retardationof the maturation variable and a distributed temporal delay. Our aim is to prove thatthe behaviour of primitive cells inuences the global behaviour of the population.1 Introduction and motivationThis paper analyses a general model of the blood production system based on a modelproposed by Mackey and Rey in 1993 [12]. The initial form of this model is a time-age-maturity structured system and it describes the dynamics of proliferative stem cells andprecursors in the bone marrow. It consists in a population of cells which are capable of bothproliferation and maturation.In this model, the period of life of each cell is divided into a resting phase and a pro-liferating phase (see [4]). 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 proliferatingphase, if they do not die by apoptosis, the cells are committed to divide and give birth, atthe point of cytokinesis, to two daughter cells. The two daughter cells enter immediately theresting phase and complete the cycle. In the resting phase, a cell can remains indenitely.The model in [12] has been analysed by Mackey and Rey in 1995 [13, 14], Crabb et al.in 1996 [5, 6], Dyson, Villella-bressan and Webb in 1996 [7] and Adimy and Pujo-Menjouet∗This paper has been published in Nonlinear Analysis, 54, 8, 1469-1491 (2003).†mostafa.adimy@univ-pau.fr‡fabien.crauste@univ-pau.fr

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