Existence positivity and stability for a nonlinear model of cellular proliferation

Existence, positivity and stability for a nonlinear model of cellular proliferation? Mostafa Adimy† and Fabien Crauste‡ Year 2004 Laboratoire de Mathematiques Appliquees, FRE 2570 Universite de Pau et des Pays de l'Adour, Avenue de l'universite, 64000 Pau, France Abstract In this paper, we investigate a system of two nonlinear partial differential equations, arising from a model of cellular proliferation which describes the production of blood cells in the bone marrow. Due to cellular replication, the two partial differential equations exhibit a retardation of the maturation variable and a temporal delay depending on this maturity. We show that this model has a unique solution which is global under a classical Lipschitz condition. We also obtain the positivity of the solutions and the local and global stability of the trivial equilibrium. Keywords: nonlinear partial differential equation, age-maturity structured model, blood production sys- tem, delay depending on the maturity, positivity, local and global stability. 1 Introduction We analyse, in this paper, a mathematical model arising from the blood production system. It is based on a system proposed by Mackey and Rudnicki [19], in 1994, to describe the dynamics of hematopoietic stem cells in the bone marrow. The origin of this system is a model of Burns and Tannock [7] (1970) in which each cell can be either in a proliferating phase or in a resting phase (also called G0-phase).

• univ-pau


• cells just after


• cellular proliferation


• after division


• mature stem cells


• partial differential equations


• nonlinear analysis


• universite de pau et des pays de l'adour