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Math. Model. Nat. Phenom. Vol. 1, No. 2, 2006, pp. 1-22 Asymptotic Behavior of a Discrete Maturity Structured System of Hematopoietic Stem Cell Dynamics with Several Delays M. Adimya,b, F. Craustec1 and A. El Abdllaouia a Laboratoire de Mathematiques Appliquees CNRS UMR 5142 Universite de Pau et des Pays de l'Adour, 64000 Pau, France b ANUBIS Team, INRIA Futurs c Universite de Lyon, Universite Lyon1, CNRS UMR 5208 Institut Camille Jordan F - 69200 Villeurbanne, France Abstract. We propose and analyze a mathematical model of hematopoietic stem cell dy- namics. This model takes into account a finite number of stages in blood production, char- acterized by cell maturity levels, which enhance the difference, in the hematopoiesis process, between dividing cells that differentiate (by going to the next stage) and dividing cells that keep the same maturity level (by staying in the same stage). It is described by a system of n nonlinear differential equations with n delays. We study some fundamental properties of the solutions, such as boundedness and positivity, and we investigate the existence of steady states. We determine some conditions for the local asymptotic stability of the trivial steady state, and obtain a sufficient condition for its global asymptotic stability by using a Lyapunov functional. Then we prove the instability of axial steady states.