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Abstract: In this paper, we study an unstructured model of a cellular population in the spirit of Grabosch and Heijmans [Grabosch A, Heijmans HJAM. Production, development and maturation of red blood cells, A mathematical model. AM-R 8919, ISSN 0924-2953, 1989.] model. The cellular population is described by a system of differential equations with one delay. The basic assumption is that the cell population responsible for the production of blood cells consists of three compartments: the stem cells, the precursor cells, and the blood cells. We prove that the model has two possible steady states and their dynamics (depending on time delay) are studied in term of the local stability, we illustrate these results with numerical simulations for some different values of the time delay. [Copyright &y& Elsevier]

Abstract: We consider the haematopoietic stem cells model (HSC) with one delay introduced by Mackey [M.C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, blood 51 (1978) 5; M.C. Mackey, Mathematical models of haematopoietic cell replication and control, in: The Art of Mathematical Modelling: Case Studies in Ecology, Physiology and Biofluids, H.G. Othmer, F.R. Adler, M.A. Lewis, J.C. Dallon (Eds), Prentice-Hall, New York, 1997, p. 149] and Andersen and Mackey [L.K. Andersen, M.C. Mackey, Resonance in periodic chemotherapy: a case study of acute myelogenous leukemia, J. theor. Biol. 209 (2001) 113]. There are two possible stationary states in the model. One of them is trivial and the second E ∗(τ) depending on the delay is non-trivial . This paper investigates the stability of the non-trivial state and occurrence of the Hopf bifurcation depending on time delay. We prove the existence and uniqueness of a critical values τ 0 and of the delay such that E ∗(τ) is asymptotically stable for τ < τ 0 and unstable for . We show that E ∗(τ 0) is a Hopf bifurcation critical point for an approachable model. [Copyright &y& Elsevier]