Stability and Hopf bifurcation in an approachable haematopoietic stem cells model

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 τ ¯ of the delay such that E ∗( τ) is asymptotically stable for τ < τ 0 and unstable for τ 0 < τ < τ ¯ . We show that E ∗( τ 0) is a Hopf bifurcation critical point for an approachable model.