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( *)(tau) 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 tau(0) and tau of the delay such that E( *)(tau) is asymptotically stable for tau<tau(0) and unstable for tau(0)<tau<tau . We show that E( *)(tau(0)) is a Hopf bifurcation critical point for an approachable model.
Talibi Alaoui
Hamad
H
Université Chouaib Doukkali Faculté des Sciences, Département de Mathématiques et Informatique, El Jadida, Morocco. talibi.hamad@caramail.com
Yafia
Radouane
R
D000465:Algorithms; D002452:Cell Count; D002453:Cell Cycle / Q000502:physiology*; D049109:Cell Proliferation; D002470:Cell Survival; D006412:Hematopoietic Stem Cells / Q000166:cytology*; D006801:Humans; D007700:Kinetics; D008954:Models, Biological