Mathematical Biosciences and Engineering, 2006, 3(2): 325-346. doi: 10.3934/mbe.2006.3.325.

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Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model

1. Laboratoire de Mathématiques Appliquées, FRE 2570, Université de Pau et des Pays de l'Adour, Avenue de l'université, 64000 Pau

We analyze the asymptotic stability of a nonlinear system of two differential equations with delay, describing the dynamics of blood cell production. This process takes place in the bone marrow, where stem cells differentiate throughout division in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells in the introduction of cells in cycle, we are led to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the population's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature.
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Keywords blood cell model; delay di?erential equations; stem cells.; asymptotic stability; Hopf bifurcation; delay-dependent coe?cients; characteristic equa- tion

Citation: Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences and Engineering, 2006, 3(2): 325-346. doi: 10.3934/mbe.2006.3.325

 

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  • 2. Chi Jin, Keqin Gu, Silviu-Iulian Niculescu, Islam Boussaada, Stability Analysis of Systems With Delay-Dependent Coefficients: An Overview, IEEE Access, 2018, 6, 27392, 10.1109/ACCESS.2018.2828871
  • 3. Fabien Crauste, , Complex Time-Delay Systems, 2009, Chapter 8, 263, 10.1007/978-3-642-02329-3_8
  • 4. F. Crauste, Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch, Mathematical Modelling of Natural Phenomena, 2009, 4, 2, 28, 10.1051/mmnp/20094202

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