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Mathematical Biosciences and Engineering

2006, Volume 3, Issue 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
  • Received: 01 October 2005 Accepted: 29 June 2018 Published: 01 February 2006

  • MSC : 34K20, 92C37, 34D05, 34C23, 34K99.

  • 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.

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

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  • 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.


  • This article has been cited by:

    1. Fabien Crauste, 2009, Chapter 8, 978-3-642-02328-6, 263, 10.1007/978-3-642-02329-3_8
    2. D. Efimov, W. Perruquetti, J.-P. Richard, Development of Homogeneity Concept for Time-Delay Systems, 2014, 52, 0363-0129, 1547, 10.1137/130908750
    3. F. Crauste, Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch, 2009, 4, 0973-5348, 28, 10.1051/mmnp/20094202
    4. Chi Jin, Keqin Gu, Silviu-Iulian Niculescu, Islam Boussaada, Stability Analysis of Systems With Delay-Dependent Coefficients: An Overview, 2018, 6, 2169-3536, 27392, 10.1109/ACCESS.2018.2828871
    5. Chi Jin, Islam Boussaada, Silviu-Iulian Niculescu, Keqin Gu, 2017, An overview of stability analysis of systems with delay dependent coefficients, 978-1-5386-3842-2, 430, 10.1109/ICSTCC.2017.8107072
    6. Mostafa Adimy, Fabien Crauste, Catherine Marquet, Asymptotic behavior and stability switch for a mature–immature model of cell differentiation, 2010, 11, 14681218, 2913, 10.1016/j.nonrwa.2009.11.001
    7. Mostafa Adimy, Fabien Crauste, My Lhassan Hbid, Redouane Qesmi, Stability and Hopf Bifurcation for a Cell Population Model with State-Dependent Delay, 2010, 70, 0036-1399, 1611, 10.1137/080742713
    8. L. Pujo-Menjouet, V. Volpert, Blood Cell Dynamics: Half of a Century of Modelling, 2016, 11, 0973-5348, 92, 10.1051/mmnp/201611106
    9. Aying Wan, Junjie Wei, Bifurcation analysis in an approachable haematopoietic stem cells model, 2008, 345, 0022247X, 276, 10.1016/j.jmaa.2008.04.014
    10. M. Adimy, F. Crauste, Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling, 2012, 7, 0973-5348, 1, 10.1051/mmnp/20127601
    11. S. Q. Ma, S. J. Hogan, Bifurcation in an modified model of neutrophil cells with time delay, 2024, 0924-090X, 10.1007/s11071-024-09786-3
    12. Chi Jin, Keqin Gu, Qian Ma, Silviu-Iulian Niculescu, Islam Boussaada, Stability analysis of systems with delay-dependent coefficients and commensurate delays, 2024, 0932-4194, 10.1007/s00498-024-00399-0
    13. Suqi Ma, S. J. Hogan, Generalized Hopf Bifurcation in a Delay Model of Neutrophil Cells Model, 2024, 13, 2167-9479, 11, 10.4236/ijmnta.2024.132002
    14. Ma Suqi, S. J. Hogan, Mohammad Safi Ullah, Bifurcation in a G0 Model of Hematological Stem Cells With Delay, 2024, 2024, 1085-3375, 10.1155/aaa/1419716
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