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Delayed nonlocal reaction–diffusion model for hematopoietic stem cell dynamics with Dirichlet boundary conditions

Mostafa Adimy 1, 2, 3 Abdennasser Chekroun 4 Toshikazu Kuniya 5
1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
CGPhiMC - Centre de génétique et de physiologie moléculaire et cellulaire, Inria Grenoble - Rhône-Alpes, ICJ - Institut Camille Jordan [Villeurbanne]
3 MMCS - Modélisation mathématique, calcul scientifique
ICJ - Institut Camille Jordan [Villeurbanne]
Abstract : The paper focuses on the mathematical analysis and modeling of hematopoietic stem cell (HSC) dynamics that lead to the production and regulation of blood cells in the bone morrow. The HSC population is seen as a continuous medium structured in age and space. Using the method of characteristics, we reduce the age structured system to a reaction–diffusion equation containing a nonlocal spatial term and a time delay. Firstly, we give some properties on the existence, uniqueness and positivity of the solution. Secondly, we obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional and we prove that if it is not globally asymptotic stable then, it is unstable. Thirdly, we give sufficient conditions for the existence and uniqueness of the positive steady state by using the sub-and super-solutions method. Finally, we prove the uniform persistence of the system when the trivial steady state is unstable. Throughout the paper, we provide some numerical simulations to illustrate our results. Mathematics Subject Classification. 34B18, 35B35, 37N25, 92C37.
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Submitted on : Sunday, January 14, 2018 - 2:43:00 PM
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Mostafa Adimy, Abdennasser Chekroun, Toshikazu Kuniya. Delayed nonlocal reaction–diffusion model for hematopoietic stem cell dynamics with Dirichlet boundary conditions. Mathematical Modelling of Natural Phenomena, EDP Sciences, 2017, 12 (6), pp.1 - 22. ⟨10.1051/mmnp/2017078⟩. ⟨hal-01683636⟩

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