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

¹ Inria, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, 69200 Villeurbanne Cedex, France
² Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen 13000, Algeria
³ Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

^* e-mail: mostafa.adimy@inria.fr

Received: 30 October 2017
Accepted: 13 November 2017

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.