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

• Published in: Mathematical modelling of natural phenomena Vol. 12; no. 6; pp. 1 - 22
• Main Authors: Adimy, M., Chekroun, A., Kuniya, T.
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.01.2017
• Subjects: 34B18; 35B35; 37N25; 92C37; Age-space-structured PDE; Asymptotic properties; Blood cells; Boundary conditions; cell dynamics; Computer simulation; Dirichlet boundary conditions; Dirichlet problem; Dynamical Systems; Economic models; existence and uniqueness of positive steady state; Hematopoietic stem cells; Lyapunov functional; Mathematical models; Mathematics; Method of characteristics; Reaction-diffusion equations; Steady state; Stem cells; Time lag; time-delayed reaction–diffusion equation; Uniqueness; time-delayed reaction-diffusion equation; Lyapunov functional; existence and uniqueness of positive steady state; Dirichlet boundary conditions; Age-space-structured PDE; cell dynamics
• ISSN: 0973-5348; 1760-6101
• DOI: 10.1051/mmnp/2017078

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.