Coupled reaction-diffusion and difference system of cell-cycle dynamics for hematopoiesis process with Dirichlet boundary conditions

Abstract

In this paper, we propose a mathematical model describing the dynamics of hematopoietic stem cell (HSC) population during the cell-cycle. We take into account the spatial diffusion in the bone marrow. We consider for each cell, during its cell-cycle, which is a series of events that take place in the cell leading to its division, two main phases: quiescent phase and dividing phase. At the end of dividing phase, each cell divides and gives two daughter cells. A part of these daughter cells enters to the quiescent phase and the other part returns to the proliferating phase to divide again. Then, we obtain two nonlinear age-space-structured partial differential equations. Using the method of characteristics, we reduce this system to a coupled reaction-diffusion equation and difference one containing a nonlocal spatial term and a time delay. We start by studying the existence, uniqueness, positivity and boundedness of solutions for the reduced system. We then investigate the stability analysis and obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional. We also obtain a sufficient condition for the existence and uniqueness of positive steady state by using the method of lower and upper solutions. Finally, we investigate the questions of persistence of the solution of the system when the trivial steady state is unstable.