Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model

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.