  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 divisions in blood cells. Taking into account an explicit role of
the total population of hematopoietic stem cells on the introduction of cells
in cycle, we are lead 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.
