  We introduce and analyze several aspects of a new model for cell
differentiation. It assumes that differentiation of progenitor cells is a
continuous process. From the mathematical point of view, it is based on partial
differential equations of transport type. Specifically, it consists of a
structured population equation with a nonlinear feedback loop. This models the
signaling process due to cytokines, which regulate the differentiation and
proliferation process. We compare the continuous model to its discrete
counterpart, a multi-compartmental model of a discrete collection of cell
subpopulations recently proposed by Marciniak-Czochra et al. in 2009 to
investigate the dynamics of the hematopoietic system. We obtain uniform bounds
for the solutions, characterize steady state solutions, and analyze their
linearized stability. We show how persistence or extinction might occur
according to values of parameters that characterize the stem cells
self-renewal. We also perform numerical simulations and discuss the qualitative
behavior of the continuous model vis a vis the discrete one.
