  Erythropoiesis is a mechanism for the production of red blood cells by
cellular differentiation. It is based on amplification steps due to an
interplay between renewal and differentiation in the successive cell
compartments from stem cells to red blood cells. We will study this mechanism
with a stochastic point of view to explain unexpected fluctuations on the red
blood cell numbers, as surprisingly observed by biologists and medical doctors
in a rest erythropoiesis. We consider three compartments: stem cells,
progenitors and red blood cells. The dynamics of each cell type is
characterized by its division rate and by the renewal and differentiation
probabilities at each division event. We model the global population dynamics
by a three-dimensional stochastic decomposable branching process. We show that
the amplification mechanism is given by the inverse of the small difference
between the differentiation and renewal probabilities. Introducing a parameter
$K$ which scales simultaneously the size of the first component, the
differentiation and renewal probabilities and the red blood cell death rate, we
describe the asymptotic behavior of the process for large $K$. We show that
each compartment has its own size scale and its own time scale. Focussing on
the third component, we prove that the red blood cell population size,
conveniently renormalized (in time and size), is expanded in an usual way
inducing large fluctuations. The proofs are based on a fine study of the
different scales involved in the model and on the use of different convergence
and average techniques in the proofs.
