  We consider the modeling of the dynamics of the chemostat at its very source.
The chemostat is classically represented as a system of ordinary differential
equations. Our goal is to establish a stochastic model that is valid at the
scale immediately preceding the one corresponding to the deterministic model.
At a microscopic scale we present a pure jump stochastic model that gives rise,
at the macroscopic scale, to the ordinary differential equation model. At an
intermediate scale, an approximation diffusion allows us to propose a model in
the form of a system of stochastic differential equations. We expound the
mechanism to switch from one model to another, together with the associated
simulation procedures. We also describe the domain of validity of the different
models.
