  In a chemostat, bacteria live in a growth container of constant volume in
which liquid is injected continuously. Recently, Campillo and Fritsch
introduced a mass-structured individual-based model to represent this dynamics
and proved its convergence to a more classic partial differential equation. In
this work, we are interested in the convergence of the fluctuation process. We
consider this process in some Sobolev spaces and use central limit theorems on
Hilbert space to prove its convergence in law to an infinite-dimensional
Gaussian process.As a consequence, we obtain a two-dimensional Gaussian
approximation of the Crump-Young model for which the long time behavior is
relatively misunderstood. For this approximation, we derive the invariant
distribution and the convergence to it. We also present numerical simulations
illustrating our results.
