title: Gaussian approximations for chemostat models in finite and infinite dimensions.
abstract: 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.
authors.affiliation: INRA Montpellier UMR MISTEA, 2 place Pierre Viala, 34060, Montpellier, France.
authors.forename: Bertrand
authors.initials: B
authors.lastname: Cloez
authors.affiliation: CMAP, École Polytechnique, UMR CNRS 7641, route de Saclay, 91128, Palaiseau Cedex, France. coralie.fritsch@inria.fr.|Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, 54506, Vandœuvre-lès-Nancy, France. coralie.fritsch@inria.fr.|Inria, TOSCA, 54600, Villers-lès-Nancy, France. coralie.fritsch@inria.fr.
authors.forename: Coralie
authors.initials: C
authors.lastname: Fritsch
mesh_terms: D001419:Bacteria / Q000254:growth & development* / Q000378:metabolism; D001431:Bacteriological Techniques; D018533:Biomass; D019149:Bioreactors; D001709:Biotechnology; D003198:Computer Simulation; D003470:Culture Media; D008390:Markov Chains; D055641:Mathematical Concepts; D008954:Models, Biological; D016011:Normal Distribution
chemical_list: D003470:Culture Media
keywords: Central limit theorem on Hilbert-space; Chemostat model; Crump–Young model; Individual-based model; Stationary and quasi-stationary distributions; Weak convergence