Gaussian approximations for chemostat models in finite and infinite dimensions

TL;DR

This paper develops Gaussian approximations for chemostat models, analyzing fluctuations in both finite and infinite dimensions, and provides insights into the long-term behavior of the system through theoretical and numerical methods.

Contribution

It introduces an infinite-dimensional Gaussian process approximation for chemostat models and analyzes their long-term dynamics, extending previous convergence results.

Findings

Convergence of fluctuation process to an infinite-dimensional Gaussian process.

Derivation of invariant distribution for the Gaussian approximation.

Numerical simulations supporting theoretical results.

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.