Gaussian approximations for chemostat models in finite and infinite dimensions

Gaussian approximations for chemostat models in finite and infinite dimensions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Biology Springer Journals

Gaussian approximations for chemostat models in finite and infinite dimensions

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical and Computational Biology; Applications of Mathematics
ISSN
0303-6812
eISSN
1432-1416
D.O.I.
10.1007/s00285-017-1097-6
Publisher site
See Article on Publisher Site

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.

Journal

Journal of Mathematical BiologySpringer Journals

Published: Jan 27, 2017

References

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