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.
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Notes
The simulations of the 1000 runs in large population size was made on the babycluster of the Institut Élie Cartan de Lorraine: http://babycluster.iecl.univ-lorraine.fr/.
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Acknowledgements
The authors thank Sylvie Méléard about some discussions on tightness on Hilbert spaces. This work was partially supported by the Chaire “Modélisation Mathématique et Biodiversité” of VEOLIA Environment, École Polytechnique, Muséum National d’Histoire Naturelle and Fondation X and by the project PIECE (Piecewise Deterministic Markov Processes) of ANR (French national research agency).
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Cloez, B., Fritsch, C. Gaussian approximations for chemostat models in finite and infinite dimensions. J. Math. Biol. 75, 805–843 (2017). https://doi.org/10.1007/s00285-017-1097-6
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DOI: https://doi.org/10.1007/s00285-017-1097-6
Keywords
- Chemostat model
- Central limit theorem on Hilbert-space
- Individual-based model
- Weak convergence
- Crump–Young model
- Stationary and quasi-stationary distributions