[Comportement quasi-stationnaire d’un modèle markovien déterministe par morceaux de chemostat : le modèle de Crump–Young]
The Crump–Young model consists of two fully coupled stochastic processes modeling the substrate and micro-organisms dynamics in a chemostat. Substrate evolves following an ordinary differential equation whose coefficients depend of micro-organisms number. Micro-organisms are modeled though a pure jump process whose jump rates depend on the substrate concentration.
It goes to extinction almost-surely in the sense that micro-organism population vanishes. In this work, we show that, conditionally on the non-extinction, its distribution converges exponentially fast to a quasi-stationary distribution.
Due to the deterministic part, the dynamics of the Crump–Young model are highly degenerated. The proof is therefore original and consists of technically precise estimates and new approaches for quasi-stationary convergence.
Le modèle de Crump–Young se compose de deux processus stochastiques entièrement couplés modélisant la dynamique du substrat et des micro-organismes dans un chemostat. Le substrat évolue selon une équation différentielle ordinaire dont les coefficients dépendent du nombre de micro-organismes. Les micro-organismes sont modélisées via un processus de saut pur dont les taux de saut dépendent de la concentration en substrat.
Ce processus s’éteint presque sûrement dans le sens où la population de micro-organismes s’éteint presque sûrement. Dans cet article, nous démontrons que, conditionnellement à la non-extinction, la loi du processus converge exponentiellement vite vers une distribution quasi-stationnaire.
En raison de la partie déterministe du modèle, la dynamique du modèle de Crump–Young est fortement dégénérée. La preuve est donc originale et consiste en des estimées précises et de nouvelles approches pour démontrer la convergence vers des distributions quasi-stationnaires.
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Keywords: Quasi-stationary distribution, Chemostat model, Lyapunov function, Crump–Young model, Piecewise Deterministic Markov Process (PDMP)
Cloez, Bertrand 1 ; Fritsch, Coralie 2
CC-BY 4.0
@article{AHL_2023__6__1371_0,
author = {Cloez, Bertrand and Fritsch, Coralie},
title = {Quasi-stationary behavior for a piecewise deterministic {Markov} model of chemostat: the {Crump{\textendash}Young} model},
journal = {Annales Henri Lebesgue},
pages = {1371--1427},
year = {2023},
publisher = {\'ENS Rennes},
volume = {6},
doi = {10.5802/ahl.191},
language = {en},
url = {https://www.numdam.org/articles/10.5802/ahl.191/}
}
TY - JOUR AU - Cloez, Bertrand AU - Fritsch, Coralie TI - Quasi-stationary behavior for a piecewise deterministic Markov model of chemostat: the Crump–Young model JO - Annales Henri Lebesgue PY - 2023 SP - 1371 EP - 1427 VL - 6 PB - ÉNS Rennes UR - https://www.numdam.org/articles/10.5802/ahl.191/ DO - 10.5802/ahl.191 LA - en ID - AHL_2023__6__1371_0 ER -
%0 Journal Article %A Cloez, Bertrand %A Fritsch, Coralie %T Quasi-stationary behavior for a piecewise deterministic Markov model of chemostat: the Crump–Young model %J Annales Henri Lebesgue %D 2023 %P 1371-1427 %V 6 %I ÉNS Rennes %U https://www.numdam.org/articles/10.5802/ahl.191/ %R 10.5802/ahl.191 %G en %F AHL_2023__6__1371_0
Cloez, Bertrand; Fritsch, Coralie. Quasi-stationary behavior for a piecewise deterministic Markov model of chemostat: the Crump–Young model. Annales Henri Lebesgue, Tome 6 (2023), pp. 1371-1427. doi: 10.5802/ahl.191
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