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Stochastic Models for a Chemostat and Long-Time Behavior

Part of: Markov processes

Published online by Cambridge University Press:  04 January 2016

Pierre Collet ,
Servet Martínez ,
Sylvie Méléard  and
Jaime San Martín
Pierre Collet*
Affiliation:
École Polytechnique
Servet Martínez*
Affiliation:
Universidad de Chile
Sylvie Méléard*
Affiliation:
École Polytechnique
Jaime San Martín*
Affiliation:
Universidad de Chile
*
∗ Postal address: Centre de Physique Théorique, CNRS UMR 7644, École Polytechnique, F-91128 Palaiseau Cedex, France. Email address: collet@cpht.polytechnique.fr
∗∗ Postal address: Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile.
∗∗∗∗ Postal address: CAMP, École Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France. Email address: sylvie.meleard@polytechnique.edu
∗∗ Postal address: Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile.
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Abstract

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We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

Keywords

Population processchemostatextinctionquasistationary distribution

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 822 - 836
Copyright
© Applied Probability Trust 

References

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