Stochastic models for a chemostat and long time behavior

TL;DR

This paper develops stochastic models of a chemostat to analyze long-term bacterial population behavior, proving extinction, existence of quasi-stationary distributions, and properties of nutrient distribution conditioned on population size.

Contribution

It introduces novel stochastic chemostat models with coupled population-nutrient dynamics and establishes their long-term behavior and distribution properties.

Findings

Population almost surely goes extinct without conditioning.

Existence of quasi-stationary distributions is proven.

Nutrient distribution conditioned on population size is absolutely continuous with smooth densities.

Abstract

We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending 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 the non-extinction. We prove the global existence of the process and its almost sure extinction. The existence of quasi-stationary 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.