Stochastic models of the chemostat

TL;DR

This paper develops stochastic models for the chemostat at different scales, bridging microscopic jump processes, diffusion approximations, and classical deterministic ODEs, with methods for transitioning between models.

Contribution

It introduces a stochastic modeling framework for the chemostat that connects microscopic, intermediate, and macroscopic scales, detailing model transitions and validity domains.

Findings

Microscopic pure jump stochastic model reproduces ODE dynamics at large scale.

Diffusion approximation provides a stochastic differential equation model.

Guidelines for switching between models and their domains of validity.

Abstract

We consider the modeling of the dynamics of the chemostat at its very source. The chemostat is classically represented as a system of ordinary differential equations. Our goal is to establish a stochastic model that is valid at the scale immediately preceding the one corresponding to the deterministic model. At a microscopic scale we present a pure jump stochastic model that gives rise, at the macroscopic scale, to the ordinary differential equation model. At an intermediate scale, an approximation diffusion allows us to propose a model in the form of a system of stochastic differential equations. We expound the mechanism to switch from one model to another, together with the associated simulation procedures. We also describe the domain of validity of the different models.