# Stochastic analysis of a full system of two competing populations in a chemostat

## TL;DR

This paper develops a stochastic model for two competing microbial populations in a chemostat, analyzes their dynamics near coexistence, and finds that one population tends to dominate over time, supported by numerical solutions and simulations.

## Contribution

It introduces a novel stochastic differential equation model with two noise sources for competing populations and performs an asymptotic reduction to analyze long-term behavior.

## Key findings

- One population becomes dominant at large times.
- Reduced 2D system accurately describes population dynamics.
- Numerical solutions align with stochastic simulations.

## Abstract

This paper formulates two 3D stochastic differential equations (SDEs) of two microbial populations in a chemostat competing over a single substrate. The two models have two distinct noise sources. One is general noise whereas the other is dilution rate induced noise. Nonlinear Monod growth rates are assumed and the paper is mainly focused on the parameter values where coexistence is present deterministically. Nondimensionalising the equations around the point of intersection of the two growth rates leads to a large parameter which is the nondimensional substrate feed. This in turn is used to perform an asymptotic analysis leading to a reduced 2D system of equations describing the dynamics of the populations on and close to a line of steady states retrieved from the deterministic stability analysis. That reduced system allows the formulation of a spatially 2D Fokker-Planck equation which when solved numerically admits results similar to those from simulation of the SDEs. Contrary to previous suggestions, one particular population becomes dominant at large times. Finally, we brie y explore the case where death rates are added.