Approximation of the Fokker-Planck equation of the stochastic chemostat

TL;DR

This paper models the stochastic chemostat as a diffusion process, derives its Fokker-Planck equation including washout boundary conditions, and proposes a finite difference scheme for its approximation.

Contribution

It introduces a stochastic chemostat model with washout phenomena, derives the associated Fokker-Planck equation, and develops a numerical approximation method.

Findings

Derived the Fokker-Planck equation with boundary conditions for washout

Proposed a finite difference scheme for the equation

Provided a framework for simulating stochastic chemostat dynamics

Abstract

We consider a stochastic model of the two-dimensional chemostat as a diffusion process for the concentration of substrate and the concentration of biomass. The model allows for the washout phenomenon: the disappearance of the biomass inside the chemostat. We establish the Fokker-Planck associated with this diffusion process, in particular we describe the boundary conditions that modelize the washout. We propose an adapted finite difference scheme for the approximation of the solution of the Fokker-Planck equation.