Asymptotic Behavior of a Chemostat Model with Stochastic Perturbation on the Dilution Rate

Chaoqun Xu,Tonghua Zhang,Sanling Yuan

doi:10.1155/2013/423154

Abstract

We present a stochastic simple chemostat model in which the dilution rate was influenced by white noise. The long time behavior of the system is studied. Mainly, we show how the solution spirals around the washout equilibrium and the positive equilibrium of deterministic system under different conditions. Furthermore, the sufficient conditions for persistence in the mean of the stochastic system and washout of the microorganism are obtained. Numerical simulations are carried out to support our results.

Highlights

Modeling microbial growth is a problem of special interest in mathematical biology and theoretical ecology
To derive a stochastic model they considered a discrete-time Markov process with jumps corresponding to the deterministic system added with a centered Gaussian term, letting the time step converges to zero leads to a system of stochastic differential equations
They proved that random effects may lead to extinction in scenarios where the deterministic model predicts persistence; they established some stochastic persistence results

Summary

Introduction

Modeling microbial growth is a problem of special interest in mathematical biology and theoretical ecology. Taking into account the effect of randomly fluctuating environment, we introduce randomness into model (1) by replacing the dilution rate D by D → D + αḂ(t), where Ḃ(t) is a white noise (i.e., B(t) is a Brownian motion) and α ≥ 0 represents the intensity of noise This is only a first step in introducing stochasticity into the model. We only mention a recent paper by Imhof and Walcher [7] They introduced a variant of the deterministic single-substrate chemostat model for which the persistence of all species is possible. To derive a stochastic model they considered a discrete-time Markov process with jumps corresponding to the deterministic system added with a centered Gaussian term, letting the time step converges to zero leads to a system of stochastic differential equations They proved that random effects may lead to extinction in scenarios where the deterministic model predicts persistence; they established some stochastic persistence results.

The Model and Some Fundamental Results

Washout of the Organism in the Chemostat

Simulations and Discussions