General Nonlinear Stochastic Systems Motivated by Chemostat Models: Complete Characterization of Long-Time Behavior, Optimal Controls, and Applications to Wastewater Treatment

TL;DR

This paper analyzes a stochastic chemostat model with environmental noise, fully characterizing its long-term behavior, including extinction and persistence, and providing insights applicable to wastewater treatment processes.

Contribution

It offers a complete classification of the asymptotic behavior of a hybrid switching diffusion model, including the critical case where the growth rate equals zero.

Findings

System's long-term behavior depends on the sign of λ

Exponential convergence to invariant measure when λ>0

Bacteria extinction when λ≤0

Abstract

The paper considers a chemostat model describing an activated sludge process in wastewater treatment. The model is assumed to be subject to environment noise in terms of both white noise and color noise. The paper fully characterizes the asymptotic behavior of the model that is a hybrid switching diffusion. We show that the long-term properties of the system can be classified using a value $\lambda$. More precisely, if $\lambda\leq 0$, the bacteria in the sewage will die out, which means that the process does not operate. If $\lambda>0$, the system has an invariant probability measure to which the transition probability of the solution process converges exponentially fast. One of the distinctive contributions of this paper is that the critical case $\lambda=0$ is considered. Numerical examples are given to illustrate our results.