Dynamics of stochastic microorganism flocculation models
Alexandru Hening, Nguyen T. Hieu, Dang H. Nguyen, Nhu Nguyen
TL;DR
The paper develops a stochastic, environment-switching model for microorganism flocculation that combines Brownian fluctuations with finite-state environmental shifts, and analyzes long-term outcomes when the system is not in Kolmogorov form. It defines an invasion rate $\Lambda$ via invariant measures and explicit linear-algebraic constructions, establishing extinction (when $\Lambda<0$) or stochastic persistence (when $\Lambda>0$) for the microorganism density, complemented by rigorous moment bounds and well-posedness. The authors extend the framework to nonlinear perturbations using the HNC20 approach, deriving a threshold $\lambda(\boldsymbol{\nu})$ that governs persistence versus extinction and employing a Girsanov transformation and control-theoretic accessibility to obtain almost-sure results. Overall, the work provides a rigorous, multi-scale methodology for predicting the fate of microbial flocs under environmental variability, with explicit computations and broad extensions including nonlinear noise.
Abstract
In this paper we study the dynamics of stochastic microorganism flocculation models. Given the strong influence of environmental and seasonal fluctuations that are present in these models, we propose a stochastic model that includes multiple layers of stochasticity, from small Brownian fluctuations, to possibly large changes due to environmental `shifts'. We are able to give a full classification of the asymptotic behavior of these models. New techniques had to be developed to prove the persistence and extinction of the process as the system is not in Kolmogorov form and, as a result, the analysis is significantly more involved.