This paper presents long-time analysis of a stochastic chemostat model with instantaneous nutrient recycling. We focus on the investigation of the sufficient and almost necessary conditions of the exponential extinction and persistence for the model. The convergence to the invariant measure is also established under total variation norm. Our work generalizes and improves many existing results. One of the interesting findings is that random disturbance can suppress microorganism growth, which can provide us some useful control strategies to microbiological cultivation. Finally, some numerical simulations partly based on the stochastic sensitive function technique are given to illustrate theoretical results.

This study focuses on analyzing the exponential stability in the pth moment for a class of random impulsive delayed nonlinear systems (RIDNSs) influenced by multiple randomly delayed impulses. The underlying continuous-time dynamics are governed by random delay differential equations subjected to stochastic perturbations modeled as second-order moment processes. To establish new stability conditions, we employ tools such as the average random impulsive estimation (ARIE), average impulsive delay (AID), average delay time (ADT), and Lyapunov-based techniques. The developed criteria are applicable not only to inherently stable or unstable systems, but also to scenarios where impulses with stabilizing and destabilizing effects appear simultaneously. Several illustrative examples are included to verify the applicability and advantages of the proposed approach.

We investigate the stability of stochastic delay differential equations (SDDEs) that describe systems affected by both memory effects and stochastic disturbances. By adopting a comparison principle approach, we derive unified criteria for $p$-th moment, asymptotic, and exponential stability under conditions that relax the conventional requirement of a negative diffusion operator. The proposed results broaden the applicability of classical methods and are validated through illustrative numerical examples, demonstrating their potential relevance to engineering and applied sciences.