In the microorganism cultivation process, delay and stochastic perturbations are inevitably accompanied, which results in complicated dynamical behaviors for microorganisms. In this paper, a mathematical model with discrete delay and random perturbation is constructed to understand how the dynamics of microorganisms in the turbidostat can be characterized. The existence, uniqueness and boundedness of the positive solution are first determined for the mathematical model. Furthermore, sufficient conditions for microorganism extinction and permanence in the turbidostat are obtained with the theory of stochastic differential equations. The system has the stationary distribution under a low-level intensity of stochastic perturbation from the environment; that is, microorganism in the turbidostat is persistent fluctuating around a positive value. On the contrary, microorganisms will be extinct with a strong enough intensity of noise. Several numerical simulations are applied to validate the theoretical results for the dynamics of the system.

在微生物培养过程中，不可避免地会伴随着延迟和随机扰动，这导致微生物复杂的动力学行为。在本文中，建立了具有离散时滞和随机扰动的数学模型，以了解如何表征turbidostat中微生物的动力学。首先为数学模型确定正解的存在性，唯一性和有界性。此外，利用随机微分方程的理论，获得了在灭螺器中微生物灭绝和持久存在的充分条件。该系统在来自环境的低水平随机干扰强度下具有平稳分布；就是说，在turbidostat中的微生物持续在正值附近波动。相反，微生物将以足够强的强度灭绝。应用了几个数值模拟，以验证系统动力学的理论结果。