In this paper, stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied, and we use distribution delay to simulate the delay in nutrient conversion. By the linear chain technique, we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations. First, we state that this model has a unique global positive solution for any initial value, which is helpful to explore its stochastic properties. Furthermore, we prove the stochastic ultimate boundness of the solution of system. Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established, which means the microorganism will be extinct. Moreover, we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions. Finally, we provide some numerical examples to illustrate theoretical results, and some conclusions and analysis are given.

中文翻译：

---

具有分布延迟和随机扰动的恒化器模型解的随机性质

本文研究了具有分布延迟和随机扰动的恒化器模型解的随机特性，并利用分布延迟来模拟养分转换的延迟。通过线性链技术，我们将具有弱核的随机恒化器模型转换为包含三个方程的等效简并系统。首先，我们声明该模型对于任何初始值都有一个独特的全局正解，这有助于探索其随机属性。此外，我们证明了系统解的随机极限有界性。然后建立了系统以指数速率趋于边界平衡点的解的充分条件，这意味着微生物将灭绝。而且，我们还通过构造一些合适的随机Lyapunov函数得到了该系统解的遍历性的一些充分条件。最后，我们提供了一些数值例子来说明理论结果，并给出了一些结论和分析。