We consider a model of multi-species competition in the chemostat that includes demographic stochasticity and discrete delays. We prove that for any given initial data, there exists a unique global positive solution for the stochastic delayed system. By employing the method of stochastic Lyapunov functionals, we determine the asymptotic behaviors of the stochastic solution and show that although the sample path fluctuate, it remains positive and the expected time average of the distance between the stochastic solution and the equilibrium of the associated deterministic delayed chemostat model is eventually small, i.e. we obtain an analogue of the competition exclusion principle when the noise intensities are relatively small. Numerical simulations are carried out to illustrate our theoretical results.

中文翻译：

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具有离散时滞的多种群随机Chemostat模型的全局渐近行为

我们考虑在恒化器中进行多物种竞争的模型，其中包括人口统计的随机性和离散的延迟。我们证明对于任何给定的初始数据，对于随机时滞系统都存在唯一的全局正解。通过使用随机Lyapunov泛函方法，我们确定了随机解的渐近行为，并表明，尽管样本路径波动，但它仍然是正值，并且随机解与相关确定性平衡之间的距离的预期时间平均延迟了。 chemostat模型最终很小，即当噪声强度相对较小时，我们获得了竞争排斥原理的类似物。进行数值模拟以说明我们的理论结果。