We study a single-species chemostat model with variable nutrient input and variable dilution rate with delayed (fixed) response in growth. The first goal of this article is to prove that persistence implies uniform persistence. Then we concentrate on the particular case with periodic nutrient input and same periodic dilution with delayed response in growth. We obtain a threshold that allows us to determine whether the system is uniformly persistent or every positive solution is washed out. Furthermore, we prove that (uniform) persistence is equivalent to the existence of a unique non-trivial periodic solution. We also prove that this solution is attractive. We remark in no case we need to impose any restrictions on the size of the delay.

中文翻译：

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可变输入趋化器模型的统一持久性标准，具有生长延迟响应和周期性情况的完整分析


我们研究了一种具有可变营养输入和可变稀释速率的单物种恒温器模型，具有生长延迟（固定）响应。本文的第一个目标是证明持久性意味着均匀的持久性。然后我们专注于具有周期性营养输入和相同的周期性稀释以及生长反应延迟的特殊情况。我们得到一个阈值，使我们能够确定系统是均匀持续的还是每个正解都被洗掉了。此外，我们证明（均匀）持久性等价于存在一个唯一的非平凡周期解。我们还证明了这种解决方案是有吸引力的。我们声明，在任何情况下，我们都不需要对延误的大小施加任何限制。