We introduce a model of the growth of a single microorganism in a self-cycling fermentor in which an arbitrary number of resources are limiting, and impulses are triggered when the concentration of one specific substrate reaches a predetermined level. The model is in the form of a system of impulsive differential equations. We consider the operation of the reactor to be successful if it cycles indefinitely without human intervention and derive conditions for this to occur. In this case, the system of impulsive differential equations has a periodic solution. We show that success is equivalent to the convergence of solutions to this periodic solution. We provide conditions that ensure that a periodic solution exists. When it exists, it is unique and attracting. However, we also show that whether a solution converges to this periodic solution, and hence whether the model predicts that the reactor operates successfully, is initial condition dependent. The analysis is illustrated with numerical examples.

我们介绍了一种自循环发酵罐中单个微生物的生长模型，其中限制了任意数量的资源，当一种特定底物的浓度达到预定水平时就会触发脉冲。该模型采用脉冲微分方程系统的形式。如果反应堆在没有人工干预的情况下无限循环，我们认为该反应堆的运行将是成功的，并为此创造条件。在这种情况下，脉冲微分方程系统具有周期解。我们证明成功等于解决方案收敛到该周期解。我们提供确保定期解决方案存在的条件。当它存在时，它是独特的并且吸引人。但是，我们还表明，一个解是否收敛于该周期解，因此模型是否预测反应堆成功运行取决于初始条件。通过数值示例说明了该分析。