基于脉冲微分方程的自循环发酵罐中多源相互作用的生长
Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach
摘要：
我们介绍了一个模型的自循环发酵罐中的一个单一的微生物的生长，其中任意数量的资源是有限的，和脉冲被触发时，一个特定的底物的浓度达到预定的水平。该模型的形式是一个系统的脉冲微分方程。我们认为，如果反应堆在没有人为干预的情况下无限循环，并推导出发生这种情况的条件，则反应堆的操作是成功的。在这种情况下，脉冲微分方程系统具有周期解。我们表明，成功是等价于这个周期解的解决方案的收敛。我们提供的条件，确保周期解的存在。当它存在时，它是独特的和吸引人的。然而，我们也表明，无论是解决方案收敛到这个周期性的解决方案，因此，无论是模型预测，反应堆成功运行，是初始条件依赖。最后用数值例子说明了分析结果。
英文摘要：
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.
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