自循环发酵罐中两种限制性必需源的生长
Growth on Two Limiting Essential Resources in a Self-Cycling Fermentor
摘要：
利用状态依赖脉冲的脉冲微分方程系统，建立了自循环发酵罐中两种有限必需资源上单种群的生长模型。潜在的应用包括水净化和生物废物修复。自循环发酵过程是一个半间歇过程，该模型是一个混杂系统的例子。在这种情况下，充分搅拌的罐被部分排空，随后当两种资源（假设为污染物）的浓度低于某个可接受的阈值时，使用新鲜介质重新填充。我们认为，如果可以无限期地达到排空/再填充反应堆的阈值，而连续排空/再填充之间的时间没有限制，并且没有操作员的干扰，则该过程是成功的。我们证明了，只要过程是成功的，该模型预测的人口和资源的浓度收敛到一个正的周期解。我们推导出的过程中，被证明是初始条件依赖的成功操作的条件，并证明，如果这些条件不满足，那么反应器失败。我们数值计算表明，有一个最佳的比例，从坦克在每个脉冲，最大限度地提高输出的过程中排出的介质。
英文摘要：
A system of impulsive differential equations with state-dependent impulses is used to model the growth of a single population on two limiting essential resources in a self-cycling fermentor. Potential applications include water purification and biological waste remediation. The self-cycling fermentation process is a semi-batch process and the model is an example of a hybrid system. In this case, a well-stirred tank is partially drained, and subsequently refilled using fresh medium when the concentration of both resources (assumed to be pollutants) falls below some acceptable threshold. We consider the process successful if the threshold for emptying/refilling the reactor can be reached indefinitely without the time between successive emptying/refillings becoming unbounded and without interference by the operator. We prove that whenever the process is successful, the model predicts that the concentrations of the population and the resources converge to a positive periodic solution. We derive conditions for the successful operation of the process that are shown to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails. We show numerically that there is an optimal fraction of the medium drained from the tank at each impulse that maximizes the output of the process.
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