Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

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.