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. 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 falls below some prescribed threshold. We consider the process successful if the threshold for emptying and 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 show to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails after at most finitely many impulses. 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. Potential applications include water purification and biological waste remediation.

This is joint work with Tyler Meadows, Lin Wang, and Gail S.K. Wolkowicz.