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A system of impulsive differential equations with state-dependent impulses isused to model the growth of a single population on two limiting essentialresources in a self-cycling fermentor. Potential applications include waterpurification and biological waste remediation. The self-cycling fermentationprocess 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 subsequentlyrefilled using fresh medium when the concentration of both resources (assumedto be pollutants) falls below some acceptable threshold. We consider theprocess successful if the threshold for emptying/refilling the reactor can bereached indefinitely without the time between successive emptying/refillingsbecoming unbounded and without interference by the operator. We prove thatwhenever the process is successful, the model predicts that the concentrationsof the population and the resources converge to a positive periodic solution.We derive conditions for the successful operation of the process that are shownto be initial condition dependent and prove that if these conditions are notsatisfied, then the reactor fails. We show numerically that there is an optimalfraction of the medium drained from the tank at each impulse that maximizes theoutput of the process.

Growth on Two Limiting Essential Resources in a Self-Cycling Fermentor