  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.
