  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.
