Time-varying control laws with guaranteed persistence for a class of multi-species chemostats

We study classical families of chemostat models with an arbitrary number of species competing for a single limiting substrate. For families of models, we present some obstructions to the existence of asymptotically stabilizable periodic trajectories. For other families of growth rates, we design a dilution rate and input substrate time-varying feedback controllers so that a positive trajectory of the chemostat model becomes globally asymptotically stable. That way, the control laws ensure persistence of all the species. A local version of this result is given in the case where only the substrate concentration is available by measurement.