A periodic-parabolic Droop model for two species competition in an unstirred chemostat

We study a periodic-parabolic Droop model of two species competing for a single-limited nutrient in an unstirred chemostat, where the nutrient is added to the culture vessel by way of periodic forcing function in time. For the single species model, we establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear periodic eigenvalue problem. In particular, when diffusion rate is sufficiently small or large, the sign can be determined. We then show that for the competition model, when diffusion rates for both species are small, there exists a coexistence periodic solution.