$N$-species competition in a periodic chemostat

Abstract

A threshold result on the global dynamics of the scalar asymptotically periodic Kolmogorov equation is proved and then applied to models of single-species growth and $n$-species competition in a periodically operated chemostat. The operating parameters and the species-specific response functions can be periodic functions of time. Species-specific removal rates are also permitted. Sufficient conditions ensure uniform persistence of all of the species and guarantee that the full system admits at least one positive, periodic solution. In the special case when the species-specific removal rates are all equal to the dilution rate, the single-species growth model has a threshold between global extinction and uniform persistence, in the form of a positive, periodic coexistence state. Improved results in the case of 3-species competition are also given, including an example illustrating competition-mediated coexistence of three species.