Dynamics of a chemostat with periodic nutrient supply and delay in the growth

Abstract

This paper introduces a new consideration in the well known chemostat model of a one-species with a periodic input of single nutrient with period ω, which is described by a system of differential delay equations. The delay represents the interval time between the consumption of nutrient and its metabolization by the microbial species. We obtain a necessary and sufficient condition ensuring the existence of a positive periodic solution with period ω. Our proof is based firstly on the construction of a Poincaré type map associated to an ω-periodic integro-differential equation and secondly on the existence of zeroes of an appropriate function involving the fixed points of the above mentioned map, which is proved by using Whyburn’s Lemma combined with the Leray–Schauder degree. In addition, we obtain a uniqueness result for sufficiently small delays.