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This article revisits and extends to the nonautonomous framework the results about the dynamics of a discrete and nonlinear matrix model describing the growth of a size–structured single microbial population in an autonomous chemostat, which has been introduced by T.B. Gage et.al and H.L. Smith. The first and the second result provide a threshold determining either the extinction or the persistence of the total biomass. The main result establishes a set of sufficient conditions ensuring the existence, uniqueness and global attractiveness of an $ \omega $–periodic solution. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper proposes an ω -periodic version of the Ellermeyer model of delayed chemostat. We obtain a sufficient condition ensuring the existence of a positive ω -periodic solution. Our proof is based on the application of the generalized continuation theorem. In addition, as a consequence of the implicit function theorem, we obtain a uniqueness result for sufficiently small delays. [ABSTRACT FROM AUTHOR]