A discrete one-species chemostat model with delayed response in the growth and non-constant supply

TL;DR

This paper analyzes a discrete delayed chemostat model with non-constant nutrient supply, establishing conditions for species persistence or extinction, and demonstrating the existence of periodic solutions in periodic supply scenarios.

Contribution

It introduces a novel discrete delayed model for a one-species chemostat with non-constant supply, providing new criteria for persistence and extinction based on Bohl exponents.

Findings

Conditions for species persistence and extinction are derived.

Existence of attractive periodic solutions in periodic supply cases.

Complete characterization of dynamics in periodic nutrient supply scenarios.

Abstract

A non-autonomous discrete delayed system for a one-species chemostat based on an Ellermeyer model for the continuous case is studied. Conditions for the persistence or the extinction of the solutions are obtained respectively in terms of the lower and upper Bohl exponents for a scalar linear equation associated to the problem. Furthermore, the condition for persistence also implies the attractiveness, that is, the existence of a bounded solution that attracts all the others. As a special case, when the nutrient supply is $\omega$-periodic, the picture is complete: the condition for persistence implies the existence of an attractive non-trivial $\omega$-periodic solution, while non-persistence implies extinction.