Uniform persistence criteria for a variable inputs chemostat model with delayed response in growth and complete analysis of the periodic case

We study a single-species chemostat model with variable nutrient input and variable dilution rate with delayed (fixed) response in growth. The first goal of this article is to prove that persistence implies uniform persistence. Then we concentrate on the particular case with periodic nutrient input and same periodic dilution with delayed response in growth. We obtain a threshold that allows us to determine whether the system is uniformly persistent or every positive solution is washed out. Furthermore, we prove that (uniform) persistence is equivalent to the existence of a unique non-trivial periodic solution. We also prove that this solution is attractive. We remark in no case we need to impose any restrictions on the size of the delay.