  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 in the particular case with periodic nutrient input and same
periodic dilution with delayed response in growth. We obtain a threshold for
either the (uniform) persistence of the model or that
  the biomass of every solution tends to vanish. Furthermore, we prove that
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.
