  Phenotypically structured equations arise in population biology to describe
the interaction of species with their environment that brings the nutrients.
This interaction usually leads to selection of the fittest individuals. Models
used in this area are highly nonlinear, and the question of long term behaviour
is usually not solved. However, there is a particular class of models for which
convergence to an Evolutionary Stable Distribution is proved, namely when the
quasi-static assumption is made. This means that the environment, and thus the
nutrient supply, reacts immediately to the population dynamics. One possible
proof is based on a Total Variation bound for the appropriate quantity. We
extend this proof to several cases where the nutrient is regenerated with
delay. A simple example is the chemostat with a rendering factor, then our
result does not use any smallness assumption. For a more general setting, we
can treat the case with a fast reaction of nutrient supply to the population
dynamics.
