  We study a parabolic Lotka-Volterra type equation that describes the
evolution of a population structured by a phenotypic trait, under the effects
of mutations and competition for resources modelled by a nonlocal feedback. The
limit of small mutations is characterized by a Hamilton-Jacobi equation with
constraint that describes the concentration of the population on some traits.
This result was already established in Barles-Perthame 2008,
Barles-Mirrahimi-Perthame 2009, Lorz-Mirrahimi-Perthame 2011 in a
time-homogenous environment, when the asymptotic persistence of the population
was ensured by assumptions on either the growth rate or the initial data. Here,
we relax these assumptions to extend the study to situations where the
population may go extinct at the limit. For that purpose, we provide conditions
on the initial data for the asymptotic fate of the population. Finally, we show
how this study for a time-homogenous environment allows to consider temporally
piecewise constant environments.
