  We study the long time behavior of a parabolic Lotka-Volterra type equation
considering a time-periodic growth rate with non-local competition. Such
equation describes the dynamics of a phenotypically struc-tured population
under the effect of mutations and selection in a fluctuating environment. We
first prove that, in long time, the solution converges to the unique periodic
solution of the problem. Next, we describe this periodic solution
asymptotically as the effect of the mutations vanish. Using a theory based on
Hamilton-Jacobi equations with constraint, we prove that, as the effect of the
mutations vanishes, the solution concentrates on a single Dirac mass, while the
size of the population varies periodically in time. When the effect of the
mutations are small but nonzero, we provide some formal approximations of the
moments of the population's distribution. We then show, via some examples, how
such results could be compared to biological experiments.
