Survival criterion for a population subject to selection and mutations ; Application to temporally piecewise constant environments

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 [PB08, BMP09, LMP11] in a constant environment, when the asymptotic persistence of the population was ensured. Here, we relax the assumptions on the growth rate and the initia data to extend the study to situations where the population goes 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 constant environment allows to consider temporally piecewise constant environments. This applies to several applications in biology such as the adaptation to a pharmacological treatment, and the interaction between two populations evolving on different ecological timescales