Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

In this talk, I will present to you a dummy proof talk intoducing our model to study the adaptation of a diffusing population facing two different dynamics.  On  one hand, the population growth  is  time and space dependent, thus modelling strong heterogeneities of the environment. On the other hand, the environment itself is dynamic. It can both change size and shift over time.  The reasons for such moving range boundaries could be the consequences of flooding, forest fire, etc.
We will first investigate the fixed domain case, in particular estimating the principal eigenvalue of the underlying periodic parabolic operator. This estimate is crucial to construct sub and supersolutions on the moving domain. We then address the problem of extinction vs. persistence, taking into account the interplay between the moving habitat and the selection. Finally if we have time, to explore these dynamics, we construct a stable space-time finite elements scheme using upwind test functions in order to gain some insight on the dynamics of this problem. These will unravel some significant differences with classical results on fixed domains.