When analyzing persistence/extinction of a species in population dynamics, one is led to consider the principal eigenvalue of some indefinite weighted problems in a bounded domain. The minimization of such eigenvalue, to foster population persistence, translates into a shape optimization problem involving the subregion of the habitat which is favorable to the species.
We perform the analysis of the singular limit of this problem, associated with either Dirichlet or Neumann boundary conditions, in case of arbitrarily small favorable region. We show that, in this regime, the favorable region is connected, and it concentrates at points depending on the boundary conditions. Moreover, we investigate the interplay between the location of the favorable region and its shape. Joint works with Lorenzo Ferreri, Dario Mazzoleni and Benedetta Pellacci.