  We consider a shape optimization problem related to the persistence threshold
for a biological species, the unknown shape corresponding to the zone of the
habitat which is favorable to the population. Analytically, this translates in
the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with
respect to a bang-bang indefinite weight. For such problem, we provide a full
description of the singularly perturbed regime in which the volume of the
favorable zone vanishes, with particular attention to the interplay between its
location and shape.
  First, we show that the optimal favorable zone shrinks to a connected, nearly
spherical set, in $C^{1,1}$ sense, which aims at maximizing its distance from
the lethal boundary. Secondly, we show that the spherical asymmetry of the
optimal favorable zone decays exponentially, with respect to a negative power
of its volume, in the $C^{1,\alpha}$ sense, for every $\alpha<1$. This latter
property is based on sharp quantitative asymmetry estimates for the
optimization of a weighted eigenvalue problem on the full space, of independent
interest.
