  We carry on our study of the connection between two shape optimization
problems with spectral cost. On the one hand, we consider the optimal design
problem for the survival threshold of a population living in a heterogenous
habitat $\Omega$; this problem arises when searching for the optimal shape and
location of a shelter zone in order to prevent extinction of the species. On
the other hand, we deal with the spectral drop problem, which consists in
minimizing a mixed Dirichlet-Neumann eigenvalue in a box $\Omega$. In a
previous paper arXiv:1811.01623 we proved that the latter one can be obtained
as a singular perturbation of the former, when the region outside the refuge is
more and more hostile. In this paper we sharpen our analysis in case $\Omega$
is a planar polygon, providing quantitative estimates of the optimal level
convergence, as well as of the involved eigenvalues.
