  We consider a weighted eigenvalue problem for the Dirichlet laplacian in a
smooth bounded domain $\Omega\subset \mathbb{R}^N$, where the bang-bang weight
equals a positive constant $\overline{m}$ on a ball $B\subset\Omega$ and a
negative constant $-\underline{m}$ on $\Omega\setminus B$. The corresponding
positive principal eigenvalue provides a threshold to detect
persistence/extinction of a species whose evolution is described by the
heterogeneous Fisher-KPP equation in population dynamics. In particular, we
study the minimization of such eigenvalue with respect to the position of $B$
in $\Omega$. We provide sharp asymptotic expansions of the optimal eigenpair in
the singularly perturbed regime in which the volume of $B$ vanishes. We deduce
that, up to subsequences, the optimal ball concentrates at a point maximizing
the distance from $\partial\Omega$.
