  In this paper, we are interested in the analysis of a well-known free
boundary/shape optimization problem motivated by some issues arising in
population dynamics. The question is to determine optimal spatial arrangements
of favorable and unfavorable regions for a species to survive. The mathematical
formulation of the model leads to an indefinite weight linear eigenvalueproblem
in a fixed box $\Omega$ and we consider the general case of Robin boundary
conditions on $\partial\Omega$. It is well known that it suffices to consider
{\it bang-bang} weights taking two values of different signs, that can be
parametrized by the characteristic function of the subset $E$ of $\Omega$ on
which resources are located. Therefore, the optimal spatial arrangement is
obtained by minimizing the positive principal eigenvalue with respect to $E$,
under a volume constraint. By using symmetrization techniques, as well as
necessary optimality conditions, we prove new qualitative results on the
solutions. Namely, we completely solve the problem in dimension 1, we prove the
counter-intuitive result that the ball is almost never a solution in dimension
2 or higher, despite what suggest the numerical simulations. We also introduce
a new rearrangement in the ball allowing to get a better candidate than the
ball for optimality when Neumann boundary conditions are imposed. We also
provide numerical illustrations of our results and of the optimal
configurations.
