  In this paper, we investigate an optimal design problem motivated by some
issues arising in population dynamics. In a nutshell, we aim at determining the
optimal shape of a region occupied by resources for maximizing the survival
ability of a species in a given box and we consider the general case of Robin
boundary conditions on its boundary. Mathematically, this issue can be modeled
with the help of an extremal indefinite weight linear eigenvalue problem. The
optimal spatial arrangement is obtained by minimizing the positive principal
eigenvalue with respect to the weight, under a L 1 constraint standing for
limitation of the total amount of resources. The specificity of such a problem
rests upon the presence of nonlinear functions of the weight both in the
numerator and denominator of the Rayleigh quotient. By using adapted
symmetrization procedures, a well-chosen change of variable, as well as
necessary optimality conditions, we completely solve this optimization problem
in the unidimensional case by showing first that every minimizer is unimodal
and bang-bang. This leads to investigate a finite dimensional optimization
problem. This allows to show in particular that every minimizer is (up to
additive constants) the characteristic function of three possible domains: an
interval that sticks on the boundary of the box, an interval that is
symmetrically located at the middle of the box, or, for a precise value of the
Robin coefficient, all intervals of a given fixed length.
