  In this article, we consider a species whose population density solves the
steady diffusive logistic equation in a heterogeneous environment modeled with
the help of a spatially non constant coefficient standing for a resources
distribution. We address the issue of maximizing the total population size with
respect to the resources distribution, considering some uniform pointwise
bounds as well as prescribing the total amount of resources. By assuming the
diffusion rate of the species large enough, we prove that any optimal
configuration is bang-bang (in other words an extreme point of the admissible
set) meaning that this problem can be recast as a shape optimization problem,
the unknown domain standing for the resources location. In the one-dimensional
case, this problem is deeply analyzed, and for large diffusion rates, all
optimal configurations are exhibited. This study is completed by several
numerical simulations in the one dimensional case.
