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In this article, we consider a species whose population density solves thesteady diffusive logistic equation in a heterogeneous environment modeled withthe help of a spatially non constant coefficient standing for a resourcesdistribution. We address the issue of maximizing the total population size withrespect to the resources distribution, considering some uniform pointwisebounds as well as prescribing the total amount of resources. By assuming thediffusion rate of the species large enough, we prove that any optimalconfiguration is bang-bang (in other words an extreme point of the admissibleset) meaning that this problem can be recast as a shape optimization problem,the unknown domain standing for the resources location. In the one-dimensionalcase, this problem is deeply analyzed, and for large diffusion rates, alloptimal configurations are exhibited. This study is completed by severalnumerical simulations in the one dimensional case.

Optimal location of resources maximizing the total population size in logistic models