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

Idriss Mazari (1) , Grégoire Nadin (1) , Yannick Privat (2, 3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 IRMA - Institut de Recherche Mathématique Avancée
3 TONUS - TOkamaks and NUmerical Simulations

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.

Mots clés: shape optimization; optimal control; diffusive logistic equation; rearrangement inequalities; symmetrization; optimality conditions

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Idriss Mazari, Grégoire Nadin, Yannick Privat. Optimal location of resources maximizing the total population size in logistic models. Journal de Mathématiques Pures et Appliquées, 2020, 134, pp.1-35. ⟨10.1016/j.matpur.2019.10.008⟩. ⟨hal-01607046v4⟩