A variational model with finite carrying capacity for segregated multi-species populations in the whole Euclidean space

Abstract

We study a variational problem motivated by models of multiple segregated species in a nonhomogeneous environment. The analysis combines tools from potential theory, in particular free boundary methods. The main novelty is that we formulate the variational problem in the whole Euclidean space while imposing a finite carrying capacity constraint, which allows us to investigate saturated regions where the species attain their maximal density. In addition, we provide a modeling interpretation of the constrained energy. We show that it arises as the variational limit of energies with increasingly strong crowding and interspecific competition penalties, in the sense that, regardless of their relative rates, every sequence of minimizers of the penalized energies admits a subsequence that converges strongly to a minimizer of the constrained energy.