Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

The subject of this paper is inspired by Cantrell and Cosner (1989) and Cosner, Cuccu and Porru (2013). Cantrell and Cosner (1989) investigate the dynamics of a population in heterogeneous environments by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem −Δu = λmu in a bounded smooth domain (Formula Presented), under homogeneous Dirichlet boundary conditions, where λ ∈ R and (Formula Presented). The domain represents the environment and m(x), called the local growth rate, says where the favourable and unfavourable habitats are located. Then, Cantrell and Cosner (1989) consider a class of weights m(x) corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive.