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 −∆u = λmu in a bounded smooth domain Ω ⊂ RN , N ≥ 1, under homogeneous Dirichlet boundary conditions, where λ ∈ R and m ∈ L ∞(Ω). The domain Ω 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. In our work we consider a sort of refinement of the result above. We write the weight m(x) as sum of two (or more) terms, i.e. m(x) = f1(x) + f2(x), where f1(x) and f2(x) represent the spatial densities of the two resources which contribute to form the local growth rate m(x). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of f1 (x) and f2 (x) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if Ω is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions).

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

#### Abstract

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 −∆u = λmu in a bounded smooth domain Ω ⊂ RN , N ≥ 1, under homogeneous Dirichlet boundary conditions, where λ ∈ R and m ∈ L ∞(Ω). The domain Ω 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. In our work we consider a sort of refinement of the result above. We write the weight m(x) as sum of two (or more) terms, i.e. m(x) = f1(x) + f2(x), where f1(x) and f2(x) represent the spatial densities of the two resources which contribute to form the local growth rate m(x). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of f1 (x) and f2 (x) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if Ω is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions).
##### Scheda breve Scheda completa Scheda completa (DC) Population dynamics, eigenvalue problem, indefinite weight, optimization, Steiner symmetry.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11584/323610`
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