# Optimal location of resources for biased movement of species: the 1D case

## TL;DR

This paper solves a one-dimensional optimization problem for resource placement to maximize species survival, revealing that optimal configurations are unimodal, bang-bang, and can be characterized as specific interval types under Robin boundary conditions.

## Contribution

It provides a complete solution to the 1D resource placement optimization problem with nonlinear eigenvalue constraints, including explicit characterization of optimal shapes.

## Key findings

- Optimal resource regions are unimodal and bang-bang.
- Minimizers are characteristic functions of specific intervals.
- Explicit solutions depend on Robin boundary parameters.

## Abstract

In this paper, we investigate an optimal design problem motivated by some issues arising in population dynamics. In a nutshell, we aim at determining the optimal shape of a region occupied by resources for maximizing the survival ability of a species in a given box and we consider the general case of Robin boundary conditions on its boundary. Mathematically, this issue can be modeled with the help of an extremal indefinite weight linear eigenvalue problem. The optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to the weight, under a L 1 constraint standing for limitation of the total amount of resources. The specificity of such a problem rests upon the presence of nonlinear functions of the weight both in the numerator and denominator of the Rayleigh quotient. By using adapted symmetrization procedures, a well-chosen change of variable, as well as necessary optimality conditions, we completely solve this optimization problem in the unidimensional case by showing first that every minimizer is unimodal and bang-bang. This leads to investigate a finite dimensional optimization problem. This allows to show in particular that every minimizer is (up to additive constants) the characteristic function of three possible domains: an interval that sticks on the boundary of the box, an interval that is symmetrically located at the middle of the box, or, for a precise value of the Robin coefficient, all intervals of a given fixed length.