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

## TL;DR

This paper investigates how to optimally distribute resources in a heterogeneous environment to maximize the total population size of a species, using mathematical analysis and numerical simulations, especially focusing on the case of large diffusion rates.

## Contribution

It proves that optimal resource distributions are bang-bang configurations and characterizes these configurations in one-dimensional settings for large diffusion rates.

## Key findings

- Optimal configurations are bang-bang (extreme points).
- In one dimension, all optimal configurations are characterized for large diffusion.
- Numerical simulations support theoretical results.

## Abstract

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.