Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems

TL;DR

This paper investigates how the fractional order of the Neumann spectral fractional Laplacian affects the principal eigenvalue related to population survival thresholds, revealing optimal dispersal strategies in heterogeneous habitats.

Contribution

It provides the first analysis of the optimization of the principal eigenvalue with respect to the fractional order in Neumann problems, linking dispersal strategies to habitat fragmentation.

Findings

For habitats not too fragmented, the eigenvalue has no local minima for 0<s<1.

Optimal survival strategies are either Brownian motion (s=1) or long jumps (s approaching 0).

Results extend to the fractional Laplacian in periodic environments in $\\mathbb{R}^N$.

Abstract

We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for $0<s<1$. As a consequence, the best strategy for survival is either following the diffusion with $s=1$ (i.e. Brownian diffusion), or with the lowest possible $s$ (i.e. diffusion allowing long jumps), depending on the size of the domain. In addition, we show that analogous results hold for the standard fractional Laplacian in $\mathbb{R}^N$, in periodic environments.