L\'evy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal L\'evy walk strategy for random searches

TL;DR

This paper rigorously demonstrates that inverse square Le9vy walks optimize search efficiency for random searches within spherical shells with absorbing boundaries, providing strong theoretical support for their optimality in foraging.

Contribution

It provides a mathematically rigorous proof that inverse square Le9vy walks are optimal for search strategies in bounded spherical domains, confirming prior hypotheses.

Findings

Inverse square Le9vy walks optimize search within spherical shells.

Mathematically rigorous proof for the optimality of inverse square Le9vy walks.

Supports the Le9vy flight foraging hypothesis with formal analysis.

Abstract

The L\'evy flight foraging hypothesis states that organisms must have evolved adaptations to exploit L\'evy walk search strategies. Indeed, it is widely accepted that inverse square L\'evy walks optimize the search efficiency in foraging with unrestricted revisits (also known as non-destructive foraging). However, a mathematically rigorous demonstration of this for dimensions $D \geq 2$ is still lacking. Here we study the very closely related problem of a L\'evy walker inside annuli or spherical shells with absorbing boundaries. In the limit that corresponds to the foraging with unrestricted revisits, we show that inverse square L\'evy walks optimize the search. This constitutes the strongest formal result to date supporting the optimality of inverse square L\'evy walks search strategies.