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

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.