F. Bartumeus | Searcharxiv

arXiv2026

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

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.

cond-mat.stat-mech↗

arXiv2021

Comment on "Inverse Square Lévy Walks are not Optimal Search Strategies for d $\geq$ 2" [Phys. Rev. Lett. 124, 080601 (2020)]

It is widely accepted that inverse square Lévy walks are optimal search strategies because they maximize the encounter rate with sparse, randomly distributed, replenishable targets when the search restarts in the vicinity of the previously visited target, which becomes revisitable again with high probability, i.e., non-destructive foraging [Nature 401, 911 (1999)]. The precise conditions for the validity of this Lévy flight foraging hypothesis (LFH) have been widely described in the literature [Phys. Life Rev. 14, 94 (2015)]. Nevertheless, three objecting claims to the LFH have been raised recently for $d \geq 2$: (i) the capture rate $η$ has linear dependence on the target density $ρ$ for all values of the Lévy index $α$; (ii) "the gain $η_{max}/η$ achieved by varying $α$ is bounded even in the limit $ρ\to 0 $" so that "tuning $α$ can only yield a marginal gain"; (iii) depending on the values of the radius of detection $a$, the restarting distance $l_c$ and the scale parameter $s$, the optimum is realized for a range of $α$ [Phys. Rev. Lett. 124, 080601 (2020)]. Here we answer each of these three criticisms in detail and show that claims (i)-(iii) do not actually invalidate the LFH. Our results and analyses restore the original result of the LFH for non-destructive foraging.

cond-mat.stat-mech↗