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

TL;DR

This paper refutes recent criticisms of the inverse square Lévy walk hypothesis, reaffirming its validity as an optimal search strategy in higher dimensions through detailed analysis and counterexamples.

Contribution

It provides a detailed response to three recent objections, demonstrating that the Lévy flight foraging hypothesis remains valid for non-destructive search in multiple dimensions.

Findings

Criticisms do not invalidate the LFH.

The capture rate depends linearly on target density.

Optimality of Lévy walks is restored under detailed conditions.

Abstract

It is widely accepted that inverse square L\'evy 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 L\'evy 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 $\eta$ has linear dependence on the target density $\rho$ for all values of the L\'evy index $\alpha$; (ii) "the gain $\eta_{max}/\eta$ achieved by varying $\alpha $ is bounded even in the limit $\rho \to 0 $" so that "tuning $\alpha$ 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 $\alpha$ [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.