Intermittent Inverse-Square Lévy Walks are Optimal for Finding Targets of All Sizes

Lévy walks are random walk processes whose step-lengths follow a long-tailed power-law distribution. Due to their abundance as movement patterns of biological organisms, significant theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantag...

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Bibliographic Details
Published in:arXiv.org 2021-07
Main Authors: Guinard, Brieuc, Korman, Amos
Format: Article
Language:English
Subjects:
Diameters
Lower bounds
Motion perception
Optimization
Random walk
Stochastic processes
Strategy
Target detection
Toruses
Online Access:Get full text
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Summary:Lévy walks are random walk processes whose step-lengths follow a long-tailed power-law distribution. Due to their abundance as movement patterns of biological organisms, significant theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the \emph{Lévy foraging hypothesis}, and are thus expected to impact future experiments on animals performing Lévy walks.
ISSN:2331-8422
DOI:10.48550/arxiv.2003.13041