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

Abstract

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.

References in corpus (2)

• Lévy walks
• Inverse square Lévy walks are not optimal search strategies for $d\ge 2$

Cited by in corpus (5)

• Fluctuating landscapes and heavy tails in animal behavior
• Search efficiency of discrete fractional Brownian motion in a random distribution of targets
• Optimizing the random search of a finite-lived target by a Lévy flight
• Non-Markovian Rock-Paper-Scissors games
• Defining and finding lifelike entities with a lazy filter