Walk model that continuously generates Brownian walks to L\'evy walks depending on destination attractiveness

TL;DR

This paper introduces a walk model where agents adapt their movement strategy between Brownian and Le9vy walks based on destination attractiveness, explaining observed migratory behaviors and offering insights for optimization problems.

Contribution

The study presents a novel model linking destination attractiveness to movement patterns, bridging Brownian and Le9vy walks, and explaining the emergence of Cauchy walks in uncertain conditions.

Findings

Attractive destinations induce Brownian search behavior.

Unattractive destinations lead to Le9vy walks with exponents less than two.

Uncertain destination attractiveness results in Cauchy walks with inverse-distance search probability.

Abstract

The L\'evy walk, a type of random walk characterized by linear step lengths that follow a power-law distribution, is observed in the migratory behaviors of various organisms, ranging from bacteria to humans. Notably, L\'evy walks with power exponents close to two, also known as Cauchy walks, are frequently observed, though their underlying causes remain elusive. This study proposes a walk model in which agents move toward a destination in multi-dimensional space and their movement strategy is parameterized by the extent to which they pursue the shortest path to the destination. This parameter is taken to represent the attractiveness of the destination to the agents. Our findings reveal that if the destination is very attractive, agents intensively search the area around it using Brownian walks, whereas if the destination is unattractive, they explore a distant region away from the point using L\'evy walks with power exponents less than two. In the case where agents are unable to determine whether the destination is attractive or unattractive, Cauchy walks emerge. The Cauchy walker searches the region with a probability inversely proportional to the distance from the destination. This suggests that it preferentially searches the area close to the destination, while concurrently having the potential to extend the search area much further. Our model, which can change the search method and search area depending on the attractiveness of the destination, has the potential to be utilized for exploring the parameter space of optimization problems.