Walk model that continuously generates Brownian walks to Lévy walks depending on destination attractiveness
This provides a mechanistic explanation for observed animal migration patterns and could be applied to optimization problems, though it is incremental in extending random walk theory.
The study tackled the problem of understanding why organisms exhibit Lévy walks, particularly Cauchy walks, by proposing a model where agents' movement strategies depend on destination attractiveness, finding that attractive destinations lead to Brownian walks for local search, unattractive ones to Lévy walks for distant exploration, and uncertain attractiveness results in Cauchy walks with distance-proportional search probabilities.
The Lévy 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, Lévy 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 Lévy 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.