AUTHOR=Ishikawa Atushi , Fujimoto Shouji , Mizuno Takayuki 

TITLE=Statistical relationship between the enclosed area and trajectory length of animal movement trajectories

JOURNAL=Frontiers in Physics

VOLUME=Volume 13 - 2025

YEAR=2025

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2025.1617160

DOI=10.3389/fphy.2025.1617160

ISSN=2296-424X

ABSTRACT=All animals, humans included, generate closed trajectories—or loops—by repeatedly leaving and returning to the same location. In this study we statistically analyse such closed paths by measuring their total length L and the area S that they enclose, and by quantifying the scaling law S∼Lα. Our data comprise GPS tracks for nine taxa archived in Movebank—common kestrel, demoiselle crane, tortoise, blue whale, reindeer, elephant, wildebeest, lion and nomadic humans—together with smartphone-based GPS logs of people moving in Urayasu, Japan. Daily loops extracted from these records reveal a two-regime geometry: for short displacements (<5km) both humans and kestrels display nearly two-dimensional behaviour with α≈2, whereas for longer distances the exponent drops to α≈1.5, indicating a transition toward one-dimensional excursions. At the annual scale every species shows seasonal round-trip movement, yet the trajectory exponent diverges by taxon: nomadic humans, demoiselle cranes, tortoises and blue whales yield α≈2, wildebeest, elephants, and lions fall between 1 and 2, and reindeer approach α≈1. These results suggest that open environments such as sky, ocean or plain foster two-dimensional roaming, while strong social or environmental constraints—for example, herd mobility—compress movement toward a one-dimensional pattern. A Lévy-flight simulation that incorporates a return potential and bounds the turning angle reproduces the observed α clusters, demonstrating that the strength of directional constraints is a key determinant of geometric dimensionality. Our findings establish the exponent α as a simple, quantitative metric for comparing movement patterns across species and across spatial and temporal scales.