Statistical Relationship Between the Enclosed Area and Trajectory Length of Animal Movement Trajectories

Atushi Ishikawa^1*Shouji Fujimoto¹Takayuki Mizuno²

• ¹Kanazawa Gakuin University, Kanazawa, Japan
• ²National Institute of Informatics, Chiyoda-ku, Tôkyô, Japan

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 ∼ L α . 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 cluster around α ≈ 1., 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 L évy-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.