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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math> and the area <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:math> that they enclose, and by quantifying the scaling law <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>S</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>&lt;</mml:mo><mml:mspace width="-0.2em"/><mml:mn>5</mml:mn><mml:mi mathvariant="normal">k</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> both humans and kestrels display nearly two-dimensional behaviour with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mrow><mml:mi>α</mml:mi><mml:mo>≈</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> , whereas for longer distances the exponent drops to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"><mml:mrow><mml:mi>α</mml:mi><mml:mo>≈</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"><mml:mrow><mml:mi>α</mml:mi><mml:mo>≈</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> , wildebeest, elephants, and lions fall between 1 and 2, and reindeer approach <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m9"><mml:mrow><mml:mi>α</mml:mi><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m10"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math> clusters, demonstrating that the strength of directional constraints is a key determinant of geometric dimensionality. Our findings establish the exponent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m11"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math> as a simple, quantitative metric for comparing movement patterns across species and across spatial and temporal scales.