Detecting Lévy walks without turn designation

Abstract

Many organisms have been reported to have movement patterns that are well approximated as Lévy walks. This is typically because distributions of straight line distances between consecutive significant turns in movement paths have heavy power law tails. This diagnostic tool has been called into question because there is currently no standard, unambiguous way to identify significant turns. Even if such a way could be found, statistical analyses based on significant turns cannot account for actual movements made between turns and as a consequence cannot distinguish between true Lévy walks and other fractal random walks such as Lévy modulated correlated random walks where organisms randomly meander rather than move in straight lines between consecutive reorientation events. Here, I show that structure functions (i.e. moments of net displacements made across fixed time intervals) can distinguish between different kinds of Lévy walks and between Lévy walks and random walks with a few scales such as composite correlated random walks and correlated random walks. Distinguishing between these processes will lead to a better understanding of how and why animals perform Lévy walks and help bridge the apparent divide between correlated random walks and Lévy walks. Structure functions do not require turn identification and instead take account of entire movement paths. Using this diagnostic tool, I bolster previous claims that honeybees use a movement strategy that can be approximated by Lévy walks when searching for their hive. I also show how structure functions can be used to establish the extent of self-similar behaviour in meandering Lévy walks.