How Many Animals Really Do the Lévy Walk?

Abstract

Lévy walks (LW) are superdiffusive and scale-free random walks that have recently emerged as a new conceptual tool for modeling animal search paths. They have been claimed to be more efficient than the "classical" random walks, and they also seem able to account for the actual search patterns of various species. This suggests that many animals may move using a LW process. LW patterns look like the actual search patterns displayed by animals foraging in a patchy environment, where extensive and intensive searching modes alternate, and which can be generated by a mixture of classical random walks. In this context, even elementary composite Brownian walks are more efficient than LW but may be confounded with them because they present apparent move-length-heavy tail distributions and superdiffusivity. The move-length "survival" distribution (i.e., the cumulative number of moves greater than any given threshold) appears to be a better means to highlight a LW pattern. Even once such a pattern has been clearly identified, it remains to determine how it was actually generated, because a LW pattern is not necessarily produced by a LW process but may emerge from the way the animal interacted with the environment structure through more classical movement processes. In any case, emergent movement patterns should not be confused with the processes that gave rise to them.

Environmental context explains Lévy and Brownian movement patterns of marine predators

“…Hence, controversy remains over whether Lévy behaviour occurs in nature 6,9,17 , despite many empirical studies 1,18 . Furthermore, long time series of movements (over weeks to months) derived from animalattached electronic tags will very probably capture complex movement data resulting from different types of behaviour (for example searching, travelling and resting) as animals respond to various biotic and abiotic factors over time.…”

Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion

“…This composite BM approach-from randomly mixing bouts of scale-specific movement at different scales-is feasibly described as intra-patch movement mixed with less frequent inter-patch movement [24,28]. Plank & Codling [29] extended Benhamou's [24] exemplification of Lévy look-alike processes from composite BM, by exploring a broader range of scale constants and conditions for the scale-specific components under a range of sampling lags on the generated series. They found that a strong directional persistence (correlated random walk) for the coarser-scale process in combination with low frequency of appearance of this component relative to the finer-scaled process led to an LW-like pattern.…”

Predicting oscillatory dynamics in the movement of territorial animals

“…For large m, one may approximate [30] m 0m ≃ ðm À 1=4Þp as well as the Bessel function for large values of its argument and rewrite the series as P 1 m¼1 ðÀ1Þ m ðm À 1=4Þ À3=2 cos½ðm À 1=4Þpjt=T . This series can be recast in terms of combination of Lerchphi function [33] and shown to possess the first discontinuous derivative at t/T ¼ j 21 . We have tested numerically that indeed the location of the first maximum in the MSD follows this inverse proportionality relation as function of j, with greater accuracy as we increase j.…”

“…Two such examples are correlated random walks [9 -12] and Lévy walks [13,14]. Whereas the former has a long mathematical history [15], the latter has been proposed more recently [16] and only in the last decade has it been used as a tool to analyse and interpret animal movement data [17][18][19]; however, its use has not been immune to criticisms [20][21][22][23][24][25]. Here, we are interested in exploring the dynamics of a territorial animal in two dimensions when its movement possesses some degree of persistence.…”