Article,
How many animals really do the Lévy walk? Comment
Ecology, 89 (8): 2347-2351 (August 2008)
Abstract
Two major challenges in spatial ecology are understanding the effects of landscape heterogeneity on movement, and translating observations taken at small spatial and temporal scales into expected patterns at greater scales (Morales and Ellner 2002). Correlated random walk (CRW) models emerged from the analysis of short-scaled movement data acquired in experiments usually lasting less than an hour and performed in arenas extending over no more than several meters (Kareiva and Shigesada 1983, Bovet and Benhamou 1988, Turchin 1991). Analysis of animal movements over much larger spatial scales and/or longer temporal scales has given rise to Lévy walk models (LW, often referred to as Lévy flight models; Viswanathan et al. 1996, 1999, Atkinson et al. 2002). Recently, however, Edwards et al. (2007) questioned the claim of Lévy flight behavior in the Wandering Albatross, first reported by Viswanathan et al. (1996). Edwards et al. (2007) also identified some other data sets (Viswanathan et al. 1999) in which the analysis was unable to definitively discriminate the presence of Lévy flights. Bartumeus et al. (2005) argued that CRW can be interpreted as being the by-product of local scanning mechanisms, whereas LW have fundamental properties (super-diffusivity and scale invariance) that allow for higher search efficiencies in random search scenarios. This prompted them to propose that some animals may have evolved the ability to perform LW when confronted with uncertainty. In a recent Ecology report, Benhamou (2007) enriched the debate. He stressed two points: (1) that LW can look like foraging patterns in patchy environments, but in such an environment LW are far from being an optimal searching strategy; and (2) with the usual methodology it is easy to find apparent LW in composite Brownian random walks movements (hereafter referred to as composite CRW movements in accordance with standard terminology). Here I point out that Benhamou's (2007) composite CRW model can, in fact, be interpreted as being an adaptive LW model and that, as a consequence, he has demonstrated that adaptive LW are better than nonadaptive LW when searching in patchy environments. I then generalize Benhamou's second point by showing that heavy-tailed distributions of move lengths, a hallmark of LW, are almost inevitable when collating movement data acquired at observational scales that encompass heterogeneity. This elucidates the notion that the scale of observation influences the description of the pattern (Levin 1992). The emergence of an inverse-square power-law tail does, however, require rather special combinations of local movements and patterns of heterogeneity. This is at variance with the prevalence of such scaling in a diverse range of animals moving within a diverse range of environments (Atkinson et al. 2002, Bartumeus et al. 2003, Reynolds 2007a, b, Reynolds and Frye 2007). Finally, I demonstrate that intrinsic LW characteristics are quite robust with respect to subsampling and that, as a consequence, reports on LW stemming from analyses based on moves between arbitrary location fixes remain secure.
