Snapshot From the Lévy Controversy

Are animals moving in a scale-specific or a scale-free manner? Are they toggling between these modes under different environmental conditions (basically represented by the Lévy flight foraging hypothesis)? This theme has emerged as a hot potato, which does not seem to cool down anytime soon.

For example, Andy Reynolds’ recently replied to no less than nine (!) responses to his recent paper on this topic. The following reference provides an overview:

Reynolds, A. 2015. Venturing beyond the Lévy flight foraging hypothesis: Reply to comments on “Liberating Lévy walk research from the shackles of optimal foraging”. Physics of Life Reviews 14:115–119.

In my book I summarize elements of the debate on scale-free space use, and put it in context of the parallel processing (PP) conjecture. PP is conceptualized as representing the ceiling in the Scaling Cube, which I briefly described in another post.

In the present post, however, I want to draw the attention to another interesting paper:

Breed et al. 2014. Apparent power-law distributions in animal movements can arise from intraspecific interactions. J. R. Soc. Interface 12: 20140927.

Baltimore_Checkerspot. TaylorsCheckerspot

Females of the two butterfly species Baltimore checkerspot and Taylor’s checkerspot were tracked during foraging and other activities, and their locations were collected every 15-20 seconds. The authors provide a very thorough statistical analysis of the respective distributions of step lengths (fix-to-fix lengths during the constant sampling interval). They conclude that these individuals are not performing scale-free (i.e., Lévy-like) movement but a mixture of scale-specific movement bouts with different parameter characteristics. Such “composite walks” imply in this case a superposition of two exponential functions for the pooled step length distribution with different mean step lengths. Consider area-restricted search (the classic ARS model) as an analogy, often referred to as toggling between intensive and extensive moves. However, in the present scenario the more extended type of moves were frequently triggered by harassment by males. The total distribution of both types of movement may then give the wrong impression of a “fat tail”, which is considered an indication of scale-free (power law) movement.

This work is a very nice and well executed study, which will probably ignite some responses. The potato will be kept warm. Here are my own preliminary thoughts on the results and conclusions in Breed et al. (2014):

  • The authors have chosen to analyze the entire distribution of step lengths, including the relatively high frequency of very small ones. They argue sensibly for their choice: “…short steps were not attributable to observation error and they are numerically important. Thus, we could not justify ignoring some or all of these short steps—often fitting procedures have excluded the shorter steps and only fit the distribution tails.” (p4). My comment to this choice: butterflies can be expected to have spent a considerable amount of time sucking nectar during short displacements (calculated as the distance between successive moves at 15-20 sec. intervals). Such episodes are obviously less dominant during medium and longer range displacements. In other words, the animals may on average have spent a disproportionally smaller part of available time on actual locomotion during intervals with small displacements. Consequently, the number of short steps may then tend to be inflated during periods when the individual is moving slowly, relative to when it is moving faster, when measured at a fixed time scale. This may lead to artificially increased statistical weight of the smallest step lengths and thus to a stronger support for a two-phased kind of scale-specific moves (composite random walk) relative to – for example – power law (Lévy-like) movement.
  • This effect could perhaps easily be adjusted for? If – for example – the butterflies on average spent x% of the time nectar-feeding during displacements belonging in a given distribution bin, the number of steps in this bin could artificially be reduced by x% (random removal) prior to performing the statistical analysis. If x is substantially larger for small-step bins than for larger bins such an adjustment may be critically important.
  • A long-lasting challenge with respect to differentiating between a composite random walk and a true scale-free walk is the fact that the former may easily appear scale-free in statistical tests if the exponential function parameters and the respective components’ frequency of occurrence are “tuned” towards fitting a scale-free distribution of steps (Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88: 1962-1969). However, in my book I summarize two very simple methods to distinguish a composite walk  – “a Lévy walk in disguise” – from a true scale-free walk; the coarse-graining method (Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122: 612-620) and the parallel shift method (Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9: 2332-2340). More statistically sophisticated methods have also been proposed (Auger-Methe et al. 2015. Differentiating the Lévy walk from a composite correlated random walk. Methods in Ecology and Evolution 6: 1179–118). However, a prerequisite for applying these tests is a large data set, particularly since the coarse-graining method and the parallel shift method depends on studying the distribution of step lengths as the sampling interval is varied (in practice, the original set of relocations is sub-sampled). This requirement probably exclude the application of these tests on the present material, where each series is relatively short (still impressive from a field ecologist’s perspective, I may add).

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