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).

The Beautiful Anatomy of a Home Range

In an upcoming paper I explore the intrinsic network topology of individual space use, under the premises of site fidelity and multi-scaled habitat use. In short – the process that leads to the emergence of a home range.

networktopologyThe illustration above provides a glimpse into this inner structure of nodes (re-visited locations), their respective statistical weight (how often they have been re-visited) and how they are inter-connected.

Inter-connectivity describes how some nodes are more closely linked (showing a higher frequency of inter-node commuting) than other nodes. Thus, the inter-node distance in these graphs are expressing the relative degree of connectivity, and not the actual spatial distance between the nodes. For example, two nodes that are close  in topological terms – neighbour nodes in the graph above – may in fact happen to be distant in space.

The most intriguing aspect of the two graphs above regards how the topology matures over time. In the early phase after the model individual has settled in an area, there is a relatively “shallow” hierarchy of revisited nodes (Figure 1A). In other words, most nodes are revisited once or just a few times. However, as time is progressing so is the scaling complexity of the network (Figure 1B). Some nodes become dominant, and the “depth” of the hierarchical structure of revisited locations is increasing. The space use becomes more multi-scaled.

In the paper I describe this beautiful process from several statistical–physical angles, and discuss some ecological implications. For example, while a deeper network topology of space use implies a more efficient habitat utilization and a more resilient home range – this maturity may also make the animal vulnerable to disturbance if the dominant node(s) are destroyed by surgical precision.

Since this is a paper in prep. I cannot go into further details here. However, in the book I describe both the simulation model, and several implications of self-reinforcing use of patches that may emerge from site fidelity.


The Scaling Cube

For the first time three directions of animal space use research are unified into a common conceptual model: a three-dimensional biophysical continuum, involving (1) spatial memory, (2) temporal memory, and (3) the animal’s hierarchical perception of the environment. The model is described in detail in Chapter 7 of my book.


The spatial memory aspect regards the degree of memory map utilization. The temporal memory aspect regards degree of mixture of high and low frequency of locomotion, like the classic ARS model (area-restricted search) and related kind of composite space use. Hierarchical processing regards simultaneous (in contrast to sequential!) mixture of tactics and strategy (Figure 70, page 197).

The common theoretical framework for ecological research – providing the majority of models and statistical procedures – is located in the lower left corner, marked as BM/RW (Brownian motion, classic random walk).

  • Are you applying methods like the kernel density estimation or a Brownian bridge model to quantify animal space use? Then you are in the lower left corner of the cube. Like it or not!
  • Is your animal under study utilizing spatial memory or multiple scales of its environment, then you may need to adjust your research methods away from the lower left corner. Like it or not!

The biophysical universality class which represents classic modelling of space use is in the book coherently presented in concert with the more recent theoretical extensions as referred to above. A “biophysical” universality class regards animal space use as we may observe it from – for example – collections of individual GPS locations or from estimates of population density; in other words, when the system is studied at a statistical and/or dynamic meso-scale of time and/or space.

“For example, when studying individual space use, this level is reflected in a sample of GPS fixes, where behavioural modes and movement-influencing events are hidden at finer temporal and spatial resolutions than the sampled path. The temporal scale interval from the fine-resolved movement path to the sampled path (leading to a set of relocation dots on the map, rather than a continuous line) is referred to as “the hidden layer” in this book. At the population level the hidden layer is best reflected by the spatial resolution of the study. This resolution determines local population density; number of individuals per spatial unit at this resolution; and at a chosen temporal resolution (a day, week, or year, depending on context). Again, the actual biological events and interactions like individual searching, feeding, courting, resting, and a myriad of other aspects are spatially and temporally fine-grained processes being executed by the population’s constituents at micro-scale below the resolution for the study; i.e., below the hidden layer.”
(from Preface, page iii)

In the lower left corner of the cube we have the location of the class that embeds classic models, basically containing standard random walk, correlated random walk and simple variants of biased random walk (universality class: Brownian motion and diffusion-compliant biophysics).

In short, this is the corner for space use dynamics where (a) the animal is self-crossing its path by chance only, and (b) it responds to its environment in a purely tactical manner. How are these conditions met by your animal under study?

As all of you are aware, two directions which bring us away from the comfort zone of the lower left corner are now receiving much attention, and is subject to rapid theoretical progress. The first extension regards contemporary modelling of space use that is influenced by spatial memory (the x-axis of the cube). This involves concepts like site fidelity and the emergence of a home range. The second extension involves so-called Multi-scaled space use. Along one line of research (the y-axis of the cube) this regards composite random walk, with area-restricted search (ARS) as a prime example. Along another line of research; represented by the z-axis, we have hierarchical scaling of spatio-temporal dynamics (“parallel processing” compliant processes). This includes hot topics like Levy walk/flight in the upper left corner, and Multi-scaled random walk (space use that involves all three axes) at the upper right corner. The x-y plane; marked by an “M“, is collectively embedding the theoretically well-established Markovian process framework. The z-dimension brings in the parallel processing kind of dynamics; “PP“, which is a qualitatively different ballgame.

The scaling cube brings these directions of research together under a coherent biophysics framework. It also forces upon us a need to differentiate between mechanistic dynamics (the M-floor) and non-mechanistic dynamics (the PP-ceiling).

I’m looking forward to your comments to these statements, and the scaling cube concept in general!