Fractal Compliant Space Use: Intrinsic Scaling or Extrinsic Habitat Heterogeneity?

A growing number of analyses of spatial scatter of GPS positions of animals (fixes) verify a fractal dimension (D) substantially smaller than what should be expected from standard models. Specifically, D tends to be close to 1, which reflects a very heterogeneous fix dispersion over a wide range of spatial resolutions. A recurring question among analysts of animal space use is thus: is this so-called scale-free pattern – statistically speaking – reflecting a matching heterogeneous habitat that happens to satisfy a self-similar dispersion (the “environmental forcing” explanation), or is the scale-free space use a manifestation of intrinsic, cognitive processes (the “emergent property” explanation)? In this post I illustrate the challenge by showing an analysis of large mammals' space use.

The image above, showing the spatial accumulation of fixes from GPS-sampling a red deer Cervus elaphus individual during the summer season illustrates this key question (see Gautestad et al. 2013 for details, where 52 individuals are included). As shown below, when counting number of fix-containing virtual grid cells at different spatial resolutions for respective individuals, the entire set of data covering all individuals showed D=1.18 on average. Due to the log-log linearity the pattern satisfies a statistical fractal. Two individuals with particularly large sample sizes illustrate that also the individual sets of fixes show close coherence with self-similarity (open symbols; both sets confirming linear regression slope on log-log scale). The actual analysis of the SF17s series – see scatter of fixes above – shows D=1.12 (red triangles).

So far, the multi-faceted statistical scaling analysis of these red deer data (Gautestad et al. 2013) represents the broadest test of scale-free space use by terrestrial animals ever published. The pressing question is thus: is the scale-free space use formed by an environment that happens to be distributed in a fractal-compliant manner (“self-similar” heterogeneity), or does the pattern emerge as a by-product of scale-free habitat utilization? Three arguments support the latter interpretation, for logical reasons alone:

  • Any individual shows space use that is reflecting a multitude of habitat attributes and internal modes of the animal during the GPS sampling period. For example, even when considering foraging behaviour alone, the individual relates to several “layers” (mapping of environmental conditions) in parallel, similar to layers in a GIS map. In short, different food/prey categories have respective spatial distributions; i.e., local probability of occurrence, based on the target individual’s perception and experience). When these layers are superimposed, even if (hypothetically) the respective layers are fractal-confirming with D≈1, the superposition will tend towards D≈2; i.e., space-filling at all resolutions covered by the analysis. In other words, empty spaces (lack of presence) of a given habitat attribute at respective spatial resolutions will tend to fill up all space in the superposition, due to a larger or smaller extent of independence of the respective layers. Broadening the set of layers to include attributes like resting spots, overlapping space use by conspecifics, local predator risk, etc., will tend to smear out any hypothetical fractal pattern of D≈1 in respective environmental factor even further towards D≈2 in the over-all space use pattern. 
  • Many of the respective habitat attributes’ spatial mapping are temporally variable – a shifting mosaic. Again, this effect will tend to smear out an environmental fractal pattern of D≈1 towards D≈2 during the period of fix sampling of the target individual’s whereabouts. 
  • Even if 1) and 2) do not apply (the attributes are spatially overlapping with similar presence/absence, and they are also temporally stationary during fix sampling) the animal that is steered primarily by extrinsic factors must consistently avoid the “repelling patch” mosaic at respective resolutions over the entire range of actual scales. The dimension of a statistically fractal pattern is estimated from presence/absence of fixes in respective locations (typically, virtual grid cells) at respective resolutions. Hence, if there is some probability of locating the target individual in “forbidden” space (repelling areas of the environmental fractal), compliance with D≈1 from environmental forcing will not emerge.
When it comes to red deer, the area that is embedded in the uppermost illustration obviously contain – for the most part – heterogeneous patches that are distributed somewhere on the gradient from good to bad. “Forbidden space”, like lakes, rivers, steep slopes, etc., neither dominates the home range nor show a scale-free D≈1 pattern.

To conclude from ecological reasoning alone, the D=1 pattern that was verified for the red deer should be considered an emerging property of intrinsic origin (multi-scaled cognitive processing) as this kind of behaviour is expressed by GPS-sampled space use through the season.

In our 2013 paper we went beyond the simple logical reasoning in 1-3 above by expanding the analysis with several complementary statistical approaches, which all supported the same conclusion. In statistical-mechanical terms, the pattern satisfied what is expected from Multi-scaled random walk (MRW) in the scaling cube.


Gautestad, A. O., L. E. Loe, and A. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82:572-586.