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!

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