### A Statistical-Mechanical Perspective on Site Fidelity – Part III

**In Part I of this group of posts I introduced physical concepts like entropy and ergodicity, and I contrasted these concepts under classic and extended (complex) system conditions of animal space use. In Part II I followed up on this theme by adding a novel piece of theory to the jigsaw puzzle, where “inward contraction” of the magnitude of entropy towards finer spatial scales of a home range was counterbalanced by “outward expansion” of entropy as a reflection of the area expansion under increasing sample size N under the Home range ghost concept. In this manner I showed how the MRW model was coherent with a key premise for a sound statistical-mechanical framework despite its qualitatively different structure in comparison to classic Boltzmann-Gibbs theory. Basically, the extended theory rests on a conjecture about parallel processing rather than Markov-compliant mechanics at micro-scales. In this Part III follow-up take additional steps towards a theory for complex statistical mechanics of animal space use.**

A statistical-mechanical clarification of the parallel processing conjecture is important for many reasons, also from an ecological modelling perspective. For example, it may provide a theoretical justification for the concept I introduced in another post – the individual’s characteristic scale of space use (represented by the parameter c in the Home range ghost equation, A = c√N). In short,

why does a given individual’s aggregated movement over the actual fix sampling period self-organize towards a balancing level – the characteristic scale of space use? What is so special about this scale, which obviously is larger for an elephant than for a mouse, and – for a given individual – larger in a harsh environment versus a more optimal habitat? Why approximately square root expansion of observed space use (power law, with exponent 0.5) with increasing sample size of fixes, N, as we have verified in many data sets covering a wide range of species?

The illustration to the left recapitulates the general property of the Multi-scaled random walk model (the home range ghost, due to exponent >0) from the perspective of observed area accumulation as a function of N. In this case, area regards number of non-empty virtual grid cells, incidence, I, at a chosen grid resolution. This resolution is indicated by the respective squares on the right hand side. “The characteristic scale of space use” is marked as CSSU. Both axes are log-transformed to better visualize the model’s power law property (green line). Log(c) is found as the intercept with log(I).

First a trivial clarification. The green line does not expand with N “forever”, despite a lack of asymptote in the model formulation. Within a given time frame T (for example, a season), the animal has obviously limited its space use to a given range. Thus, if sampling frequency is increased sufficiently during a fixed T the green line will ultimately have to level off. However, due to the “absorbing” effect of inward area expansion and the small power exponent of 0.5, N typically has to be very large to reveal this limit.

In fact, the frequency of fix collection 1/Δt within T will have to be increased to a magnitude where the hidden layer (see definition under The scaling cube) becomes too narrow for a proper statistical-mechanical system description. In that case one has approximated the micro-scale path resolution, which implies that there is insufficient “degrees of freedom” left on the animal’s true movement path to make the animal’s next position sufficiently uncertain in statistical terms. In short, the ergodic property – explained in Post I of this group – is no longer satisfied at sampling interval Δt under this condition.

Returning to the conceptual graph above, the slope z=0.5 at resolution CSSU (green line) illustrates the standard model. At this pixel scale, inward contraction equals outward expansion (see above), and additionally expresses the condition where the animal has distributed its temporally scaled goals with equal statistical weight over the scale range (see Part I-II of this set of posts). In other words, the animal has on average over the actual period T and temporal resolution Δt executed short-, medium- and longer term goals in a specific and self-organized manner that has led to geometric scaling rather than just an arithmetic accumulation of spatial displacements (a power law, rather than an asymptotic growth of incidence with increasing N). I refer to the book and other blog posts for conceptual details on the parallel processing conjecture.

CSSU is the key “balancing point” of space use with respect to spatial scale level, and it provides a great potential for ecological interpretation. This is how to estimate it. If the chosen pixel size is is too fine- or coarse-scaled relative to the true CSSU, the relationship I(N) breaks down unless N is very large. At fine resolutions new fixes will all tend to fall outside the existing set of non-empty cells (red line), until a sufficiently large N is reached and z=0.5 is achieved. Thus, I(N) grows proportionally with N rather than with √N in this range of N. The given sample size is not sufficiently large to reveal the animal’s fractal space use pattern from “inward” entropy contraction at these fine scales. Further, the intercept log(c) becomes superficially inflated (see dotted line with arrow towards the y-intercept). On the other hand, if grid resolution is chosen too coarse relative to the true CSSU, new fixes will all tend to fall inside the the existing home range demarcation (blue line). Thus, I(N) is independent of N in this range of N (zero slope). Again, a large N will be needed for the fractal dispersion of fixes to escape this zone of statistical artifacts. However, the estimated y-intercept is in this case too low relative to the true CSSU.

**As outlined in my book, optimizing grid resolution for I(N) to estimate CSSU is a matter of “zooming” towards the virtual grid cell (pixel) size where log(c) is close to zero. At this scale, the power exponent z is predicted to be close to 0.5 (MRW under the default condition β=2 for exploratory moves).**

In a follow-up post (Part IV) I plan to bring forward yet another novel brick in the extended statistical-mechanical theory for complex space use, by bringing in the characteristics of micro- and macrostates as we move from a Markovian condition (the standard theory) to a parallel processing system.