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

**One particular class of complex space use – Multi-scaled random walk (MRW) – implies that movement is influenced both by spatially explicit memory utilization and temporally multi-scaled goal execution; in short, it covers spatially explicit strategic locomotion, which may be processed over a range of temporal scales (the parallel processing conjecture; see the December post that summarizes the scaling cube).**

In this Part II I continue to focus on statistical-mechanical properties I elaborate on the entropy aspect of movement. Specifically, I describe how this key property may be coherently maintained – albeit in a surprising manner – even in the non-classic kind of space use, as represented by the MRW model.

According to the standard framework, one should not expect a change of entropy when “zooming” in and out of a given system of a given extent. This property will now be rattled as we move to the scale-free condition.

First, the classic system condition. Consider that a virtual grid is superimposed on the spatial scatter of serially non-auto-correlated GPS fixes (see Part I of this group of posts). Whether the grid has small or large cell sizes, it is a matter of trivial re-calculation to express the home range’s utilization distribution at respective pixel sizes. Half as large pixel size (√2 smaller spatial scale) implies half as large density of fixes at this finer scale, calculated in terms of the new scale unit.

In short, “zooming in” towards finer resolutions does not improve information about the animal’s pixel location at the next relocation in the series. A finer grid reveals more details of the utilization distribution, given that the sample size of fixes is sufficiently large to resolve these details. However, the embedded entropy is expected to be trivially maintained at different pixel sizes for the defined macroscopic system – the home range.

Now, be prepared for a surprise as we change from a classic to a complex kind of space use.

The illustration above is copied from Gautestad and Mysterud (2005). It shows three replicates of two model conditions of complex space use, represented by the MRW model. The upper row shows a scenario where the model animal on average has put stronger weight on short-term relative to longer term goals (MRW, β=3; see the post for the scaling cube). The lower row illustrates an animal that has executed goals more equally distributed over a range of temporal scales (MRW, β=2). As a consequence, in the latter case the space use on one hand seems to become less “dense” over the range of pixel sizes due to the higher level of fragmentation, and on the other hand space use seems more dense when the fragments alone are considered (more empty grid cells in the virtual grid at a given resolution, leading to fix concentration in the others). This difference between upper and lower row can be quantified by the fractal dimension of the spatial pattern of presence/absence (non-empty grid cells, called incidence, at respective resolutions), changing from approximately 1.4

Then, finally – what about entropy in this comparison of scenaria? Above I have reproduced Figure 5 from Gautestad and Mysterud (2005). It shows the local density of fixes in the “interior” part of the home range as a function of spatial scale (square root of cell area), using two of the fix dispersion patterns in the first illustration above; one from MRW with β=3 and one from the fully expressed complexity scenario, MRW with β=2. Both axes are log-transformed for the sake of stressing the respective scenaria’s power law (scale-free) properties. The local density of fixes equals the number of locations per grid cell, calculated over cells with nonzero abundance of locations (incidence). When the spatial resolution is increased from right to left, the density decreases because the given number of fixes is spread over an increasing number of smaller grid cells, and density is calculated as locations pr. non-empty cell at the scale of the new cell size.

The data from the top left panel in the first illustration above show a slope close to 2 (filled circles). This slope is trivially compliant with the classic entropy expectation, to be explained in more detail below. This result verifies that “disorder”; i.e., entropy, is maintained at the same level over a range of spatial resolutions. In other words, when the slope equals 2, entropy is both “space-filling” and resolution-independent within a given area extent, and thus complies with standard statistical mechanics. In the present scenaria the environment was defined to be homogeneous, to emphasize the fractal dimension’s relevance in the context of intrinsically driven space use behaviour (see this post for the relevance of simulating under a homogeneous condition).

**The non-standard property is shown by the smaller slope of magnitude 1.06 for the D=1 condition. Here, density per non-empty cell is increasing non-trivially relative to the given area unit for density calculation.**

The discrepancy, marked by the Δ character, emerges in local density of locations, and shows how local intensity of habitat use – even in these scenaria of site fidelity in a homogeneous environment – increases in a heterogeneous manner towards finer resolutions: a given total number of locations inside the grid arena is concentrated on a smaller and smaller number of cells when cell size is reduced. Consequently, entropy – lack of information of the animal’s whereabouts – is apparently reduced towards finer resolutions.

By the way, the transition towards a common slope of 2 at the coarsest resolutions in this illustration is an artifact from choosing a grid extent that does not cover the total set of relocations in the given set (i.e., some parts of the distribution at the fringes of the home range were left out, for the sake of studying the home ranges’ “interior”). This effect could, to a large extent, have been avoided by choosing an area extent that covered all fixes.

**However, here comes an even more interesting – and biophysically novel – observation: the slope ≈1 (in log-log transformation) for complex space use in the illustration above means that density N/(incidence area) – where incidence area equals the sum grid cells with at least one relocation – is increasing proportionally with the square root of increased resolution (decreased cell size). I am now referring to the aggregated area of non-empty grid cells at respective cell sizes where density is transformed to a specific reference scale (like N/m**

^{2}), rather than being “re-calibrated” to actual cell size at respective resolutions. This implies that the entropy reduction that is surprisingly revealed when observing the system towards finer spatial resolutions is exactly compensated for by entropy expansion from the Home range ghost concept: observed home range area (e.g., non-empty grid cells at a given resolution) expanding proportionally with I(N) ∝ √N! Simultaneously, the entropy reduction from increased observation intensity (larger N) happens in parallel and “non-observed” within each non-empty cell at the given resolution for the observed I(N).Why this harmonic log-linear relationship between “inward” versus “outward” space use expansion? Entropy is a so-called extensive statistical-mechanical property, which means that it is expected to change proportionally with the log-transformed magnitude of system extent (in this case, home range size). In the present context of complex space use the sum of log(inward entropy reduction at smaller pixel scales) and log(outward expansion of system that follows from increased sample size of fixes that is necessary to reveal the increased order at the spatially finer resolution) equals:

“inward” change of entropy + “outward” change of entropy = -0.5 + 0.5 = 0

The sum of the power law exponents cancels out. Entropy theory is consequently maintained, otherwise it would have been a dubious extension of statistical-mechanical theory!

In this extended frame of reference for statistical mechanics, entropy (or information content) is distributed non-trivially over a scale range due to the parallel processing postulate for the MRW model. Within this extended system the zooming paradox I referred to above when such a system is interpreted from the perspective of the classic scenario is resolved by considering the totality of change of entropy towards finer resolutions and towards coarser resolutions in tandem.

Contrast this with the classic system, where there is no paradox in the first place, since the power exponents for change of entropy (represented here by change of standardized density of space use, like N/m

^{2}) are both zero; i.e.,

“inward” change of entropy + “outward” change of entropy = 0 + 0 = 0

In this case there is no fine-pixel surprise factor in the form of reduced entropy if the observer is zooming in (within a given area extent), and no non-trivial area expansion with increasing sample size, N. In the latter case the only area expansion comes from a small-sample artifact – the trivial N-dependency – as the observed area is approaching the true home range size asymptotically with increasing N.

Under conditions of complex space use, simulations are -at least for the time being – a prerequisite for exploring the system’s dynamic and statistical properties. Theory from classic statistical mechanics simply does not suffice to predict the model’s behaviour. Consequently, simulations contribute to extending the theoretical framework for complex space use through an interplay between inductive and deductive reasoning.

In statistical-mechanical terms our empirical tests have generally shown closer compliance with the lower row of the first illustration above. Consequently, a range of alternative methods for ecological inference related to home range data are presented in my book.

In a follow-up Part III-post on the present theme I show for the first time a fascinating “point of balance” scale between what I referred to above as inward contraction versus outward expansion of entropy. Hence, an additional bridge between GPS-based ecological studies on animal space use and the theory for complexity-extended statistical-mechanics is emerging.