### Emergent properties of Animal Space Use - Part II

**In Gautestad (2022) I explored the dual nature of Multi-scaled Random Walk (MRW), both from the network-topological and the spatio-temporal (Eulerian) angle. The results add additional weight to alternative methods to study behavioral-ecological aspects of site fidelity and habitat selection under influence of memory. In this follow-up post I toggle to the model's spatio-temporal aspects under various levels of stressed site fidelity. One of the new results of practical importance is application of a novel way to describe magnitude of serial autocorrelation in series of fix samples. A theoretical framework to study cognitive ****movement ecology under condition of spatial memory and scale-free ****habitat utilization continues to mature****.**

Site fidelity that follows from an individual entering a locality that the individual prefer. The animal’s home range is growing in spatial extent over time due to the mixture of exploratory moves and occasional return events, but with a much slower rate in comparison to movement in the absence of site fidelity. The present question is thus focusing on how the observed statistics from the model animal behaves if the pattern of site fidelity is drifting over space under otherwise constant habitat conditions. This analysis explore this important aspect under influence of erratic, external perturbations ("stress").

As an example from Gautestad (2022), a tendency for drifting home range is apparent from the lower degree of spatial overlap between samples of 100 fixes from early, middle and late part of the total series of spatial locations, presented by, respectively, colored dots.

*I*(N), which is the preferred approach in MRW analyses**. The sample size of fixes, N, can be drawn incrementally from the total series in two ways; either by adding new fixes in a time-incremental manner (continuous sampling; a sample size that is proportional with sampling time) or by increasing sampling frequency within the total time period for the simulation (including every n

^{th}fix within the total time period, by increasing n until n = N). Owing to the strong spatial autocorrelation in this scenario a wide difference between log(

*I*) for a given log(N) over the mid-range of N appears from these two fix sampling methods (open icons). By averaging over the two series the slope becomes log-linear with log-transformed axes (filled icons) and thus power law compliant with power exponent 0.5. The theoretical drilling into this fascinating feature of MRW will have to wait till a later post (or paper).

**The y-intercept, log(c) from**

*I*(N) ≈*c*N^{0.5}after normalizing to unit scale, from such averaging seems quite resilient, and - crucially - the Characteristic Scale of Space Use (CSSU; search this blog) seems to be little influenced even under this strongly shifting pattern of site fidelity, given the novel method to averaging over the two sampling schemes to cancel out the autocorrelation effect.With respect to site fidelity stress, one should expect the return probability to specific sites to decline as a function of increased uncertainty of site profitability or increased risk in connection with return to historic locations; e.g., due to increased environmental variability and unpredictability, or due to a predator’s local search map being influenced by learning the prey’s habits. On the other hand, site familiarity provides crucial benefits with respect to utilizing a memory map (Piper, 2011).

To simulate a varying environment with respect to influencing stability of site fidelity and—in particular—whether this environmental heterogeneity is influencing c and the power exponent z (or conversely, how resilient these parameters are under increased environmental complexity), three conditions are explored in Gautestad(2022) by varying strength of so-called “punctuated site affinity.” At regular intervals (the punctuations) the model individual is narrowing its time horizon for memory-influenced movement by disregarding utilization of the older parts of its historic path during return events. At these intervals the movement path is thus simplistically split into “sections.” Older parts of potential nodes for returns are not any longer included in the process of return decisions.

To summarize the model structure, a premise of scale-free property of movement steps (as illustrated here by the log-linearity above) follows from an an animal that is postulated to relate to its environment at many spatio-temporal scales in parallel over a given scale range ("parallel processing"). In contrast, the classical approaches to use-availability analysis of habitat selection is based on a premise of independent revisits to respective sections of a home range; i.e., a memory-less and area-constrained process in cognitive movement terms (Boyce et al., 2002). Conequently, the behavior is ttraditionally assumed to comply with some variant of standard Brownian motion-like or Lévy walk-like random walk properties in statistical-physical terms. This classic paradigm premise is neither compatible with an evolving network of nodes (see Part I), nor compatible with the MRWmodel*, which is formulated to be compliant with evolving memory map utilization and a scale-free kind of space use at the statistical-physical level.

Three main arguments that support the choice of MRW as the basic statistical-physical model for memory-implemented space use; how observed area use expands as a function of observation intensity, how the scatter of fixes satisfy a statistcal fractal, and to what extent the distribution of inter-fix step lengths satisfy a power law**).

The topological/Eulerian duality that is summarized in Part I / II illustrates how statistical physics of complex phenomena may point towards a new roadmap and a potential for a more realistic interpretation and analysis of how animals move and choose where to live and when to leave. The MRW approach seek to resolve some intrinsic theoretical paradoxes with practical consequences from the perspective of the traditional models. Studying MRW beaviour under site fidelity stress is the latest attempt to reveal degree of parameter resilience related to previously described statistical properties of MRW.

Claiming that the MRW approach is potentially more realistic*** that the current thoretical paradigm that has dominated the field in more than a century is quite a mouthfull. However, after 30 years of stepwise construction and testing of the model from our side (both by simulations and by inspiring tests on real data) I dare to raise this subjective claim. However, as always in science, novel hypotheses need to be scutinized by peers. Dozens of papers in peer-reviewed and high-ranking jounals now supports the MRW approach.

NOTES

*) MRW simulates movement to be studied at a coarsened temporal resolution; i.e., at a temporal unit scale which is coarse enough to ensure that successive steps are randomly distributed in directional terms. This satisfies the premise of a statistical-physical observations of the process in a more simplified mathematical context, relative to studying the process at finer (“hybrid”) temporal resolutions where deterministic, “tactical” behavior and directional step persistence becomes more influential on the movement path (e.g., correlated random walk). The return steps are memory-dependent and describe site fidelity. What regards the statistical-physical aspect, analysis of individual space use is typically based on re-sampling of spatial locations (fixes) that are collected at large time intervals relative to the temporally fine-grained deterministic behavioral response time for successive movement-influencing events within the animal’s current field of perception. For example, GPS fixes from vertebrate space use may be collected at intervals of 1–2 h or larger, embedding much intermediate, tactical and unobserved movement behavior. Thus, theoretical simulation and the accompanying analysis of the space use process at this coarsened “strategic” temporal scale is statistical-physical by nature and in compliance with common empirical protocols from analysis of fixes.

**) First, based on analyses of real data, area demarcation (home range, A, using various demarcation methods) has generally been shown to satisfy the MRW-characteristic power law A≈ N^{0.5} for all species we have studied so far (see references in Gautestad 2021, 2022). Second, by superimposing a virtual grid on the spatial scatter of relocations and counting the number of grid cells containing one or more fixes (incidence, *I*, as a specific variant of A, as a function of grid resolution (a common approach to observe complex space use from a statistical-physical perspective), we have also consistently found a power law relationship, from which we could estimate the fix scatter’s fractal dimension, D. Typically, we find D ≈ 1, which indicates that fixes are statistically distributed in a scale-free (self-similar) manner. In other words, fixes tend to show strong and consistent aggregations over a range of spatial resolutions. Third, when the successive fix distances from red deer movement were analyzed (“step lengths,” L, at 2 hour time resolution), we found that a power law fitted the distribution F(L) better than the negative exponential, where the latter would be expected from a scale-specific and classic random walk-like kind of movement rather than scale-free space use. (Gautestad et al. 2013)

***) In short, the dominating framework of animal space use, both at the individual and the population level, is based on statistical-mechanical principles of intrinsically memory-less reshuffling of particles; i.e., diffusion-like and under external influence of forces like advection and convection. Mathematically, models of animal space use may be summarized as low order Markovian processes. In terms of behavioual ecology, one may describe it as an individual's random self-crossing of its historic movement path. This may be exemplified by an important premise for well-stablished models of habitat selection: statistically, each revisit to a patch should be inddependent of previous visits. In contrast, MRW allows for a potential for non-random self-crossing of the individual’s movement path, as a result of spatio-temporal memory utilization.

REFERENCES

Boyce, M. S., P. R. Vernier, S. E. Nielsen,F. K. A. Schmiegelow. (2002). Evaluating resource selection functions. Ecological Modelling 157,281-300.

Gautestad, A. O. (2021). Animal Space Use, Second Edition: Memory Effects, Scaling Complexity and Biophysical Model Coherence. Newcastle upon Tyne: Cambridge Scholars Publishing.

Gautestad A.O (2022) Individual Network Topology of Patch Selection Under Influence of Drifting Site Fidelity. Front. Ecol. Evol. 10:695854. doi: 10.3389/fevo.2022.695854

Gautestad, A. O., L. E. Loe, 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.

Piper, W. H. (2011). Making habitat selection more "familiar": a review. Behav. Ecol. Sociobiol. 65,1329-1351.