Roe Deer Comply With the MRW Model

Animal space use expresses a balance between exploring new localities and returning to the known. However, as stated by Ranc et al. (2020), the link between spatio-temporal resource patterns and animal movement has so far found limited experimental support. In their paper they found that following the loss of the individuals' preferred resource, roe deer Capreolus capreolus actively tracked resource dynamics leading to more exploratory movements, and larger, spatially-shifted home ranges. They also demonstrated the return of individuals to their familiar, preferred resource patches after local resource restoration. On the background of this interesting and dynamic habitat manipulation, I have allowed myself to drill into the raw GPS data as provided by their paper's Supplementary material. How does space use by roe deer fit the Multi-scaled random walk (MRW) property of scale-free, memory-utilizing space use?

Choosing a biophysical universality class of movement that best complies with empirical data is key to achieving the best results from analysis of behavioural ecology from these data.  In the present context of roe deer, I studied two key aspects of the Parallel processing universality class, as expressed by the MRW model: 
  1. the Home range ghost formula for observed incidence (non-empty grid cells), I, as a function of sample size; I(N) ≈ cN0.5, and 
  2. the complementary compliance with scale-free distribution of I for a given set of fixes over a range of spatial resolutions, k [I(k) ∝ k-D with D ≈ 1]; i.e., a statistical fractal.
The first aspect regards memory-influenced,  scale-free movement at a balancing scale, "the Characteristic space use" (CSSU), as expressed by c in the Home range ghost formula (fixed grid scale, variable N), and the second aspect regards this behaviour's pattern upon "zooming" into the dispersion of a set of fixes (fixed N, variable grid scale). As already described in numerous blog posts, empirical compliance with these two model properties will support the MRW model and its biophysical framework (the Parallel processing conjecture) in contradiction of the traditional assumption for space use. The former assumes that animals have the capacity to utilize long term spatial memory to return to previous locations. The latter implies that each successive visit to a local site is independent of previous visits (the universality class of Markov-compliant movement; i.e. random walk-like in statistical-mechanical terms). 

The two MRW aspects above are related by the fractal dimension D (Gautestad and Mysterud 2010):

Ik = ckN(1-D/2)   | k = √(CSSU)

The default MRW condition of D ≈ 1 depends on several system aspects; infinite spatio-temporal memory and equal weight of habitat utilization over the actual range of spatio-temporal scales (these aspects will be treated in an upcoming post).

The following results are not considering directly the specific questions raised by Ranc et al. (2020). Instead, the principal aim is to seek the most realistic statistics to analyze space use pattern. If the MRW/Parallel processing framework provides the best compliance with the data, then one should formulate test procedures under this protocol (for example, with respect to habitat selection) rather than applying the classic methods.

In the following Figures the error bars have been omitted in all plots, simply because the errors were very small and thus negligible.

First, consider the first aspect, I(N). Below, I include a simulation output of MRW under condition of  serial autocorrelation of the fixes*. As shown to the right, the average I(N)  from sampling N fixes proportionally with time (diamonds) and fixes that are sampled with increasing frequency within the given total time span for the series (squares) satisfies the Home range ghost formula (filled circles). In absence of autocorrelation the three sets of plots would overlap.
Next, I show the similar analysis of the roe deer data, using the first five series as provided in Supplemental data of Ranc et al. (2020). As shown to the right, the result complies well with MRW expectation under premise of serial autocorrelation! The I(N) regression expresses a power law - not an an approach towards an area asymptote. I conclude that the roe deer confirm MRW compliant space use, and thus compliance with the Parallel processing universality class of movement (search Archive: "The Scaling cube").

Next, consider the second aspect of MRW, I(k) from zooming over spatial resolutions. As shown by the simulation example to the right, the magnitude of incidence expands power law-like towards finer resolutions, but with much smaller slope; about -1 in log-log terms, than expected from the standard universality class of movement. Since a given data set of fixes has a limited capacity to reveal fine-grained space  use pattern, the slope will eventually flatten as log(k) is stretched beyond this N-dependent limit. This is called the Dilution effect (Gautestad and Mysterud 2010). At the coarsest resolution one will generally find that all grid cells contain fixes at the next-coarsest resolution and thus a slope of -2 (the Space fill effect). Hence, a larger sample of fixes provides a large spatial range where the pattern complies with a statistical fractal.

As explained in the blog post "CSSU – the Alternative Approach" (September 16, 2016), the "balancing" CSSU scale can be found close the the log-scaled mid-point between the resolutions of dilution and space fill effects. This property of a dual expression of CSSU underscores the complementary expression of observation intensity, I(N) and I(k).

How do the roe deer data comply with this predicted dualism? Again, the result is closely coherent with a MRW pattern (right). In particular, the CSSU can be estimated from the midpoint between the zones of dilution and space fill effects. Around this "balancing scale", the animal will tend to self-organize the pattern of I(k) towards a self-similar, fractal statistical object in two scale directions, towards finer and coarser grained details respectively. In other words, the range with slope -1 in the regression will expand with the magnitude of the number of fixes available to explore this property. Scale-free space use implies that  the log[I(k)] regression provides a CSSU estimate of similar magnitude as CSSU that was expressed by c in the Home range ghost formula for I(N) above. This relationship was nicely approximated by the roe deer; average CSSU both from I(N) and from I(k) was estimated to ca 1:12 of the area extent that was applied in the analyses [log2(k) ≈ -3.6].

The alternative "MRW home range" visualization (using the first individual's set of fixes; left) may not be as graphically appealing as the classic Kernel density-based method, since it lacks multi-modal smoothing of the fitted utilization distribution. In short, it will be theoretically wrong to smooth a MRW compliant spatial dispersion of fixes, due to its fractal properties. However, by superimposing CSSU-sized tiles onto each relocation of the animal an important property of the utilization - the balancing scale between outwards and inwards expansion of the statistical fractal pattern - becomes prominent. Additional fixes are expected to fall outside the existing tiles in proportion to √N (outward expansion of space use) while inwards improvement of finer-grained patch use is hidden below the CCSU scale; i.e., inside existing area of tiles. However, the overall size of the tiles themselves  are expected to remain constant unless the local or temporal conditions are changing.

The traditional home range model implies that home range size is a function of biological and ecological conditions. The alternative MRW-based model implies that home range size will vary non-trivially with observation intensity (sample size of fixes**), while the CSSU scale rather than the home range size reflects biological and ecological conditions.



NOTES

*) The model animal has gradually changes its preferred base of return steps, leading to a temporally shifting mosaic of space use. Consequently this behaviour also leads to serially autocorrelated fixes. 

**) Non-trivial I(N) implies that a small-N statistical artifact on observed incidence is not expected.


REFERENCES

Gautestad, A. O., and I. Mysterud. 2010. The home range fractal: from random walk to memory dependent space use. Ecological Complexity 7:458-470.

Ranc, N., P. R. Moorcroft, K. Whitney Hansen, F. Ossi, T. Sforna, E. Ferraro, A. Brugnoli, and F. Cagnacci. 2020. Preference and familiarity mediate spatial responses of a large herbivore to experimental manipulation of resource availability. Scientific Reports 10:11946.