### The Binary Home Range

**In my foregoing post I criticized the Burt legacy (Burt 1943) for hampering progress in analysis of animal space use on a local scale. Analyses of the spatial pattern of GPS fixes – when this has been explicitly explored by multi-scale methods – consistently confirm a statistical fractal with dimension 0.9 < D < 1.2 rather than a paradigm-confirming to-dimensional area demarcation (albeit with fuzzy borders; see below). In an ideal world the D≈1 result should lead to hefty follow-up tests from the community of animal ecologists for the sake of verifying or falsifying the “home range as a fractal”-model and its behavioural-ecological implications. After all, the home range concept is a cornerstone of animal ecology. Nope. The Burt legacy still appears impenetrable. However, things finally seem to start rolling.**

After a quarter of a century long invitation period following the initial papers on the topic (Loehle 1990; Gautestad and Mysterud 1993, 1994) the application of fractal based approach that has spun off from the home range ghost concept (Gautestad and Mysterud 1995) has begun in small steps (Morellet et al. 2013; Campos et al. 2014). Also from the theoretical angle a development is finally surfacing along biophysically fascinating paths (Song et al. 2010; Boyer et al. 2012; Boyer and Romo-Cruz 2014; Boyer and Solis-Salas 2014; Boyer and Pineda 2016).

The D≈1 pattern, which represents “fix aggregation within fix aggregation within…”, needs to be understood in a multi-scaled context of animal space use, which is in conflict with basic assumptions under the standard home range concept and statistical methods. How should a fractal analysis proceed? For example, starting with a set of GPS fixes and analyzing it in a two-dimensional histogram, let the zero-count columns represent lack of incidence of GPS fixes (binary zeros), as opposed to non-zero columns (binary ones). The sum of the latter represents the home range representation, called incidence (I), at the actual grid resolution. The similar sums at other resolutions give a set of finer and coarser-scale representations of this “binary home range”. The initial analysis is simple:

- The box counting method (Feder 1988). By studying the incidence function F(k) for all N fixes at linear resolutions k=1, 1/2, 1/4, 1/8, …, where k
^{2}=1 is the unit square that embeds all fixes, one gets the first result: is the pattern fractal-compliant or not? A power law-compliant regression; i.e., the regression line for log[F(k)] over a range of log(k) is approximately linear, will satisfy this criterion. As a rule-of-thumb, such linearity (power law compliance, called “self-similar”) should persist over a relative scale range of 100 or larger. Thus, N should be large to allow for this test. Thus, be aware of the dilution effect and the space fill effect which both tend to artificially narrow the scale-free range (Gautestad and Mysterud 2012). - The fractal dimension. The negative of the slope of log[F(k)] over a range of log(k) gives an estimate of D. For an area-confirming kind of home range; i.e., an object satisfying D=2, one should expect number of non-empty squares to increase proportionally with 1/k
^{2}. If the behaviour is compliant with a space-constrained random walk of the classical kind (a common modelling approach) and thus leading to quite fuzzy area demarcations, one should expect 1.4 < D < 2. See examples in Chapter 3 of my book. If D≈1; i.e, incidence increases proportionally with 1/k^{1}= 1/k, , the paradigmatic HR concept has a problem. Under this condition, the pattern shows core areas within core areas within …, even in a hypothetically homogeneous environment (Gautestad and Mysterud 2005; see also this post). - The home range ghost. As an alternative analysis to F(k), study incidence at a given resolution (k < 1) as a function of sample size of fixes N=1, 2, 4, 8, 16, … N
_{max}. In other words, study F(N) at a given k rather than F(k) at a given N. Then repeat the procedure for different scales k, as explained above. If the given individual’s space use is both scale-free (spatially self-similar) and also influenced by site fidelity (resulting in a home range-emerging kind of movement) you should find a “balancing” resolution k* where log-transformed incidence log(I) expands approximately linear with log(N). The basic home range ghost model predicts I ∝ N^{0.5};*i.e*., at this scale the binary home range area expands proportionally with the square root of number of fixes in the sample. This “paradoxical” N-dependency persists after factors like serial auto-correlation in the fixes and drifting space use has been accounted for. It is an emergent property from an animal that is utilizing its home range in a scale-free manner. The regression line’s intercept with N=1; i.e., at log(N)=0, then gives “the characteristic scale of space use” (CSSU), which is this theory’s replacement of the traditional home range area concept. By the way, the CSSU can also be inferred in a complementary manner from the box counting method for estimating D, as explained in a previous post (Archive: "CSSU - an alternative approach").

- In the classical framework, an area demarcation makes sense, since it represents a given percentage of a two-dimensional stationary statistical distribution of space use.
- In the binary home range framework (based on the parallel processing-compliant space use) the home range area is not intrinsically stationary since it is N-dependent (lacking an area asymptote). Instead, k* represents the stationary parameter.
- Since k* represents a linear scale, squaring it makes (k*)
^{2}proportional with the parameter c in I=cN^{z}(where the power exponent z≈0.5 in the ideal model). - As shown in Eq. 17 in my book, he relationship between D from F(k) and k* from F(N)=I(N) is reflected in the formula

I(N) = cN^{(1-D/2)}.

**At first sight a binary representation of a home range may appear to imply information loss in comparison to a continuous variable like a utilization distribution (“density surface”). However, D, k*, c and z in the binary modelling approach are all continuous quantities. Further, this framework’s analogue to the utilization distribution is a surface where the local variation of intra-home range density of fixes is replaced by intra-home range variation of (1/k*)**

^{2}= 1/c. All these novel system descriptors provide the alternative toolbox for ecological research on animal space use.In the example to the right (simulated data) a large set of “fixes” is collected from a scenario where intra-home range intensity of space use varies between the four quadrats at k=1/2 relative to total area k=1. The parameter c is then estimated from subsets of fixes in respective quadrats. Details will become available upon publication (Gautestad

*in prep.*).

The latter analysis – local variation in 1/c – implies splitting the spatial GPS scatter of fixes into spatial sub-sections (e.g., simple squares, or polygon demarcations of local habitat patches based on some ecological criteria). The embedded subset of fixes within each subsection of the home range is then subject of fractal analysis for the respective intra-home range locations. A preliminary pilot test of the latter involving empirical data rather than simulations was presented in this post.

As shown by the references above, applications and theoretical developments of the fractal home range model has only recently commenced beyond our own series of papers. Why is the D≈1 property of animal space use so difficult to relate to in the community of wildlife ecologists? It can’t be due to lack of proposals for concrete methodological guidelines (Gautestad and Mysterud 2005, 2012; Gautestad 2012, Gautestad and Mysterud 2013; Gautestad et al. 2013). Further, it shouldn’t be due to lack of theory-supporting empirical results (Gautestad and Mysterud 1993, 1995; Gautestad et al.1998, 2013). The basic statistical methods to test or apply the theory are also quite straight-forward, as summarized above. In particular, the key method to translate the spatial scatter of GPS fixes as a binary presence/absence of fixes in respective virtual grid cells and repeating this analysis over a range of grid resolutions is easy to implement both in R and other statistical packages.

Optimistically, I’m still waiting for the ketchup effect with respect to frenetic research on the home range as a fractal; the binary and multi-scaled representation of space use. This requires a critical evaluation of the Burt legacy. Thus, I may still have to wait for a while…

REFERENCES

Boyer, D., M. C. Crofoot, and P. D. Walsh. 2012. Non-random walks in monkeys and humans. Journal of the Royal Society Interface 9:842-847.

Boyer, D., and J. C. R. Romo-Cruz. 2014. Solvable random walk model with memory and its relations with Markovian models of anomalous diffusion. Physical review E. 90:1-12.

Boyer, D., and C. Solis-Salas. 2014. Random walks with preferential relocations to places visited in the past and their application to biology. arXiv 1403.6069v1:1-5.

Boyer, D., and I. Pineda. 2016. Slow Lévy flights. arXiv:1509.01315v2

Burt, W.H., 1943. Territoriality and home range concepts as applied to mammals. J. Mammal. 24:346-352.

Campos, F. A., M. L. Bergstrom, A. Childers, J. D. Hogan, K. M. Jack, A. D. Melin, K. N. Mosdossy et al. 2014. Drivers of home range characteristics across spatiotemporal scales in a Neotropical primate, Cebus capucinus. Animal Behaviour 91:93-109.

Feder, J. 1988, Fractals. New York, Plenum Press.

Gautestad, A. O., and I. Mysterud. 1993. Physical and biological mechanisms in animal movement processes. J. Appl. Ecol. 30:523-535.

Gautestad, A. O., and I. Mysterud. 1994. Are home ranges fractals? Landscape Ecol. 2:143-146.

Gautestad, A. O., and I. Mysterud. 1995. The home range ghost. Oikos 74:195-204.

Gautestad, A. O., and I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 165:44-55.

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

Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9:2332-2340.

Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.

Gautestad, A. O., and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.

Gautestad, A. O., and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.

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

Loehle, C. 1990. Home range: a fractal approach. Landscape Ecology 5:39-52.

Morellet, N., C. Bonenfant, L. Börger, F. Ossi, F. Cagnacci, M. Heurich, P. Kjellander et al. 2013. Seasonality, weather and climate effect home range size in roe deer across a wide latitudinal gradient within Europe. Journal of Animal Ecology 82:1326-1339.

Song, C., T. Koren, P. Wang, and A.-L. Barabási. 2010. Modelling the scaling properties of human mobility. Nature Physics 6:818-823.