MRW and Ecology – Part V: Black Bear Home Ranges Revisited

Back in 1994 I enjoyed an unforgettable and extremely inspiring 2-month stay at University of Tennessee, visiting professor Stuart L. Pimm (Department of Ecology and Evolutionary Biology)  and Professor Mike L. Pelton (Department of Forestry, Wildlife and Fishery). During some hectic weeks I worked on transforming the mathematical formulation of the Zoomer model for complex population dynamics into a spatially explicit simulation model (Stuart’s lab) in parallel with interaction with many dedicated students of the biology and ecology of black bear Ursus americanus (Mike’s lab).The Zoomer model is published in my book and already commented on this blog. Regarding the stay at Mike’s lab we published a test on the bears’ general space use, where we found close compliance with the Multi-scaled random walk model, MRW (Gautestad et al. 1998). In this post I revisit the black bear data and find this model’s additional potential to cast light on behavioural ecology in a wildlife management context.

In the 1998 paper we applied the so-called re-scaled range analysis, R/SD (Mandelbrot 1983:pp247-25), to study home range area as a function of fix sample size. Details on the method and the raw data can be found in Gautestad et al. 1998. Below I revisit these bear data and re-test the Home range ghost function with the presently preferred method. Following this exercise I show a new result, which indicates that a collared bear’s characteristic scale of space use may have been inflated during the initial period of fix sampling of its path. Implicitly, the result rises the question if the experience of being collared and getting used to bearing the collar is influencing the bear’s space use towards a more coarse-grained habitat utilization on average; i.e., a larger CSSU, during the first months following the capture/release. The home range ghost regards the “paradoxical” pattern whereby home range area, A, apparently expands non-asymptotically with sample size of fixes, N, in compliance with the power law A = cNz. and z≈0.5. The parameter c is the Characteristic scale of space use, CSSU, emerging from memory-dependent tendency to return to familiar locations.

The inset exemplifies the Home range ghost for one of the black bear individuals. The apparent area asymptote [inset, showing arithmetic axes for A(N)] disappears under log-transformation of the axes; i.e., the expansion with N is non-asymptotic and power law compliant with exponent 0.5. Thus, the area expands with the square root of N.

In the present results to the right and below the A(N) pattern was analyzed with the most recent MRW method, using incidence (number of virtual grid cells of size, I, embedding at least one location) at the estimated grid resolution of CSSU as a representative for home range utilization intensity. The respective (N,I) plots were calculated as the average from two sampling schemes; frequency and time-continuous (see this post). The advantage over R/SD is the I(N) method’s ability to estimate the parameters c (representing CSSU) and power exponent (z) even for autocorrelated series of fixes.

In the present pilot test using a subset of 15 individual data sets (the first part of the total database of 77 series) the result shows strong coherence for estimates of z between the previous R/SD method and the present I(N) method with respect to how home range area responded to sample size, N. Once more I stress the advantage of using a model which accounts for the non-trivial N-dependency in home range size estimates, making CSSU – where the N-dependency is filtered out – a superior ecological proxy relative to the traditional method, direct area estimate.

As expected from scale-free space use, the distribution of CSSU is approximately log-normal (observe the log-scaling x-axis).

On this background it’s time to move in the direction of ecological analysis of respective individual’s space use.The histogram to the right shows a strongly variable CSSU between the 15 individuals. This raises interesting ecological questions. For example, why has the leftmost individual (female F170) 1:240 magnitude of CSSU – ca 1:15 in terms of linear rather than area scale – relative to the rightmost score (female F060)? In other words, according to the present analysis F170 utilized its habitat with substantially higher intensity than F060 (intensity ∝ 1/CSSU). F170 showed a relatively small I(N) for a given N, meaning that F170 utilized its habitat in a more area-restricted manner. Unfortunately I don’t have access to individual behaviour details nor environmental GIS data for the respective space use patterns.

However, consider the relationship between sample size, N, and the CSSU (right). F060 and F170 are found as the largest and smallest CSSU in the scatterplot. The average CSSU for all 15 individuals is ca 866,000 m2 (0.866 km2). Under constant conditions one should expect CSSU to be stationary under larger sample size (larger N) and non-autocorrelated series. However, the plot indicates a negative relationship between N and CSSU in these data, which is unexpected from the standard MRW model a priori (as shown by simulated data in this post)*.

Why? Many hypotheses may be invoked and tested – here I indicate one in particular. The capturing, radio collaring and subsequent tracking might have stressed the bears for some time after release. To study this, I re-analyzed the three individuals with the largest fix sampling period, including the first 100 fixes only. As shown by open symbols, all individuals (F182, F201 and F243) showed larger CSSU during the initial period for 100 fixes following release – a period of ca 1-2 years (!) – relative to the total fix sampling period of almost four years**), and thereby became similar to the other bears with respect to average spatial scale.

Is this an indication of a more restless space use behaviour in the first period following capture, release with radio collar and subsequent telemetry tracking 1-2 times/week (distance observer-animal is unknown)? To explore this aspect, several alternative hypotheses should also be considered (the three individuals were collared in August, June and June, and were 7, 3 and 3 years of age) – but the present pilot test does at least rise an interesting hypothesis. It also illustrates how the MRW model’s CSSU parameter may be applied to cast light on a potentially concerning aspect of data collection and its interpretation in the context of wildlife management***).


*) Update: see also this post. which illustrates a transient effect in the early phase of home range establishment, and stationary CSSU for mature home range utilization.  In the present context, is it reasonable to assume mature home ranges even for the initial 1-2 years of fix sampling.

**) Alternative methods to test for inter-sample difference in space use intensity abound, but have drawbacks. For example, the mean displacement length in a set of fix sampling intervals (lags) will be influenced by intermediate return events. Further, due to the very long-tailed (leptocurtic) distribution of step lengths in data from black bear, which has been verified to comply with the MRW space use model, comparing the relative difference in median displacement size in two or more samples is subject to large error terms due to the extreme outlier issue (“occasional sallies”). CSSU resolves these obstacles.

***) It should bee mentioned that  the radio telemetry collars for black bear back in 1978 were substantially heavier than today’s standard. Triangulation also required close stalking to get a fix, in contrast to modern GPS.


Gautestad, A. O., I. Mysterud, and M. R. Pelton. 1998. Complex movement and scale-free habitat use: testing the multi-scaled home range model on black bear telemetry data. Ursus 10:219-234.

Mandelbrot, B. B. 1983, The fractal geometry of nature. New York, W. H. Freeman and Company.


Statistical-mechanical Details on Space Use Intensity

While stronger intensity of space use in the standard (Markovian/mechanistic) biophysical model framework is equal to the proxy variable fix density, density=N/area, the complex system analogue is 1/c. This alternative expression for intensity is derived from from the Home range ghost formula cN0.5 c√N). Below I illustrate the biophysical difference between the two intensity concepts by a simple Figure and some basic mathematics of the respective processes. The extended statistical mechanics of complex space use underscores the importance of estimating and applying a realistic spatial resolution, close to the magnitude of CSSU, when analyzing individual habitat utilization within various habitat classes. The traditional density variable for space use intensity will invoke a large noise term and even spurious results in ecological use/availability analyses of home range data.

A spatial dispersion of a small and a large sample of fixes is shown in the upper and lower row, respectively. Two resolutions (spatial scales) are shown; the spatial extent (large squares) and a virtual grid scale (dotted lines, shown in the upper right square only). For interpretation of low and high intensity of complex space use, 1/c, see the main text.

In statistical-mechanical terms, one of the main discrepancies between the traditional space use models (mechanistic modelling) and complex movement (MRW) regards the representation of locally varying intensity of space use.

Classical space use intensity may be calculated from a single scale, and trivially extrapolated to a coarser resolution up to the full area extent.

Why is this “freedom to zoom” feasible and mathematically allowed? Consider an example where the system extent is represented by the demarcation of a specific habitat type within a home range, simplified by a square under four conditions in the Figure to the right. Due to assumed compliance with standard statistical mechanics under classical space use analysis, we are specifically assuming finite system variance within the given spatial extent,

Var(X1) + Var(X2) + … +  Var(Xn) = σ2

where [X] is the set of spatial elements from sectioning a system’s extent into sub-sets 1, 2 , 3 .., n; and sub-sets into sub-sub-sets to find respective sub-set variances. Thus, Var(Xi) is the i‘th element’s second moment variability (variance). For example, σ2 could be the intrinsic variances of the spatial inter-cell number of fixes in the virtual grid cells in the Figure above’s upper right scenario (sub-sub grid cells not shown).

The variance also changes proportionally with density. In other words, variance is stationary upon scaling and can thus be assumed to change proportionally with grid scale and density. This implies compliance with the central limit theorem. Even if intra-cell variance is not constant between grid cells at a given resolution within the given extent, as is expected in a heterogeneous habitat where local density varies, the sum of variance of these local parts is independent of this finer-scale variability between sub-components. Once again I underscore that this enormously simplifying system property regards scenaria under the standard statistical-mechanical framework!

On the other hand, the local variability of fix density from complex space use does not comply with the central limit theorem. Intensity of use needs to be calculated over a scale range – from “grain” to extent – rather than any scale, and the grain scale must be chosen with care. 

Traditionally, space use may be quantified by the magnitude of “free space” (area/N) in a sample of N relocations (fixes) of an individual, due to compliance with the central limit theorem, as explained above. On the other hand we have complex space use; i.e., scale-free movement under influence of spatial memory and under compliance with the parallel processing postulate. Under this biophysical framework free space is expressed by the ratio area/√N, rather than the ratio area/N, and quantified by the characteristic scale of space use (CSSU). CSSU is a function of average movement speed and average return rate to previous locations. The  system complexity from CSSU implies that the sum of the system parts’ standard deviation – rather than variance – is stationary upon re-scaling; i.e., 

s.d.(X1) + s.d.(X2) + … +  s.d.(Xn) = √σ2

In other words, by default the spatial statistics follow a Cauchy distribution with scale parameter γ=1, rather than the classical Gaussian distribution. CSSU is proportional with the parameter c in the Home range ghost formula I=c√N, where I is number of fix-embedding virtual grid cells at spatial scale c≈CSSU. 

What if we “lose focus” by studying the system (applying the grain scale) at coarser of finer resolutions than the CSSU? In the illustration above it is assumed that the superimposed virtual grid in the upper right corner reflects a spatial resolution that is close to this system’s true CSSU. If the system’s CSSU had been higher (1/c implicitly lower, as in the upper left-hand scenario), applying the same observer-defined grid resolution as in the upper right scenario would show deviance from the Cauchy distribution. The Cauchy scale parameter and the Home range ghost exponent are both inflated*) due to this “out of focus” situation; i.e., γ>>1 and z>>0.5<1.

In short, by superimposing a virtual grid at scale <<CSSU, we will observe I≈cNz with z≈1 rather than z≈0.5. The parameter c and thus the true CSSU has been erratically estimated. The power exponent z → 1 as grid scale is successively decreased by using cells that are smaller than the true CSSU scale under this condition. However, compliance with z=0.5 may be regained under a “Low 1/c” scenario (upper left) by sufficiently increasing the grid scale relative to cell sizes in the “large 1/c” scenario shown in the upper right example. We can then re-estimate CSSU by such scale zooming towards a coarser resolution and find that z → 0.5 as the coarse-graining is approaching the true CSSU. By comparing scenaria with low and high CSSU; i.e., high and low intensity of space use (1/CSSU), we can rise behavioural-ecological hypotheses about these differences. One obvious example regards strength of intra-home range habitat selection, but where intensity of space use is expressed by habitat-specific 1/c rather than density of fixes.

On the other hand, starting with a too coarse grid cell scale to estimate CSSU will lead to 0<z<<0.5. Defining the scale for I for system observation substantially larger than the true CSSU scale means that I will be seen to increase extremely slowly or not at all with increasing N. In Cauchy terms, the scale parameter 0<γ<1. Hence, the chance to need an extra grid cell to cover all fixes when increasing sample size to N+1 is very small, but not negligible! Occasional sallies of surprising magnitude happen! Surprising from the standard statistical-mechanical framework, but just part of the picture in a space use system that obeys parallel processing principles.

To summarize, while a “Gauss-compliant” (non-complex) kind of space use allows the average intensity of space use to be considered trivially constant upon zooming and linear rescaling over a scale range within the system extent, “Cauchy-compliant” space use requires a search for the correct grain scale to find the system’s average CSSU at this scale within the given extent. 

More details on the statistical-mechanical system description of complex space use is found in my book.


*) Apparently but erroneously, the variability under too fine-grained pixel resolution (grid cell scale) leading to z≈1 and Cauchy scale parameter γ≈2 may be interpreted as Gauss-compliant statistics. However, the Cauchy distribution does not have finite moments of any order. Thus, in strict terms, the reference to √σ2 under the γ=1 scenario is not correct since variance is a term under standard statistical mechanics, but represents a commonly applied approximation (Mandelbrot 1983, Schroeder 1991).


Mandelbrot, B. B. 1983, The fractal geometry of nature. New York, W. H. Freeman and Company.

Schroeder, M. 1991, Fractals, Chaos, Power Laws – Minutes from an Infinite Paradise. New York, W. H. Freedman and Company.