Alternatively, 1/C can be calculated in a manner that uses different scales. Superimpose a grid on a map of a study area with cell size such that one cell contains all the locations of given animal. The area of the single can be con sidered as A and 1/C = n1l2IA. Now divide the single cell into four equal cells and calculate 1/C for each cell, letting A be the area of each new cell and n the number of locations in each new cell. The cells can be divided again each into four equal cells and the new 1/C calculated for each. In either of these approaches, a utility distribution can be calculated on different scales appropriate for different questions.
Gautestad and Mysterud (1993:526) also argued that the fractal approach to animal movements shows that "it is just as meaningless to calculate [home range] areas or perimeters as it is to calculate specific lengths of a rugged coastline." They concluded that home range areas cannot be measured because the number of data points needed for an accurate estimate exceeds the number that can be collected on most studies. Unfortunately, Gautestad and Mysterud overstate their point. Clearly, home range boundaries and areas are simple and usually poor measures of animals' home ranges. The important aspects an animal's home range relate to the intensity of use and the importance of areas on the interior of the home range (Horner and Powell 1990). So Gautestad and Mysterud are correct in playing down the importance of boundaries and areas.
Nonetheless, boundaries and areas can be estimated. Animals' home ranges have indistinct boundaries, just as the coastline of an island becomes indistinct when viewed using several different scales. But an island whose perimeter cannot be measured accurately nonetheless has a finite limit to its area, and that limit can be estimated. Likewise, animals who confine their movements to local areas (exhibit site fidelity) do have home ranges whose areas can be estimated, even if those areas must be estimated as a range between upper and lower limits, and even if the home range boundaries may never be known precisely. In addition, a useful estimate of the internal structure of a home range may be estimated with fewer data than needed to obtain reasonable estimates of the home range boundary or area.
In fact, during a finite period of time, an animal must confine its movements to a finite area and limits to that area can be estimated. The black bears I have studied do confine their movements to finite areas. Fixed kernel estimates of the areas of the annual home ranges of all bears located more than 300 times reached asymptotes after at most 300 chronological locations (131 ± 90, mean ± SD, n = 7; Powell, unpublished data; asymptote at 300 for a bear located more than 450 times, 95 percent home ranges). However, equation 3.1 states that the estimated home range area must increase infinitely as the number of location data points used to estimate the home range increases. Clearly, this is a contradiction. The solution to the contradiction lies, I believe, with whether one includes unused areas within an animal's home range and whether one uses sta ble measures of the interiors of home ranges or uses unstable measures of the periphery.
Gautestad and Mysterud (1993, 1995) appear to have run their simulations using simulated utility distributions so large that their simulated animals could not use their whole "home ranges" within biological meaningful time periods. When this is the case, estimates of home ranges should increase in size as more and more simulated data points are used for the estimates. Indeed, after thousands of data points were used, the estimated home range areas do reach asymptotes at the areas of the utility distributions (Gautestad and Mys-terud, personal communication), but note that this implies that equation 3.1 is not accurate for large n.
Some real animals may not use within a single year (or within some other biologically meaningful period) all the areas with which they are familiar. This raises the question of whether areas not used by an animal during a biologically meaningful period of time should be included in the estimate of its home range. Perhaps Gautestad and Mysterud's simulated utility distributions actually represent animals' cognitive maps. Is an animal's cognitive map its home range? Or is its home range only the areas with which it is familiar and that it uses? No definitive answers exist for these questions. Equation 3.1 may be true for some animals. It is most likely to be true for animals that are familiar with areas far larger than they can use in a biologically meaningful period of time. And if equation 3.1 is true, then the time periods over which we estimate home ranges may be as important as the numbers of locations. The time periods must be biological meaningful periods. To obtain accurate estimates of animals' home ranges, we may need to collect as many data as possible, organized into biologically meaningful time periods.
Another solution exists to the contradiction (not necessarily an independent solution). Gautestad and Mysterud estimated home range areas using 100 percent minimum convex polygons (but using the fudge factor Q (n)), which use only extreme, unstable data and must increase whenever an animal reaches a new extreme location. They purposefully incorporated occasional sallies into their model but did not exclude them from their home range calculations. Small changes in sampling points at the extremes of animals' home ranges can lead to huge differences in calculated home range areas although the animals may not have changed use of the interiors of their home ranges. I calculated home ranges areas for black bears using a kernel estimator, which emphasizes central tendencies, which are stable; home range estimates from kernel estimators do not change each time an animal explores a new extreme location.
Finally, Gautestad and Mysterud's model may be unrealistic. Any model of animal movement must be a simplification, so Gautestad and Mysterud's model does simplify animal movements. It does incorporate multiscale aspects of movement and appears to be a better model than, say, random walk models. Nonetheless, the multiscale random walk model still lacks important characteristics of true animal movements, and may thereby cause equation 3.1 to give a false prediction.
Even if equation 3.1 is false, the fractal utility distribution based on 1/ C may still provide insight into use of space by animals. Unfortunately, by calculating C for each cell in a grid, one loses multiscale information that is available from an entire data set. In addition, 1/C provides no insight into estimated use of interstitial cells because it is only a transformation of the frequencies per cell (n1/2 instead of n). Finally, Vandermeer's (1981) cautions concerning grid dimensions must be addressed. One gains equal insight by calculating kernel home ranges and examining the probabilities for animals to be in cells of different sizes (scales), and kernel estimators are free of grid size constraints.
Fractal approaches to animal movements may provide new insights into animals' home ranges, but their utility is still uncertain.
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