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x-y plane. The Fourier transform estimator smooths the spikes into a surface
that estimates an animal's utility distribution. I developed a similar method
using spline smoothing techniques (Powell 1987). Both of these estimators
accurately show multiple centers of activity that may be considerably removed
from the arithmetic mean of the x and y data, but both behave poorly near the
edges of home ranges, probably because the location data do not meet assump-
tions needed to make the transformations. To address the problem of poor esti-
mates of home range peripheries, Anderson (1982) recommended using ani-
mals' 50 percent home ranges (the smallest area encompassing a 50 percent
probability of use) rather than 95 percent home ranges. Fifty percent is no less
arbitrary than 95 percent, but it departs completely from the basic concept of
a home range (Burt 1943) or stretches that concept to its limit by assuming
that an animal is on an “occasional sally” 50 percent of the time.
HARMONIC MEAN DISTRIBUTION
Human population densities fall in an inverse harmonic mean fashion from
centers of urban areas through rural areas. Consequently, Dixon and Chap-
man (1980) proposed using a harmonic mean distribution to describe animal
home ranges. Contours for a utility distribution are developed from the har-
monic mean distance from each animal location to each point on a superim-
posed grid. The harmonic mean estimator may accurately show multiple cen-
ters of activity, but each estimated utility distribution is unique to the position
and spacing of the underlying grid. Spencer and Barrett (1984) modified the
method to reduce the problem of grid placement but a large problem with grid
size remains. When a very fine grid is used, the resulting utility distribution
becomes a series of sharp peaks at each animal location. When a coarse grid is
used, the utility distribution lacks local detail and is overly smoothed. For
many data sets, the harmonic mean estimator actually appears both to exag-
gerate peaks at animal locations and to oversmooth elsewhere. In addition, the
estimator calculates values for all grid points, provides no outline for a home
range, and does not provide a utility distribution. Most researchers choose for
the home range outline the contour equal to the largest harmonic mean dis-
tance from an animal location to all other animal locations (Ackerman et al.
1988) and from this a utility distribution can be calculated. Although this is an
objective criterion, it is affected by sample size. Finally, for animal home ranges
that have geographic constraints that confine shapes (e.g., lakes, mountains;
Powell and Mitchell 1998; Reid and Weatherhead 1988; Stahlecker and Smith
1993), much area not actually in an animal's home range will be included in
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