## Utility Distributions

From location data such as those shown in figure 3.2, most home range estimators produce a utility distribution describing the intensity of use of different areas by an animal. The utility distribution is a concept borrowed from economics. A function, the utility function, assigns a value (the utility, which can be some measure of importance) to each possible outcome (the outcome of a decision, such as the inclusion of a place within an animal's home range; Ellner and Real 1989). If the utility distribution maps intensity of use, then it can be transformed to a probability density function that describes the probability of an animal being in any part of its home range (Calhoun and Casby 1958; Hayne 1949; Jennrich and Turner 1969; White and Garrott 1990; van Winkle 1975), as shown in figure 3.2. Utility distributions need not be probability density functions, although they usually are. A utility distribution could map the fitness an animal gains from each place in its home range, or it could map something else of importance to a researcher.

The approach using a utility distribution as a probability density function provides one objective way to define an animal's normal activities. A probability level criterion can be used to eliminate Burt's (1943) occasional sallies. Including in an animal's home range the area in which it is estimated to have a 100 percent probability of having spent time would include occasional sallies. Including only, say, the smallest area in which the animal spent 95 percent of its time could exclude occasional sallies or areas the animal will never visit again. Using a utility distribution, one can arbitrarily but operationally define the home range as the smallest area that accounts for a specified proportion of the total use. Most biologists use 0.95 (i.e., 95 percent) as their arbitrary but

Figure 3.2 Location estimates (circles) and contours for the probability density function for adult female black bear 87 studied in 1985. The lightly dotted black line marks the study area border.

repeatable probability level; the smallest area with a probability of use equal to 0.95 is defined as an animal's home range. No strong biological logic supports the choice of 0.95 except that one assumes that exploratory behavior would be excluded by using this probability level; to my knowledge, this assumption has never been tested. An alternative approach is to exclude from consideration the 5 percent of the locations for an animal that lie furthest from all others. Eliminating these locations might also eliminate occasional sallies. A strong statistical argument exists for excluding some small percentage of the location data, the utility distribution, or both; extremes are not reliable and tend not to be repeatable. However, this argument does not specify that precisely 5 percent should be excluded. Using 95 percent home ranges may be widely accepted because it appears consistent with the use of 0.05 as the (also) arbitrary choice for the limiting p-value for judging statistical significance.

Once home range has been defined as a utility distribution, a reliable method must be sought to estimate the distribution. Estimating utility distributions has been problematic because the distributions are two- or three-dimensional, observed utility distributions rarely conform to parametric mod els, and data used to estimate a distribution are sequential locations of an individual animal and may not be independent observations of the true distribution (Gautestad and Mysterud 1993, 1995; Gautestad et al. 1998; Seaman and Powell 1996; Swihart and Slade 1985a, 1985b). However, lack of independence of data may not be a great problem for some analyses (Andersen and Rongstad 1989; Gese et al. 1990; Lair 1987; Powell 1987; Reynolds and Laun-dre 1990). After all, data that are not statistically autocorrelated are nonetheless biologically autocorrelated because animals use knowledge of their home ranges to determine future movements. Boulanger and White (1990), Harris et al. (1990), Powell et al. (1997), Seaman and Powell (1996), and White and Garrott (1990) reviewed many home range estimators and Larkin and Halkin (1994) summarized computer software packages for home range estimators.