Particular parts of an animal's home range must be more important than other parts. In general, foods and other resources are patchily distributed (Curio 1976; Frafjord and Prestrud 1992; Goss-Custard 1977; Mitchell 1997; Powell et al. 1997), so the parts of a home range with greater density of critical resources ought, logically, to be more important than areas with few resources. For years, biologists have conceived the core as the part of an animal's home range that is most important to it (Burt 1943; Ewer 1968; Kaufmann 1962; Samuel et al. 1985; Samuel and Green 1988). To understand home ranges well, identifying cores is important, if cores do indeed exist.

My understanding of a home range core has two major parts (Powell et al. 1997; Seaman and Powell 1990). First, a core must be used more heavily than the apparent clumps of heavy use that occur from uniform random use of space within a home range. Random use of space leads to some areas being used more than others, even though no place is more important to the animal than any other. Therefore, random use of space leads to apparent clumps of use in some places and little use of other places. Consequently, the core of a home range must be used more than expected by random use, which means that for a home range to have a core, use of space within that home range must be statistically clumped and not random or even. Testing for clumped versus random (or for clumped versus even) use of space is usually straightforward (Horner and Powell 1990; Mitchell 1997). In a uniform random distribution, the mean equals the variance. If an animal uses space at random, then the mean number of locations in each cell in its home range equals the variance. If the variance is significantly greater than the mean, then use is clumped; if the variance is less than the mean, then use is even. Note that many nonrandom distributions have means equal to their variances. Therefore, equal mean and variance does not prove uniform random use of space, but unequal mean and variance does disprove random use of space.

Second, a core must not be strictly determined by home range area. Animals with home ranges of equal size but with different patterns of home range use (e.g., central place foragers, strongly territorial animals, extensive wanderers) should have differently sized cores. Any technique developed to identify the core of a home range must reflect this biological understanding of what a core is.

Most definitions of cores have been ad hoc or subjective. Many define the core as the smallest area with an arbitrary probability of use (e.g., the smallest area enclosing 25 percent of total use). A crucial problem with such defini tions, beyond their subjectivity, is that the cores so defined are not strictly tied to intensity of use of space. In fact, animals that use their home ranges randomly or in an even fashion have cores by these definitions. Samuel et al. (1985) developed an objective method for identifying the maximum possible core: all parts of the home range used more heavily than would result from evenly distributed use. Although this definition is objective, it is still arbitrary, it allows cores to incorporate space used little more than adjacent space, and it defines cores for animals that use space randomly. Seaman and I (1990; Powell et al. 1997) introduced a technique for identifying cores that is objective, not arbitrary, and that allows the animals themselves to define their cores. Our technique is based on the logic used to identify behavior bouts (Fagen and Young 1978; Slater and Lester 1982). Bingham and Noon (1997) used the same method and Harris et al. (1990) may have, but their explanation is not clear.

If an animal's use of space is random within its home range, a plot of home range area at a certain percentage use versus probability of use should yield a straight line going from the 100 percent home range at probability of use equal to 0 (100 percent of the home range has probability of use 0 or above) to the 0 percent home range at probability of use greater than the maximum probability of use (0 percent of the home range has probability of use greater than the greatest probability of use; figure 3.6a). If probability of use is transformed to percentage of largest probability of use, then both x andy axes on the graph range from 0 to 100 (figure 3.6a) and the descending line representing random use has a slope of —1. If use of space by an animal is clumped, however, its curve will sag below the line of random use (figure 3.6b) and if use of space is even (all areas used with equal intensity), the graph will remain as a high plateau from x equals 0 probability of use up to x equal some large probability of use and then plummet (figure 3.6c).

When use of space by animals is clumped (figure 3.6b), Seaman and I defined as an animal's core the areas in its home range that are used most intensively. The parts of the home range mapped onto the steeply descending slope of the area—probability curve along the y-axis are used least and constitute the periphery of the home range. The parts of the home range mapped onto the nearly horizontal slope along the x-axis are used most intensively and constitute the core. The curve can be divided into two pieces at the point whose tangent has a slope of—1, that is, whose tangent is parallel to the line for random use. This is also the point that is furthest from the line with a slope of—1 (figure 3.6b). Plots of actual data may not yield smooth curves; these plots can be fit with reasonable curves.

Figure 3.6 For an animal's home range, possible relationships between probability of use and percentage of home range with the probability of use or greater. The x-axis is probability of use for areas within an animal's home range calculated as the percentage of maximum probability of use. The y-axis is the percentage of the home range with the given probability of use of higher use. (A) Relationship for random use by an animal of the area within its home range, (B) clumped or patchy use of the home range, (C) even or overdispersed use of the home range.

Figure 3.6 For an animal's home range, possible relationships between probability of use and percentage of home range with the probability of use or greater. The x-axis is probability of use for areas within an animal's home range calculated as the percentage of maximum probability of use. The y-axis is the percentage of the home range with the given probability of use of higher use. (A) Relationship for random use by an animal of the area within its home range, (B) clumped or patchy use of the home range, (C) even or overdispersed use of the home range.

This criterion for a home range core clearly identifies the most intensively used areas within an animal's home range, and it allows the data (i.e., the animal) to decide where the boundary between core and periphery should be located. The criterion is objective and, for me, intuitive (figure 3.7; Powell et al. 1997; Seaman and Powell 1990). By this objective, each animal's core, if it has one, is at a different probability of use. In addition, some animals have large cores and some have small cores.

■ Quantifying Home Range Overlap and Territoriality

Home ranges of conspecifics often overlap, sometimes extensively. For a population, hypotheses can be tested regarding simultaneous use of areas of home range overlap. Subsets of a population may exhibit different patterns of simultaneous use. Relatives, for example, may use their areas of overlap more than expected from random use, whereas nonrelatives may avoid each other and use areas of overlap less than expected. Intrasexually territorial animals may spend less time in their areas of overlap with members of the opposite sex than expected.

For some species, territorial behavior has been documented objectively. Extensive experimentation with limiting resources and territorial displays and calls has defined territoriality clearly in many birds (e.g., nectarivorous birds, reviewed by Hixon 1980; Hixon et al. 1983; blackbirds, reviewed by Orians 1980). For many species, however, apparent lack of overlap of home ranges is the only evidence of territorial behavior. For many carnivorous mammals, territory defense and responses to scent marks are difficult to document. For such species, home range overlap can be quantified in an objective manner that weighs probability of use of different parts of a home range or territory. Home range overlap can then be compared statistically among populations or species that appear to differ in territorial behavior but for whom territorial behavior has not been manipulated experimentally.

Doncaster (1990) defined two types of overlap, called static and dynamic interactions. Static interaction is the spatial overlap of two home ranges and dynamic interaction involves interdependent movements of the two animals whose home ranges overlap. These types of overlap can be quantified in several different ways.

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