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Landscape Ecology vol. 9 n is number of points, and r is the search radius. C will indeed be dubious if data are recorded so fre- quently that the animal is still + standing and chewing at both t and t At. For C to be useful, At must be sufficiently large so that successive circles do not overlap when the animal is foraging, though clearly no such condition is relevant if the predator uses a perch. The objection of Gautestad and Mysterud that ‘C is a dependent variable even when the loca- tions are non-autocorrelated. The reason is that, for large n, making C n-dependent.’ In reply, I note first that the degree of overlap with previous circles is precisely what C is meant to measure: how much the same area is used (searched) repeatedly. It is true that C will decrease as non-cor- related, and is intended to vary with r, which reflects the search radius and is a biologically meaningful difference between species. Gautestad and Mysterud proposed an alternative ‘graininess’ parameter C as (Gau- 148

where A is the area of the average minimum convex polygon and n is reasonably large (n > 50- 100). For their C, I note (1) that it assumes that all of the area within the minimum convex polygon is used (whereas a lake may in fact be in the middle of a bear’s range), (2) that it neglects the search area, and (3) that A will tend to stabilize at large n, lead- ing to a decrease in C with increasing n. Further- more, the minimum convex polygon area is very sensitive to excursions (occasional long trips away measures generally referred to as the dimension of natural measure). Here we use the information dimension from the central territory) and is meaningless for non-territorial or migrating species. A young eagle may wander over several states, but is this its terri- tory? Both my index C and my fractal characteriza- tion work equally well for all of these cases. Gautestad and Mysterud next pointed out that my method of calculating the fractal dimension has problems. This is true, but the problem has nothing to do with the introduction of the disks. The disks represent the area searched, and we do in fact wish to quantify this pattern. The disks only ‘fill Mys- terud are correct that my method of calculation in 1990 was in error. The proper calculation should be based on a frequency approach, as follows. we are interested in computing the fractal dimension of the map of pseudoelevations in order to quantify the roughness. The difficulty here is that ordinary fractals require that the x, y, and z dimensions all be in the same metric al. (1983) provided a solution to this problem. They pointed out that dimension has several definitions, which fall into two general classes: (1) those that depend only on metric properties (capacity) and (2) those that de- pend on the frequency with which a function visits different regions of the space (including several Pi=zi/ i= 1 zi Pi measure the relative number of disks over- lapping a pixel, and M c Zi values within a box Pi as a probabil- ity at each scale, the P value within a box E must be recalculated as M i=c 1 K(E) is the total number of boxes of size E. For example, for 2 x 2 boxes we would sum the four Wein (1994).

Given these probabilities as a function of loca- tion and scale, we may define the information dimension Pi i= 1 (1/E)

d, is well defined for a home range, because it is not necessarily d, is not constant across scales). We therefore compute the information dimension dis- cretely at a series of scales by using K(E) MY) dA&)= i=l Pi log Pj log [1/&] - log (7) d, as the slope of the line, at each scale, given by the information measure versus log Wein (1994). We can see that the information dimension Pi log O’Neill et al. (1988). The index (df). For a flat map where all pixels are assigned to a type (value = 1) or not (value = 0), giving a black and white image, gener- ally df Thus, a uniform map with a distinct linear feature such as a river will have a df dI df because in fact n because it is probability (frequency) based. It is also insensi- tive to rare outlier points because they are not weighted heavily, whereas even a single long excur- sion by an animal will result in a huge increase in the 149 minimum convex polygon size. The concern that the fractal dimension is not well defined at very small box sizes, discussed by Gautestad and was prepared in connection with work performed under contract W-3 1-109-ENG-38 with the U.S. Department of Energy, Office of Energy Research, Office of Health and Environmental Research. References Farmer, J.D., Ott, E. and Yorke, J.A. 1983. The dimension of chaotic attractors. Gautestad, A.O. and Mysterud, I. 1994. Are home ranges frac- Loehle, C. 1990. Home range: A fractal approach. Landscape Ecology Loehle, C. and Wein, G. 1994. Landscape habitat diversity: A Krummel, J.R., Gardner, R.H., Sugihara, G., Jackson, B., 1: 153-162. Sommerer, J.C. and Ott, E. 1993. Particles floating on a moving fluid: A dynamically comprehensible physical fractal. Science 259: 335-339.