## Fractal Estimators

Bascompte and Vila (1997), Gautestad and Mysterud (1993, 1995), and Loehle (1990) modeled animal movements as multiscale random walks and analyzed the patterns of locations as fractals. Bascompte and Vila (1997) explained that D, the fractal dimension, can be estimated as

where n is the number of steps along a trace of an animal's movements (1 less than the number of locations), L is the sum of the lengths of all steps (total length of the movement), and d is the planar diameter, which can be estimated as the greatest distance between two locations. For a movement that is a straight line, d = L, so D = 1; a line has one dimension. For a random walk, D = 2; a random walk spreads over a plane and has two dimensions.

For the animals studied by Bascompte and Vila (1997) and Gautestad and Mysterud (1993, 1995), the fractal dimensions, D, for movements averaged less than 2. Finding D < 2 means that as they scrutinized their animal location data on smaller and smaller scales, they found clumps of locations within clumps within clumps ad infinitum. The movements of the animals did not spread randomly across the landscape. Gautestad and Mysterud (1993, 1995) argued, therefore, that animals use their home ranges in a multiscale manner, which makes ultimate sense. Optimality modeling (giving up time) and empirical data show that animals who forage in patchy environments are predicted to and, indeed, do change their movements dependent on both fine-scale and large-scale characteristics of food availability (Curio 1976; Krebs and Kacelnik 1991). Thus an animal's decision to remain in or to leave a food patch depends not just on the availability of food within the patch but also on the availability of food across its home range and on the locations of the other patches of food.

In addition, Gautestad and Mysterud (1993, 1995) showed that if animals move in a manner described by a multiscale random walk that incorporates the multiscale, fractal nature of animal movements, then the estimated home range area should increase infinitely in proportion with the square root of the number of location estimates used to estimate the area of the home range using a minimum convex polygon. Indeed, the home ranges of several species, quantified using minimum convex polygons, do appear to increase in area as predicted (Gautestad and Mysterud 1993, 1995; Gautestad et al. 1998). The predicted relationship between home range area (Amcp, for minimum convex polygons) and the number of locations (n) is

where C is the constant of proportionality, or the scaling factor, and Q(n) is a function that adjusts the relationship for underestimates of AMCP because of small sample size. Curve fitting indicates that

for n > 5. When not calculating home range area from minimum convex polygons, Q(n) should not be used.

Gautestad and Mysterud (1993) interpret C to be a measure of how an animal perceives the grain of its environment. When a grid is superimposed over a plot of an animal's locations, C can be calculated for each cell and 1/C is a descriptor of the intensity of use for each cell (Gautestad 1998).

1/ C can be calculated in two ways. Superimpose a grid on a map of a study area such that no cells have fewer than five locations for a target animal (cells with fewer than five locations might alternatively be ignored). Calculate the area of the minimum convex polygon formed by all locations within each cell and use that for Amcp in equation 3.1. Calculate 1/C as 