Most of us are familiar with fractals from computer graphics and special effects. They are infinitely complex, sometimes beautiful shapes that can be used to immitate the complex textures we see in the real world. Fractal geometry. (Mandelbrot 1982) has infiltrated the worlds of design, music , literature, and the arts, influencing the way people percieve and think about the world. Because landscape ecology is concerned with the perceived spatial heterogeneity of the natural world, fractal geometry is primarily useful in this field as an analytical method akin to other statistical and geostatistical tools. Bruce Milne's Applications of Fractal Geometry to Wildlife Biology provides a more technical introduction and set of examples, but as a start I will scratch the surface here.
Fractals can be thought of as objects or patterns that have non-integer dimensions. In standard Euclidean geometry, we learn that a point has a dimension (D) of 0, a line has D=1, a plane has D=2, and so on. But what about a set of disconnected lines, like a string of morse code, or a jagged, convoluted line tracing a rough coastline, or a set of irregular polygons like a landscape mosaic? What is the geometric dimension of these objects? A string of morse code is not just a series of 0D points but neither does it make continuous line. Likewise the coastline and landscape mosaic defy the limitations of one dimensional characterization but they do not completely fill the higher dimension space in which they are imbedded. Thus the string of morse code can be thought of as having a dimension between 0 and 1, say 0.4, and the coastline and landscape have dimensions between 1 and 2. If in our landscape we include some complex topography, it would exhibit 2<D<3, characterizing how the landscape begins to fill the 3D space it resides in.
This is not a formal definition of fractals, but it does give an intuitive feeling for a concept that many find somewhat daunting. It also begs the question, "So what?" True fractals are abstractions, but so are circles, ellipses, and triangles. However, fractals are very rich abstractions that bring to light two essential qualities that must be addressed in any attempt to understand complex spatial patterns like those found in nature: self-similarity and scale-dependence.
Any experienced mountaineer will tell you that if a mountain appears jagged from far away, the climbers will likely find many footholds, while if it is smooth from a distance the mountain will be a more difficult climb. He understands the self-similarity of natural patterns: qualities of the pattern at coarse scales are repeated at finer and finer scales. Large pinnacles and outcroppings translate into small irregularities on which the climber can find purchase. So the intuitive concepts that form the base of fractal geometry are not new. It is the formalism involved in modelling these concepts mathematically that is revolutionary.
A correlate of the inherent self-similarity of fractals is the idea of scale-dependence. The classic illustrative concept here is a coastline. Say you wish to measure the length of the coast of England. The coast is fractal, with the large penninsulas and inlets depicted on the map encompassing myriad smaller and undepicted features. That the coastline is scale-dependent means that its length will depend on the length of the ruler you use to measure it. Looking at the map of England, we can take a caliper and set its opening to 10 km on the map scale. We then "step" around the coast, and the number of steps multiplied by 10, the caliper length, gives us the coast length. Then repeat the process at successively larger openings. As the scale of measurement gets larger, features of the coastline will effectively be eliminated as the caliper merely steps over them. Over several orders of magnitude in length scale (here meaning the caliper opening), the measured length will decrease exponentially. The relationship can be formally represented as
where is the length of the coastline measured at scale L and the D term is the fractal dimension or scaling exponent estimated from the slope of the number of steps a caliper of length L makes along the coast, C(L), graphed against L on logarithmic axes (Voss 1988, Milne in review).
The same sorts of measurements can be used for describing the scaling behavior of higher dimensional fractal objects or patterns by using windows or cubes of varying lengths instead of just one dimensional rulers. The scaling behavior that is described by the exponents is of interest to ecologists because the patterns we see in nature are the outcome of endless biotic and abiotic phenomena interacting across a wide range of spatial and temporal scales. Fractal geometry provides a way to discern the often obscured regularities present in complex natural patterns. Thus far, fractal geometry has been used primarily as a method for developing indices of landscape pattern that are often used in regression models. Below are some more elaborate fractal methods that provide predictive power for landscape level analyses, rather than descriptive indices.
where D is the mass dimension of the forest and k is a constant. The slope of a doubly logarithmic regression of M(L) against L is a good estimate of this value. Because large windows will be limited by the extent of the map, only the data sampled by all window sizes can be included in this and all subsequent analyses.
From the geometric measures described above, we can define a probability density function, P(m,L), describing the probability of finding m pixels in a window of length L. This probability function is analogous to a standard statistical distribution, and from it we can calculate moments analogous to the mean, variance, and skewness of a standard distribution. This is the qth moment of the P(m,L) distribution.
Values of q can be either positive or negative, which has powerful analytical implications. Values of q >> 1 will emphasize the contribution of densely filled windows, because m is exponentially increased before being multiplied by the probability density function. Conversely, values of q << 1 weight the contribution of sparsely filled windows. In this way the pattern under analysis can be partitioned based on relative density or aggregation, and the scaling behavior of these partitions can be quantified and compared (Voss 1988, Milne 1991, Milne 1992). The variation in scaling behavior between the partitions is what characterizes a multifractal (Feder 1988, Milne 1995 in review).
The first moment is the mass dimension from above. This same relationship can be generalized to give us a spectrum of scaling exponents, Dq. The qth roots of the moments, averaged over all values of L, scale as a power of L.
These scaling exponents quantify the statistical behavior of each density partition of the map. The validity of treating natural distributions as multifractals can be checked by graphing the qth roots of the moments against L on double logarithmic axes to make sure that the moments scale as a power of L over several orders of magnitude (Scheuring and Riedi 1994). The Dq spectrum of can be graphed against q to show how the scaling behavior of the pattern varies across its density partitions. Sole' and Manrubia (1995) were able to detect changes in the probability of mortality in a cellular automata forest model through changes in the Dq spectrum of the resulting pattern of forest canopy gaps. This implies that ecological processes will manifest themselves in the scaling behavior of ecological pattern.
Scheuring and Riedi (1994) demonstrated a powerful method of applying
multifractals in ecology by extending the analysis to encompass
patterns made up of two or more different types of points. The general
nature of the analysis is as follows. We count the number of pixels of
type j=1 that occur in a window of length L which is
also occupied by at least one pixel of types j=2,3,...,S,
where S is the number of types in the study. The set of
points of each type, lj, corresponds to the map of that
type. Set is referred to as the
focal set while are the
constraint sets. We next find the number of windows centered on type
j=1 pixels and also containing pixels of types j=2
... S. For each window location i, the counting
procedure gives us
the number of pixels of type j=1 conditional on the presence of all the other types. The conditional multifractal moments are then derived, as above, from the P(m,L) function where
(Milne in review).
It is then useful to compare the Dq spectrum of the conditional data with those of the two (or more) original data patterns. Values of q where the spectra converge would hypothetically indicate some form of ecological constraint, one pattern taking on the scaling behavior of another. Because the geometric measures (m's) have geographic coordinates, the areas represented by specific density partitions can be remapped at a desired length scale. By partitioning the data by density at multiple scales, we can avoid the assumption of spatial independence and test whether an ecological pattern is constrained by common factors at all scales and in all parts of its distribution.
I am currently applying this type of analysis to the movements of the Florida panther to understand the extent to which the distribution of panthers conforms to the distribution of preferred habitat, and to find ecologically relevant scales for evaluating critical habitat and potential reintroduction sites.
This is not meant to be an exhaustive index of fractal-related websites. With the constantly changing nature of the web, such an index would be out of date in a month anyway. Instead, I'm just providing a few introductory sites that are interesting and well-linked in each of three general categories:
The following are general mathematical as well as ecological references for understanding fractal geometry and its application in landscape ecology.