Biology 576: Landscape Ecology & Macroscopic Dynamics

Fractal Geometry in Landscape Ecology

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.

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
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.

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 *q*th moment of the
*P(m,L)* distribution.

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 *q*th roots of the moments,
averaged over all values of *L*, scale as a power of
*L*.

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.

more to come....

Other good introductions to fractal geometry and scale include David Green's tutorial from the Australian National University and a Los Alamos National Lab page on fractals and chaos.

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:

- aesthetic explorations
- mathematical sites
- applications and software

- The Fractal Microscope allows you to zoom in and out on the Mandelbrot set and explore its self-similarity.
- This French fractal art gallery has a wide variety of beautiful images that reward the patient browser, while
- The Fractals calender gives the afficianado three years of monthly fractal pinups.

- Math 480 is an introductory course in fractal geometry at the University of Pennsylvania.
- The fractal geometry of the Mandelbrot set is explored at Boston University. The information is presented in two parts: I. The periods of the bulbs and II. How to count and how to add.
- The Fractal Microscope people also offer a good mathematical intro to fractal geometry.

- Fractint is a freeware fractal generator created for IBMPC's and compatible computers.

The following are general mathematical as well as ecological references for understanding fractal geometry and its application in landscape ecology.

- Feder, J. 1988. Fractals. Plenum Press, New York.
- Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. W.H. Freeman and Co. New York.
- Milne, B.T. 1991. lessons from applying fractal models to
landscape patterns. pp. 199-235
*In*Turner, M.G. and R.H. Gardner, (eds.) Quantitative Methods in Landscape Ecology. Springer-Verlag, New York. - Milne, B.T. 1992. Spatial Aggregation and neutral models in
fractal landscapes.
*American Naturalist*139:32-57. - Scheuring, I. and R.H. Riedi. 1994. Application of multifractals
to the analysis of vegetation pattern.
*Journal of Vegetation Science*5;489-495. - Sole', R.V. and S.C. Manrubia. 1995. Are rainforests
self-organized in a critical state?
*Journal of Theoretical Biology*173:31-40. - Voss, R.F. 1988. Fractals in nature: from characterization to
simulation. pp. 21-70
*In*Peitgen, H.O. and D. Saupe (eds.) the Science of Fractal Images. Springer-Verlag, New York.

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Andrew J. Kerkhoff, Department of Biology, University of New Mexico, Albuquerque, NM 87131 kerkhoff@algodones.unm.edu

Faculty sponsor:

Dr. Bruce T. Milne,Department of Biology, University of New Mexico, Albuquerque, NM 87131