The Application of Fractal Geometry to Ecology


Principles of Ecology 310L
Victoria Levin
7 December 1995

Abstract

New insights into the natural world are just a few of the results from the use
of fractal geometry. Examples from population and landscape ecology are used to
illustrate the usefulness of fractal geometry to the field of ecology. The
advent of the computer age played an important role in the development and
acceptance of fractal geometry as a valid new discipline. New insights gained
from the application of fractal geometry to ecology include: understanding the
importance of spatial and temporal scales; the relationship between landscape
structure and movement pathways; an increased understanding of landscape
structures; and the ability to more accurately model landscapes and ecosystems.
Using fractal dimensions allows ecologists to map animal pathways without
creating an unmanageable deluge of information. Computer simulations of
landscapes provide useful models for gaining new insights into the coexistence
of species. Although many ecologists have found fractal geometry to be an
extremely useful tool, not all concur. With all the new insights gained through
the appropriate application of fractal geometry to natural sciences, it is clear
that fractal geometry a useful and valid tool.

New insight into the natural world is just one of the results of the increasing
popularity and use of fractal geometry in the last decade. What are fractals and
what are they good for? Scientists in a variety of disciplines have been trying
to answer this question for the last two decades. Physicists, chemists,
mathematicians, biologists, computer scientists, and medical researchers are
just a few of the scientists that have found uses for fractals and fractal
geometry.

Ecologists have found fractal geometry to be an extremely useful tool for
describing ecological systems. Many population, community, ecosystem, and
landscape ecologists use fractal geometry as a tool to help define and explain
the systems in the world around us. As with any scientific field, there has been
some dissension in ecology about the appropriate level of study. For example,
some organism ecologists think that anything larger than a single organism
obscures the reality with too much detail. On the other hand, some ecosystem
ecologists believe that looking at anything less than an entire ecosystem will
not give meaningful results. In reality, both perspectives are correct.
Ecologists must take all levels of organization into account to get the most out
of a study. Fractal geometry is a tool that bridges the "gap" between different
fields of ecology and provides a common language.

Fractal geometry has provided new insight into many fields of ecology. Examples
from population and landscape ecology will be used to illustrate the usefulness
of fractal geometry to the field of ecology. Some population ecologists use
fractal geometry to correlate the landscape structure with movement pathways of
populations or organisms, which greatly influences population and community
ecology. Landscape ecologists tend to use fractal geometry to define, describe,
and model the scale-dependent heterogeneity of the landscape structure.

Before exploring applications of fractal geometry in ecology, we must first
define fractal geometry. The exact definition of a fractal is difficult to pin
down. Even the man who conceived of and developed fractals had a hard time
defining them (Voss 1988). Mandelbrot\'s first published definition of a fractal
was in 1977, when he wrote, "A fractal is a set for which the Hausdorff-
Besicovitch dimension strictly exceeds the topographical dimension" (Mandelbrot
1977). He later expressed regret for having defined the word at all (Mandelbrot
1982). Other attempts to capture the essence of a fractal include the following
quotes:

"Different people use the word fractal in different ways, but all agree that
fractal objects contain structures nested within one another like Chinese boxes
or Russian dolls." (Kadanoff 1986)

"A fractal is a shape made of parts similar to the whole in some way."
(Mandelbrot 1982)

Fractals are..."geometric forms whose irregular details recur at different
scales." (Horgan 1988)

Fractals are..."curves and surfaces that live in an unusual realm between the
first and second, or between the second and third dimensions." (Thomsen 1982)

One way to define the elusive fractal is to look at its characteristics. A
fundamental characteristic of fractals is that they are statistically self-
similar; it will look like itself at any scale. A statistically self-similar
scale does not have to look exactly like the original, but must look similar. An
example of self-similarity is a head of broccoli. Imagine holding a head of
broccoli. Now break off a large floret; it looks similar to the whole head. If
you continue breaking off smaller and smaller florets, you\'ll see that each
floret is similar to the larger ones and to the original. There is, however, a
limit to