Term paper:Principles of Ecology 310L 

                      New Ecological Insights:
       The Application of Fractal Geometry to Ecology


                           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 how
small you can go before you lose the self- similarity. 

Another identifying characteristic of fractals is they usually have a non- integer
dimension. The fractal dimension of an object is a measure of space-filling ability and
allows one to compare and categorize fractals (Garcia 1991). A straight line, for example,
has the Euclidean dimension of 1; a plane has the dimension of 2. A very jagged line,
however, takes up more space than a straight line but less space then a solid plane, so it
has a dimension between 1 and 2. For example, 1.56 is a fractal dimension. Most fractal
dimensions in nature are about 0.2 to 0.3 greater than the Euclidean dimension (Voss 1988). 

Euclidean geometry and Newtonian physics have been deeply rooted traditions in the
scientific world for hundreds of years. Even though mathematicians as early as 1875 were
setting the foundations that Mandelbrot used in his work, early mathematicians resisted the
concepts of fractal geometry (Garcia 1991). If a concept did not fit within the boundaries
of the accepted theories, it was dismissed as an exception. Much of the early work in
fractal geometry by mathematicians met this fate. Even though early scientists could see the
irregularity of natural objects in the world around them, they resisted the concept of
fractals as a tool to describe the natural world. They tried to force the natural world to
fit the model presented by Euclidean geometry and Newtonian physics. Yet we


