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

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