Essay/Term paper: The application of fractal geometry to ecology

Essay, term paper, research paper:  Science

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 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 all know that "clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor does lightning travel in
a straight line" (Mandelbrot 1982).

The advent of the computer age, with its sophisticated graphics, played an
important role in the development and acceptance of fractal geometry as a valid
new discipline in the last two decades. Computer-generated images clearly show
the relevance of fractal geometry to nature (Scheuring and Riedi 1994). A
computer- generated coastline or mountain range demonstrates this relevance.
Once mathematicians and scientists were able to see graphical representations of
fractal objects, they could see that the mathematical theory behind them was not
freakish but actually describes natural objects fairly well. When explained and
illustrated to most scientists and non-scientists alike, fractal geometry and
fractals make sense on an intuitive level.

Examples of fractal geometry in nature are coastlines, clouds, plant roots,
snowflakes, lightning, and mountain ranges. Fractal geometry has been used by
many sciences in the last two decades; physics, chemistry, meteorology, geology,
mathematics, medicine, and biology are just a few.

Understanding how landscape ecology influences population ecology has allowed
population ecologists to gain new insights into their field. A dominant theme of
landscape ecology is that the configuration of spatial mosaics influences a wide
array of ecological phenomena (Turner 1989). Fractal geometry can be used to
explain connections between populations and the landscape structure.
Interpreting spatial and temporal scales and movement pathways are two areas of
population ecology that have benefited from the application of fractal geometry.

Different tools are required in population ecology because the resolution or
scale with which field data should be gathered is attuned to the study organism
(Wiens et al. 1993). Insect movements, like plant root growth, follow a
continuous path that may be punctuated by stops but the tools required to
measure this continuous pathway are very different. Plant movement is measured
by observing root growth through photographs, insect movement by tracking
insects with flag placement, and animal movement by using tracking devices on
larger animals (Gautestad and Mysterud 1993, Shibusawa 1994, Wiens et al. 1993).

Spatial and temporal scale are important when measuring the home range of a
population and when tracking animal movement (Gautestad and Mysterud 1993, Wiens
et al. 1993). Animal paths have local, temporal, and scale-specific fluctuations
in tortuosity (Gautestad and Mysterud 1993) that are best described by fractal
geometry. The mapping of insect movement also required use of the proper spatial
or temporal scale. If too long of a time interval is used to map the insect's
progress, the segments will be too long and the intricacies of the insect's
movements will be lost. The use of very short intervals may create artificial
breaks in behavioral moves and might increase the sampling effort required until
it is unmanageable (Wiens et al. 1993).

Movement pathways are one of the main characteristics influenced by the
landscape. Movement pathways are influenced by the vegetation patches and patch
boundaries (Wiens et al. 1993). Root deflection in a growing plant is similar to
an animal pathway being changed by the landscape structure. Paths of animal
movement have fractal aspects.

In a continuously varying landscape, it is difficult to define the area of a
specie's habitat (Palmer 1992). Application of fractal geometry has given new
insights into animal movement pathways. For example, animal movement determines
the home range. Because animal movement is greatly influenced by the fractal
aspect of the landscape, home range is directly influenced by the landscape
structure (Gautestad and Mysterud 1993). Animal movement is not random but
greatly influenced by the landscape of the home range of the animal (Gautestad
and Mysterud 1993). Structural complexity of the environment results in tortuous
animal pathways (Gautestad and Mysterud 1993), which in turn lead to ragged home
range boundaries.

Gautestad and Mysterud (1993) found that home range can be more accurately
described by its fractal properties than by the traditional area-related
approximations. Since demarcation of home range is a difficult task and home
range can't be described in traditional units like square meters or square
kilometers, they used fractal properties to better describe the home range area
as a complex area utilization pattern (Gautestad and Mysterud 1993). Fractals
work well to describe home range because as the sample of location observation
increases, the overall pattern of the position plots takes the form of a
statistical fractal (Gautestad and Mysterud 1993).

Fractal dimensions are used to represent the pathways of beetle movement because
the fractal dimension of insect movement pathways may provide insights not
available from absolute measures of pathway configurations (Wiens et al. 1993).
Using fractal dimensions allowed ecologists to map the pathway without creating
an unmanageable deluge of information (Wiens et al. 1993).

Insect behavior such as foraging, mating, population distribution, predator-
prey interactions or community composition may be mechanisticly determined by
the nature of the landscape. The spatial heterogeneity in environmental features
or patchiness of a landscape will determine how organisms can move around (Wiens
et al. 1993). As a beetle or an other insect walks along the ground, it does not
travel in a straight line. The beetle might walk along in a particular direction
looking for something to eat. It might continue in one direction until it comes
across a bush or shrub. It might go around the bush, or it might turn around and
head back the way it came. Its path seems to be random but is really dictated by
the structure of the landscape (Wiens et al. 1993).

Another improvement in population ecology through the use of fractal geometry is
the modeling of plant root growth. Roots, which also may look random, do not
grow randomly. Reproducing the fractal patterns of root systems has greatly
improved root growth models (Shibusawa 1994).

Landscape ecologists have used fractal geometry extensively to gain new insights
into their field. Landscape ecology explores the effects of the configuration of
different kinds of environments on the distribution and movement of organisms
(Palmer 1992). Emphasis is on the flow or movement of organism, genes, energy,
and resources within complex arrangements of ecosystems (Milne 1988). Landscapes
exhibit non-Euclidean density and perimeter-to-area relationships and are thus
appropriately described by fractals (Milne 1988). New insights on scale,
increased understanding of landscape structures, and better landscape structure
modeling are just some of the gains from applying fractal geometry.

Difficulties in describing and modeling spatially distributed ecosystems and
landscapes include the natural spatial variability of ecologically important
parameters such as biomass, productivity, soil and hydrological characteristics.
Natural variability is not constant and depends heavily on spatial scale.
Spatial heterogeneity of a system at any scale will prevent the use of simple
point models (Vedyushkin 1993).

Most landscapes exhibit patterns intermediate between complete spatial
independence and complete spatial dependence. Until the arrival of fractal
geometry it was difficult to model this intermediate level of spatial dependence
(Palmer 1992, Milne 1988).

Landscapes present organisms with heterogeneity occurring at a myriad of length
scales. Understanding and predicting the consequences of heterogeneity may be
enhanced when scale-dependent heterogeneity is quantified using fractal geometry
(Milne 1988). Landscape ecologists usually assume that environmental
heterogeneity can be described by the shape, number, and distribution on
homogeneous landscape elements or patches. Heterogeneity can vary as a function
of spatial scale in landscapes. An example of this is a checker board. At a very
small scale, a checker board is homogeneous because one would stay in one square.
At a slightly larger scale, the checker board would appear to be heterogeneous
since one would cross the boundaries of the red and black squares. At an even
larger scale, one would return to homogeneity because of the pattern of red and
black squares (Palmer 1992).

An increased understanding of the landscape structures results from using the
fractal approach in the field of remote sensing of forest vegetation. Specific
advantages include the ability to extract information about spatial structure
from remotely sensed data and to use it in discrimination of these data; the
compression of this information to few values; the ability to interpret fractal
dimension values in terms of factors, which determine concrete spatial
structure; and sufficient robustness of fractal characteristics (Vedyushkin
1993).

Computer simulations of landscapes provide useful models for gaining new
insights into the coexistence of species. Simulated landscapes allow ecologists
to explore some of the consequences of the geometrical configuration of
environmental variability for species coexistence and richness (Palmer 1992). A
statistically self-similar landscape is an abstraction but it allows an
ecologist to model variation in spatial dependence (Palmer 1992). Spatial
variability in the environment is an important determinant of coexistence of
competitors (Palmer 1992). Spatial variability can be modeled by varying the
landscape's fractal dimension.

The results of this computer simulation of species in a landscape show that an
increase in the fractal dimension increases the number of species per microsite
and increases species habitat breadth. Other results show that environmental
variability allows the coexistence of species, decreases beta diversity, and
increases landscape undersaturation (Palmer 1992). Increasing the fractal
dimension of the landscape allows more species to exist in a particular area and
in the landscape as a whole; however, extremely high fractal dimensions cause
fewer species to coexist on the landscape scale (Palmer 1992).

Although many ecologists have found fractal geometry to be an extremely useful
tool, not all concur. Even scientists who have used fractal geometry in their
research point out some of its shortcomings. For example, Scheuring and Riedi
(1994) state that "the weakness of fractal and multifractal methods in
ecological studies is the fact that real objects or their abstract projections
(e.g., vegetation maps) contain many different kinds of points, while fractal
theory assumes that the natural (or abstract) objects are represented by points
of the same kind."

Many scientists agree with Mandelbrot when he said that fractal geometry is the
geometry of nature (Voss 1988), while other scientists think fractal geometry
has no place outside a computer simulation (Shenker 1994). In 1987, Simberloff
et al. argued that fractal geometry is useless for ecology because ecological
patterns are not fractals. In a paper called "Fractal Geometry Is Not the
Geometry of Nature," Shenker says that Mandelbrot's theory of fractal geometry
is invalid in the spatial realm because natural objects are not self-similar
(1994). Further, Shenker states that Mandelbrot's theory is based on wishing and
has no scientific basis at all. He conceded however that fractal geometry may
work in the temporal region (Shenker 1994). The criticism that fractal geometry
is only applicable to exactly self-similar objects is addressed by Palmer (1982).
Palmer (1982) points out that Mandelbrot's early definition (Mandelbrot 1977)
does not mention self-similarity and therefore allows objects that exhibit any
sort of variation or irregularity on all spatial scales of interest to be
considered fractals.

According to Shenker, fractals are endless geometric processes, and not
geometrical forms (1994), and are therefore useless in describing natural
objects. This view is akin to saying that we can't use Newtonian physics to
model the path of a projectile because the projectile's exact mass and velocity
are impossible to know at the same time. Mass and velocity, like fractals, are
abstractions that allow us to understand and manipulate the natural and physical
world. Even though they are "just" abstractions, they work quite well.

The value of critics such as Shenker and Simberloff is that they force
scientists to clearly understand their ideas and assumptions about fractal
geometry, but the critics go too far in demanding precision in an imprecise
world.

With all the new insights and new knowledge that have been gained through the
appropriate application of fractal geometry to natural sciences, it is clear
that is a useful and valid tool.

The 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.

One of the most valuable aspects of fractal geometry, however, is the way that
it bridges the gap between ecologists of differing fields. By providing a common
language, fractal geometry allows ecologists to communicate and share ideas and
concepts.

As the information and computer age progress, with better and faster computers,
fractal geometry will become an even more important tool for ecologists and
biologists. Some future applications of fractal geometry to ecology include
climate modeling, weather prediction, land management, and the creation of
artificial habitats.


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