Patterns within patterns: fractals and landscape ecology
It should be noted that due to copyright laws, no actual graph or image from any
book or journal appears on this webpage. All figures below are simplified
representations of the actual images.
What are fractals?
A central goal of landscape ecology is to detect patterns through space or time and
to extrapolate those patterns across multiple scales (Urban et al. 1987, Turner 1989,
Turner et al., 2001). Fractal geometry is a method used to aid in achieving these goals.
This approach is valuable to the field of landscape ecology not only because few tools are
currently available that explicitly address and measure scale, but it additionally allows
investigators to generalize a pattern across a range of scales. This detection of patterns
can then enable investigators to entertain questions concerning the processes creating the
observed patterns and to look for these processes at the appropriate scales.
Analytically, the fractal dimension (D) is derived from the slope of a double log
plot and describes shape complexity. It is a statistical descriptor like the mean and mode
and therefore does not necessarily explain the process in question. Rather, it provides a
way of describing a geometric shape. Any linkages made between D and an ecological
process results from experiments studying the process of interest, not from D alone.
For an object to be considered fractal, two requirements must be met. The first
requirement is that a fractal must be “a shape made of parts similar to the whole in some
way” (Mandelbrot 1982). As the grain of measurement increases or decreases for the
self-similar object, the shape or pattern tends to repeat itself. Mandelbrot developed a
broad definition for fractal geometry because of its usefulness as an analytical tool in
fields such as physics, biology, mathematics and ecology. Furthermore, the definition
purposely does not differentiate between exactly self-similar objects or statistically selfsimilar objects. Exact self-similar structures, such as the Sierpinski carpet and triangle
(figure 1), only exist in the mathematical sense. In nature, there is an inherent
randomness that prevents perfect mathematical fractals from forming. What we can
observe in landscapes are fractal shapes and dimensions that are statistically self-similar.
In statistical self-similarity, a portion of an object or phenomenon will look similar to the
whole if enlarged and/or reduced to the proper scales (Leduc et al. 1994). This definition
implies that a fractal relationship found in nature will not hold across all scales but rather
will only pertain to a certain range of scales.
Figure 1. Recreated Sierpinski Carpet and Sierpinski Triangle. Both are
examples of mathematical fractals.
The second requirement for a fractal object is that it must have a non-integer
dimension. In simple Euclidean space, a point has a dimension of zero, a line a
dimension of one (length), a polygon a dimension of two (length and width), and a
volume has three dimensions (length, width and height). Many objects in nature,
however, do not fit so neatly into one of these categories. A landscape surface with
ridges is more than two dimensional, but since those ridges are not found everywhere on
the surface, it is not entirely three-dimensional either (figure 2) (Li 2000). Its true
dimension is some fraction between two and three, which is where the term fractal
originated (Mandelbrot 1982).
Figure 2. The surface above fills more than just the two-dimensional plane, but
the mountains do not fill the entire three-dimensional space. This surface can
therefore be described by some fractional dimension between two and three.
Fractals and Landscape Ecology: A Land Use Case Study
Using fractal geometry, patterns can be described across multiple spatial scales.
Any generalizations made across scales, however, only hold true where D does not
change. As the slope changes D will also change, indicating that a new set of processes
is becoming important in determining what is observed in the landscape.
Krummel et al. (1987) used this idea of changing slopes as an indicator of
different processes operating in the landscape. The perimeter/area method (link to a page
of links) was used to derive the fractal dimensions of forest patches in Mississippi and
Louisiana. Once D values were plotted against the forest patch size, it became apparent
that two distinct processes were affecting the size of the deciduous patches (figure 3).
Figure 3. The fractal dimension (D), as a function of forest area. The break in D
at approximately 22 m2, illustrates how different processes are affecting the size
of patches found. Image adapted from Krummel et al., 1987.
Below a patch size of approximately 22 m2, the values of D are quite low. Low D values
signify that geometric shapes are relatively simple and were most likely a product of
anthropogenic processes constraining the patch shape and size. After 22 m2, D increases
rapidly before becoming relatively constant again reflecting larger patches with more
complex shape. These patches are a result of natural processes and topography creating
landscape patterns as opposed to the smaller more simple patches shaped by human
influences (Krummel et al. 1987, De Cola 1989, O’Neill et al. 1988).
The use of fractals to describe a landscape is not limited to only spatial scales.
Fractal geometry can also involve the temporal study of changes to a landscape. For
example, fractal dimensions may provide insight into the human impact of urban sprawl
(link to Heather) on natural ecosystems. Using the perimeter/area method (link to page
of links), D can be found for different land use categories by using a time series of two
remotely-sensed images (link to Ken) at a given location. This method of change
detection (link to Jon) can then be used to examine changes in the shape complexity of
land use types over many temporal scales (link to Jennie) (De cola 1989).
Fractals and Landscape Ecology: Organismal Case Studies
A fox may view an entire forest as its landscape while an insect, whose entire life
cycle begins and ends on a singe tree within that forest, perceives the landscape at a much
finer scale. As a landscape ecologist, it is important to understand at what scale the
organism of interest perceives the landscape and responds to landscape features. Finding
and defining the proper scales for organisms has implications for many fields in ecology
including restoration ecology (link to Polly), conservation biology (link to Jenn) and even
the modeling of avian communities (link to Alison).
To better understand how organisms view their surroundings, Kimberly With
(1994) measured and plotted the path length of three different sized grasshopper species
using different ruler lengths. From these plotted points, the fractal dimension for each
species was determined. The three species of grasshopper had different values of D,
indicating that they took different paths of varying complexity. Higher D values signify
that a grasshopper species took a more convoluted path, while lower values of D reflect a
more linear path through the grassland habitat. The different D values for the three
grasshopper species imply that each species views the landscape differently. Species with
high values of D may consider a portion of their habitat inhospitable which could be why
they took a more convoluted path.
Alternatively, because the different species were of different body sizes, a blade
of grass that a small species may have considered too large to travel over might be
ignored by a larger species. Not only does this mean that a species with a lower fractal
dimension can travel through a landscape more quickly and directly, but it also suggests
that larger species are integrating landscape information at a broader scale (With 1994).
That the species with the lowest D did not travel below –3 cm (this is a natural log
number) indicates that it may be assimilating landscape information at a scale above –3
cm and does not consider habitat below this number. It should be noted, however, that
because the fractal relationship in Kimberly With’s study is only over a small spatial
scale, it may not truly describe a scaling relationship. It has been suggested that a scaling
relationship must be constant over two orders of magnitude to be considered statistically
self similar (Milne 1991).
In a different application of fractals to organisms, Milne (1997) mapped habitat
availability at different scales to better understand how two different species that varied
in body size may view the same landscape. With an equation containing a fractal
exponent relating body size to home range area, Milne determined the home range area of
a pocket gopher and a black tailed jack rabbit. Using a remotely sensed image (link to
Ken), a moving window (link to page of links) was placed over the digital map, and each
pixel was classified as either grassland habitat or forest habitat. Moving window sizes
were based on each species’ characteristic home range length, as found from the body
size/home range area equation. The visual result of such an experiment provides a
striking difference between how both rabbits and gophers might view the same habitat
(figure 5).
Figure 5. Habitat as viewed by the gopher (A) and rabbit (B). Areas in white
represent available habitat while dark areas represent forest/inhospitable habitat.
Image adapted from Milne, 1997.
The boundary between grassland and forest is much more distinct for the gopher
(figure 5A) as compared to the rabbit (figure 5B). Because the window used to map the
gopher habitat was similar to the resolution of the image, the grassland habitat boundary
is much more defined. The sharp boundary indicates that this particular gopher species
may be more constrained by the forest edge habitat than the rabbit species. Since the
rabbit is a larger animal, its home range incorporates a much larger area and it might be
able to utilize some of the edge habitat that the gopher may find inhospitable. Therefore,
the boundary for the rabbit species is softer, resulting in a more fuzzy definition of rabbit
edge habitat. Similar analyses may be of use in other studies of herbivory by different
species (link to Fleur) to better ascertain how different herbivores view the same area and
consequently how much habitat each species could have available to it.
Fractals and Landscape Ecology: Multifractals
Although multifractals, an extension of the concept of fractal geometry, are
becoming more common, they are only briefly mentioned here. Multifractals assume
that no single D describes patterns observed in natural landscapes. Rather, there is a
spectrum of scaling exponents creating the observed landscape pattern (Scheuring et al.
1994, Feder 1988, Milne 1991). Different agents of disturbance, for example, can have
different scales at which they affect the landscape. Ecologists could therefore look at
fractal moments, analogous to the mean, variance, skewness and higher moments based
on a probability distribution, to see how the various pattern forming factors change with
length scale (Milne 1991).
Fractals do not explain all the patterns that are observed in nature, but they do
identify the presence of patterns at multiple scales. Part of the appeal of fractals is that a
single statistic can be used to describe potentially complex patterns in natural
environments. Furthermore, because shapes with fractal dimensions are prevalent in
many disciplines, D also provides a means for scientists in various disciplines to
communicate with each other. The addition of fractal geometry should enable landscape
ecologists to better entertain questions relating scale to various patterns, processes and
predictions. The use of fractal geometry, however, should only be viewed as another tool
to be used by landscape ecologists to aid in answering questions relating to scale, not the
answers in and of themselves.
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