Fractals and Applications in Landscape Ecology

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Fractals and Applications in Landscape
Ecology Jiquan Chen University of Toledo Feb. 18,
2009

• How long is the costal line on earth?
• Would the pathways of a species through different
  patches be similar? why?
• Can one use the knowledge of fractals to simulate
  different landscape patterns and processes?

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Fractals and Applications in Landscape
Ecology Jiquan Chen University of Toledo Feb. 18,
2009

• How long is the costal line on earth?
• Would the pathways of a species through different
  patches be similar? why?
• Can one use the knowledge of fractals to simulate
  different landscape patterns and processes?

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The Euclidean dimension of a point is zero, of a
line segment is one, a square is two, and of a
cube is three. In general, the fractal dimension
is not an integer, but a fractional dimensional
(i.e., the origin of the term fractal by
Mandelbrot 1967)
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Sierpinski Carpet generated by fractals
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So what is the dimension of the Sierpinski
triangle? How do we find the exponent in this
case? For this, we need logarithms. Note that,
for the square, we have N2 self-similar pieces,
each with magnification factor N. So we can write
http//math.bu.edu/DYSYS/chaos-game/node6.html
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Self-similarity One of the basic properties of
fractal images is the notion of self-similarity.
This idea is easy to explain using the Sierpinski
triangle. Note that S may be decomposed into 3
congruent figures, each of which is exactly 1/2
the size of S! See Figure 7. That is to say, if
we magnify any of the 3 pieces of S shown in
Figure 7 by a factor of 2, we obtain an exact
replica of S. That is, S consists of 3
self-similar copies of itself, each with
magnification factor 2.
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Triadic Koch Island
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• r11/2, N12
• R21/4, N24
• D0

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http//mathworld.wolfram.com/Fractal.html
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• A geometric shape is created following the same
  rules or by the same processes inducing a
  self-similar structure
• Coastal lines
• Stream networks
• Number of peninsula along the Atlantic coast
• Landscape structure
• Movement of species

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Wiens et al. 1997, Oikos 78 257-264
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Vector-Based
Raster-Based
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Figure 11 The Sierpinski hexagon and pentagon
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n mice start at the corners of a regular n-gon of
unit side length, each heading towards its
closest neighboring mouse in a counterclockwise
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