Wavelet Analysis of Shannon Diversity (H’)

1. 1500 Scale 1000 500 1000 0 Resolution (m) 1.2 0.6 0 W. Var. Access Road 500 Sand Road- Mod. Use E Old Harvest Landing W 0 Sand Road- Light Use H’ 0 1.0 2.0 Clearing OPB MA YA2 JPO SPB OPB SPB CC YA2 H1 PA BOPB OPB H2 0 1000 2000 3000 Distance (m) Wavelet Analysis of Shannon Diversity (H’) 1500

2. MA YA2 H2 H1 PA CC OPB OPB SPB YA2 BOPB SPB JPO ORP15 JPO RP15 MP POA NCC OCC OCC F2 OBCC H F H H OCC RP5 Pine Barrens Large-Block PO Access Road Sand Road- Mod. Use Old Harvest Landing Sand Road- Light Use 1500 1500 Resolution (m) Small-Block Pine POA Forest 1000 1000 Clearing ATV Trail 500 500 Dry Streambed E W 0 0 Grassy Roadside 0 0 500 500 1000 1000 1500 1500 2000 2000 2500 2500 3000 3000 MP TRP60 RP7 OCC RP7 OCC OCC RP60 F2 H2 H2 RP12 RRP TRP60 RJP H MP H C H H F H F2 C C Distance (m) Wavelet Analysis Comparison

3. Wavelet Variance of litter cover for the four study transects

6. Fractal Dimension and Applications in Landscape Ecology Jiquan Chen University of Toledo Feb. 26, 2007 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)

8. Sierpinski Carpet generated by fractals

9. 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 N^2 self-similar pieces, each with magnification factor N. So we can write: http://math.bu.edu/DYSYS/chaos-game/node6.html

11. 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.

12. Triadic Koch Island

13. r1=1/2, N1=2 • R2=1/4, N2=4 • D=0

15. http://mathworld.wolfram.com/Fractal.html

16. 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 • …

20. Wiens et al. 1997, Oikos 78: 257-264

21. Vector-Based Raster-Based

22. Figure 11: The Sierpinski hexagon and pentagon

23. 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