1. Wavelet Analysis of Shannon Diversity (H’) 1500 Old Harvest Landing Sand Road- Mod. Use Sand Road- Light Use Clearing Access Road 0 1000 500 Resolution (m) OPBBOPBOPBPASPBCCYA2 H2 H1JPOSPB OPB MA YA2 1000020003000 Distance (m) H’ 0 1.0 2.0 1000 1500 500 0 0 0.6 1.2 W. Var. Scale W E

2. Wavelet Analysis Comparison 100050001500200025003000 BOPBOPB PA SPB CC YA2 H2 H1 JPO SPB OPB MA YA2 OCC OBCCOCCHHHF F2 JPO RP5 MP NCC POA RP15 ORP15 100050001500200025003000 0 500 1000 1500 0 500 1000 1500 Resolution (m) Distance (m) Pine Barrens POA Forest Small- Block Pine Large- Block PO MP F2 HH H H H H2 F C TRP60RJP RP12 RRP OCCRP7 RP60 OCC MP OCC TRP60 CC W E Old Harvest Landing Sand Road- Mod. Use Sand Road- Light Use Clearing Access Road ATV Trail Dry Streambed Grassy Roadside

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. 1)r 1 =1/2, N 1 =2 2)R 2 =1/4, N 2 =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-BasedRaster-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