J .A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
139
boundary-grid method Sugihara and May, 1990 was used to determine the fractal dimension of each profile. Eight grids i
5 1, 2, . . . , 8 were superimposed on the first 20-cm length of each cross-section. Each grid contained squares with a side length of n
i
pixels, where n 5 3 3 2 5 6, 12, 24, 48, 96, 192, 384, or 768, resulting in squares with
side lengths of 1.44–200 mm. The number of squares entered by each profile N was counted for each grid. Fractal dimension, D, was determined from the following
equation where k represents a constant:
2D
N 5 kn
1 where D equals the negative of the slope from the linear regression of log n against
2
log N.
2
2.4. Statistical analysis To gain some understanding of how fractal dimension varied at the within-cast spatial
scale, the nonparametric runs test was used to determine if the D values for consecutive surface profiles in each cast had random variability or, alternatively, were serially
correlated Zar, 1984. To test for differences in D among the five casts, the nonparametric Kruskal–Wallis test one-way ANOVA on ranks and Dunn’s a posteriori
multiple comparison test were used Zar, 1984.
3. Results
3.1. Mussel density and size–class structure
22
Mean mussel density in the bed was 14.88 63.69 S.E. individuals 0.02 m .
Size–class structure for live mussels was essentially unimodal, with the mode at 32–36 mm and the largest individual in the 60–64-mm class Fig. 2. For the death assemblage,
the size–class structure was similar, with the mode shifted somewhat to the right and the largest valve in the 88–92 mm class Fig. 2.
3.2. Fractal dimensions of surface profiles A representative cross-section with its regression graph is shown for each of the three
mussel bed surface types Figs. 3 and 4. For every one of the 88 surface profiles, the
2
regression of log n versus log N was highly significant P ,0.001, with r values
2 2
ranging from 0.984 to 0.999. All the mussel bed profiles were fractal. There was never a break in a regression line indicating a change in slope, meaning that the fractal
dimension was the same for a given profile over the entire range of grid square sizes from 1.44 to 200 mm.
Within each of the five casts, D values were not serially correlated. The runs test results demonstrated that consecutive profiles in all the casts had random variability
P .0.25. There were significant between-cast differences in D Table 1. Although the
largest mean and median D values were from a high percent cover cast, and the smallest
140 J
.A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
Fig. 2. Size–class histograms for live individuals and death assemblage of Mytilus edulis at Bob’s Cove.
were from the zero percent cover cast, there was not a consistent relationship between percent cover and D Fig. 5.
3.3. Comparison of surface profiles with the Koch curve Six profiles from high percent cover Cast 1 and four profiles from intermediate
percent cover Cast 1 had D .1.26, the value for the Koch curve. These 10 D values
were only slightly larger than 1.26, ranging from 1.264 to 1.310, and represented just 11.4 of all the profiles. The remaining 88.6 of the profiles had D
,1.26. Every cast had mean and median values of D
,1.26 Table 1, the largest being high percent cover Cast 1 mean
51.24260.046, median51.251. Thus, the first hypothesis was supported: the typical fractal dimension of the bed surface profile was slightly less than 1.26.
3.4. Comparison of surface profiles with horizontal spatial pattern Fractal dimension values for individual profiles ranged from 1.049 to 1.310. Every
one of the 88 profiles, and thus the mean and median values for every cast, had D ,1.36
Table 1, the smallest aerial view D value reported for any 25 325-cm quadrat by
Snover and Commito 1998. Thus, the second hypothesis was supported: bed surface topography was less complex than bed spatial pattern.
J .A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
141
Fig. 3. Examples of profiles from bed surface casts made at locations with high, intermediate, and zero mussel cover. The regressions to calculate fractal dimension from the same three profiles are shown in Fig. 4.
142 J
.A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
Fig. 4. Boundary-grid method regressions from the same profiles shown in Fig. 3. Fractal dimension is equal to the negative of the slope of the regression line. Note the correspondence between apparent surface
roughness and the fractal dimension in each of the three examples.
J .A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
143 Table 1
Summary statistics for five mussel bed surface casts Mussel cover
Number of Fractal dimension
profiles
1
Range Mean
6S.E. Median
Significance High
Cast 1 16
1.164 61.310
1.242 60.046
1.251 a
Cast 2 16
1.115 61.218
1.175 60.026
1.177 a
Intermediate Cast 1
16 1.124
61.289 1.202
60.058 1.190
a Cast 2
17 1.031
61.227 1.122
60.040 1.130
b Zero
Cast 1 23
1.049 61.172
1.106 60.033
1.110 b
1
Same letters indicate no significant difference Dunn’s a posteriori multiple comparison test after Kruskal–Wallis one-way ANOVA on ranks, P
.0.05.
4. Discussion
