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
The results from Bob’s Cove supported the two hypotheses about the fractal geometry of mussel bed surface topography. They demonstrated that the bed surface was fractal
over a spatial scale of more than two orders of magnitude grid squares ranging from 1.44 to 200 mm in side length, a factor of nearly 140 times. The upper bound of the bed
surface fractal dimension was determined empirically from the previously determined fractal dimension of the bed spatial pattern. Furthermore, a close estimate of the fractal
Fig. 5. Mean 6S.E. fractal dimension for each of the five bed surface casts.
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.A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
dimension was predicted theoretically from the Koch curve. The results indicate that fractal dimension is a parameter that can be useful in characterizing the irregular bed
surface by incorporating the size, shape, and scale of roughness elements into a simple, numerical metric.
Soft-bottom mussel beds are fractal in profile this study and in aerial view Snover and Commito, 1998. They show self-similarity across spatial scales, so inferences from
studies on spatial assembly in other systems may be applicable to this benthic species. For example, Kaandorp 1991, 1994, 1999 developed relatively simple fractal
generators that predicted growth patterns and shapes of model sponges and corals under different water flow and light regimes. His field experiments verified the predictions and
demonstrated, for example, that sponge branching complexity and fractal dimension are greater in fast flow than slow flow. Mussel beds are an excellent model system for
Kaandorp’s approach because, like his colonial organisms, they are made up of many small units of similar shape assembled into larger, yet still similar, shapes. Unlike the
Koch curve, mussels are not perfect triangles, and mussel beds do not consist of triangles deployed in a precisely regular spatial arrangement. That the mussel bed profile
D values in this study could be predicted from the Koch curve suggests that future fractal generators more sophisticated than this first attempt could lead to useful bed
shape predictions. Hypotheses about bed growth, fragmentation, infill between patches, and other aspects of spatial dynamics might be derived from simple assembly rules.
Furthermore, these hypotheses can be tested because mussel beds are amenable to successful experimental manipulation in the laboratory and field.
4.1. Fractal dimension and other surface complexity studies Fractal geometry has rarely been employed to characterize bottom topography. Over a
large spatial scale, Schwinghamer et al. 1996 quantified the impact of trawling on benthic habitat structure in an area of sandy sediment on the Grand Banks of
Newfoundland using Hilbert-transformed high resolution acoustic signal profiles from 12
330-cm rectangular areas on the bottom. Fractal dimensions were calculated for the shapes of the electronic signals as a proxy for the actual surface profiles. Post-trawling D
values were smaller than pre-trawling values, indicating that trawling made the bottom smoother within each trawl path.
Working at much smaller spatial scales, Le Tourneux and Bourget 1988 showed on plastic settlement panels that fractal dimension values were higher for locations selected
by barnacle Semibalanus balanoides larvae than for nearby unselected sites. The investigators measured surface microheterogeneity profiles with a diamond stylus that
was moved over the surface. Hills et al. 1999 demonstrated that settlement density of S. balanoides larvae was positively correlated with both Euclidean and fractal surface
complexity indices of plastic panels they created with different surface textures. They scanned the surface with a laser device to measure surface profiles. Kostylev et al.
1997 used contour gauges with 1-mm wide pins to create profiles of the rocky shore covered with mussels Mytilus galloprovincialis and barnacles Chthamalus stellatus.
They related abundances of different sizes and morphs of the snail Littorina saxatilis to the fractal dimensions of the profiles. Erlandsson et al. 1999 used the same technique
J .A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
145
to show that the fractal dimension of the rocky shore surface was a good predictor of snail Cellana grata movement patterns. Beck 1998 used aerial view stereophotog-
raphy to derive surface profiles of rocky shore and mangrove Avicennia marina habitats. Of the four indices of structural complexity he calculated, fractal dimension
was the best predictor of density for the five gastropod species under study.
These intertidal investigations estimated surface profiles with ingenious, non-destruc- tive, and time-saving ‘down-from-above’ techniques. Their results must be interpreted
with caution, however. For some surfaces, the techniques are as accurate as the profiles derived from cross-sections. For a surface like the mussel bed at Bob’s Cove with tilted,
overhanging, or undercut objects, they would simplify the actual surface profiles by ignoring the undersides of these shapes, resulting in an underestimate of D Fig. 6.
However, the loss of accuracy with a down-from-above technique may be more than offset by the savings in time compared to making surface casts that are then cut into
sections to produce true profiles.
4.2. Fractal dimension and percent cover Kostylev et al. 1997 observed a monotonically increasing relationship between
mussel cover and the fractal dimension of the mussel- and barnacle-covered rock surface at their site, although variability in D was quite large at high percent cover values. We
did not find a clear positive relationship in our study, nor is there any reason to believe that D will automatically increase with increasing mussel cover. For example, Snover
and Commito 1998 found that the largest D values for the mussel spatial pattern in aerial view occurred at intermediate values of mussel cover and density. Similarly, for
the bed surface, an intermediate percent cover of patchy mussels of different ages and lengths could have a highly irregular surface and a large D, while 100 cover of tightly
packed mussels might have a relatively smooth surface with a small D. The latter would be especially likely if all the individuals were the same size or if small individuals filled
Fig. 6. Top panels illustrate a profile that is accurately characterized with a ‘down-from-above’ technique. Bottom panels illustrate a profile that is simplified with a ‘down-from-above’ technique because of undetected
complexity under the overhanging portions, leading to an underestimate of fractal dimension.
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.A. Commito, B.R. Rusignuolo J. Exp. Mar. Biol. Ecol. 255 2000 133 –152
Fig. 7. Top panels illustrate a mussel bed with intermediate percent cover, complex profile, and large fractal dimension. Bottom panels illustrate a mussel bed with high percent cover, simple profile, and small fractal
dimension.
in the gaps between the larger individuals Fig. 7. One possible reason for the different results between Kostylev et al. 1997 and our study might be that maximum cover at
their site was higher than at Bob’s Cove, reaching 100. Furthermore, theirs was a rocky shore site with epifaunal Mytilus galloprovincialis, while ours was a soft-bottom
site with semi-infaunal M
. edulis. Despite these differences, the D values in both studies were similar. At 0 mussel cover, their D
51.031 and our D51.110. At 50 mussel cover, their D
51.122 and our D51.130 and 1.190. At 85 mussel cover our highest percent cover, their D
51.186 and our D51.177 and 1.251. The fact that their D values were generally lower than ours is consistent with the idea that the down-from-above
profiling technique produces smaller D values than does cross-section profiling. The rocky shore sites of Beck 1998 and Erlandsson et al. 1999 had 0 mussel cover and
down-from-above D values of approximately 1.02, in close agreement with Kostylev et al. 1997. The similar results from the three hard-bottom studies and our soft-bottom
bed suggest that fractal dimension may be useful in relating surface topography from different locations and bottom types.
4.3. Fractal dimension, roughness elements, and flow Water flow calculations require information about the height of roughness elements
above the bottom Eckman, 1990; Ke et al., 1994; Green et al., 1998. Few studies have focused specifically on water flow over Mytilus edulis beds. Knowledge of the fractal
dimension of bed surfaces might be useful in designing and interpreting laboratory and field studies of this type. For example, Butman et al. 1994 used live Mytilus edulis to
simulate a mussel bed in a laboratory flume. They were able to determine the contours of turbulent stress and food depletion over the bed at different flow velocities. The mussel
bed increased turbulent stress by a factor of up to three in slow flow and a factor of more than 10 in fast flow. The mussels they used were all young and similar in size mean
61
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147
22
S.D. 52.760.2 cm, with a density of 75.5611.4 individuals 100 cm . Based on this
information alone i.e., small, same-sized, tightly packed individuals, it is likely that the fractal dimension was smaller than what was found at Bob’s Cove with its patchy
distribution of different-sized mussels. It is interesting to consider what the turbulent stress contours over the flume bed would look like with surface topographies of different
fractal dimensions. And just as flow properties and food depletion might be influenced by the fractal dimension of the bed, the fractal dimension values almost certainly depend
on flow rate and food concentration. For example, Butman et al. 1994 discovered that mussels changed position and orientation so that mean mussel bed height in the flume
was approximately 70 higher after their phytoplankton enhancement experiment compared to experiments with natural phytoplankton concentrations. It is likely that
fractal dimension is a dynamic feature of mussel beds, meaning that the development of non-destructive, in situ techniques to measure D could provide a way to monitor mussel
feeding rates over space and time.
Other studies that have measured water flow over Mytilus edulis beds have demonstrated that roughness elements play a key role in controlling a number of
ecologically important processes, including biodeposition and erosion rates Widdows et al., 1998; food-regulated growth and vertical gradients of particulate organic material
´ ´
´ Frechette and Bourget, 1985a,b; Frechette et al., 1989; Frechette and Grant, 1991; and
removal of individuals by current-induced dislodgment Dolmer and Svane, 1994. Topographic complexity also regulates the flow environment in other bivalve species
O’Riordan et al., 1993; Weissburg and Zimmer-Faust, 1993; Breitburg et al., 1995; Green et al., 1998; Lenihan, 1999. In all these examples, fractal dimension could serve
as a useful measure of surface roughness when considering hydrodynamically controlled phenomena.
4.4. Fractal dimension and mussel bed stability Mussel beds are extremely dynamic and can undergo rapid changes in percent cover
and density see review in Commito and Dankers, in press; Mytilus edulis: Nehls and Thiel, 1993; Reusch and Chapman, 1995, 1997; Dreissena polymorpha: Ricciardi et al.,
1997; Musculista senhousia: Crooks, 1998; Crooks and Khim, 1999. Sugihara and May 1990 have suggested that fractal dimension is related to stability in ecological systems.
According to their argument, a stable system should have uniformity over a large area, resulting in a relatively low D value. As the system destabilizes, it breaks up and
becomes fragmented and more complex, causing D to increase. Thus, fractal dimension should be inversely proportional to persistence. This idea has not been explicitly tested,
but so far the results from benthic systems would not seem to support it. For example, Mouritsen et al. 1998 observed that persistent populations of the tube-building
amphipod Corophium volutator created an irregular sediment surface of plateaus and pools. The bed became smoother and more uniform after mass mortality of the
amphipods. Aronson and Precht 1995 found that coral reef topographic complexity estimated with the chain technique was not inversely proportional to persistence. Surface
complexity their C value was highest at intermediate levels of disturbance. Similarly, in no mussel bed studies was D lowest at the highest level of cover and density, as
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predicted by the Sugihara and May model. Instead, it increased as cover and density rose to intermediate or high levels Kostylev et al., 1997; Snover and Commito, 1998; Fig. 5
and Table 1 from this study. In some soft-bottom mussel beds in Maine, the local abundance can be so high
because of high rates of recruitment and growth that clumps of mussels form vertical columns taller than they are wide, even if percent cover is less than 100 because of
nearby bare patches Commito, personal observation. This hummocking lifts the mussels up off the sediment surface, resulting in very complex bed topography. The
Koch curve no longer describes such a bed, and we predict that the fractal dimension would be higher than what was found at Bob’s Cove. If so, then fractal dimension may
serve as a useful indicator of high rates of mussel recruitment and rapid body growth.
Hummocked soft-bottom mussel beds may be particularly susceptible to fragmenta- tion due to dislodgment of mussels by water currents Harger and Ladenberger, 1971.
But even beds without hummocks are subjected to damage, and clumps of mussels carried away from beds are a major form of dispersal Reusch and Chapman, 1995,
1997. Fractal dimension is probably high for fragmented beds and newly dispersed clumps. They have high perimeter: area ratios with a surface topography that creates
important edge effects at the margins, where water flow changes abruptly as it meets mussels projecting above the substrate Butman et al., 1994. Rates of larval recruitment
and body growth are higher at the edges of mussel patches than in the interior Bertness and Grosholz, 1985; Okamura, 1986; Lin, 1991; Stiven and Gardner, 1992; Svane and
Ompi, 1993. Growth of mussels out from the edges fills in the gaps between patches Petraitis, 1995; Reusch and Chapman, 1997, most likely lowering the fractal
dimension.
On the other hand, mussel mortality rates due to ice scour and predation are often higher at the edges, demonstrating the trade-offs that occur with respect to position
Bertness and Grosholz, 1985; Lin, 1991; Stiven and Gardner, 1992; McGrorty and Goss-Custard, 1993; but see Reusch and Chapman, 1997, and Dolmer, 1998. In fact, the
presence of predators causes individual mussels to move towards each other. There is reduced per mussel predation risk because of smaller perimeter: area ratios as they
coalesce into large clumps with relatively simple, smooth shapes Reimer and Teden-
ˆ ´ gren, 1997; Dolmer, 1998; Cote and Jelnikar, 1999; see also Leonard et al., 1999.
Fractal dimension could be used to measure this response to predators. A bed comprising a few large clumps with simple outlines and small perimeter: area ratios would have a
lower fractal dimension than a bed with a few large, irregularly shaped clumps or many small clumps.
Mussel bed fragmentation, clump dispersal, predation, recruitment, growth, and movement all affect and are affected by bed shape and surface topography. Knowledge
of the fractal geometry of mussel beds might have predictive power concerning these ecological processes because fractal dimension detects and quantifies the edges and
surface irregularities that play a role in controlling them.
5. Conclusion
