255 (2000) 133–152
www.elsevier.nl / locate / jembe
Structural complexity in mussel beds: the fractal geometry of
surface topography
*
John A. Commito , Brian R. Rusignuolo
Environmental Studies Program and Biology Department, Gettysburg College, Gettysburg, PA 17325, USA Received 20 April 2000; received in revised form 27 June 2000; accepted 27 August 2000
Abstract
Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob’s Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44–200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D ) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.24260.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D,1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Fractals; Maine; Mussel bed; Mytilus edulis; Scale; Soft-bottom; Surface topography
*Corresponding author. Tel.:11-717-337-6030; fax:11-717-337-6666.
E-mail address: [email protected] (J.A. Commito).
1. Introduction
The seafloor is not flat. Soft-bottom surface topography is irregular because intertidal and subtidal sites have a variety of current-induced bedforms as well as burrows, pits, and mounds created by infaunal and epibenthic animals (see Commito et al., 1995a; Cutter and Diaz, 1998, for examples). Cobbles and boulders produce additional surface relief (Guichard and Bourget, 1998), as do structures built by animals and plants that project above the bed, including polychaete tubes, bivalve shells, seagrasses, marsh grasses, and mangrove pneumatophores and aerial roots (see Beck, 1998, for examples). Similarly, rock bottoms (Erlandsson et al., 1999) and the animals (Petraitis, 1991; Aronson and Precht, 1995; Kaandorp, 1999) and plants (Gee and Warwick, 1994a,b) that live on them have complex shapes. Human activities such as commercial dredging and trawling also affect the three-dimensional structure of the bottom (Schwinghamer et al., 1996).
Surface complexity is important for a variety of reasons. It provides habitat structure for juvenile and adult animals (Raffaelli and Hughes, 1978; Gee and Warwick, 1994a,b; Dittmann, 1996; Kostylev et al., 1997; Beck, 1997, 1998). It plays a role in regulating foraging patterns (Dolmer, 1998; Erlandsson et al., 1999). Perhaps most significantly, it alters boundary-layer flow over the bottom (Butman et al., 1994; Ke et al., 1994; Green et al., 1998). This interaction of flow and substrate heterogeneity affects larval settlement (see Lapointe and Bourget, 1999; and Hills et al., 1999, for many examples) and subsequent population performance because it controls delivery of food, oxygen, and chemical cues (Weissburg and Zimmer-Faust, 1993; Bertness et al., 1998; Leonard et al., 1998, 1999; Lenihan, 1999). Surface complexity also influences the flow-mediated erosion, transport, and deposition of sediment (Widdows et al., 1998), meiofauna (Fleeger et al., 1995), and macrofauna (Commito et al., 1995a,b).
The purpose of this investigation was to utilize fractal geometry to characterize the surface topography of a soft-bottom mussel bed (Mytilus edulis L.) in Maine, USA. Mussel beds of several species dominate low intertidal and shallow subtidal hard and soft bottoms worldwide. They have some of the highest densities, biomass, and respiration rates of all marine systems (Nixon et al., 1971; Seed, 1976). Soft-bottom mussel beds control species composition and abundance by providing habitat space, outcompeting other surface dwellers, altering flow regimes, and creating a sedimentary environment high in organic material and low in oxygen, thus favoring some species and eliminating others (Radziejewska, 1986; Asmus, 1987; Commito, 1987; Commito and Boncavage, 1989; Commito and Dankers, in press; Dittmann, 1990; Svane and Setyobudiandi, 1996; Warwick et al., 1997; Cummings et al., 1998; Widdows et al., 1998; Ragnarsson and Raffaelli, 1999). Moreover, exotic mussel species have devastat-ing impacts when introduced to new localities (Ricciardi et al., 1997; Crooks, 1998; Crooks and Khim, 1999).
Flume and field studies have demonstrated that mussel bed surface complexity creates ´
turbulent flow along the bottom (Frechette et al., 1989; Butman et al., 1994; Green et al., 1998) and strongly influences local ecological conditions such as food supply and rates
´ ´
1989; Widdows et al., 1998). Experiments with non-living mussel mimics have shown that their surface structure can be as important as the biological activities of the mussels themselves in regulating species composition and abundance within mussel bed communities (Ricciardi et al., 1997; Crooks and Khim, 1999). In addition, surface roughness influences the rates at which mussels are eaten by predators (Dolmer, 1998) and dislodged by water currents (Dolmer and Svane, 1994).
Despite the importance of surface topography in mussel beds and other benthic systems, there is no consensus in the literature about how to define and measure the complexity of seafloor structure. Often a dichotomous ‘rough versus smooth’ or ‘structure versus no structure’ characterization is made (Raffaelli and Hughes, 1978; Petraitis, 1990, 1991; Mouritsen et al., 1998). Other workers have attempted more quantitative estimates of structure by extracting information from cross-sectional profiles of the bottom (Rhoads and Young, 1970; Le Tourneux and Bourget, 1988; Ke et al., 1994; Aronson and Precht, 1995; Schwinghamer et al., 1996; Kostylev et al., 1997; Beck, 1997, 1998; Cutter and Diaz, 1998; McCormick-Ray, 1998; Erlandsson et al., 1999; Hills et al., 1999; Perez et al., 1999). The seabed roughness calculation (Ke et al., 1994; Green et al., 1998), chain technique (Aronson and Precht, 1995; Beck, 1997, 1998), and other methods (summarized in Beck, 1998) utilize profile irregularity data to produce a numerical value that characterizes topographic complexity.
The application of fractal geometry can also utilize profile data. Le Tourneux and Bourget (1988) pioneered this approach in their investigation of barnacle settlement, using it to quantify surface roughness profiles of plastic settling panels. Since then, Kaandorp (1991, 1994, 1999), Gee and Warwick (1994a,b), and Davenport et al. (1996) have employed fractals to describe the branching structure of hard-bottom organisms, including sponges, hydrozoans, scleractinian corals, and macrophytic algae. But few investigators have attempted to use fractal geometry to quantify surface complexity of the bottom (subtidal sandy sediment: Schwinghamer et al., 1996; rocky intertidal shores: Kostylev et al., 1997; Beck, 1998; Erlandsson et al., 1999; mangrove shores: Beck, 1998; sandblasted resin fouling tiles: Hills et al., 1999). It has not been used to characterize some important intertidal systems, including soft-bottom mussel beds.
many small irregularities and one that has a few large ones. It allows investigators to determine if large surface irregularities are similar in shape to small ones and to aggregated copies of small ones. These are important descriptors for mussel beds, kelp beds, boulder fields, coral reefs, and other seafloor habitats made up of multiple units with different sizes and similar shapes.
Snover and Commito (1998) recently used fractal geometry to characterize the spatial pattern of Mytilus edulis in the soft-bottom mussel bed at Bob’s Cove, Maine. They calculated fractal dimensions of mussel aggregation outlines in quadrats using the boundary-grid technique (Sugihara and May, 1990) and discovered that the complex, patchy distribution of mussels within the bed was spatially ordered. Over two orders of magnitude in spatial scale, the fractal dimension values varied predictably with percent
2
cover (concave-downward second-order regression curve, r 50.94), density
(concave-2
downward second-order regression curve, r 50.92), and Morisita’s index of spatial
2
dispersion (negative-slope regression line, r 50.82). The mussel bed was fractal in the horizontal plane (i.e., aerial view), but the vertical profile was not examined.
In the present investigation, the surface of the mussel bed at Bob’s Cove was investigated. Two hypotheses were tested:
1.1. H : the bed surface profile is fractal, with 11 ,D,1.26
A mussel bed is composed of multiple copies of individual mussels of different sizes but with the same approximate shape. Thus, a bed surface is a likely candidate for self-similarity. If so, then bed surface profiles should be fractal. By definition, this means that profiles have fractal dimension values lying somewhere between 1 (Euclidian straight line or smooth curve) and 2 (complex curve so convoluted that in the limit it is a completely space-filling Euclidian plane).
More specifically, the familiar, well-studied Koch curve (Fig. 1; Hastings and Sugihara, 1993) can serve as a model for the mussel bed surface at Bob’s Cove. The Koch curve is made up of repeated, identically shaped triangular units of different sizes that grow out from the existing profile, not unlike the settlement and growth of new mussels in a soft-bottom bed of existing semi-infaunal mussels. The Koch curve has a fractal dimension of 1.26. Mussel beds might display what are termed ‘statistical’ fractal patterns (Hastings and Sugihara, 1993) rather than the perfect fractal qualities of the algorithm-derived Koch curve, so their surface fractal dimension should be similar to, but generally smaller than, that of the Koch curve. If so, the bed surface profiles should have fractal dimensions approaching 1.26.
1.2. H : mussel bed surface topography is less complex than the mussel bed spatial2
pattern
Fig. 1. Generation of the Koch curve, a self-similar shape with fractal dimension of D51.26.
mud-mussel boundary in aerial view at the 25325-cm scale ranged from 1.36 to 1.86. If bed surface topography is less complex than the bed spatial pattern, then over distances of about 25 cm, surface profile fractal dimension values should be smaller than those obtained in the horizontal plane by Snover and Commito (1998).
2. Methods
2.1. Study site
temperature from 0 to 108C. In the summer, shallow water moving on to or off the flat can reach 218C. Salinity is usually about 30‰.
2.2. Field and laboratory procedures
To obtain background information on the density and size–class distribution of live
2
and dead mussels at the site, on 27 May 1994, eight 20 cm30.02 m cores were taken at uniform positions 5 m apart, along two parallel transects 4 m apart from each other and 4 m in from the upper margin of the mussel bed. Core contents were sieved in situ on 0.5-mm mesh. The residue was placed in buffered formalin, stained with rose bengal, and sorted. The lengths of all live mussels and whole disarticulated valves were measured to the nearest 0.1 mm with vernier calipers or an ocular micrometer.
Casts of the mussel bed surface were made 8–11 June 1994 approximately 30 m east of the site where cores had been taken. Casts were made of three surface types found in a patchy mosaic within a 5310-m area: zero percent cover (no live mussels; empty, disarticulated whole and broken valves covered the bottom); intermediate percent cover (approximately 50% live mussel cover); and high percent cover (approximately 85% live mussel cover; in nearly 20 years of observations at soft-bottom mussel beds in eastern Maine, 100% cover has sometimes been observed at other sites, but not at Bob’s Cove). Cast locations were spaced as far apart as possible to achieve overdispersion. Fifteen coffer dams of thin aluminum sheeting were constructed in a range of sizes with nominal width3length3height dimensions up to 30330330 cm. Plaster of paris mixed in situ with seawater was poured to a thickness of approximately 10 cm into dams that had been inserted several centimeters into the bed. Casts were made at low tide, so they did not capture the positions of mussels while feeding, nor was there any attempt to determine the effects of the casting process on mussel behavior. Casts were left in place to harden until low tide the next day and then carefully retrieved with all live mussels and other surface objects embedded within the plaster. Casts were immediately returned to the laboratory and placed in an oven for approximately 4 h at 1508C to aid in the plaster curing process and kill any live organisms adhering to the plaster. Casts were re-measured and did not change size or shape during curing. In test casts, surface objects removed from pre-cured casts fit exactly back into their same locations after the casts had been cured.
To obtain the longest surface profiles possible, only the five largest casts were used in the analysis: two high, two intermediate, and one zero percent cover. The central area 2.5 cm in from the perimeter of each cast was utilized in order to avoid possible edge effects. To preserve cast topography, all mussels and whole and broken valves were left embedded in the plaster. The surface of each cast was coated with black graphite. An electric bandsaw with a 1.6-mm thick blade was used to cut each cast into 1.3-cm thick cross-sections. Each of the resulting 88 cross-sections was scanned into a computer. The black graphite and gray shells contrasted sharply with the white plaster exposed on the sides of the cross-sections, allowing the bed surface profile to be readily determined.
2.3. Calculating fractal dimension
boundary-grid method (Sugihara and May, 1990) was used to determine the fractal dimension of each profile. Eight grids (i51, 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 n5332 56, 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
N5kn (1)
where D equals the negative of the slope from the linear regression of log n against2
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 14.8863.69 (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 (P2 2 ,0.001), with r values 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.
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 (mean51.24260.046, 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
Table 1
Summary statistics for five mussel bed surface casts Mussel cover Number of Fractal dimension
profiles
1 Range Mean6S.E. Median Significance
High
Cast 1 16 1.16461.310 1.24260.046 1.251 a
Cast 2 16 1.11561.218 1.17560.026 1.177 a
Intermediate
Cast 1 16 1.12461.289 1.20260.058 1.190 a
Cast 2 17 1.03161.227 1.12260.040 1.130 b
Zero
Cast 1 23 1.04961.172 1.10660.033 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
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 12330-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.
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. 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 D51.031 and our D51.110. At 50% mussel cover, their D51.122 and our D51.130 and 1.190. At 85% mussel cover (our highest percent cover), their D51.186 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
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
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
soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. Our results demonstrated that the bed surface was fractal over a spatial scale of more than two orders of magnitude and supported two hypotheses about fractal dimension. First, the bed surface (in vertical profile) was less complex than the bed spatial pattern (in horizontal plane). Second, bed surface fractal dimension was similar to the theoretical value of the Koch curve (D51.26). We predict that fractal dimension will be larger than this value when mussel hummocking occurs, meaning that fractal dimension can serve as an indicator of high mussel settlement rate and rapid growth. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.
Acknowledgements
We thank Gettysburg College Environmental Studies Program students J. Abraham-son, D. Risso, W. Steel, and R. Winklemann for assistance in the field. W. Ambrose, B. Beal, A. Commito, P. Fong, and an anonymous reviewer made valuable comments on an earlier version of the manuscript. This research was supported in part by grants to J.A.C. from the Gettsyburg College Grants Advisory Commission and to B.R.R. from the Gettysburg College Senior Research Fund. [SS]
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