Structural complexity in mussel beds: the fractal geometry ...directory.umm.ac.id/Data...

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Journal of Experimental Marine Biology and Ecology255 (2000) 133–152

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Structural complexity in mussel beds: the fractal geometry ofsurface topography

*John A. Commito , Brian R. RusignuoloEnvironmental 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 structureand alters boundary-layer flow over the bottom. Despite its importance, there is little agreement onhow to define and measure surface complexity. The purpose of this investigation was to utilizefractal geometry of vertical cross-section profiles to characterize the surface topography of thesoft-bottom mussel bed (Mytilus edulis L.) at Bob’s Cove, ME, USA. Mussels there have beenshown previously to have spatially ordered fractal characteristics in the horizontal plane. Twohypotheses were tested. The first was that the bed surface is fractal over the spatial scale of1.44–200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, ourmodel for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is lesscomplex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Bothhypotheses 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 inthe 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) valuessimilar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than thebed spatial pattern because every profile had D , 1.36, the smallest value previously obtainedfrom aerial views of the bed. The investigation demonstrated for the first time that surfacetopography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamicprocesses 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 dimensionthat can result from other profiling methods reported in the literature. The results demonstrate thatfractal dimension can be useful in the analysis of habitat space and water flow over any irregularseafloor surface because it incorporates the size, shape, and scale of roughness elements into asimple, 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).

0022-0981/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PI I : S0022-0981( 00 )00285-9

134 J.A. Commito, B.R. Rusignuolo / J. Exp. Mar. Biol. Ecol. 255 (2000) 133 –152

1. Introduction

The seafloor is not flat. Soft-bottom surface topography is irregular because intertidaland 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 surfacerelief (Guichard and Bourget, 1998), as do structures built by animals and plants thatproject above the bed, including polychaete tubes, bivalve shells, seagrasses, marshgrasses, 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) thatlive on them have complex shapes. Human activities such as commercial dredging andtrawling 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 structurefor 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 regulatingforaging patterns (Dolmer, 1998; Erlandsson et al., 1999). Perhaps most significantly, italters boundary-layer flow over the bottom (Butman et al., 1994; Ke et al., 1994; Greenet al., 1998). This interaction of flow and substrate heterogeneity affects larvalsettlement (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; Leonardet al., 1998, 1999; Lenihan, 1999). Surface complexity also influences the flow-mediatederosion, 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 thesurface 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 andsoft bottoms worldwide. They have some of the highest densities, biomass, andrespiration rates of all marine systems (Nixon et al., 1971; Seed, 1976). Soft-bottommussel beds control species composition and abundance by providing habitat space,outcompeting other surface dwellers, altering flow regimes, and creating a sedimentaryenvironment high in organic material and low in oxygen, thus favoring some species andeliminating others (Radziejewska, 1986; Asmus, 1987; Commito, 1987; Commito andBoncavage, 1989; Commito and Dankers, in press; Dittmann, 1990; Svane andSetyobudiandi, 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´ ´of sedimentation and resuspension (Frechette and Bourget, 1985a,b; Frechette et al.,

J.A. Commito, B.R. Rusignuolo / J. Exp. Mar. Biol. Ecol. 255 (2000) 133 –152 135

1989; Widdows et al., 1998). Experiments with non-living mussel mimics have shownthat their surface structure can be as important as the biological activities of the musselsthemselves in regulating species composition and abundance within mussel bedcommunities (Ricciardi et al., 1997; Crooks and Khim, 1999). In addition, surfaceroughness 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 benthicsystems, there is no consensus in the literature about how to define and measure thecomplexity 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 morequantitative estimates of structure by extracting information from cross-sectional profilesof 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 toproduce a numerical value that characterizes topographic complexity.

The application of fractal geometry can also utilize profile data. Le Tourneux andBourget (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 fewinvestigators have attempted to use fractal geometry to quantify surface complexity ofthe 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 tocharacterize some important intertidal systems, including soft-bottom mussel beds.

A fractal shape is a jagged, non-differentiable curve, often associated with self-similarstructure (objects with similar shape) that is repeated at different scales (Hastings andSugihara, 1993). Mussel beds fit this description because they are composed of multiplecopies of individual mussels that are different in size but similar in shape. Fractalgeometry allows the investigator to gain insights into surface topography that theabove-mentioned techniques cannot. For example, the chain technique (Aronson andPrecht, 1995; Beck, 1997) does not differentiate between one large surface irregularityand two smaller ones whose profile length adds up to that of the large one. Theroughness calculation (Ke et al., 1994; Green et al., 1998) uses the average obstacleheight above the bottom. It does not take into account their variability in height, and itignores the spatial array of objects. These techniques provide extremely valuableinformation, but they were not designed to characterize shape per se. Fractal geometry isuseful because it pinpoints the spatial scale at which roughness elements changeconfiguration. The fractal dimension (D) can distinguish between a surface that has


many small irregularities and one that has a few large ones. It allows investigators todetermine if large surface irregularities are similar in shape to small ones and toaggregated copies of small ones. These are important descriptors for mussel beds, kelpbeds, boulder fields, coral reefs, and other seafloor habitats made up of multiple unitswith different sizes and similar shapes.

Snover and Commito (1998) recently used fractal geometry to characterize the spatialpattern of Mytilus edulis in the soft-bottom mussel bed at Bob’s Cove, Maine. Theycalculated fractal dimensions of mussel aggregation outlines in quadrats using theboundary-grid technique (Sugihara and May, 1990) and discovered that the complex,patchy distribution of mussels within the bed was spatially ordered. Over two orders ofmagnitude in spatial scale, the fractal dimension values varied predictably with percent

2cover (concave-downward second-order regression curve, r 5 0.94), density (concave-2downward second-order regression curve, r 5 0.92), and Morisita’s index of spatial

2dispersion (negative-slope regression line, r 5 0.82). The mussel bed was fractal in thehorizontal 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 wasinvestigated. Two hypotheses were tested:

1.1. H : the bed surface profile is fractal, with 1 , D , 1.261

A mussel bed is composed of multiple copies of individual mussels of different sizesbut with the same approximate shape. Thus, a bed surface is a likely candidate forself-similarity. If so, then bed surface profiles should be fractal. By definition, this meansthat profiles have fractal dimension values lying somewhere between 1 (Euclidianstraight line or smooth curve) and 2 (complex curve so convoluted that in the limit it is acompletely space-filling Euclidian plane).

More specifically, the familiar, well-studied Koch curve (Fig. 1; Hastings andSugihara, 1993) can serve as a model for the mussel bed surface at Bob’s Cove. TheKoch curve is made up of repeated, identically shaped triangular units of different sizesthat grow out from the existing profile, not unlike the settlement and growth of newmussels in a soft-bottom bed of existing semi-infaunal mussels. The Koch curve has afractal dimension of 1.26. Mussel beds might display what are termed ‘statistical’ fractalpatterns (Hastings and Sugihara, 1993) rather than the perfect fractal qualities of thealgorithm-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 shouldhave fractal dimensions approaching 1.26.

1.2. H : mussel bed surface topography is less complex than the mussel bed spatial2

pattern

Over relatively short horizontal distances, the vertical relief within the mussel bed atBob’s Cove is approximately the height of the largest mussel. But in aerial view, musselbeds have more highly irregular shapes with gaps, invaginations, and evaginations manymeters in length. Snover and Commito (1998) found that fractal dimension values of the


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 25 3 25-cm scale ranged from 1.36 to 1.86. Ifbed surface topography is less complex than the bed spatial pattern, then over distancesof about 25 cm, surface profile fractal dimension values should be smaller than thoseobtained in the horizontal plane by Snover and Commito (1998).

2. Methods

2.1. Study site

The study site was at the eastern end of the Mytilus edulis bed at Bob’s Cove, anintertidal flat of sandy mud in Jonesboro, Washington County, Maine, USA (448339N:678359W), near the mussel bed locations used in earlier studies describing the soft-bottom community inside and outside of the bed (Commito, 1987; Commito andBoncavage, 1989) and the fractal geometry of the mussel spatial pattern (Snover andCommito, 1998). The bed extends across the mouth of the cove in the lower intertidaland shallow subtidal zones. Air temperature ranges from 2 35 to 358C, and water


temperature from 0 to 108C. In the summer, shallow water moving on to or off the flatcan 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 live2and dead mussels at the site, on 27 May 1994, eight 20 cm 3 0.02 m cores were taken

at uniform positions 5 m apart, along two parallel transects 4 m apart from each otherand 4 m in from the upper margin of the mussel bed. Core contents were sieved in situon 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 weremeasured 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 eastof the site where cores had been taken. Casts were made of three surface types found ina patchy mosaic within a 5 3 10-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% livemussel cover; in nearly 20 years of observations at soft-bottom mussel beds in easternMaine, 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. Fifteencoffer dams of thin aluminum sheeting were constructed in a range of sizes with nominalwidth 3 length 3 height dimensions up to 30 3 30 3 30 cm. Plaster of paris mixed in situwith seawater was poured to a thickness of approximately 10 cm into dams that hadbeen inserted several centimeters into the bed. Casts were made at low tide, so they didnot capture the positions of mussels while feeding, nor was there any attempt todetermine the effects of the casting process on mussel behavior. Casts were left in placeto harden until low tide the next day and then carefully retrieved with all live musselsand other surface objects embedded within the plaster. Casts were immediately returnedto the laboratory and placed in an oven for approximately 4 h at 1508C to aid in theplaster curing process and kill any live organisms adhering to the plaster. Casts werere-measured and did not change size or shape during curing. In test casts, surface objectsremoved from pre-cured casts fit exactly back into their same locations after the castshad been cured.

To obtain the longest surface profiles possible, only the five largest casts were used inthe analysis: two high, two intermediate, and one zero percent cover. The central area2.5 cm in from the perimeter of each cast was utilized in order to avoid possible edgeeffects. To preserve cast topography, all mussels and whole and broken valves were leftembedded in the plaster. The surface of each cast was coated with black graphite. Anelectric bandsaw with a 1.6-mm thick blade was used to cut each cast into 1.3-cm thickcross-sections. Each of the resulting 88 cross-sections was scanned into a computer. Theblack graphite and gray shells contrasted sharply with the white plaster exposed on thesides of the cross-sections, allowing the bed surface profile to be readily determined.

2.3. Calculating fractal dimension

The profile from one face of each cross-section was included in the analysis. The


boundary-grid method (Sugihara and May, 1990) was used to determine the fractaldimension of each profile. Eight grids (i 5 1, 2, . . . , 8) were superimposed on the first20-cm length of each cross-section. Each grid contained squares with a side length of n

ipixels, where n 5 3 3 2 5 6, 12, 24, 48, 96, 192, 384, or 768, resulting in squares withside lengths of 1.44–200 mm. The number of squares entered by each profile (N) wascounted for each grid. Fractal dimension, D, was determined from the followingequation (where k represents a constant):

2DN 5 kn (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 spatialscale, the nonparametric runs test was used to determine if the D values for consecutivesurface profiles in each cast had random variability or, alternatively, were seriallycorrelated (Zar, 1984). To test for differences in D among the five casts, thenonparametric Kruskal–Wallis test (one-way ANOVA on ranks) and Dunn’s a posteriorimultiple comparison test were used (Zar, 1984).

3. Results

3.1. Mussel density and size–class structure

22Mean 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 32–36mm 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 thelargest 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 threemussel bed surface types (Figs. 3 and 4). For every one of the 88 surface profiles, the

2regression of log n versus log N was highly significant (P,0.001), with r values2 2

ranging from 0.984 to 0.999. All the mussel bed profiles were fractal. There was never abreak in a regression line indicating a change in slope, meaning that the fractaldimension was the same for a given profile over the entire range of grid square sizesfrom 1.44 to 200 mm.

Within each of the five casts, D values were not serially correlated. The runs testresults 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 thelargest mean and median D values were from a high percent cover cast, and the smallest


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 betweenpercent 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 intermediatepercent cover Cast 1 had D.1.26, the value for the Koch curve. These 10 D valueswere only slightly larger than 1.26, ranging from 1.264 to 1.310, and represented just11.4% of all the profiles. The remaining 88.6% of the profiles had D,1.26. Every casthad mean and median values of D,1.26 (Table 1), the largest being high percent coverCast 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

Fractal dimension values for individual profiles ranged from 1.049 to 1.310. Everyone 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 25325-cm quadrat bySnover and Commito (1998). Thus, the second hypothesis was supported: bed surfacetopography was less complex than bed spatial pattern.


Fig. 3. Examples of profiles from bed surface casts made at locations with high, intermediate, and zero musselcover. The regressions to calculate fractal dimension from the same three profiles are shown in Fig. 4.


Fig. 4. Boundary-grid method regressions from the same profiles shown in Fig. 3. Fractal dimension is equalto the negative of the slope of the regression line. Note the correspondence between apparent surfaceroughness and the fractal dimension in each of the three examples.


Table 1Summary statistics for five mussel bed surface casts

Mussel cover Number of Fractal dimensionprofiles

1Range Mean6S.E. Median Significance

HighCast 1 16 1.16461.310 1.24260.046 1.251 aCast 2 16 1.11561.218 1.17560.026 1.177 a

IntermediateCast 1 16 1.12461.289 1.20260.058 1.190 aCast 2 17 1.03161.227 1.12260.040 1.130 b

ZeroCast 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 afterKruskal–Wallis one-way ANOVA on ranks, P.0.05).

4. Discussion

The results from Bob’s Cove supported the two hypotheses about the fractal geometryof mussel bed surface topography. They demonstrated that the bed surface was fractalover a spatial scale of more than two orders of magnitude (grid squares ranging from1.44 to 200 mm in side length, a factor of nearly 140 times). The upper bound of the bedsurface fractal dimension was determined empirically from the previously determinedfractal 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.


dimension was predicted theoretically from the Koch curve. The results indicate thatfractal dimension is a parameter that can be useful in characterizing the irregular bedsurface 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 (Snoverand Commito, 1998). They show self-similarity across spatial scales, so inferences fromstudies on spatial assembly in other systems may be applicable to this benthic species.For example, Kaandorp (1991, 1994, 1999) developed relatively simple fractalgenerators that predicted growth patterns and shapes of model sponges and corals underdifferent water flow and light regimes. His field experiments verified the predictions anddemonstrated, for example, that sponge branching complexity and fractal dimension aregreater in fast flow than slow flow. Mussel beds are an excellent model system forKaandorp’s approach because, like his colonial organisms, they are made up of manysmall units of similar shape assembled into larger, yet still similar, shapes. Unlike theKoch curve, mussels are not perfect triangles, and mussel beds do not consist oftriangles deployed in a precisely regular spatial arrangement. That the mussel bed profileD values in this study could be predicted from the Koch curve suggests that futurefractal generators more sophisticated than this first attempt could lead to useful bedshape 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 tosuccessful 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 alarge spatial scale, Schwinghamer et al. (1996) quantified the impact of trawling onbenthic habitat structure in an area of sandy sediment on the Grand Banks ofNewfoundland using Hilbert-transformed high resolution acoustic signal profiles from12330-cm rectangular areas on the bottom. Fractal dimensions were calculated for theshapes of the electronic signals as a proxy for the actual surface profiles. Post-trawling Dvalues were smaller than pre-trawling values, indicating that trawling made the bottomsmoother within each trawl path.

Working at much smaller spatial scales, Le Tourneux and Bourget (1988) showed onplastic settlement panels that fractal dimension values were higher for locations selectedby barnacle (Semibalanus balanoides) larvae than for nearby unselected sites. Theinvestigators measured surface microheterogeneity profiles with a diamond stylus thatwas moved over the surface. Hills et al. (1999) demonstrated that settlement density ofS. balanoides larvae was positively correlated with both Euclidean and fractal surfacecomplexity indices of plastic panels they created with different surface textures. Theyscanned 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 shorecovered with mussels (Mytilus galloprovincialis) and barnacles (Chthamalus stellatus).They related abundances of different sizes and morphs of the snail Littorina saxatilis tothe fractal dimensions of the profiles. Erlandsson et al. (1999) used the same technique


to show that the fractal dimension of the rocky shore surface was a good predictor ofsnail (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 dimensionwas 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 interpretedwith caution, however. For some surfaces, the techniques are as accurate as the profilesderived 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 byignoring 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 thanoffset by the savings in time compared to making surface casts that are then cut intosections to produce true profiles.

4.2. Fractal dimension and percent cover

Kostylev et al. (1997) observed a monotonically increasing relationship betweenmussel cover and the fractal dimension of the mussel- and barnacle-covered rock surfaceat their site, although variability in D was quite large at high percent cover values. Wedid not find a clear positive relationship in our study, nor is there any reason to believethat D will automatically increase with increasing mussel cover. For example, Snoverand Commito (1998) found that the largest D values for the mussel spatial pattern inaerial view occurred at intermediate values of mussel cover and density. Similarly, forthe bed surface, an intermediate percent cover of patchy mussels of different ages andlengths could have a highly irregular surface and a large D, while 100% cover of tightlypacked mussels might have a relatively smooth surface with a small D. The latter wouldbe 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 undetectedcomplexity under the overhanging portions, leading to an underestimate of fractal dimension.


Fig. 7. Top panels illustrate a mussel bed with intermediate percent cover, complex profile, and large fractaldimension. Bottom panels illustrate a mussel bed with high percent cover, simple profile, and small fractaldimension.

in the gaps between the larger individuals (Fig. 7). One possible reason for the differentresults between Kostylev et al. (1997) and our study might be that maximum cover attheir site was higher than at Bob’s Cove, reaching 100%. Furthermore, theirs was arocky shore site with epifaunal Mytilus galloprovincialis, while ours was a soft-bottomsite with semi-infaunal M. edulis. Despite these differences, the D values in both studieswere similar. At 0% mussel cover, their D51.031 and our D51.110. At 50% musselcover, their D51.122 and our D51.130 and 1.190. At 85% mussel cover (our highestpercent cover), their D51.186 and our D51.177 and 1.251. The fact that their D valueswere generally lower than ours is consistent with the idea that the down-from-aboveprofiling technique produces smaller D values than does cross-section profiling. Therocky shore sites of Beck (1998) and Erlandsson et al. (1999) had 0% mussel cover anddown-from-above D values of approximately 1.02, in close agreement with Kostylev etal. (1997). The similar results from the three hard-bottom studies and our soft-bottombed suggest that fractal dimension may be useful in relating surface topography fromdifferent locations and bottom types.

4.3. Fractal dimension, roughness elements, and flow

Water flow calculations require information about the height of roughness elementsabove the bottom (Eckman, 1990; Ke et al., 1994; Green et al., 1998). Few studies havefocused specifically on water flow over Mytilus edulis beds. Knowledge of the fractaldimension of bed surfaces might be useful in designing and interpreting laboratory andfield studies of this type. For example, Butman et al. (1994) used live Mytilus edulis tosimulate a mussel bed in a laboratory flume. They were able to determine the contours ofturbulent stress and food depletion over the bed at different flow velocities. The musselbed increased turbulent stress by a factor of up to three in slow flow and a factor of morethan 10 in fast flow. The mussels they used were all young and similar in size (mean61


22S.D.52.760.2 cm), with a density of 75.5611.4 individuals 100 cm . Based on thisinformation alone (i.e., small, same-sized, tightly packed individuals), it is likely that thefractal dimension was smaller than what was found at Bob’s Cove with its patchydistribution of different-sized mussels. It is interesting to consider what the turbulentstress contours over the flume bed would look like with surface topographies of differentfractal dimensions. And just as flow properties and food depletion might be influencedby the fractal dimension of the bed, the fractal dimension values almost certainly dependon flow rate and food concentration. For example, Butman et al. (1994) discovered thatmussels changed position and orientation so that mean mussel bed height in the flumewas approximately 70% higher after their phytoplankton enhancement experimentcompared to experiments with natural phytoplankton concentrations. It is likely thatfractal dimension is a dynamic feature of mussel beds, meaning that the development ofnon-destructive, in situ techniques to measure D could provide a way to monitor musselfeeding rates over space and time.

Other studies that have measured water flow over Mytilus edulis beds havedemonstrated that roughness elements play a key role in controlling a number ofecologically important processes, including biodeposition and erosion rates (Widdows etal., 1998); food-regulated growth and vertical gradients of particulate organic material

´ ´ ´(Frechette and Bourget, 1985a,b; Frechette et al., 1989; Frechette and Grant, 1991); andremoval 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 serveas a useful measure of surface roughness when considering hydrodynamically controlledphenomena.

4.4. Fractal dimension and mussel bed stability

Mussel beds are extremely dynamic and can undergo rapid changes in percent coverand density (see review in Commito and Dankers, in press; Mytilus edulis: Nehls andThiel, 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 andbecomes fragmented and more complex, causing D to increase. Thus, fractal dimensionshould 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-buildingamphipod Corophium volutator created an irregular sediment surface of plateaus andpools. The bed became smoother and more uniform after mass mortality of theamphipods. Aronson and Precht (1995) found that coral reef topographic complexityestimated with the chain technique was not inversely proportional to persistence. Surfacecomplexity (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


predicted by the Sugihara and May model. Instead, it increased as cover and density roseto intermediate or high levels (Kostylev et al., 1997; Snover and Commito, 1998; Fig. 5and Table 1 from this study).

In some soft-bottom mussel beds in Maine, the local abundance can be so highbecause of high rates of recruitment and growth that clumps of mussels form verticalcolumns taller than they are wide, even if percent cover is less than 100% because ofnearby bare patches (Commito, personal observation). This hummocking lifts themussels up off the sediment surface, resulting in very complex bed topography. TheKoch curve no longer describes such a bed, and we predict that the fractal dimensionwould be higher than what was found at Bob’s Cove. If so, then fractal dimension mayserve 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 musselscarried away from beds are a major form of dispersal (Reusch and Chapman, 1995,1997). Fractal dimension is probably high for fragmented beds and newly dispersedclumps. They have high perimeter: area ratios with a surface topography that createsimportant edge effects at the margins, where water flow changes abruptly as it meetsmussels projecting above the substrate (Butman et al., 1994). Rates of larval recruitmentand body growth are higher at the edges of mussel patches than in the interior (Bertnessand Grosholz, 1985; Okamura, 1986; Lin, 1991; Stiven and Gardner, 1992; Svane andOmpi, 1993). Growth of mussels out from the edges fills in the gaps between patches(Petraitis, 1995; Reusch and Chapman, 1997), most likely lowering the fractaldimension.

On the other hand, mussel mortality rates due to ice scour and predation are oftenhigher at the edges, demonstrating the trade-offs that occur with respect to position(Bertness and Grosholz, 1985; Lin, 1991; Stiven and Gardner, 1992; McGrorty andGoss-Custard, 1993; but see Reusch and Chapman, 1997, and Dolmer, 1998). In fact, thepresence of predators causes individual mussels to move towards each other. There isreduced per mussel predation risk because of smaller perimeter: area ratios as theycoalesce 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 comprisinga few large clumps with simple outlines and small perimeter: area ratios would have alower fractal dimension than a bed with a few large, irregularly shaped clumps or manysmall clumps.

Mussel bed fragmentation, clump dispersal, predation, recruitment, growth, andmovement all affect and are affected by bed shape and surface topography. Knowledgeof the fractal geometry of mussel beds might have predictive power concerning theseecological processes because fractal dimension detects and quantifies the edges andsurface irregularities that play a role in controlling them.

5. Conclusion

This investigation demonstrated for the first time that surface topography of a


soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processesand habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractaldimension that can result from other profiling methods reported in the literature. Ourresults demonstrated that the bed surface was fractal over a spatial scale of more thantwo 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 (inhorizontal plane). Second, bed surface fractal dimension was similar to the theoreticalvalue of the Koch curve (D51.26). We predict that fractal dimension will be larger thanthis value when mussel hummocking occurs, meaning that fractal dimension can serve asan indicator of high mussel settlement rate and rapid growth. The results demonstratethat fractal dimension can be useful in the analysis of habitat space and water flow overany irregular seafloor surface because it incorporates the size, shape, and scale ofroughness 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 anearlier 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 theGettysburg College Senior Research Fund. [SS]

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