(Spatial scales of variation)
I. Fu
Abstract:
The spatial scales of variation of three intertidal invertebrates were determined using a
nested sampling design and optimal quadrat size for each species. Sampling was done at sites,
which were kilometers apart; subsites within sites, which were roughly 100 meters apart; transects
within subsites, which were about 10 meters apart; and quadrats, which were on the scale of
meters apart. An analysis of variance revealed that Lottia limatula is patchy on the spatial scale of
tens of meters, while Serpulorbis squamigerus is patchy on the spatial scale of hundreds of meters,
with considerable spatial variation within the patches themselves. For Littorina planaxis, there
were no significant differences at any of the three levels of nesting, indicating that the snail is fairly
evenly distributed in the sites sampled. The index of departure from the Poisson distribution and
the index of aggregation for S. squamigerus and L. planaxis indicate that both species tend to
have clumped distributions, which agree with the natural histories of these organisms. The indices
are more ambiguous for L. limatula, but the presence of a few high and many low counts, as well
as aspects of the natural history of this organism, indicate a clumped distribution. Determination of
the spatial scales of variation for an organism to be studied permits the formation of a sampling
design that maximizes accuracy and precision while minimizing the total costs in terms of time,
effort, and money.
I. Fu
(Spatial scales of variation)
Introduction:
One problem often encountered in sampling is that the patchiness of the distribution of an
organism is often not known before a study is conducted. A common method to assess differences
between sites is to take replicate samples, on a spatial scale of meters, within sites that themselves
may be separated by kilometers. The problem with this sampling design is that it will not be able
to detect variation on spatial scales between meters and kilometers. If an organism is patchy on the
scale of hundreds of meters, all of the one-meter replicates may fall between two patches, resulting
in a low mean abundance for that organism in that site. On the other hand, the replicates for the
other site may all fall within one large patch, resulting in a high mean abundance at the second site.
Using this sampling method, the two sites would incorrectly appear to have very different mean
abundances, when they are actually fairly similar. One solution to this problem is to employ a
nested sampling design, in which replicate samples are taken at different spatial scales so that
variations at intermediate spatial scales can be distinguished.
In this study, sampling for three intertidal invertebrates was done following a fully nested
design. The species studied were Lottia limatula, a limpet found in the middle to low intertidal
zone; Serpulorbis squamigerus, a tube snail found low in the intertidal zone or subtidally; and
Littorina planaxis, a periwinkle abundant on the rocks of the high intertidal zone. Although
numerous studies have been conducted on the limpet, little work has been done on its abundance
and distribution. Some studies have determined the number of limpets at one or several particular
sites in the Monterey Peninsula area (Ruth 1948, Hahn 1985), but the overall patterns of
distribution for this well-studied limpet are not known. S. squamigerus occurs in concentrations
of up to 650 m 2 in southern California (Pequegnat 1964) and has been seen singly or in clusters in
central California (Morris et al. 1980). In the last decade or so, the abundance of the tube snail has
increased dramatically in the Monterey area (C. Baxter, pers. comm.). Although the species is
noted as gregarious by a number of studies (Holmes 1900, Morton 1965, MacGinitie &
MacGinitie 1968), the larger spatial scales of variation for this organism are not known. L.
planaris group together in cracks or other irregularities in the rock surface, and it has been
I. Fu
(Spatial scales of variation)
theorized that this clumping behavior helps reduce desiccation (Miyamoto 1968). Like the other
two species, the larger spatial scales of variation have not been studied.
Use of a fully nested sampling design enabled the determination of the spatial scales of
variation for these three invertebrates. This study illustrates the importance of doing such a nested
sampling design to obtain accurate estimates of the abundance and distribution of organisms.
Materials and Methods:
Determination of optimal quadrat size
This portion of the study was conducted in the rocky intertidal area at Hopkins Marine
Station (HMS) in Pacific Grove, California (Fig. 1). Sampling procedures consisted of placing
20-meter transects in areas of comparable vertical height and then counting the number of desired
organisms present in randomly spaced quadrats. Notes were also taken on the microhabitat of each
quadrat.
For L. limatula, two transects were located in the mid-intertidal zone of an area protected
from wave action, with one crossing predominantly horizontal surfaces and the other passing
primarily over vertical surfaces. A third transect was situated in a mid-intertidal area that was more
exposed to wave action, and it encompassed both vertical and horizontal surfaces. For each
transect, five 0.0625 m2 quadrats were sampled. The counts were repeated at the same locations
using 0.25 m2 and 1.0 m2 quadrats. For S. squamigerus, one transect was run in the low-intertidal
zone, and the numbers of tube snails were counted in 0.0625 m2 and 0.25 m2 quadrats. Finally,
one transect was run in the high-intertidal zone for L. planaris, with 0.0625 m2 and 0.25 m¬
quadrats.
The precision of the counts for each transect was calculated using the following formula:
P =SE/X,
where p is the precision of a set of counts, SE is the standard error of the counts for one transect,
and Xis the average count per quadrat for that transect (Underwood 1981). Then, setting the
desired precision at the lowest value obtained for the different quadrat sizes, the number of
I. Fu
(Spatial scales of variation)
quadrats of each size that would need to be counted to obtain this level of precision was determined
for each quadrat size and species. Based on the assumption that a 1.0 m2 quadrat takes 16 times
the effort needed to examine a 0.0625 m2 quadrat, the cost of counting the number of quadrats
needed to get the desired precision was calculated for each quadrat size using the following
equation:
C=nc,
where Cr is the total cost of the sampling effort (time), n'is the number of quadrats of a particular
size needed to obtain the desired precision level, and cis the cost in terms of time and effort needed
to count a quadrat of a particular size (Underwood 1981). The quadrat size that resulted in the
lowest total cost for each species was then used in the second part of the study.
Determination of the spatial variation of three species
Sampling methods for the second part of the experiment were similar to those used in the
preliminary study, except that the transects were 15 rather than 20 meters in length and only the
optimal quadrat size for each species was used. For all the species, the study involved sampling at
the following four spatial scales: sites, which were kilometers apart; subsites within sites, which
were separated by roughly 100 meters; transects within subsites, which were about 10 meters from
each other; and quadrats, which were on the scale of meters apart. This was a fully nested
sampling design, with a specific number of quadrats counted for each transect, three transects
counted in each subsite, and two subsites examined for every site that was chosen (Fig. 2).
For L. limatula, the four sites chosen were HMS, Carmel, Point Pinos, and the Great Tide
Pool (Fig.1). Along each transect, 20 0.0625 m2 quadrats were counted. For S. squamigerus,
three sites were chosen-HMS, the area between Fisherman's Wharf and the Coast Guard wharf,
and Point Pinos (Fig. 1). Five 0.25 m2 quadrats were counted for each transect. For L. planaris,
the three sites chosen were HMS, Carmel, and Point Pinos (Fig. 1), and eight 0.0625 m2 quadrats
were counted per transect.
I. Fu
(Spatial scales of variation)
Data Analysis
Cochran’s tests were run on the data to detect heterogeneity of variances (Underwood
1981). For all three species, the Cochran’s test were significant (pæ0.05), indicating that the
variances were too heterogeneous to meet the assumptions of an ANOVA. To increase the
homogeneity of the transect variances, the limpet and tube snail counts were transformed by taking
the square root of the count for each quadrat and then adding the result to the square root of the
count plus one (Sokal & Rohlf 1995). L. planaris data was transformed by taking the square roots
of the counts. After transformation, only the limpet data remained heterogeneous. Subsequent
tests for these data were conducted at a-0.0l rather than 0.05. The transformed data were then
analyzed using a three factor nested ANOVA.
To determine whether the distributions of the species were random, regular, or aggregated,
the variance to mean ratio, the index of departure from the Poisson curve, and the index of
aggregation were calculated for each of the species (Hurlbert 1990). The index of departure was
calculated using the following equation:
Dp=1-2 min (qy0, qk0.
were Dp is the index of departure, w is the largest number of individuals observed in any quadrat,
k is the number of individuals per quadrat, qup is the number of quadrats expected to contain k
individuals in a Poisson distribution, and qua is the number of quadrats containing k individuals in
the observed distribution (Hurlbert 1990). The index of aggregation was calculated using the
following formula:
IM-(X/X-1) (1/X) (Var /X+X-1),
where IM is the index of aggregation, Xis the total number of individuals that were counted, Xis
the average count for all the quadrats sampled, and Var is the variance. IM measures how many
times more likely it is that two randomly selected individuals will be from the same quadrat than if
the X individuals in the population were distributed at random (Hurlbert 1990).
(Spatial scales of variation)
I. Fu
Results:
Mean limpet densities per transect ranged from 0.05-1.45 m2 (Fig. 3). There were
significant (pe0.05) differences among transects within sites, but not at either of the larger spatial
scales (Table la). A significant difference at the transect level indicates that, in at least one subsite
within one site, there were significantly different mean limpet densities between transects. The
largest differences among the transect averages were in Subsite 1 in Carmel, which most likely
contributed to the significant result at the transect level (Fig. 3). There was some variation between
the average limpet counts for the subsites at HMS, Carmel, and the Great Tide Pool, but these
differences were not significant (p»0.05).
Mean tube snail densities per transect ranged from 0-584 m22 (Fig. 4). There were
significant differences among transects and subsites for S. squamigerus but not among sites
(Table 1b). Large differences among transects and subsites at HMS most likely contributed to the
significant differences at those levels of nesting (Fig. 4, Table 1). An apparently large difference
between the average tube snail abundances at the Fisherman's Wharf and Point Pinos sites was not
significant (p2O.05).
For L. planaris, mean densities per transect ranged from 145.6 606.4 m2 (Fig. 5). There
were no significant differences at any of the three spatial scales.
Residual variance contributed 84.3% of the total variance for L. planaxis and 92.8% of the
total variance for L. limatula. (Fig. 6). For the tube snail, the four sources of variation contributed
roughly equal percentages of the total variance (Fig. 6).
The variance to mean ratios and indices of aggregation calculated for the three species were
all greater than one (Table 2). The indices of departure from the Poisson curve ranged from a high
of 0.844 for S. squamigerus to a low of 0.125 for L. limatula, with an intermediate value of 0.632
obtained for L. planaxis (Table 2). These values indicate that the observed distribution for the
limpet has a fairly large degree of overlap with a Poisson distribution of the same mean (Fig. 7a).
while the observed distribution for the tube snail has a relatively small degree of overlap (Fig. 7b).
I. Fu
(Spatial scales of variation)
The observed distribution for L. planaris has an intermediate degree of oversap with the
corresponding Poisson distribution (Fig. 7c).
Discussion:
Because L. limatula had significant differences among transects (within subsites), it is
apparently patchy on a scale of tens of meters. Optimal sampling efforts should thus concentrate
on this spatial scale to detect changes among sites. In contrast, S. squamigerus varied significantly
among both transects and subsites. This indicates that the tube snail is patchy on a scale of
hundreds of meters and that there is a great deal of spatial variation within the patches themselves.
For L. planaxis, there were no significant differences at any of the three spatial scales, indicating
that it is spread out fairly evenly throughout the sites. The large contribution of residual variance to
the total variances for L. limatula and L. planaxis indicate that there was considerable variation
among quadrats (Fig. 6).
One difficulty with a nested sampling design is that the degrees of freedom (and hence the
power of the test) decrease at increasingly higher levels of nesting. For example, in this study, the
mean abundances of tube snails at Fisherman's Wharf and Point Pinos appeared to be fairly
different, yet the analysis of variance indicated this difference was not significant. This surprising
result is due partly to the small power of the test at that level of nesting (3 and 4 degrees of
freedom). Another problem with this sampling design is that a large variance at a lower level in the
hierarchy tends to obscure variation at higher levels. For the tube snail, 25% of the total variance
was contributed at the subsite level for this species, meaning that only a very large difference in
mean abundances at the sites would have been detected as significant.
The large Dp for S. squamigerus shows that very little of the tube snail's observed
distribution conformed to a Poisson distribution. The HI indicates a clumped distribution, which is
supported by the natural history of the organism. Because S. squamigerus is a sessile snail, the
differences in mean abundances at the subsite and transect level most likely resulted from patterns
of larval settlement. Based on the gregarious nature of the tube snail, the larvae are thought to
(Spatial scales of variation)
I. Fu
settle on or near adults of the same species (Hadfield 1966). This continual addition of new
individuals to areas near established populations could result in the larger spatial scale of patchiness
that was observed for this species. Also, if some of the larvae settle in the existing large patches,
then variation could occur within the patches at smaller spatial scales. A particle feeder, the tube
snail spins delicate mucus webs to trap its food and therefore prefers more quiet water (Keen
1960). At HMS, the tube snails had a much higher mean density in the subsite that was more
protected from wave action than in the subsite that was more exposed (Fig. 4). Thus, localized
variation in water motion could also contribute to spatial patchiness.
The observed distribution of L. planaris did not closely follow that of a Poisson
distribution. An I of 1.55 for the organism suggests a clumped distribution, which is supported
by previous studies on the natural history of this snail. Miyamoto (1964) theorized that L. planaxis
tends to cluster in cracks in an attempt to reduce desiccation, and Bush (1964) has shown that
groups of snails desiccate at a slower rate than single individuals in both laboratory and field
conditions. If the snails are grouped together in cracks or irregularities in the rocks, their
distribution will be clumped, with most of the variation occurring on the spatial scale of meters.
Mating behavior in L. planaxis could also affect its patterns of distribution, for males can often be
found sitting on the shells of females throughout the year, although reproduction in mass seems to
be confined to spring and summer (Ricketts et al. 1952, Gibson 1964). Other possible factors
include the abundance and distribution of the predatory snail Acanthina spirata, which releases a
waterborne chemical stimulus that elicits an avoidance reaction in the L. planaxis; the tendency of
the snails to follow mucus trails laid down by other snails (Peters 1964); and wave action, which
appears to selectively knock larger snails off of rock faces (Bigler 1964). All or a combination of
these factors may have contributed to the variation observed at a spatial scale smaller than tens of
meters for L. planaxis.
Limpet distribution did not depart strongly from a Poisson distribution (Dp-O. 13). This
appears to contradict the l, which indicates a rather clumped distribution (I-3.6). The apparent
conflict may be due to the overall low abundance of limpets. Iy is very sensitive to the Var/X
I. Fu
(Spatial scales of variation)
ratio. Thus, the presence of outliers will greatly increase the variance and inflate M, especially if X
is small. In contrast, Dp is less sensitive to departures from the Poisson distribution when the
overall abundances are very low. In the case of L. limatula, which had most of the quadrat counts
at zero with a few outliers, Dp will not weight the outliers as heavily, and the observed distribution
could fit a Poisson distribution fairly well. Two of the quadrats had counts of seven or higher,
showing that there were clumps in the limpet distribution. Aspects of the natural history of the
limpet could result in a patchy distribution. For example, L. limatula exhibits an escape response
to predatory sea stars, especially Pisaster ochraceus, by moving upward on vertical rocks or
downstream on horizontal ones in response to a sea star scent in the water (Phillips 1974). Thus,
if the distribution of sea stars is patchy on the scale of tens of meters, then the variation at that
spatial scale observed in the limpets may have been due to this avoidance behavior. In terms of
food availability, the limpets feed on microscopic algae, as well as the encrusting algae
Hildenbrandia and Peyssonnelia (Kitting 1978, Eaton 1968). If the abundances of these algae vary
on the scale of tens of meters, then they may have been partly responsible for the significant
differences among transects. Patchy larval settlement and juvenile survival could also have
contributed to the significant differences at that spatial scale.
With limited time, money, and effort available for conducting research, it is imperative that
any sampling done on an organism provide an accurate estimate of the abundance and distribution
of the species of interest. Without knowing beforehand what the spatial scales of variation for the
organism are, however, any sampling effort cannot be depended on for reliable results. This study
illustrates the value of doing a preliminary study, using a nested sampling design, on an organism
to determine the patchiness of that animal’s distribution before an extensive and costly sampling
effort is made.
Literature Cited
Andrew, N. L. and B. D. Mapstone. 1987. Sampling and the description of spatial
pattern in marine ecology. Oceanography and Marine Biology Annual Review.
5:39-90.
Bigler, E. 1964. Attrition on the Littorina planaxis population. Final papers, Biol. I75H.
library, Hopkins Marine Station of Stanford University, Pacific Grove, California.
Bush, P. S. 1964. Effects of desiccation on Littorina planaris. Final papers, Biol. 175H,
library, Hopkins Marine Station of Stanford University, Pacific Grove, California.
Eaton, C. M. 1968. The activity and food of the file limpet, Acmaea limatula.. Veliger.
11:5-12.
Gibson, D. G. 1964. Mating Behavior in Littorina planaxis Philippi. Veliger. Il
(Suppl.):134-137.
Hadfield, M. G. 1966. The reproductive biology of the California vermetid gastropods
Serpulorbis squamigerus (Carpenter, 1857) and Petaloconchus montereyensis Dall,
1919. Doctoral thesis, Biological Sciences, Stanford University, Stanford,
California 174 pp.
Hahn. T. P. 1985. Effects of predation hy black oystercatchers (Haematopus bachmani
audubon) on intertidal limpets. Masters thesis, Biological Sciences, Stanford
University, Stanford, California 70 pp.
Holmes, S. J. 1909.The early cleavage and formation of the mesoderm of Serpulorbis
squamigérus Carpenter. Biological Bulletin. 1:115-121.
Hurlbert, S. H. 1990. Spatial distribution of the montane unicorn. Oikos. 58:257-271.
Keen, A. M. 1960. Vermetid gastropods and marine intertidal zonation. Veliger 3:1-2.
Kitting, C. L. 1979. Foraging of individuals within the limpet species Acmaea
(Notoacmea) scutum at Monterey Bay, California, and the consequences on their
mid-intertidal algal foods. Doctoral thesis, Biological Sciences, Stanford
University, Stanford, California 192 pp.
MacGinitie, G. E., and N. MacGinitie. 1968. Natural history of marine animals 2nd ed.
McGraw-Hill, New York, USA.
Mivamoto, A. 1964. Clustering in Littorina planaxis. Final papers, Biol. 175H, library
Hopkins Marine Station of Stanford University, Pacific Grove, California.
Morris, R. H., Abbott, D. P., and E. C. Haderlie. 1980. Intertidal invertebrates of
California. Stanford Universtiy Press, Stanford, California, USA.
Morton, J. E. 1965. Form and function in the evolution of the Vermetidae. Bulletin of the
British Museum (Natural History). 11:585-630.
Pequegnat, W. E. 1964. The epifauna of a California siltstone reef. Ecology 45:272-
283.
Peters, R. S. 1964. Function of the cephalic tentacles in Littorina planaris Philippi.
Veliger. 11(Suppl.):143-148.
Phillips, D. W. 1974. Distance chemoreception and avoidance of the predatory starfish
Pisaster ochraceus by the gastropods Acmea (Collisella) limatula and Acmaea
(Notoacmaea) scutum. Doctoral thesis, Biological Sciences, Stanford University,
Stanford, California 156 pp.
Ricketts, E. F., Calvin, J. C., and D. W. Phillips. 1985. Between Pacific Tides Sth ed.
Stanford University Press, Stanford, California, USA.
Ruth, F. S. 1948. Studies of the natural history of limpets of the family Acmaedde.
Masters thesis. Department of Zoology, University of the Pacific, Stockton,
Calfornia 153 pp.
Sokal, R. R. and F. J. Rohlf. 1995. Biometry 3rd ed. W. H. Freeman and Company,
New York, USA.
Underwood, A. J. 1981. Techniques of analysis of variance in experimental marine
biology and ecology. Oceanography and Marine Biology Annual Review. 25:513.
605.
lable 1: Results of the analysis of variance for A) Lottia limatula B) Serpulorbis
squamigerus and C) Littorina planaxis
Source
Mean Squares
F-ratio
P-value
Sites
1.441
0.569
0.665
Subsites
2.534
1.437
0.267
Transects
16
1.763
2.225
0.004
Residual
456
0.793
Source
Mean Squares
F-ratio
P-value
Sites
863.799
2.495
0.230
Subsites
346.166
4.063
0.033
Transects
85.204
3.884
0.000
Residual
21.936
72
Source
Mean Squares
F-ratio
P-value
Sites
14.025
1.376
0.377
Subsites
10.192
2.253
0.135
Transects
4.523
1.597
0.100
Residual
126
2.832
Table 2: Index of departure from the Poisson distribution (Dp), index of aggregation (ID,
and variance to mean ratios of the three species. Dp refers to the degree of overlap between
the observed distribution and the Poisson distribution. M measures how many times more
likely it is that two randomly selected individuals will be from the same quadrat than it
would be if the individuals in the population were distributed at random.
Species
Variance: Mean
Index of Departure
Index of aggregation
Lottia limatula
2.24
0.125
3.626
Serpulorbis squamigerus
0.844
5.93.
Littorina planaxi
1.55
12.99
0.63.
Figure legends:
Fig. 1: Map of the Monterey Peninsula area, showing the four sites that were sampled in
this study
Fig. 2: Nested sampling design used in the study. Three or four sites were sampled per
species, with two subsites nested within each site, three transects nested within each
subsite, and a particular number of quadrats for each transect. The number of quadrats
varied for the different species
Fig. 3: Mean densities of Lottia limatula at the different spatial scales. Each vertical bar
represents the mean limpet density for a particular transect. The arrows point to the mean
limpet densities for subsites, and the mean limpet density for a site is seen by finding the
halfway point between the two arrows for the subsites of that site. The error bars are the
standard errors.
Fig. 4: Mean densities of Serpulorbis squamigerus at the different spatial scales. Each
vertical bar represents the mean limpet density for a particular transect. The arrows point to
the mean limpet densities for subsites, and the mean limpet density for a site is seen by
finding the halfway point between the two arrows for the subsites of that site. The error
bars are the standard errors.
Fig. S: Mean densities of Littorina planaxis at the different spatial scales. Each vertical bar
represents the mean limpet density for a particular transect. The arrows point to the mean
limpet densities for subsites, and the mean limpet density for a site is seen by finding the
halfway point between the two arrows for the subsites of that site. The error bars are the
standard errors.
Fig. 6: Percentage of the total variance contributed by each level of nesting in the sampling
design.
Fig. 7: A) Graph of the observed distribution of Lottia limatula and the Poisson
distribution for the same mean, showing a large degree of overlap between the two plots.
B) Graph of the observed distribution of Serpulorbis squamigerus and the Poisson
distribution for the same mean, showing very little overlap between the two plots. C
Graph of the observed distribution of Littorina planaxis and the Poisson distribution for the
same mean, showing some overlap between the two plots.
Fig. 1:
V
Great
Tide Pool
Point Pinos

Pacific
Grove
MONTEREY
PENINSULA
Carmel
Hopkins Marine Station
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Monterey
Fisherman's Wharf

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Fig. 7:
0.8-
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0.6
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Count per quadrat
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0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
Count per quadrat