LJmrtol. Oceanogr., 32(3), 1987, 673-680
© 1987, by the American Society of Limnology and Oceanography, Inc.
Effect of interreplicate variance on zooplankton sampling
design and data analysis1
Abstract—The variance of replicate marine and
freshwater zooplankton samples is related (r2 >
0.99) to the mean population density and the
volume of the sample taken. A general variance
relationship calculated from 1,189 sets of repli
cate samples taken throughout the world is pre
sented. This function can predict the variance of
both marine and freshwater samples taken with
several types of sampling gear. Only 12% of in
dividual taxa examined showed variance rela
tionships that departed significantly from this
general function. The variance function shows
how the requisite number of samples decreases
with increased population density, sampler vol
ume, and lowered precision requirements. The
X0-2 power transformation is recommended on
empirical grounds, although caution is advised
in the use of all data transformations.
Most agree that zooplankton sampling
variability is large (e.g. de Bernardi 1984;
Fasham 1978; Malone and McQueen 1983;
Omori and Ikeda 1984). Quantitative pre
diction of sampling variance is important
for two practical reasons. First, all zoo
plankton population ecologists must decide
how many samples and what volume of
sample to take. The requisite number of
samples is determined by the precision re
quired and the sampling variance (s2). Pre
diction of s2 would allow efficient sampling
design. Second, many researchers use para
metric statistical methods because, these
1 A contribution to the Group d'Ecologie des Eaux
Douces of PUniversite de Montreal. This research was
supported by the Natural Sciences and Engineering Re
search Council of Canada, and the Department of Ed
ucation of the Province of Quebec (FCAR).
methods are powerful, familiar, and widely
available in computer packages. The vari
ance ofraw population estimates varies with
the population density, and data must be
transformed to equalize variances, alleviate
skewness, and increase the probability of
additivity (Snedecor and Cochran 1967;
Southwood 1966). Knowledge of the rela
tive size of the variance and mean (m) of
replicate zooplankton population estimates
can allow effective power transformation of
raw data (Taylor 1961).
The purpose of this study is to assemble
information on interreplicate variance in
marine and freshwater zooplankton sam
ples and to develop an equation to char
acterize this variability. Further, we show
how these analyses can be used to choose
sample number and size and to choose
proper transformations for zooplankton
data.
We thank H. Cyr, C. Plante, L. Rath, and
three anonymous reviewers for criticism of
the manuscript and B. Pinel-Alloul for help
ful discussion.
Data on m and s2 (n — 1 weighting) of
replicate zooplankton population estimates
were collected from the published literature.
We tried to gather data covering a range of
environments in many geographical areas,
collected with a variety of sampling gear
(Table 1). Population estimates were usu
ally made at the genus or species level; 161
taxa were surveyed (Table 2). We collected
1,189 sets of replicate observations («'),
covering a range of m from 5 x 10~7 to
674
Table 1.
Notes
Description of data on interreplicate variance in zooplankton population estimates, n' is the number
of sets of replicate data found in each source.
Location
Source
n'
Sampler
Freshwater
Windermere, England
(2 dates)
Colebrook 1960
6
Pump
Mary Lake, Minn.
Donk Lake, Belgium
(5 dates)
Comita and Comita 1957
Dumont 1967
5
6
Clarke-Bumpus
Plankton trap
Reservoir in east Belgium
Dumont 1968
27
(3 depths)
Eglwys Nynydd res., Wales
(ca. 42 dates, 3 depths)
Four Norwegian lakes
George 1974
272
Laneland and Rognerud 1974
Plankton trap
Van Dorn
55
Schindler-Patalas,
14
Fnedinger
Juday trap and Toronto
Clarke-Bumpus,
Algonquin Park lakes, Canada
Langford 1953
Three Algonquin Park lakes
(5 depths, 16 dates)
Mirror Lake, N.H.
Two south Ontario lakes
(4 depths)
Cultus Lake, B.C.
Langford and Jermolajev 1966
trap
258
Juday trap
Likens and Gilbert 1970
Malone and McQueen 1983
4
32
Van Dorn
Ricker 1938
18
Pump, Juday trap, tow
Schindler-Patalas,
pump
net
Marine
Firth of Clyde
(3 dates, 4 depths)
Port Nicholson, N.Z.
Bay near Great Barrier I., N.Z.
Port Nicholson, N.Z.
Pacific Ocean, near Baja Calif.
(4 series, 6 net diameters)
Atlantic Ocean near Woods Hole,
North Sea off Tynemouth,
Barnes and Marshall 1951
Cassie 1959a
Cassie 19596
Cassie 1960
McGowan and Fraundorf 1966
Winsor and Clarke 1940
20
Pump
6
15
12
405
Jars
Pump
Tow net
34
Tow net
Pump
England
1.6 x 103org liter-', s2 from 1 x 10~12to
3.2 x 106, number of replicate samples (n)
from 2 to 400, and sampler volume (V)
from 0.8 to 2 x 106 liters. The mean and
median n were 11 and 4, the median sam
pler volume was 10 liters, and the median
number of organisms found per sample was
20. Sampler shape or towing configuration
were not examined in this study although
they have some effect on s1 (e.g. McGowan
and Fraundorf 1966; Wiebe 1972). The ef
fect of subsampling was also ignored be
cause it should add little to intersample
variance if it has been done properly (e.g.
Winsor and Clarke 1940). Sampling cov
erage was less complete for marine zoo
plankton than freshwater. A full list of the
data is available, at a nominal charge, from
the Depository of Unpublished Data, CIS-
TI, National Research Council of Canada,
Ottawa, Ontario K1A 0S2.
Sampling variability is usually well cor
related with the density of organisms in
samples. Interreplicate s2 usually rises as a
power-function of m:
s2 = amb
(1)
where a and b are constants fitted by leastsquares regression of log .s2 on log m (Taylor
1961). Equation 1 is often determined for
individual species or taxa because Taylor
(1965,1984) has suggested that b constitutes
a "species characteristic." We obtained >3
sets of replicate population estimates for 104
freshwater and marine zooplankton taxa and
thus could estimate b for many species. Fit
ted b values for significant relationships
ranged from 0.97 to 3.69. The frequency
Notes
675
Table 2. Number of taxa (genera or species) of each
group included in the analysis of interreplicate vari
ance.
Group
Cephalopods
Chaetognaths
Cirripedes
Cladocerans
Copepods
Euphausiids
Fish larvae
Gastropods
Heteropoda
Nudibranchia
Thecosomata
Lamellibranchs
Polychaetes
Rotatoria
Tunicates
Freshwater
—
Marine
7
—
1
—
2
12
20
—
1
19
11
48
—
_
_
14
2
13
1.2
1
—
8
—
2.0
24
SPECIES-SPECIFIC
1
_
1
distribution of all significant species-specif
ic b values has a median of 2 (Fig. 1).
Taxon-specific s2: m relationships are
quite similar, even though b values seem
quite variable. When each of these speciesspecific relationships is plotted within the
range of observed mean densities, individ
ual relationships appear to cluster around a
central s2: m axis (Fig. 2). The break in the
data near m = 10~3 roughly represents den
sity differences between marine and fresh
water systems. The data for all taxa of zooplankton (Fig. 3) suggest that one common
algorithm might describe the interreplicate
variance of all types of zooplankton popu
lations. This overall variance function («' =
1,189, r2 = 0.99, F= 110,500) is
s2 = 0.296m1-849
1.6
2-8
3.2
"b"
Fig. 1. Distribution of b values of species-specific
relationships between s2 and m (Eq. 1). Data are only
shown for significant (P < 0.05) relationships. Regres
sions where all observations in all sets of replicates are
isolated in one replicate are omitted. Frequencies were
calculated for b intervals of 0.2 and rectangles are cen
tered on the interval. The frequency centered on 3.2
includes all b values >3.1.
samples, and the interreplicate s2 of most
taxa (88-96%) tends to rise with the mean
population density at a similar proportional
rate.
Some investigators (e.g. Taylor 1961;
Southwood 1966; Elliott 1977) have pro
posed that a in Eq. 1 is a characteristic of
the sampler. Others (e.g. Downing 1979;
Downing and Anderson 1985; Downing and
(2)
where m is the mean number of organisms
liter"1. Simple /-tests for the difference of
species-specific b values from the overall
coefficient (b in Eq. 2) show that 91 of 104
taxa were not significantly different at P <
0.05 and 100 taxa were not significantly dif
ferent at P < 0.01 (Table 3). A similar rarity
of intertaxonomic difference in spatial vari
ability was found for a more specific data
set by Winsor and Clarke (1940). In addi
tion, we found no significant (P < 0.05)
difference between b for marine vs. fresh
water zooplankton samples. Thus, Eq. 2
characterizes the s2 of most zooplankton
-6-4-2
0
2
4
LOG MEAN ZOOPLANKTON DENSITY (liter'1)
Fig. 2. Species-specific trends indicated by linear
regression analysis (Eq. 1) of log s2 (n - 1 weighting)
and log m (mean No. liter"1), calculated within 104
species and plotted within the observed range of m for
that species.
Notes
676
-6-4-2
0
2
4
LOG MEAN ZOOPLANKTON DENSITY (liter'1)
Fig. 3. Relationship between the log s2 (n - 1
weighting) and log m (mean No. liter-') of replicate
zooplankton population estimates. Observations are
plotted for 1,189 sets of replicated estimates.
Cyr 1985) have found that a varies system
atically with the size of sampler used. There
fore, we expect that s2 in zooplankton sam
ples should vary as
s2 = amhVc
(3)
where V is the sample volume in liters. We
found that interreplicate sz of zooplankton
samples follows the function
s2 = 0.745m1-622 V-°-267
(4)
(«= 1,189, r2 > 0.99, F= 68,320), and the
partial lvalues for both m and Kwere high
ly significant (P <c 0.001). This relationship
shows that zooplankton sampling s2 in
creases with increased population density
and decreases with increased sampler vol
ume. MobergandYoung(1918,p. 265) not
ed that, "In general, the more numerous the
individuals of a species, the smaller the
variations in their number." This is ex
plained by Eq. 4 which shows that the coef
ficient of variation (C.V.) of zooplankton
samples (s/m) is generally proportional to
m-o.i9 Analysis of the residuals of Eq. 4
(Draper and Smith 1981) showed no further
significant trend in the data nor gave any
indication of further curvilinearity or sig
nificant differences in fit to various
subgroups (e.g. freshwater vs. marine, sam
pler type, vertebrate vs. invertebrate, etc.).
Many of the marine data are from the
research of McGowan and Fraundorf (1966)
on the Pacific Ocean near Baja California.
At least two reviewers have suggested that
we demonstrate that Eq. 4 will make un
biased predictions ofs2 in more recent stud
ies of other marine environments. Unfor
tunately, data on sampling variability in
marine zooplankton are relatively rare. Fig
ure 4 compares the predictions of Eq. 4 to
independent observations of the spatial het
erogeneity of copepods and euphausiids in
oceanic waters off Ecuador (Wiebe 1972)
Table 3. Taxa out of 104 possible taxon-specific s2: m relationships that showed b values (Eq. 1) that were
significantly different [P < 0.01 and P < 0.05) from b of the overall s3: m relationship (Eq. 2). ri is the number
of sets of replicate data collected for each taxon and t is the computed /-statistic for the difference in exponents.
Taxon
Group
ri
b
199
78
6
6
1.25
1.41
0.98
2.09
6
12
0.87
2.00
0.73
1.39
1.37
1.40
2.41
2.22
2.46
r
P < 0.01*
Cyclops vicinus
Daphnia hyalina
Microcalanus sp.
Pomacentridaef
Copepod
Cladoceran
Copepodite
Fish larvae
-9.14
-6.13
-26.32
5.83
P < 0.05*
Atlanta fusca
Gastropod
Bosmina coregoni
Cladoceran
Cavolinia uncinata
Creseis virgula
Gastropod
Gastropod
Gastropod
Cladoceran
Fish larvae
Fish larvae
Fish larvae
Desmopterus pacificus
Diaphanosoma sp.
Diaphus sp.
Diogenichthys laternatus
Symphurus spp.
* Without correction Tor multiple comparisons.
t Unidentified species.
5
12
6
30
5
6
10
-4.34
2.60
-5.77
-2.64
-4.72
-2.23
4.81
3.13
2.72
611
Notes
Table 4. The number of zooplankton samples nec
essary to obtain a precision of p = 0.2 (SE/m = 0.2).
Predictions are from Eq. 6.
Population
(No. liter1)
xlO"6
XlO"5
xlO"4
xlO"3
xlO"2
xlO-1
1.0
I xlO
IxlO2
I xlO3
-10
-8
-6
OBSERVED
-4
-2
Vol. of replicate sample (liters)
i
10
too
1,000
3,453
1,446
606
254
107
1,867
782
1,009
423
177
546
229
96
40
45
19
8
4
2
328
137
58
24
10
5
2
2
75
31
13
6
3
2
2
17
7
3
2
2
2
10,000 100,000
296
124
52
22
9
4
2
160
2
2
2
2
2
2
67
28
12
5
3
2
0
LOG VARIANCE
Fig. 4, Comparison of independent observations of
s2 of marine zooplankton samples with the predictions
of Eq. 4. Data represent the spatial heterogeneity of
copepods and euphausiids in oceanic waters off Ec
uador (Wiebe 1972—D) and swarms of calanoid copepods and other plankton in the northwestern North
Pacific (Kawamura and Hirano 1985—A). The solid
line represents a 1:1 relationship between observa
tions and predictions. The data analyzed in this figure
were not included in the original analysis.
and swarms of calanoid copepods and other
plankton in the northwestern Pacific (Ka
wamura and Hirano 1985). Predictions fol
low observations for most zooplankton. The
available data suggest that Eq. 4 is an un
biased predictor of the variance among es
timates of replicate marine zooplankton
populations.
One of the most important and, to date,
most arbitrary decisions made by zooplank
ton ecologists is the choice of number of
samples to be taken on each date at each
sampling station. It is generally thought (e.g.
L. R. Taylor 1984; R. A. J. Taylor 1981)
that spatial heterogeneity is unpredictable,
and proper choice of sample number must
be fortuitous. An objective decision can be
made if one can estimate the interreplicate
variance to be encountered in the field
(Cochran 1977; Elliott 1977). These esti
mates have been obtained previously by
presampling or assuming that zooplankton
population estimates conform to either ran
dom or negative binomial distributions (El
liott 1977). Previous methods for estimat
ing required sample number are impractical
for zooplankton sampling. Presampling
zooplankton is expensive and single esti
mates of variance cannot be extrapolated in
time or space because variance is highly sen
sitive to variations in population density
(Eq. 4). Although early population biolo
gists assumed that plankton are randomly
distributed (Winsor and Walford 1936;
Ricker 1938), more recent research indi
cates that most zooplankton distributions
are nonrandom (Cassie 1960; Fasham 1978;
Langford and Jermolajev 1966; Malone and
McQueen 1983). The negative binomial
distribution cannot be used to deduce s2 of
nonrandom zooplankton populations be
cause k of the negative binomial is sensitive
to changes in the population density and
sampler volume; k = {amb~2Vc - m~l)~x
(Eq. 4).
The interreplicate s2 of zooplankton sam
ples can be predicted from Eq. 4; thus the
required number of replicate samples can
be forecast. The number of zooplankton
samples («) necessary to obtain a required
level of precision p (where p = SE/m, and
SE = s/n°5) can be calculated:
n = amh-2Vcp-
(5)
(Downing and Anderson 1985). For fresh
water and marine zooplankton
n = 0.745m-°-378 F-°-267/?-2.
(6)
Fewer replicate zooplankton samples are
needed with increasing population density
and sampler volume (Table 4). Relaxation
of precision requirements decreases the re-
678
Notes
quired number of samples. Equation 6 is a
good way of estimating the required level
of replication when specific information is
not available. The residual <1% of varia
tion in log,^2 in Eq. 4 is probably due to
species differences or differences among en
vironments. Researchers working in noto
riously variable regions or on species known
to possess extreme aggregative behavior
would be advised to take more samples than
Eq. 6 suggests. The general superiority of
variance functions to techniques based on
the Poisson or negative binomial distribu
tions for making predictions of s2 and n has
been demonstrated elsewhere (Downing
1979; Downing and Anderson 1985).
The choice of optimal sampler volume
and number depends on sampling and
counting costs (Cochran 1977). Sampling
cost probably increases rapidly with sam
pler volume while counting cost also in
creases or remains constant if subsampling
can be used effectively. If the total cost of
taking a sample (T= collection + counting
costs) increases with sampler volume at an
exponential rate of >0.26 (i.e. T oc V0-26 or
greater), then small samples will be most ef
ficient. If T is not related to V as a power
function, then optimalities of intermediate
sampler volumes might arise. Other studies
(Downing and Anderson 1985; Downing
and Cyr 1985) suggest that attention to interreplicate variance and sampling cost may
afford up to 30-fold savings in the cost of
sample extraction. We therefore suggest that
zooplankton researchers keep careful track
of sampling costs and use them in concert
with Table 4, Eq. 6, or ecosystem-specific
variance functions to choose optimal sam
pling designs.
Ecological data rarely lend themselves to
parametric tests of hypotheses based on the
normal distribution. One ofthe major prob
lems is that s2 of zooplankton samples is
unequal among populations or treatments.
This fact is indicated in Fig. 3 where s2 in
creases systematically with m. Many differ
ent transformations have been suggested to
alleviate this instability in s2 (see Venrick
1978). Unstable variance of sets of replicate
samples can be alleviated by transformation
of the original data (X) to X' before analysis
such that
A
—A
(/)
where b is taken from Eq. 3 (Taylor 1961;
Prepas 1984). Although this transformation
often works, all transformations should be
verified to see if they have corrected the
problem (Prepas 1984).
Following Eq. 7, a good general transfor
mation for zooplankton population esti
mates should be
X' = X0-20
(8)
because the exponent associated with m in
Eq. 4 is 1.622. This transformation corre
sponds closely to the power transform sug
gested by Frontier (1972) and will lead to
more frequent stability of variance than
either the commonly used square-root or
logarithmic transformations. Chang and
Winnell (1981) and Downing (1981) cau
tion researchers on the blind use of any sin
gle transformation. Table 3 shows that
species-specific b values vary in up to 12%
of species, thus the A'02 transformation will
not always stabilize the s2. Table 3 suggests
that the log transformation may be better
than Eq. 8 in up to 4% of the taxa examined.
On the other hand, the log transformation
will often overtransform the data where the
general exponent in the s2: m relationship
is <2, yielding unstable s2 where no prob
lem may have existed before transforma
tion (e.g. Downing 1981). We therefore rec
ommend that data be transformed with Eq.
8 unless site- or taxon-specific data indicate
a different power transformation.
The interreplicate variability of zoo
plankton has been a source of frustration
and conflict for nearly 100 yr (Lussenhop
1974). The analysis presented here suggests
a reason for the historical lack of agreement
of ecologists regarding the spatial hetero
geneity of zooplankton populations. The
same population can appear random or ag
gregated depending on the size of sample
used. If, for example, a population of 10
organisms liter-' is sampled with a 1-liter
bottle, s2 (31.2) would be greater than m
and the population would appear highly
clumped (C.V. = 0.56) (Eq. 4). If, on the
other hand, the same population is sampled
with a 50-m haul of a 0.5-m-diam Hensen
net (9,817 liters), the s2 (2.7) would be less
Notes
than m and the population would appear
randomly distributed (C.V. = 0.16). Our
analysis ofpublished data suggests that much
sampling variation can be accounted for by
mean density of plankton and the size of
the sampler used. Certainly many factors
combine to determine the spatial distribu
tion of zooplankton populations, but the
empirical regularity expressed in Eq. 4 can
be exploited to help in optimizing zooplank
ton population estimation.
John A. Downing
Martin Perusse
Yves Frenette
Departement des Sciences Biologiques
Universite de Montreal
C.P. 6128, Succursale 'A'
Montreal, Quebec H3C 3J7
References
Barnes, H., and S. M. Marshall. 1951. On the
variability of replicate plankton samples and some
applications of'contagious1 series to the statistical
distribution of catches over restricted periods. J.
Mar. Biol. Assoc. U.K. 30: 233-263.
Cassie, R. M. 1959a. An experimental study of fac
tors inducingaggregation in marine plankton. N.Z.
J. Sci. 2: 339-365.
. 19596. Micro-distribution of plankton. N.Z.
J. Sci. 2: 398-409.
1960. Factors influencing the distribution
pattern of plankton in the mixing zone between
oceanic and harbour waters. N.Z. J. Sci. 3:26-50.
Chang, W. Y. B., and M. H. Winnell. 1981. Com
ment on the fourth-root transformation. Can. J.
Fish. Aquat. Sci. 38: 126-127.
Cochran, W.G. 1977. Sampling techniques. Wiley.
Colebrook, J. M.
1960. Plankton and water move
ments in Windermere. J. Anim. Ecol. 29: 217240.
Comita,G. W.,andJ.J.Comita. 1957. The internal
distribution patterns of a calanoid copepod pop
ulation, and a description of a modified ClarkeBumpus plankton sampler. Limnol. Oceanogr. 2:
321-332.
de Bernardi, R. 1984. Methods for the estimation
of zooplankton abundance, p. 59-86. In J. A.
Downing and F. H. Rigler [eds.], A manual on
methods for the assessment of secondary produc
tivity in fresh waters. IBP Handbook No. 17, 2nd
ed. Blackwell.
Downing, J. A. 1979. Aggregation, transformation
and the design of benthos sampling programs. J.
Fish. Res. Bd. Can. 36: 1454-1463.
. 1981. How well does the fourth-root trans
formation work? Can. J. Fish. Aquat. Sci. 38:127129.
, and M. R. Anderson.
1985. Estimating the
679
standing biomass of aquatic macrophytes. Can. J.
Fish. Aquat. Sci. 42: 1860-1869.
, and H. Cyr. 1985. Quantitative estimation
ofepiphytic invertebrate populations. Can. J. Fish.
Aquat. Sci. 42: 1570-1579.
Draper, N. R., and H. Smith. 1981. Applied regres
sion analysis. Wiley.
Dumont, H. J. 1967. A five day study of patchiness
in Bosmina coregoni Baird in a shallow eutrophic
lake. Mem. 1st. Ital. Idrobiol. 22: 81-103.
. 1968. A study of a man-made freshwater
reservoir in eastern Flanders (Belgium), with spe
cial reference to the vertical migration of the zoo
plankton. Hydrobiologia 32: 97-130.
Elliott, J. M. 1977. Some methods for the statistical
analysis of samples of benthic invertebrates.
Freshwater Biol. Assoc. Sci. Publ. 25, 2nd ed.
Fasham, M. J. R. 1978. The statistical and mathe
matical analysis ofplankton patchiness. Oceanogr.
Mar. Biol. Annu. Rev. 16: 43-79.
Frontier, S. 1972. Calcul d'erreur sur un comptage
de zooplancton. J. Exp. Mar. Biol. Ecol. 8: 121132.
George, D. G. 1974. Dispersion patterns in the zoo
plankton populations of a eutrophic reservoir. J.
Anim. Ecol. 43: 537-551.
Kawamura, A., and K. Hirano. 1985. The spatial
scale of surface swarms of Calanus plumchrus Marukawa observed from consecutive plankton net
catches in the northwestern North Pacific. Bull.
Mar. Sci. 37: 626-633.
Langeland, A., and S. Rognerud. 1974. Statistical
analyses used in the comparison of three methods
of freshwater zooplankton sampling. Arch. Hydrobiol. 73: 403-410.
Langford, R. R. 1953. Methods of plankton collec
tion and a description of a new sampler. J. Fish.
Res. Bd. Can. 10: 238-252.
, and E. G. Jermolajev.
1966.
Direct effect
of wind on plankton distribution. Int. Ver. Theor.
Angew. Limnol. Verh. 16: 188-193.
Likens, G. E., and J. J. Gilbert. 1970. Notes on
quantitative sampling of natural populations of
planktonic rotifers. Limnol. Oceanogr. IS: 816—
820.
Lussenhop, J. 1974. Victor Hensen and the devel
opment of sampling methods in ecology. J. Hist.
Biol. 7:319-337.
McGowan, J. A., and V. J. Fraundorf. 1966. The
relationship between size of net used and estimates
of zooplankton diversity. Limnol. Oceanogr. 11:
456-469.
Malone, B. J., and D. J. McQueen. 1983. Horizon
tal patchiness in zooplankton populations in two
Ontario kettle lakes. Hydrobiologia 99: 101-124.
Moberg, E. G., and R. T. Young. 1918. Variation
in the horizontal distribution of plankton in Dev
il's Lake, North Dakota. Trans. Am. Microsc. Soc.
37: 239-267.
Omori, M., and T. Ikeda. 1984. Methods in marine
zooplankton ecology. Wiley.
Prepas, E. 1984. Some statistical methods for the
design of experiments and analysis of samples, p.
266-335. In J. A. Downing and F. H. Rigler [eds.],
Notes
680
A manual on methods for the assessment of sec
ondary productivity in fresh waters. IBP Hand
book No. 17, 2nd ed. Blackwell.
Ricker, W. E. 1938. On adequate quantitative sam
pling of the pelagic net plankton of a lake. J. Fish.
Res. Bd. Can. 4: 19-32.
Snedecor, G. W., and W. C. Cochran.
1967. Sta
tistical methods. Iowa State.
Southwood, T. R. E. 1966. Ecological methods with
particular reference to the study of insect popu
lations. Chapman and Hall.
Taylor, L. R. 1961. Aggregation, variance and the
mean. Nature 189: 732-735.
. 1965. A natural law for the spatial disposition
of insects, p. 396-397. In Proc. 12th Int. Congr.
Entomol.
. 1984. Assessing and interpreting the spatial
distributions of insect populations. Annu. Rev.
Entomol. 29:321-357.
Taylor, R. A. J. 1981. The behavioural basis of
redistribution 2. Simulations of the delta-model.
J. Anim. Ecol. 50: 587-604.
Venrick, E. L. 1978. Statistical considerations, p.
238-250. In A. Sournia [ed.], Phytoplankton man
ual. Monogr. Oceanogr. Methodol. 6. UNESCO.
Wiebe, P. H.
1972. A field investigation of the re
lationship between length of tow, size of net and
sampling error. J. Cons. Cons. Int. Explor. Mer
34: 268-275.
Winsor, C. P., and G. L. Clarke. 1940. A statistical
study of variation in the catch of plankton nets.
J. Mar. Res. 3: 1-34.
, and L. A. Walford. 1936. Sampling vari
ations in the use of plankton nets. J. Cons. Cons.
Int. Explor. Mer 11: 190-204.
Submitted: 16 September 1985
Accepted: 5 December 1986