\

Reprinted from

i

Ecological Studies

Jurek Kolasa

Steward T.A. Pickett

Editors

Ecological Heterogeneity © 1991 Springer-Verlag New York, Inc. Printed in United States of America.

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona

9.

Biological Heterogeneity in Aquatic Ecosystems John A. Downing

The study of heterogeneity in the biological components of aquatic eco systems has been an important part of ecology for more than a century.

Although many early ecologists (see review by Lussenhop, 1974) perceived especially the pelagic milieu to be uniform [hence the term plankton or "wanderers" for its inhabitants (Ruttner, 1953)], early quantitative limnologists such as Birge (1897) found the aquatic habitat to be highly hetero geneous in factors such as light, temperature, oxygen, and limiting nu trients. The physical heterogeneity of the aquatic habitat has long been known to be reflected in the spatial patterns of aquatic populations. It became clear early in the study of temperate aquatic habitats that they varied in composition temporally and spatially. The most important tem poral variations were perceived to be seasonal developments of pelagic communities mediated by annual cycles of temperature, and the most im portant spatial variation seemed to be related to depth. At the close of winter, lakes reheat and the renewed light availability and higher tempera

tures give rise to a rapid increase in plant and animal life. Reviews of clas sic limnology suggest that the temporal development of the pelagic com

munity follows a predictable trajectory throughout summer and autumn

until activity is decreased again during winter (Wetzel, 1983). Seasonal variation in temperature is also a major source of spatial heter

ogeneity in lake ecosystems. Owing to the physical characteristics of water

(Ruttner, 1953), temporal variation in climate yields important spatial ver-

9. Biological Aquatic Heterogeneity

161

tical and horizontal gradients in physical, chemical, and biotic components of lakes. The seasonal succession of various communities (e.g., planktonic,

benthic, fish) in lakes and the study of spatial and temporal gradients and their influence on lake ecosystems has made up a large fraction of the limnological studies performed over the last century. These studies are the backbone of limnological research and are reviewed in depth by several limnological texts (e.g., Ruttner, 1953; Cole, 1983; Goldman and Home,

1983; Wetzel, 1983). Superimposed on these spatial and temporal gradients is another kind of heterogeneity, called "stochastic variation" in temporal studies and "spatial aggregation," "contagion," or "patchiness" in spatial studies. Such

heterogeneity, not arranged in perceptible gradients, has been studied much less completely than variation along aquatic clines. This review con

centrates on the spatial heterogeneity (i.e., aggregation) of biotic compo nents of lake ecosystems not related to gradients. First to be examined are some serious methodological problems associated with its quantification that have retarded the advancement of knowledge in this field. A compara

tive analysis of spatial heterogeneity in several biological components of lake ecosystems is then presented.

Measuring Spatial Heterogeneity in Aquatic Organisms Many of the classic studies of nonclinal spatial heterogeneity were per

formed using large, slow-moving or sessile organisms such as trees, bushes, or grasses (e.g., Grieg-Smith, 1952; Pielou, 1977). It was thus easy to make interpretable and somewhat time-stable measurements of the spatial posi

tion of organisms within an ecosystem. In lakes, both the medium (e.g., water, sediments) and the organisms are highly dynamic; and wave action, turbulence, and currents render spatial patterns highly ephemeral. In aquatic ecosystems, the concept of "place" is illusory and dynamic. It is probably of greater importance to the organisms, however, how the mem

bers of the population are spaced relative to other members of the popula tion, rather than to geographical space. Such relative spacing is important to tests of theoretical questions because aggregation or heterogeneity is

believed to be important to interorganismal interactions such as reproduc

tion (Waters, 1959; Dana, 1976; Jackson, 1977; Cowie and Krebs, 1979), competition (Ryland, 1972; Keen and Neill, 1980; Veresoglou and Fitter,

1984), and predation (Anscombe 1950; Gilinsky, 1984; Sih, 1984). Another factor that renders the measurement of spatial heterogeneity difficult in aquatic ecosystems is the fact that organisms are usually small in lakes, with most plants and animals being less than a few centimeters long (fish and aquatic macrophytes are exceptions, but they are rarely numer ically dominant taxa in lakes ). Because normal swimming speed scales as around W014 (W= body mass) and length and mass are related as about

162

J.A. Downing

W«L3 (Peters 1983), the number of body lengths moved at normal swim ming speed scales as about L~0-6. That is, smaller animals can move many more body lengths per unit time than can larger animals. This fact, com bined with the difficulty of observing small, elusive animals in dark places such as deep waters or muddy sediments, has made direct measurement of the spacing of lake dwelling organisms impossible. Limnologists, like many other ecologists, have therefore relied strongly on indices of spatial heterogeneity derived from measures of variations in numbers of organisms among replicate (i.e., same sized samples taken in the same habitat) samples of population density. When several replicate samples have been taken at random or regular points within the space occupied by the population, a frequency distribution of different numbers of organisms collected in replicate samples can be produced. Two types of information are derived from this approach. The central tendency of the frequency histogram indicates the density of the population. The shape of the frequency histogram indicates the degree of heterogeneity of the population over the sampling space, greater heterogeneity being implied by more skewed frequency distributions. Because of the difficulty of observing most aquatic organisms, tests of hypotheses about spatial heter ogeneity have usually been based on synthetic indices of spatial heter ogeneity derived through such replicate sampling, and interpretation of these indices has been founded on the shape of density frequency histo grams that should be encountered in aquatic organisms exhibiting different spatial patterns.

Advances in our knowledge about factors influencing spatial aggrega tion and the ecological consequences of different levels of spatial hetero geneity have been hampered by two technical problems. First, the devel opment of indices of spatial aggregation has paid more attention to their capacity to indicate lack of randomness than to measure differences in degree of aggregation among significantly aggregated populations. Orga nisms that are spatially heterogeneous or aggregated are those that show more variability in space than would be expected from either a random

(Poisson: s2 = m), i.e., variance equals the mean, or uniform (s2 m is aggregated; and for more than one population of equal density, the population with the highest s2 the most aggregated. The problem that has daunted ecologists is that few populations exist at exactly the same density. Several indices have been proposed to permit comparison of the spatial aggregation of popula tions of unequal densities (see Elliott, 1979, for review). There is reason to suspect that some of these indices contain mathematical artifacts that lead

9. Biological Aquatic Heterogeneity

163

to systematic errors and misleading interpretations (Green, 1966; Elliott, 1979; Taylor 1984; Downing, 1986), but no systematic analysis of them has been presented.

The second technical difficulty has been that few indices have functioned generally enough to permit comparisons. The analysis of frequency his tograms of spatially replicated population estimates (e.g., Polya, 1930; Neyman, 1939; Waters, 1959; Waters and Henson, 1959) is especially notorious because entirely different distribution functions are often found to fit population data collected on consecutive dates or with different treat ments (Taylor, 1965, 1984). Similarly, methods based on measures of dis tance among individual organisms (e.g., Pielou, 1977) can be applied only under particular circumstances. Methods based on variation among spatial ly separated population counts can be applied much more frequently than those based on distance measures (Patil and Stiteler, 1974) because mobile organisms and those that cannot be observed for long periods can be ex amined only by measuring the variability among repeated population esti mates made in slightly different locations. Spatial aggregation is easy to define but has proved difficult to measure. A good index of spatial aggregation should be widely applicable, should provide measures that correspond to the definition of spatial aggregation, and should provide reliable values that can be compared among popula tions. This chapter concentrates on the universally applicable indices based on the variance (s2) and mean (m) of spatially repeated population esti mates and tests the simple hypothesis that mathematical constraints on measures and comparative analyses of spatial heterogeneity yield indices that are biased and confounded by variations in sampling design.

Indices of Spatial Aggregation Based on Replicate Counts Although many indices of spatial aggregation have been proposed, only those that have been used most frequently in aquatic studies are discussed here (Table 9.1). Historically, the first indices used were simple ratios of either standard deviation or variance of replicate counts to the average of the replicates (s/m or s2lm). Figures 9.1 and 9.2 show that s2lm of replicate samples of marine and freshwater zooplankton (example data from Down ing et al., 1987) increases with m, whereas s/m decreases with m. Several other indices attempt to compensate for systematic variation with m. The distribution coefficient of the negative binomial distribution, k (e.g., Elliott, 1979), describes the skewness of frequency histograms of popula tion counts but also varies systematically with m (Fig. 9.3). In addition, k is meaningless for s2 < m and is difficult to calculate if 0 < k < 4 (Anscombe, 1950). Lloyd's index of mean crowding (m) (Lloyd, 1967) is a variant of s2lm (Table 9.1) and tends to increase with m (Fig. 9.4). Although not illustrated here, Lloyd's index of patchiness {mini) is also correlated with

sVm

slm

m2f(s2 - m)

m + (s2lm) - 1

s2lm

CV

kb

in

m2(n —

m(n —

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fn)

1)

m

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m(n-\)

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Low Heterogeneity

1 + n — m~l

m + nm — 1

nm

High Heterogeneity

c Undefined for £ X = 1.

tion (Pieters et al., 1977); m and mim = Lloyd's (1967) index of mean crowding and index of patchiness; and Id = Morisita's (1954, 1962) index. Minimum and maximum variances are calculated after equations 2,4, and 5. "Larger k value denotes less heterogeneity.

remainder of the division of J\jf by « (/?). CV = the coefficient of variation (Elliott 1979); k — the distribution coefficient of the negative binomial distribu

aNumber of samples taken (n), mean number of organisms collected per sample (/n), total number of organisms collected in the n samples (

fnim

Calculation

Index

Limits to Indices

Table 9.1. Upper and Lower Limits to Indices of Heterogeneity and Their Dependence on Various Factors8

00

3

o

a

9. Biological Aquatic Heterogeneity

165

100000

10000

0.01

10

1000

100000

m, MEAN NUMBER COLLECTED

Figure 9.1 Variation in the variance mean ratio (s2/m) with mean density ( no.

sample"1) in 1200 sets of replicate zooplankton samples. Data are from marine and freshwater systems (Downing et al., 1987). Means and variances were calculated on a per-sampler basis and are not normalized to a common volume. The trend line is a simple moving average with a window size of 10 data points. Moving averages are shown instead of data points to facilitate examination of the trend in s2/m with m.

2.6

E

•«*

to

> O

0.2

0.1

10

1000

100000

m, MEAN NUMBER COLLECTED

Figure 9.2 Variation in the coefficient of variation {CV = slm) with mean density

(no. sample"1). Data and analyses are as in Figure 9.1.

166

J.A. Downing

0.1

10

1000

100000

m, MEAN NUMBER COLLECTED

Figure 9.3 Variation in k of the negative binomial distribution with mean density (no. sample"1), k was calculated by the product-moment method because it is the best method for small sample size (Pieters et al., 1977). Small values of k imply highest heterogeneity. Negative values of k were included in moving averages but were not plotted. Data and analyses are as in Figure 9.1.

x UJ a z

©

100000

10000-

z

o

10Q0-

O


high Posi" tive limits to k fall to negative, decreasing values. Values of k obtainable for maximum spatial aggregation generally increase with increased m. The

upper limit to m is a linear increasing function of m, whereas the lower limit tends to m — 1 as R approaches n. Empirical measurements of m must

therefore increase with increased m (see Fig. 9.4). /d is bounded above by

n, bounded below by 0 between Z-^= 1 and YjX= n, and an increasing function of m where m is an integer of more than 1. These mathematical limits combined with the structure of the indices result in the trends seen in Figures 9.1 to 9.5. The maximum possible values of s2lm, s/m, m, and /d increase with increased n, whereas the range of k must decrease. The trend in 7d seen in Figure 9.5 is probably due to the fact that low average num

bers per sample can be detected only using many samples, which can yield

higher values of /d (Table 9.1) and for the example data, s2 = am1-5 (Down

ing et al., 1987); therefore /d