Oecologia

Oecologia (Berlin) (1987) 74:272-276

9 Springer-Verlag1987

An upper limit to the abundance of aquatic organisms C.M. Duarte, S. Agusti, and H. Peters Department of Biology, McGill University, 1205 Avenue Docteur Penfield, Montreal, Quebec, Canada H3A 1B1 Summary. The maximum density achievable by aquatic organisms is an inverse linear function of their body size. As a consequence, the maximum achievable biomass is independent of body size, and is 2 orders of magnitude higher than the biomass in natural populations. The minimum interorganismic distance, calculated from the maximum density to allow comparison between aquatic and terrestrial organisms, scales as the 1/3 power of body size in both habitats. The similarities in the interorganismic distance of terrestrial and aquatic plant and animal communities suggest a fundamental regularity in the way organisms use the space. Key words: Body size - Aquatic organisms, maximum density, maximum biomass, interorganismic distance

The maximum densities of terrestrial and aquatic plants have been well studied (Gorham 1979; Duarte and Kalff 1987; Agusti et al. 1987), but not those of other organisms. However, they too must be subject to structural or metabolic constraints that limit the maximum density their populations can reach. The quantitative definition of maximal density for different populations could help identify populations which are so close to maximal density that constraints to their abundance are likely to be intrinsic to the population. Conversely, differences between maximum achievable densities and those observed in nature should reflect the magnitude of the extrinsic, ecological control of organismal abundance in different environments. The quantitative definition of maximum densities would also provide a basis for the management of experimental and commercial cultures which seak to maximize stock size prior to harvest. Consequently, we set out to establish empirical relations describing the maximum density achieved by aquatic organisms. Aquatic organisms are well suited for such analysis: Because surfaces are more irregular at finer scales, and habitable area decreases more slowly than the scale of observation (Mandelbrot 1983), it is difficult to compare areal densities for organisms of very different size, like bacteria and trees. In contrast, the volumetric densities of populations of aquatic organisms are easily defined and depend less on the scale of observation. Moreover, organisms in aqueous suspensions probably experience more homogeneous conditions than do organisms growing on interOffprint requests to: C.M. Duarte

faces, so that maximum densities are less likely to be affected by microenvironmental differences within the cultures. Finally, because we are aquatic ecologists, it was easier for us to study the maximum densities of aquatic organisms. Because larger organisms are usually rarer than small ones (Damuth 1981; Peters 1983; Peters and Wassenberg 1983; Calder 1984) we expected maximum densities to decline with body size. By analogy with other equations for animal density (Damuth 1981; Peters 1983; Peters and Wassenberg 1983; Calder 1984; Robinson and Redford 1986), we assumed that any such relation would be a power equation of body size and that the coefficients of such an equation would be best described if the largest possible range of organisms types and sizes were studied.

Methods We derived estimates of maximum density from published studies of population growth in optimal culture conditions 1. Most data came from studies which strove to maximize the biomass of cultured organisms for commercial purposes, to study density dependent phenomena, or, as in most studies of microorganisms, to compare population growth under some treatment to optimal growth. We transformed body weights to body volumes by assuming a density of one when given fresh weight, and a water content of 90% when given dry weight. If body size was not given, it was obtained from other sources (Fenchel and Finlay 1983; Nauwerck 1963). For organisms that vary greatly in size during ontogeny (i.e. fish, lobster, shrimps, and turtles), only direct estimates of size were accepted. Density data for these animals were recorded only where substantial mortality accompanied growth, because the absence of substantial mortality could indicate that the carrying capacity had not been reached. The maximum densities of smaller organisms that are less variable in size (i.e. bacteria, protozoans, rotifers, and zooplanktonic crustaceans) were taken as the densities achieved when a stationary phase replaced the exponential growth phase (Agusti et al. 1987). We used least squares linear regression to model the relationships between weight and density, because size is clearly the independent variable, and because the error involved in measuring size is much less than that related to the measurement of density. 1 Sources of data are labelled with an asterisk in the reference list

273 Results and discussion

The data represent 80 cultures, 44 species, five phyla and three kingdoms. Both size and density ranged over 15 orders of magnitude (Table 1). The relationship between the volume of individual organisms (V, gm 3) and their maximum density in culture (D . . . . m l - t ) is described by the equation: logto(Dma~) = 8.53 0.95 loglo(V)

(1)

R 2 =0.95; N = 8 0 ; F = 1558; P < 0.0001 ; S.Eqog estimate= 1.14 where the standard error of the estimate (S.E.~og ~t~m~t~)is given as an index of precision. This equation describes a very strong inverse relationship between the maximum population density of an organism and its body size (Fig. J), and has the surprising implication that maximum biomass (V*Dm,,) is independent of individual size. This was confirmed by regressing biomass on size (slope=0.014, P > 0.4). On average, the maximum biovolume of aquatic organisms is the antilog of 8.77 (i.e. 6 x 10 slam 3 m l - t ) , or roughly 0.1% of the space available regardless of body size. The relationships for protozoans, rotifers, and crustaceans did not differ significantly from each other or from Eq. (1) (Fig. 2a, Table 2, P < 0.05). No significant relationship could be found for bacteria alone, but this may simply reflect the small ranges in size and density of these organisms (Table 1) and the small sample size ( N = 12). The maximum density-size relationship for fish differed most (Fig. 2a, Table 2), having a shallower slope and a lower intercept. In other words, larger fish support a higher maximum biomass in culture than smaller larval or juvenil fish. Smaller fish also have lower maximum densities than do invertebrates of similar size. Because the ontogenetic changes in fish size are very large, the low maximum density of fish larvae could represent an adaptation to avoid otherwise catastrophic, density-dependent mortality as the fish grow. This is consistent with the suggestion that fish have evolved a strategy of density dependent growth rather than density dependent mortality (eg. Weatherley 1972), such that overstocking of fish populations results in stunting rather than extreme mortality (Swingle and Smith 1941; Weatherley 1972). Analysis of covariance also shows that warm-water fish (eg. cyprinids, catfish) reach higher densities than other species ( P < 0.0001). Despite these differences, Eq. (1) fits the data for all taxa surprisingly well. Eq. I is also remarkably similar to an equivalent equation for algae (Agusti et al. 1987), which does not differ in intercept ( P > 0.05) but has a significantly steeper slope (P < 0.05). The inverse proportionality of body size and density is not peculiar to monocultures at maximum density. It is a consistent feature of pelagic communities in both the sea (Sheldon et al. 1972; Rodriguez and Mullin 1986) and freshwaters (Sprules et al. 1983; Peters 1986; Sprules and Munawar 1986), so that this pattern also extends to natural populations in communities of mixed species. The elevations of Eq. (1) and of similar relations in Table 2 are most easily compared to each other and to other relationships in terms of biomass per unit volume. ANOVA indicated that the maximum biomasses of bacteria, protozoans, and fish do not differ significantly from one another (P>0.10), but are substantialy above those of crustaceans and rotifers ( P < 0.01). The average maximum concentrations for all groups in Table 2 are three to four orders of

Table 1. Number of observations, geometric means and ranges of size and density for the different taxa considered. All values, except N, are expressed as common logarithms Taxa

N

Bacteria Protozoans Rotifers Crustaceans Fish

12 16 7 12 32

Log Size (gin 3)

Log Density (Number/ml)

mean range

mean range

0.12-1.8 - 2.0 3.45 1.33- 6./9 5.01 2.39- 5.77 12.7 7.30-13.5 10.0 9.82-14.9

8.82 7.69- 9.84 5.81 2.13- 8.47 2.57 1.05- 6.0 -3.51 - 5 . 3 0 - -1.22 - 1.73 - 5.60-1.00

~.. 12,~. 9~

.~8-

,~

~ ~ [ ] ' ~a

a

9 fish ~ protozoans o crustaceans * bacteria ~ rotifers 9 turtles

D

a

o

~&~

o

DD

~0 E E

o~-4 -8

-2

)

~

r

1'~

78

log body size(,um 3)

Fig. I. The relationship between body size and the maximum density of aquatic organisms. The heavy line represents Equation 1

magnitude higher than the biomass of logarithmic size classes in natural communities (Sprules et al. 1983; Peters 1986; Sprules and Munawar 1986). However, each logarithmic size class of these natural communities comprises only a fraction of the total biomass, while each data point used to build Eq. (1) contains all the biomass present. Therefore natural and maximum densities may be better compared on the basis of the total biomasses achieved. The total biomass of natural pelagic communities range from 2 to 35"106 gm 3 ml 1 depending on lake trophic status (from Peters 1986, Table 2). This is still less than 1% of the maximum achievable biomass (i.e. 6,108 gm 3 ml-1). Natural populations exist well below their physiological limits, suggesting that other constraints, such as resource availability or predation, limit their abundance. However, a number of species of schooling fish attain densities close to the maxima in our analyses (Pitcher and Partridge 1979). Even if this is only coincidence, it provides a useful image of what constitutes maximum population density. Our results add to previous demonstrations of close associations between organismal size and density (Gorham 1979; Duarte and Kalff 1987; Agusti et al. 1987; Damuth 1981; Peters 1983; Peters and Wassenberg 1983; Calder 1984). However, these relationships cannot be simply compared since they comprise both areal and volumetric estimates of density. We addressed this problem by re-expressing both as the average distance between neighbouring organisms. This was done by first calculating volume or area per capita (v or a, respectively) as the inverse of density,

274 121-.,,, ~

. . . . . fish o a crustaceans ~, ~ bacteria rotifers

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1

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E=

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-8

a

. . algae

".~:L~..

-2

~

protozoans lake plankton

--

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.-.

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terrestrial

,'~

....... plants 2. . . . . . animals

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ue O-

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aquatic algae - - this study - - - plankton 9 --* plants

~

-4 log body size( Xim 3)

Fig. 2. a Comparisons of the relationships between maximum density and body size from Table 2, and for pelagic particles in lakes in Ontario (Sprules et al. 1983) b Comparison of relationships describing the average distance between organisms as a function of individual size for terrestrial and aquatic organisms (from Table 3) Table 2. Regression equations defining the maximum densities of populations from different taxa of aquatic organisms as functions of their body size. The regression equations are presented as log density (Number/ml)= a + b log size (pm 3) Taxon

a

b

R2

P

Eq.

Bacteria Protozoans Rotifers Crustaceans Fish

8.83 10.36 7.73 8.37 2.62

--0.09 --1.32 - 1.03 -1.00 -0.49

0.03 0.84 0.63 0.94 0.65

<0.50 <0.00l < 0.05 <0.001 <0.001

(2) (3) (4) (5) (6)

and then computing the average distance between neighboars as (6v/g) 1/3 for volumetric densities and (4aMc)1/2 for areal ones. This calculation assumes that each organism is in the centre of a spherical or circular individual space. It shows that the distance between organisms of similar size rises approximately as body volume 1/3 (Fig. 2b, Table 3). Since the linear dimensions of organisms also rise as body volume 1/3 (Peters 1983) interorganism distance is proportional to body length. For organisms of very different shape, a characteristic linear dimension (Lc, in pm) can be calculated as the diameter of a sphere with volume equivalent to that of the organism's body (Sheldon and Parsons 1967). This relation (Lc=1.24V ~/3) implies that, regardless of size, the interorganism distances for both animals and plants in Table 3 are about 10 to 20 times the equivalent spherical diameter for cultured organisms or organisms at their maximum density, and about 10 times further apart in nature. The size-dependent maximum implies that populations at or close to their maximum density should experience (density-dependent) mortality if the individuals are to increase in size. Such process would be analogous to selfthinning in plant populations (Westoby 1984; White 1985), and our results suggest a qualitatively similar phenomenon for animal populations (Begon et al. 1986). That the maxim u m density of aquatic organisms is proportional to their body size suggests that geometric constraints and not metabolic constraints (which characteristically scale as the 3/4 power of body size; eg. Peters 1983, Calder 1984), determine the existence of an upper limit to the abundance of aquatic organisms. The general scope of our analysis limits the precision of the relationships described (Eq. 1-6) for specific applications. However our findings may eventually prove of practical value in the management of aquatic organisms. The management of density in fish populations is a serious problem in both natural and cultured populations (Hackney 1974). Our results suggests that mortality and decreased growth (i.e. stunting) due to overcrowding could be minimized by stocking larval and juvenile fish at densities calculated from the desired size at harvest using Eqs. (1) or (6) (Table 2) or more precise, analogous, relationships developed for specific problems. Figure 2 might also help identify culture systems operating at submaximal densities which would therefore benefit from further elaboration. For example, this might be the case for rotifer culture systems. In summary, we have shown that the limit to the density of aquatic organisms is an inverse linear function of body size, so that smaller organisms are able to support greater

Table 3. Coefficients of equations relating the average distance between organisms in aquatic and terrestrial environments (Y, in pro) to individual body volume (V, pro3). Coefficients correspond to the equation: Y=aV b. The elevation and slope of the relation for terrestrial animals are the geometric means of the intercepts and slopes for equations of more taxonomically limited scope (Peters 1983) Organisms

a

b

Reference

Aquatic organisms in culture Algae in culture Crowded stands of terrestrial plants Crowded stands of aquatic plants Pelagic organisms in nature Terrestrial animals in nature

20 22 10 13 340 210

0.35 0.37 0.33 0.40 0.35 0.37

This study Agusti et al. 1987 Gorham 1979 Duarte and Kalff (1987) Sprules et al. (1983) Peters (1983)

275 densities, but not larger biomasses. The striking similarity in the way the distance between organisms, both plants and animals, scale to their body size regardless of the growing conditions, suggests the existence of a fundamental regularity that determines how organisms use the availabIe space.

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