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This document is an early draft of a paper published in The Science of the Total Environment 183: 107-114 (1996). It is not complete. Reprints may be requested from `silvert@milpah.com`. # Size-Aggregation in Models of Aquatic Ecosystems William Silvert Address at time of writing: Department of Fisheries and Oceans Habitat Ecology Section Bedford Institute of Oceanography P. O. Box 1006 Dartmouth, Nova Scotia CANADA B2Y 4A2 Current Email: `silvert@milpah.com` ## ABSTRACT Very simple models of pelagic marine and freshwater ecosystems have proved very successful and are increasingly widely used. These models are highly aggregated and based on biomass or particle size. Theoretical investigations based primarily on energy flow considerations have provided an understanding of the success of these models and suggest possible ways to extend their applicability. ## INTRODUCTION Ten years ago the author published a review on the use of particle size spectra in ecology (Silvert 1984), and during the succeeding decade there has been growing interest in this approach; there has also been a shift in emphasis from basic research to application. Size structure is now widely used as an investigative and diagnostic tool in ecological research. ## SIZE STRUCTURE OF ECOSYSTEMS One of the first ecologists to explore the size structure of ecosystems was Charles Elton (1927), who drew attention to the pyramidal ordering of terrestrial biomass shown in Figure 1a. In most cases, the biomass of primary producers is far greater than the animal biomass and, among animals, the herbivore biomass is usually far greater than that of carnivores. In some situations, there is a corresponding pyramid of individual size, with massive trees at the base of the food chain grazed by elephants and giraffes, which in turn are preyed upon by large cats. The Eltonian pyramid does not apply to pelagic aquatic ecosystems, but little was known about the size structure of these systems until the pioneering work of Sheldon et al. (1972). Based largely on a series of plankton measurements collected during the 1970 circumnavigation of the Western hemisphere by CSS Hudson and supplemented by data from other sources on macrofaunal stocks, Sheldon and his co-workers showed that over a size range of almost seven orders of magnitude in linear dimension, which is a range of approximately 20 orders of magnitude in mass, the particle concentration varies by only a single order of magnitude when expressed as the volume of particles per logarithmic size range (in simpler terms, the volume of particles in the size range 1-10 um is roughly the same as the volume of particles in the size range 1-10 cm). Thus although there is a pyramid in numbers, with 10 um plankton being roughly 1,000,000,000,000 more numerous than 10 cm fish, the biomass pyramid looks more like a vaguely Tuscan column, as shown in Figure 1b. Although there is a slight taper to the column, the ratio of biomasses at the top and at the bottom of the food chain is remarkably constant, so much so that primary production is considered by many a good predictor of fisheries production (Nixon 1988, Iverson 1990). Benthic systems do not fit either pattern and, although they exhibit a fair degree of regularity, there appear to be characteristic peaks and valleys in the biomass distribution (Schwinghamer 1981). A similar structure is present in pelagic systems, especially freshwater ones, but it varies with time and appears to involve different mechanisms (Sprules and Knoechel 1984, Boudreau et al. 1991). ## THEORY OF SIZE STRUCTURE Several theories have been developed to describe the size structure of pelagic ecosystems, all of which are based on essentially the same underlying model of energy flow through the food web (Silvert and Platt 1980). The most important assumption in the models is that the dominant feeding pattern consists of big particles eating small particles. Although there are obvious exceptions, this is so generally true that attention has recently turned to the question of why it holds as well as it does. The general pattern of feeding in zooplankton communities is shown in Figure 3, based on unpublished data (W. Greve, pers. comm.). In most cases, the predators are larger than the prey, so there is a flow of energy from small organisms to larger ones with a small reverse flow arising from the existence of some smaller predators. There is also reverse flow from spawning and excretion, but in general the flow is from small to large. Clearly this pattern does not hold as strongly in terrestrial systems. If one looks at a typical grassland ecosystem in which the principal groups are grasses, herbivorous rodents, and larger canine carnivores, the energy flow is certainly from small to large, but in the earlier example of trees grazed by giraffes and elephants which in turn are hunted by large cats (Figure 1), the opposite is the case. A likely explanation for this is the important rôle played by the substrate. On land a small animal can attack a larger one by knocking it down and pinning it to the ground. In the water this is not possible, and the usual method of predation is to engulf the prey. This works best if the prey is somewhat smaller than the predator, although if the prey is too small it can escape through gill openings or between cilia, or it may simply be energetically inefficient to chase very small prey. This is consistent with the pattern shown in Figure 3, which suggests the existence of an optimal predator/prey size ratio. Terrestrial systems can be characterized as two-dimensional because most life forms are constrained to move on the land surface, while pelagic aquatic systems are three-dimensional. Benthic systems are intermediate between the two because, as shown in Figure 4, there is a series of transitions between two and three dimensions as smaller and smaller spatial scales are examined. From the viewpoint of the macrofauna which move along the surface, it is a two-dimensional environment, but for smaller organisms which can move about in the interstitial pore water as well as for animals which tunnel through soft bottoms, the environment is three-dimensional. At smaller spatial scales, the bottom becomes two-dimensional again as it is utilized by organisms which occupy the surface of grains. It may be useful to characterize the interaction between benthic communities and the sediments in terms of fractal dimension, as shown in Figure 5. The alternation between two and three dimensions shares a curious similarity with the pattern of size structure shown in Figure 2. The idea that ecosystem dynamics might depend on the fractal dimension of the environment was advanced at least ten years ago (Silvert 1984), but so far no quantitative models relating the two have been developed. It thus appears that the size structure of ecological communities may be dominated by bioenergetic factors moderated by topological considerations, although the above argument is by no means exhaustive. In particular, the rôle of gravity has been ignored but, when considered in relation to the idea of a small carnivore pinning a large herbivore to the ground and killing it, this must play an important part. Another important factor is viscosity, which relates speed to size in aquatic environments so that large prey can easily escape small predators. Earlier work has suggested that gravity and viscosity can account for some of the observed differences between terrestrial and marine ecosystems, particularly the dependence of metabolic rates on body size, and it is likely that this would apply to predation as well (Platt and Silvert 1981). ## AGGREGATION BY SIZE The success of theories of size structure in aquatic systems has led to curiosity about how much can be learnt from highly aggregated measurements and models, and to questions about how valid this sort of aggregation can be. The idea that one can develop ecosystem models without starting from individual species is anathema to many ecologists. Functional groupings often reflect taxonomic similarities, but aggregation by size is probably as remote from taxonomic classification as is possible in ecological modelling. The potential value of size structure is clearly enormous. Sheldon et al. (1977) postulated that the observed patterns provide a means of estimating total fish production, which has been confirmed by later studies (Nixon 1988, Iverson 1990), and Sprules et al. (1979, 1984) have investigated size structure as a measure of the health of freshwater systems. An important consideration is the speed and economy of monitoring size structure, because for the important microscopic component it is possible to use a variety of automated technologies, such as the Coulter counter or various optical methods. The validity of size-structured models ultimately depends on how valid an approximation it is to aggregate on the basis of size as opposed to other properties. This issue was addressed by Silvert (1984), and the basic argument is that numerous studies have shown that in dynamic models it is best to aggregate compartments with similar turnover times, and, as shown in Figure 6, there is a strong correlation between the time scales of organisms and their sizes. ## EXTENSIONS TO SIZE STRUCTURE MODELLING There are many remaining questions in the study of size structured models. Probably the most important of these involves the changes which arise from taxonomic considerations. From Figure 6 it is evident that there is an allometric relationship between doubling time and body size of the form T ~ WK, where K is approximately 0.22. A detailed investigation of Production/Biomass (P/B) ratios by Banse and Mosher (1980) shows equivalent allometric behaviour for the P/B ratio with the same exponent, as shown in Figure 7 (P/B has dimension time-1, so the exponent of -0.22 from Figure 7 is equivalent to K = +0.22 in Figure 6). This is consistent with the hypothesis of universal time scaling proposed by Platt and Silvert (1981). The individual lines in Figure 7 show the allometric relationship between body size and P/B ratio within broad taxonomic groupings; with taxonomic disaggregation the magnitude of the allometric exponent increases drastically, in this case almost doubling. Investigations of fish mortality by Pauly (1979) reveal that the more narrowly the taxa are defined, the more the slopes increase, as shown in the shape of the oval areas for some of the taxa shown in Figure 6. This remains a fundamental issue in the theory of ecological size distributions (Boudreau et al. 1991). A possible explanation for this phenomenon is as follows: a taxon represents a type of physiological design, and each design is optimal only over a restricted size range. For example, there is an upper limit to the size of unicellular organisms, while very small animals do not need guts and central nervous systems. Thus as we consider larger organisms there is a higher level of physiological complexity with associated metabolic costs. We can express this by generalizing the allometric model to include a complexity measure C. If we write this as a multiplicative factor, the characteristic time for each organism is expressed by P/B = ACW-g where A is a universal constant. For each organism this expresses P/B as a function of complexity C and body size W. Each taxon has its own value of C, or, more generally, a range of values corresponding to the breadth of the taxon. Averaging over all taxa to estimate the characteristic time scale for organisms of size W gives <P/B> = A<C>W-g where <C> is the average complexity of organisms of this size. From the preceding arguments it is reasonable to suppose that <C> increases with W because more complex organisms are likely to require higher metabolic rates to maintain all their physiological mechanisms (compare a lethargic lizard on a rock with a mammal of the same size), and it may be assumed to be an allometric function as well. If <C>~ Wb we then obtain <P/B> = A'WbW-g = A'W(b-g) which gives us an allometric relationship for the average of all organisms which is different from that for individual taxa. Using the values from Banse and Mosher (1980) of g = 0.42 within individual taxa and g - b = 0.22 for all organisms, b = 0.20 is the exponent for complexity. For example, if the average weight of marine mammals is 1000 times that of fish (corresponding to a difference in average length of a factor of ten), this suggests that the metabolic costs of a seal would be (1000)0.2 = 4 times that of a fish of the same size. Of course the variation in metabolic rates between fish of equal size (say between sharks and tunas) may be comparable to or greater than this, but averaged over a larger size range the difference between taxa becomes significant. The difficult part in this is coming up with a meaningful definition of complexity. Bonner (1965) has proposed the number of different types of cells as a potentially useful way of classifying organisms, and that is probably the best measure to use at the present time. Various other measures have been proposed in private conversations, such as genome size, but it is difficult to find any useful quantity which is readily available for a wide range of different taxa. It is important to realize that complexity in this context means only a measure of the metabolic costs associated with a particular type of taxonomic structure and does not reflect adaptations that do not incur such costs. Examples of structures that are complex in this sense are guts, central nervous systems, and homoiothermy. ## SUMMARY Size structured models of pelagic aquatic ecosystems are examples of highly aggregated but still very successful models. There are fundamental reasons for the success of these models which reflect basic regularities of nature, as exemplified by the allometric relations which are an important part of size structure modelling. There is still a great deal of basic research to be done in this field, particularly in extending it to allow for a general pattern of increasing physiological complexity with body size. ## ACKNOWLEDGEMENT I am indebted to Dr. Wolf Greve of the Biologische Anstalt Helgoland for the use of his unpublished data on zooplankton grazing interactions. ## REFERENCES Banse, K., and S. Mosher. 1980. Adult body mass and annual production/biomass relationships of field populations. Ecol. Monogr. 50:355-379. Bonner, J. T. 1965. Size and cycle. An Essay on the Structure of Biology. Princeton University Press, Princeton, NJ, 219 pp. Boudreau, P. R., L. M. Dickie and S. R. Kerr. 1991. Body-size spectra of production and biomass as system-level indicators of ecological dynamics. J. Theor. Biol. 152:329-339. Iverson, R. L. 1990. Control of marine fish production. Limnol. Oceanogr. 35:1593-1604. Nixon, S. W. 1988. Physical energy inputs and the comparative ecology of lake and marine ecosystems. Limnol. Oceanogr. 33:1005-1025. Elton, Charles. 1927. Animal Ecology. Sidgewick and Jackson, London. Pauly, D. 1979. A new methodology for rapidly acquiring basic information on tropical fish stocks: growth, mortality, and stock recruitment relationships. In Stock Assessment for Tropical Small-Scale Fisheries, P. M. Roedel and S. B. Saila, Eds. ICMRD, Kingston, R. I. pp. 154-172. Platt, T., and W. Silvert. 1981. Ecology, physiology, allometry and dimensionality. J. Theor. Biol. 93:855860. Schwinghamer, P. 1981. Characteristic size distributions of integral benthic communities. Can. J. Fish. Aquat. Sci. 38:12551263. Sheldon, R. W., A. Prakash and W. H. Sutcliffe, Jr. 1972. The size distribution of particles in the ocean. Limnol. Oceanogr. 17:327340. Sheldon, R. W., W. H. Sutcliffe, Jr., and M.A. Paranjape. 1977. The structure of the pelagic food chain and the relationship between plankton and fish production. J. Fish. Res. Bd. Canada 34:23442353. Silvert, W. 1984. Particle size spectra in ecology. In Mathematical Ecology, S. A. Levin and T. G. Hallam, Eds. SpringerVerlag Lecture Notes in Biomathematics 54, pp. 154162. Silvert, W., and T. Platt. 1980. Dynamic energyflow model of the particle size distribution in pelagic ecosystems. In Evolution and Ecology of Zooplankton Communities, W. C. Kerfoot, ed. Univ. Press of New England, Hanover, N. H. pp. 754763. Sprules, W. G., and L. B. Holtby. 1979. Body size and feeding ecology as alternatives to taxonomy for the study of limnetic zooplankton community structure. J. Fish. Res. Board Can. 36:1354-1363. Sprules, W. G., and R. Knoechel. 1984. Lake ecosystem dynamics based on functional representations of trophic components. In Trophic Interactions within Aquatic Ecosystems, D. G. Meyers and J. R. Strickler, Eds. AAAS Selected Symposium 85. Westview Press, Boulder, CO. pp. 383-403. ## FIGURE CAPTIONS [These figures are reduced in size for ease of loading, you can click on any image to see it enlarged]
Figure 1. The Eltonian Pyramid of population biomass for terrestrial systems (a) contrasted with ... |
Figure 2. ... the more columnar biomass distribution found in pelagic aquatic systems (b). |
Figure 3. Predation intensity as a function of predator/prey size ratio in zooplankters (based on unpublished data from W. Greve). |
Figure 4. Representation of a granular benthic substrate showing the transition between two- and three-dimensional environments at different spatial scales. |
Figure 5. Fractal dimension of bottom sediments as a function of spatial scale. |
Figure 6. Relationship between doubling time and particle size (Sheldon et al. 1972). |
Figure 7. Relationship between P/B ratio and particle size (Banse and Mosher 1980). |

## Important change in this website: Bill Silvert's website has moved to http://www.milpah.com/~bill/index.html This site is no longer current! Developed and maintained by William Silvert, last updated on 01 Jul 2002. All of this material is copyrighted by William Silvert unless otherwise indicated.