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Oecologia (2005) 145: 216–224 DOI 10.1007/s00442-005-0062-z S P E C I A L T O P I C : S C A L I N G - U P I N E C O LO G Y John E. Petersen Æ Göran Englund Dimensional approaches to designing better experimental ecosystems: a practitioners guide with examples Received: 14 March 2004 / Accepted: 3 February 2005 / Published online: 11 May 2005 Springer-Verlag 2005 Abstract Enclosed, experimental ecosystems (‘‘mesocosms’’) are now widely used research tools in ecology. However, the small size, short duration and often simplified biological and physical complexity of mesocosm experiments raises questions about extrapolating results from these miniaturized ecosystems to nature. Dimensional analysis, a technique widely used in engineering to create scale models, employs ‘‘compensatory distortion’’ as a means of maintaining functional similarity in properties and relationships of interest. An earlier paper outlined a general approach to applying dimensional analysis to the construction and interpretation of mesocosm experiments (Petersen and Hastings in Am Nat 157:324, 2001). In this paper we use examples, largely drawn from the aquatic literature, to illustrate how dimensional approaches might be used to maintain key ecological properties. Such key properties include effective habitat size, environmental variability, vertical and horizontal gradients, and interactions among habitats. We distinguish both continuous and discrete approaches that can be used to achieve functional similarity through compensatory distortion. In addition to its potential as a tool for improving the realism of experimental ecosystems, the dimensional approach points towards new options for developing, Electronic Supplementary Material Supplementary material is available for this article at http://dx.doi.org/10.1007/s00442-0050062-z Communicated by Craig Osenberg J. E. Petersen (&) Oberlin College, Lewis Center for Environmental Studies, 122 Elm St, Oberlin, OH 44074, USA E-mail: [email protected] Tel.: +1-440-7756692 Fax: +1-440-7758946 G. Englund Department of Ecology and Environmental Science, Umeå Marine Science Center, Umeå University, 901 87 Umeå, Sweden testing and advancing our understanding of scaling relationships in nature. Keywords Dimensional analysis Æ Experimental design Æ Extrapolation Æ Mesocosm Æ Primary productivity Æ Scale Introduction Enclosed experimental ecosystems (‘‘mesocosms’’, ‘‘microcosms’’, ‘‘microecosystems’’, etc.) include any laboratory and field-based systems that containerize and isolate communities with their physical environment for the purpose of experimentation. These systems have become widely used tools in ecology (Ives et al. 1996) because they allow for a greater degree of control, replication, and repeatability than is achievable for experiments conducted in whole natural ecosystems or in plots that are completely open to the natural environment (Kemp et al. 1980; Kareiva 1989). Yet there are a number of potential problems with mesocosms as research tools. Specifically, relative to the ecosystems and processes they are designed to simulate, mesocosm systems are often: small, of short duration, subject to reduced biological, material and energetic exchange, exposed to reduced temporal and spatial variability, simplified in terms of both physical and biological complexity, and subject to a variety of artifacts associated with enclosure such as large edge effects. Numerous studies have demonstrated that such differences can compromise our ability to make inferences about natural systems (reviewed in Schindler 1998; Englund and Cooper 2003) How can ecologists ensure that the dynamics of large natural ecosystems are adequately represented in small experimental ecosystems? How can we extrapolate information from such experiments to nature and among natural ecosystems? Experimental research is premised on identifying clear answers to these questions. Dimensional analysis is a technique widely used by 217 engineers to design scale models (airplanes, valves, waterways, etc.) that conserve specific functional attributes of a larger systems of interest (see Langhaar 1951). In a previous paper (Petersen and Hastings 2001), a general approach was introduced for applying dimensional analysis to the design of mesocosm experiments. In this paper we draw on examples from the ecological literature to begin to outline a framework for explicit, systematic, and quantitative application in a variety of ecological and experimental settings. Dimensional considerations begin with the premise that the critics of mesocosm research (e.g., Schindler 1998; Carpenter 1999) are at least partially correct; mesocosms are inherently distorted representations of nature. The crucial question is, can we somehow compensate for these distortions in both design and interpretation of experiments in order to simulate realistic dynamics in the ecological properties of interest? The formal technique of dimensional analysis is based on the premise that certain universal relationships apply regardless of the scale and dimension of a particular system under investigation. In general, the technique involves identifying dimensionless expressions that capture a balance between processes or forces governing the dynamics of a particular system; distortions in one dimension or variable are counterbalanced by distortions in others in order to achieve ‘‘functional similarity’’. By functional similarity, we mean that the conditions, relationships and behaviors that are of interest within the experimental ecosystem are made similar to those in nature. A dimensionless expression or ‘‘nondimensional variable’’ is simply a mathematical expression in which constituent terms are combined in such a way that the units cancel. Using such variables allows researchers to directly compare and extrapolate observations made in very different types of ecological systems (Fig. 1). Applied to experimental design, dimensional analysis might be used to counteract the set of scaling distortions that are inherent in developing an enclosed experimental ecosystem (e.g., reductions in size, exchange, variability and duration) with compensatory distortions in other variables in order to conserve the value of key nondimensional variables. Our goals in this paper are: to suggest functional similarity as a critical objective for the design of experimental ecosystems; to outline a general procedure for achieving similarity; to provide empirical ecologists with examples of how dimensional considerations might be applied and expanded on to achieve functional similarity in a range of key ecological relationships; to suggest important limitations of the dimensional approach. A general procedure for conserving key ecological relationships in enclosed experimental ecosystems Five key steps can be distinguished in successful application of the dimensional approach to the design of enclosed experimental systems: 1. Carefully define and refine the research question 2. Identify the desired scale of inference and appropriate level of abstraction for the experimental system relative to the research question 3. Select key relationships that must be conserved and non-dimensional expressions that accomplish this 4. Select an experimental design that conserves these relationships 5. Analyze and extrapolate results to nature. As with other types of models, enclosed experimental ecosystems are simplifications and abstractions of nature. Selecting an appropriate level of abstraction for the system (step 2 above) has direct bearing on how we achieve and appraise functional similarity and involves tradeoffs between control, realism, generality and scale (Kemp et al. 1980). Control refers to the degree and ease with which experimental conditions can be manipulated, realism is a measure of the extent to which the experimental system and the experimental results accurately represent the dynamics of particular natural ecosystems, and generality is a measure of how broadly the findings of a particular experiment can be applied to different kinds of ecosystems. In considering tradeoffs it is useful to distinguish between two extremes in model ecosystems, generic and ecosystem-specific models. Generic mesocosms are used to test broad theories that potentially apply to many different kinds of ecosystems. These systems tend to be small, highly artificial, require minimal physical and biological complexity, and are designed to elucidate general properties, such as ecosystem development, predator–prey interactions, and relationships between stability and diversity rather than the properties of particular natural ecosystems (e.g., Cooke 1967; Fig. 1 Time series of plant biomass in experimental plankton and marsh ecosystems graphed with non-dimensional time and biomass units. Time units are the time elapsed in the experiment divided by the characteristic turnover time for primary producers. Characteristic turnover times used here are 1.4 days for phytoplankton and 3 months for marsh plants. Biomass is graphed as a percentage of maximum biomass recorded in each experiment (4 kg m2 dry weight for marsh plants and 14 lg L1 chlorophyll-a for phytoplankton). Data are from experiments conducted at the Multiscale Experimental Ecosystem Research Center at University of Maryland (see Petersen et al. 2003) 218 Luckinbill 1973; Naeem and Li 1997). Since precise correspondence with particular ecosystems is not an objective, the researcher has considerable flexibility in selecting compensatory distortions to achieve the desired state of functional similarity in these generic models. As the term implies, ecosystem-specific mesocosms are used to test hypotheses linked to particular types of ecosystems. To achieve the higher degree of realism required, these systems must incorporate the essential physical and biological features that control the dynamics in the systems that they represent. The variety of ecosystem-specific models constructed has ranged from coral reefs to rain forests (e.g., Cohen and Tilman 1996; Luckett et al. 1996). As the desired degree of specificity and desired level of realism increase, so to do the challenges associated with selecting compensatory distortions to achieve functional similarity in all of the ecological relationships of interest. Obviously a continuum exists between generic and ecosystem-specific models, with tradeoffs analogous to the successive increases in realism and experimental complexity involved in modeling human physiology with flatworms, fruit flies, rats, and chimpanzees. As we discuss in the sections that follow, different dimensional approaches are appropriate in association with different degrees of abstraction. Which of many ecological relationships to conserve (step 3) depends on the research question and the ecology of the system. As a generalization, we suggest four overlapping environmental characteristics that are key determinants of many ecological processes: 1. 2. 3. 4. Habitat size and time scale Environmental variability Horizontal and vertical gradients Interaction among adjacent habitats In addition to being crucial features of most ecosystems, these are ecological attributes that are typically distorted in the construction of experimental ecosystems. The challenge is to develop a systematic dimensional approach to designing experiments that conserves functional similarity in these key ecological relationships. Often this amounts to preserving ‘‘effective scales’’, which are calculated by standardizing scaling attributes of organisms or processes (e.g. generation time, speed, home range) to scaling attributes of the physical environment (time and space scales). Designing functional similarity in experimental ecosystems We now explore concrete examples of methods that have been used to conserve functional similarity in each of the four key environmental characteristics introduced in the previous section. In some of these examples the objective of dimensional manipulation has simply been to achieve a more realistic experimental model of nature (e.g., Margalef 1967; Adey and Loveland 1991). In other cases, dimensional manipulations have been explicitly employed as a means of investigating relationships among the counteracting variables (e.g., Huffaker 1958; Gilbert et al. 1998). In both situations, the application of dimensional thinking has often been intuitive, idiosyncratic and qualitative. Our objective here is to review and expand on these examples, point out limitations as well as opportunities, and in the process to outline a practical and quantitative procedure that might be broadly applied towards improving experimental design. Conserving effective time and space scales Manipulating organism size Perhaps the most obvious option available for conserving relatively large effective scales at reduced absolute time and space scales is to assemble mesocosm communities composed of small organisms. This approach has generally been used to construct what we have termed generic mesocosms for addressing general ecological theory. The most extreme example of this may be 1011 m3 microcapillary tubes that were successfully used as experimental systems to study spatial and temporal dynamics of competitive exclusion among protozoa feeding on herbivorous bacteria (Have 1990). More recently, this small-organism approach has received considerable attention as a means of elucidating general relationships between biological diversity and ecological function (Lawton 1995; Naeem and Li 1998). In a more applied context, Schmitz 2005 (this volume) uses cage experiments involving spiders, grasshoppers and herbs to guide management of forest ecosystems. These experiments take implicit advantage of the fact that the characteristic scales associated with organisms (i.e. lifespan, generation time, speed, home range, etc.) tend to decrease with body size (e.g. Sheldon et al. 1972). For example, if we are interested in designing a laboratory-scale experiment to study general properties of nature reserves, we might choose to match the key effective scales associated with the top predator. Suppose the organism of interest in a natural system has a generation time of G=5 years, a home range of H=1 km2, a mean squared displacement of D=4 km2 year1. Suppose further that we are interested in predicting responses over a period of T=25 years in a reserve of size A=2 km2. Two obvious nondimensional variables that capture the relationship between organism and environment are the ratios of response period to generation time (T/G=5), and of reserve area to home range (A/H=2). The overall effective temporal and spatial characteristics of the predator can be expressed in another nondimensional term GD/H, which for the organism in this example works out to 20. Imagine that we are constrained by experimental resources (time, money, and space) to an experimental system with a 219 duration (T) of 12 months and an area (A) of 1 m2. In order to conserve all three nondimensional numbers, simple algebra dictates that we seek a model predator with a generation time G 70 days, a home range H 0.5 m2, and a mean squared displacement D of 0.14 m2 day1. Although we might not find an organism with exactly these attributes, we may be able to adjust the size and duration of the experiment to achieve an acceptable level of similarity for our study. Well established allometric relationships between body size and a host of physiological and ecological attributes may provide a more general tool for identifying characteristic scales that can then be used to match effective scales between natural and experimental systems. The basic form of the allometric equation is: R ¼aW b ; ð1Þ where R, some characteristic temporal or spatial scale, W, mass or size of an organism, and a and b, are scaling coefficients. Characteristic scales of organisms amenable to allometry and relevant to experimental design include attributes such as home range, patch size, population density, generation time, gestation period, life span, speed, feeding rates, productivity, rates of succession, duration of predator–prey cycles, etc. (e.g., Peters 1983; Enriquez et al. 1996; West et al. 1999). For instance, our objective might be to select a group of organisms and/or size of experimental system so as to conserve overall relationships between home range and available habitat area relative to a natural reference. The conserved relationship can be expressed nondimensionally as either: HM HN AM ¼ ; or H M ¼ ; AM AN AN ð2Þ where H, home range, A, area of available habitat and subscripts M and N refer to the mesocosm and the natural reference ecosystem respectively. Using the basic allometric relationship (Eq. 1) HM and HN can be reexpressed as functions of body size: H M ¼ aM W bM ; H N ¼ aN W bN ; ð3Þ where WN and WM refer to the size of key organisms in nature and mesocosms respectively. Since the scaling coefficient b tends to be similar for a given variable even among disparate groups of organisms (Peters 1983), it is not given a subscript. Substituting Eq. 2 into Eq. 3 and solving for test organism size (WM) or mesocosm size (AM), we find: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W M ¼ W bN ðAM =AN ÞðaN =aM Þ; or : ð4Þ AM ¼ AN ðW M =W N Þb ðaM =aN Þ If the organisms of interest in the mesocosm and natural prototype fall within the same general groups (i.e., unicells, homeotherms, poikilotherms), then we can assume thataM aN (Peters 1983), in which case Eq. 4 can be further simplified by dropping the (aN/aM) and (aM/aN) terms. Equation 4 is robust in the sense that the AM term could be replaced with any of the other variables that conform to the basic allometric relationship (Eq. 1). An obvious and important limitation to allometric scaling, and indeed to the dimensional approach in general, is that nondimensional similarity is only one of several characteristics of model realism. In many cases idiosyncratic species-specific characteristics that are independent of dimensionality may be of equal or greater importance in determining ecological dynamics. The large scatter around many allometric regressions illustrates this point (Peters 1983). For this reason we expect allometric scaling to be more accurate for groups of species than for individual species, simply because mean dynamics of communities can be predicted more accurately than the patterns of individual organisms. Alternatively, allometric scaling may be applied to a restricted set of species that differ in size but have similar morphology and ecology. For some taxa it may even be applied to individuals within the same species, e.g., where young fish are used as a model for adult fish. Manipulating the physical environment In addition to selecting communities of small organisms, a mesocosm researcher can also compensate for reduced size by manipulating the physical characteristics of the environment within the experimental ecosystem. A common problem in enclosed experimental systems is that certain predators and prey aggregate along the walls, resulting in elevated encounter rates. Bergström and Englund showed that encounter rates could be reduced to achieve functional similarity with nature in experiments by decreasing container size, as this reduces the area from which predators and prey are recruited from (Bergström and Englund 2002, 2004). Alternatively, the length of the wall habitat could be increased. This was demonstrated in a study of an experimental mite community (Kaiser 1983), where the addition of interior walls to create a labyrinth-like arena led to reduced predation rates as a result of lowered predator and prey densities along the walls. The dilution effect observed in Kaiser’s experiment described above occurs when the scale of the added physical structure is large compared to the mobility and perceptive range of the organisms. However, in some circumstances the addition of small-scale physical structures can also reduce encounter rates because detection distances or search speeds of predators are reduced (Persson 1991). Thus there is a need for dimensionless metrics that describe physical complexity of the habitat on scales that can be related to the scale of the organisms. The equation for fractal dimension provides one of several potentially useful indices of physical complexity: LðkÞ ¼ Ck1D ; ð5Þ 220 where L, measured length, k, step length of measurement, D, fractal dimension, and C is a constant (see Sugihara and May 1990 for general discussion of application of fractals in ecology). The fractal dimension D is nondimensional and can be thought of as a measure of the degree of roughness or spatial convolution. If key ecological interactions are controlled by surface area phenomena then organism size or the perceptive grain of foragers (e.g. Hines et al. 1997) could be taken as characteristic step length (k). In such cases it may be possible to manipulate the surface roughness to achieve the same value of D experienced by organisms in the experiment and in the natural prototype or to elevate D so as to compensate for reduced total area in the experiment. mesocosms to simulate the effects of schooling planktivores on phytoplankton (Petersen et al. 2003). As another example, sinusoidal variation in light intensity has been used to simulate the variable light environment that phytoplankton experience as they move up and down through the water column (e.g., Gervais et al. 1999, also see Electronic Supplementary Material appendix S1). In general, the two variables that are controlled to achieve functional similarity are the time and intensity of exposure. Time-for-space substitutions are most appropriate when variability is controlled by physical processes over which organisms have no control, such as the variability experienced by planktonic organisms as they mix vertically and are passively transported through heterogeneous waters. Tessellated microcosm landscapes Conserving effective environmental variability Time-for-space substitution While, in some circumstances it may be desirable to elevate spatial heterogeneity in order to conserve effective habitat size, in other cases it may be the heterogeneity itself that the researcher wishes to conserve in an experimental ecosystem. Indeed, both the degree and the quality of spatial heterogeneity are increasingly recognized as important factors controlling dynamics in natural ecosystems (Hastings 1990; Duarte et al. 1992; Melbourne and Chesson 2005; Helms and Hunter 2005; Inouye 2005, this volume). Predators, disturbances, and resources are often patchy in distribution, and a significant problem for the experimentalist is that the size of these patches may be much larger than the feasible size of the experimental ecosystem. Furthermore, it is often desirable to generate a relatively homogenous environment within a mesocosm (for instance by continuous mixing in a planktonic system) so that a single sample drawn at random can be used to characterize the system as a whole. Time-for-space substitutions provide a strategy to conserve the effective environmental variability that organisms experience, while simultaneously maintaining internal homogeneity. By time-for-space substitution we mean that the spatial variability experienced by an organism as it moves through a heterogeneous environment in nature, be it predation pressure, or resource availability is substituted with temporal variation induced by the experimentalist. For example, Turpin and Harrison (1979) used pulsed additions of nutrients into a homogeneous planktonic mesocosm to study the effect of patchiness on competition between phytoplankton. They found that pulsed addition favored large species (which could store resources) while continuous addition favored a small species. Similar time-for-space substitutions have been applied to examine the effects of variability in top-down control by planktivores. For instance experiments have been conducted in which groups of fish are periodically added and removed from Natural landscapes are often subdivided into patchy habitats with various degrees of isolation from each other. An option available for achieving functional similarity in this type of spatial heterogeneity under laboratory conditions is to link multiple individual container cells with varying degrees of exchange among these cells. Studies of this type have explored the effects of manipulating a wide variety of landscape attributes including number of patches, patch size (grain), aggregate landscape size (extent), spatial configuration of patches, exchange between patches, and assembly sequence (see Electronic Supplementary Material appendix S2 for examples and references). Theoretical work on the dynamics of spatially structured populations suggests key properties that might be nondimensionalized in these experimental systems to allow for more quantitative comparison with nature. For example, an important nondimensional number for subsystems or patches is the ratio of (B+I)/(D+E), where B, I, D and E are the per capita rates of birth, immigration, death and emigration, respectively (Thomas and Kunin 1999). This ratio indicates if a patch is a net producer or consumer of individuals (source or sink). Another ratio that we may find desirable to preserve for patches in experimental landscapes is (B+D)/ (I+E), which describes the relative importance of within and between patch processes. Conserving effective gradients Biological, chemical, and physical gradients are both a cause and an effect of ecological interactions. For instance, light energy is absorbed by plant and nonliving material as it passes down through a forest canopy or water column, and this has profound impacts on ecosystem energetics and ecological zonation. The reduced length scales inherent to mesocosms constrain the space available for interactions that occur in response to many ecologically important gradients. Several dimensional approaches can be taken to compensate for reduced 221 length scale in order to maintain functional similarity with respect to natural gradients. Condensing a continuous vertical gradient: the light environment Primary productivity and other important processes in pelagic ecosystems are strongly influenced by the light gradient that organisms experience as they move up and down through the water column. For example, it appears that early measurement of primary productivity was severely flawed because the traditional method, using incubation in glass bottles at different depths, failed to reproduce the vertical movements of algae through the light gradient (Gieskes et al. 1979; Schindler 1998). The light environment, therefore, provides a useful example of how a quantitative dimensional approach can be used to condense and realistically simulate ecological gradients. Light generally exhibits an exponential decay in intensity with depth, and this can be described with the Beer-Lambert law: I Z ¼ I o ekd z ; ð6Þ where Io, surface light intensity (e.g., lmols m2 s1), Iz, light intensity at depth z, kd, the light attenuation coefficient (m1), and z, depth (m). When z is taken as the depth of the mixed layer (or depth of the water column in shallow systems), the product kdz is refered to as ‘‘optical depth’’, and provides a good description of the vertical light environment experienced by phytoplankton. The fact that optical depth is a nondimensional variable immediately suggests an option for counterbalancing the reduced depth of most aquatic mesocosms. Indeed, since the quantity is the product of only two numbers, the only way to conserve optical depth as actual depth is reduced is to increase the light attenuation coefficient. Expressed as a scaling factor : ZN K dM ¼ K dN ; ð7Þ ZM where the subscript M refers to parameter values in the mesocosm and the subscript N refers to parameter values in nature. Reports suggest that kd might be controlled by manipulating the optical properties of container walls and tank radius (W.M. Kemp, Horn Point Laboratory, unpub. data; Nixon et al. 1980; Peeters et al. 1993). Addition of different concentrations of biologically inert dye also provide a means to control light attenuation in order to simulate the optical depth of a deep water column in a shallow mesocosm. A few attempts have been made to influence the light environment by intentionally manipulating the optical properties of the wall material (Peeters et al. 1993; Rijkeboer et al. 1993). However, we are unaware of any explicit attempts to duplicate optical depth. Conserving optical depth would ensure that phytoplankton experience the same range of light intensities moving from surface waters to the bottom of the tank that they experience in nature. However, there are other important features of the light environment, such as the rate of change (fluctuations) in light intensity that phytoplankton experience as they randomly move up and down through the mixed layer. The rate of change in light intensity determines which of a variety of physiological responses phytoplankton will use (see Lewis et al. 1984 for discussion of dimensional considerations associated with photoadaptation). Unfortunately, if the vertical mixing environment (kz= vertical eddy diffusivity) is held the same in the mesocosm as in nature, then conservation of optical depth will necessarily ensure that the phytoplankton experience more rapid fluctuations in light intensity in the mesocosm than in the deeper natural system. Thus, conservation of both the variability and gradient in light entails compensatory distortion in mixing as well as light attenuation. The key variable to conserve in order to achieve functional similarity in light fluctuation is the mixing time (Tm), which is a measure of the average time it takes for a particle to circulate up and down through the water column. Mixing time can be expressed as a function of kz: Tm ¼ z2 ; 2k z ð8Þ where z, depth of the mixed layer or mesocosm (Sanford 1997). Setting mixing time equal in nature and the mesocosm and solving for eddy diffusivity, we find an appropriately scaled kz of: 2 zM ; ð9Þ k zM ¼ k zN zN Thus, to conserve the vertical light environment experienced by phytoplankton we would design our mesocosm with a light attenuation of kdM and an eddy diffusivity of kzM. This example of conserving the vertical light environment points out a number of important features of the dimensional approach. The first is that it is often necessary to simultaneously introduce several compensatory distortions (e.g., increasing kd, decreasing kz) in order to conserve the desired relationships. Second, the example demonstrates that it is important to consider the full implications of each compensatory distortion (e.g., conserving optical depth alone necessarily increases light variability). Third, it is almost certainly impossible to simultaneously conserve all variables of ecological significance, and it is therefore critical to enter into the process with a clear sense of the dominant ecological properties that need to be conserved in order to address the research question at hand. For example, in the preceding example a different set of compensatory distortions would have been warranted had our objective been to generate realistic benthic–pelagic interactions such as nutrient regeneration, benthic feeding, or sediment resuspension. 222 Simulating continuous gradients with discrete compartments The examples presented immediately above conserve effective gradients by means of compensatory distortions that compress relatively large-scale gradients that occur in nature within much smaller experimental systems. In many cases the gradient of interest may be too large to compress within a single mesocosm, or the condensed gradient may bring organisms or biogeochemical processes into artificially close proximity thereby inducing unrealistic dynamics. An alternative approach is to simulate the gradient with a series of linked modular mesocosms (similar to tessellation). Conditions within each compartment are held relatively constant, but differ among compartments in accordance with position along the gradient. This approach is analogous to the finite difference approach to solving differential equations; continuous change is approximated with discrete, incremental units. An example is the apparatus used by Estrada et al. (1987) to tackle the light gradient problem discussed above. They used an acrylic tube that was passed down through a series of black rubber sheets that segregated the water column into discrete zones, each of which was independently illuminated from the side. This apparatus is a powerful design in that it can potentially be used to simulate any vertical light profile desired, independent of phytoplankton density. For example, a water column of total depth z might be divided into p number of equally spaced partitions numbered sequentially from top to bottom. In order to simulate the light profile of a natural system with an attenuation of kd (Eq. 6), light intensity in each partition would be set at: I p ¼ I ðp - 1Þ ekdðz=pÞ ; ð10Þ where Ip, light intensity in partition p and I(p-1), light intensity in the partition above (Io for surface partition). Perhaps the most widely applied substitution of discrete for continuous gradients has occurred in models of the estuarine salinity gradient (e.g., Margalef 1967; Doering et al. 1995, Electronic Supplementary Material appendix S2). In these models, connected compartments represent different salinity zones within the estuary. Some estuarine models have been so elaborate as to include special devices that allow large organisms, such as crabs, to migrate up and down the salinity gradient (Adey et al. 1991). In many cases, particularly for ecosystem-specific models, this discrete approach may provide a more realistic model of nature than is possible with continuous approaches to compensatory distortions. Conserving interactions among adjacent habitats: ‘‘Multicosms’’ Mesocosms have typically been designed to simulate environmental conditions within a single habitat type. However, in nature biotic and abiotic interactions among functionally different habitat types are often as important as strictly internal interactions in determining overall ecological dynamics. An option for studying these among-system interactions is to control exchange among a series of discrete, coupled, mesocosms, where each mesocosm is functionally distinct. For instance, mesocosms may represent a series of adjacent habitats (e.g., upland fi marsh fi littoral fi open water), or interacting subsystems segregated by functional group (e.g., producer, consumer, decomposer, or predator, and prey). One such multicosm was designed to simulate the interactions occurring between beds of submerged aquatic vegetation (SAV) and the pelagic community that passes through on each tidal cycle. These interactions were simulated by controlling flow between linked SAV and pelagic mesocosms (Petersen et al. 2003). Retention time in the pelagic component and exchange of water between pelagic and SAV components were each scaled to simulate ranges characteristic of nearshore regions of the Chesapeake Bay. These sorts of quantitative modular approaches to linking habitats hold the promise of expanding the usefulness of mesocosm research to larger scales than are possible with single, unlinked mesocosms (see Electronic Supplementary Material appendix S2 for additional examples). Constraints and opportunities The examples assembled in this paper illustrate that a diverse variety of dimensional approaches can be brought to bear on the problem of designing experimental ecosystems that conserve key ecological relationships. Along the way we have attempted to identify challenges as well as the opportunities associated with these approaches. Limitations associated with biological constraints can be roughly parsed into four categories (Petersen et al. 2001): inflexibility, specificity, equivalency, and interdependence. Limitations associated with the inflexibility and specificity of biological variables and relationships are particularly pronounced in ecosystem-specific models where organism size, generation time, and relationships among organism are constrained by the particular ecosystem under investigation, leaving manipulation of the physical environment as the only option for achieving functional similarity. The issue of equivalency is of special concern for generic models in which we are attempting to elucidate general ecological principles that apply to many different kinds of ecological systems. Whether there really are general principles relating variables such as ecological complexity to function remains an open question that will not be easily resolved with even the cleverest of dimensional approaches. Interdependence among biological variables is problematic for both ecosystem specific and generic mesocosm experiments in that, it ensures that compensatory distortions that increase realism for one organism or one set of 223 relationships typically also decrease realism for others. For a given experiment, the dimensional compromises that provide an optimal balance among the three critical research goals of control, realism and generality are a function of the ecological processes and relationships associated with the research question at hand. Collectively, the limitations described above make it impossible to develop a single ‘recipe’ for applying dimensional approaches to experimental design. Nevertheless, we believe that the approach and examples outlined in this paper provide a useful framework for designing experimental ecosystems that use dimensional considerations to retain key functional relationships present in larger natural ecosystems. It also seems clear that the search for a quantitative dimensional approach raises questions about the ecology of scale and about the generality and realism of experimentation that are of fundamental importance to advancing ecological science. Acknowledgements John Petersen’s contributions to this work were funded by the U.S. EPA STAR program as part of the Multiscale Experimental Ecosystem Research Center (MEERC) at the University of Maryland Center for Environmental Science (Grant number R819640, Maryland U.S.A.). 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