Issues of scale, particularly extrapolations across spatial or temporal scales, are of great importance in ecotoxicology. A variety of approaches are valuable for addressing issues of scale, and in developing scaling relationships that allow for extrapolation across scales. Both theoretical and empirical approaches can be applied in deriving scaling relationships. A variety of models may be employed, including statistical models, mathematical models, computer simulations and physical models (such as microcosms and mesocosms). Specific approaches that appear promising include dimensional analysis, allometric scaling, fractal geometry and microcosm/mesocosm experiments in which scale is appropriately manipulated as a treatment variable. Further research should be directed toward assessing the strengths and weaknesses of these approaches, identifying non-linearities and thresholds in scaling relationships, integrating spatial and temporal aspects of scaling, and designing experimental studies that explicitly address scale issues.

## Introduction

Issues of scale are pervasive and of crucial importance in environmental toxicology. As in most sciences, however, issues of scale are commonly hidden from direct view. Scale may form an integral part of the conceptual and methodological framework of the discipline, but assumptions related to scale, and the consequences of those assumptions, are often not directly addressed. If you observe carefully or purposefully, things out of scale intuitively bother us. Examples from cinema come to mind: giant spiders that attack cities, or modern day dinosaurs roaming on islands (see Jurassic Park, Lost World, Gulliver's Travels, etc.). These are out of sync in size (spatial scale) or time (temporal scale), and thereby seem implausible.

Comprehensive reviews of the many and varied approaches to scale issues are available elsewhere in the literature (see, for example, Schneider, 1994; Peterson and Parker, 1998; Brown and West, 2000; Gardner et al., 2001). In this paper, we chart a path that provides an overview of major conceptual issues, with examples of their relevance to environmental toxicology. This paper addresses issues of scale, with attention to theoretical approaches, empirical relationships, and physical models, and suggests guidelines and new directions for ecotoxicological research.

It is useful to distinguish between two aspects of scale: extent and grain (Wiens, 1989; Ahl and Allen, 1996; Turner and Johnson, 2001). In the spatial domain, extent refers to physical size and is measured in units of length, area or volume. In the temporal domain, extent refers to a length of time, or duration. The extent of a physical system, or a simulation model, or an empirical data set imposes an upper limit on the size of the phenomena that can be contained or captured within the system, model or data. At the extent of a forest stand, fire acts as an exogenous perturbation that can radically alter biomass and species composition, but at the broad extent of the boreal forest, fire is an intrinsic aspect of the internal dynamics of the system. Similarly, atmospheric CO_{2} concentrations can be viewed as being relatively stable, monotonically increasing, or episodically oscillating, depending on whether the temporal extent of the data under consideration encompasses a few years, a century, or many millennia.

Another aspect of scale is grain, which specifies the resolution of data or models, or the sampling frequency associated with observations. While extent determines the largest features captured at a certain scale, grain determines the smallest (or highest frequency) phenomena to be observed. A global climate model which grids the earth's surface into large cells may be useful for predicting global trends, but inappropriate for inferring local impacts. Similarly, annual sampling of streams and rivers for contaminants may be useful for documenting long term trends, but would be inadequate for characterizing fluctuations due to episodic precipitation events.

Ecological phenomena, including the response to toxicants, encompass a wide range of scales. The data we collect and the models we use to understand or predict the effects of toxicants in ecological systems necessarily truncate this range of scales to some operational grain and extent. It is simply not feasible to collect data or construct models encompassing the entire range from molecular to global spatial scales. Rather, we collect data and construct models at scales relevant to particular questions. There is no inherently ‘right’ scale, but the scale should be chosen wisely. At times, data or models may be applied to make inferences outside the range of grain and extent to which they properly apply. This is an extrapolation across scales, and like all extrapolations, must be approached with due caution.

Scaling is often used as an adjective to refer to a mathematical relationship or function that describes how some property varies with a change in scale. Such scaling relationships may be entirely empirical, or they may be based on theoretical foundations. Alternatively, as a verb, ‘scaling’ can refer to extrapolating across scales. Such extrapolations may involve physical systems, models, or both.

## Theoretical approaches

### A simple toxicokinetic model

It is often assumed that bioconcentration of a toxicant from aqueous solution to an aquatic organism can be represented by:

where*C*

_{org}and

*C*

_{aq}are the concentrations of the toxicant in the organism and the aqueous concentration, respectively, and

*BCF*is the bioconcentration factor which is assumed to be a constant. This equation represents a steady-state condition, and is often justified by assuming a first-order, one-compartment toxicokinetic model: where

*k*

_{u}and

*k*

_{e}are uptake and elimination rate coefficients, respectively. The usual assumption is that

*C*

_{aq}is constant, which can be achieved experimentally in a flow-through system with a constant supply of toxicant. Given a constant

*C*

_{aq}, and assuming that

*C*

_{org}= 0 prior to exposure, the time course of toxicant uptake is: After sufficient time (

*t*≫

*k*

_{e}

^{−1}), the concentration of toxicant in the organism asymptotically approaches a plateau: where the bioconcentration factor (

*BCF*) is now represented as a ratio of rate coefficients.

In a static exposure system, the assumption of a constant *C*_{aq} may not be justified. Instead of the single toxicokinetic Equation (2), the situation is better modeled as a pair of equations representing the aqueous and organismal concentrations as dynamic variables:

*= 22,000 which is the bioconcentration factor measured by Carlson and Kosian (1987) for fathead minnows (*

*BCF**Pimephales promelas*) exposed to hexachlorobenzene (HCB) in a flow-through system. Since Carlson and Kosian did not determine

*k*

_{u}and

*k*

_{e}separately, we only know their ratio, and the time scale over which the plateau is reached is unspecified.

The effect of the spatial scale of the exposure system under static conditions is shown in Figure 1. Assume an aqueous concentration of HCB of 1 ppb and 1 g of fish biomass. The plateau concentrations represent an apparent *BCF* for exposure systems of various sizes, ranging from 1 to 1000 litres. Note that only in the largest system does the plateau concentration approximate the *BCF* as measured in a flow-through system. In the smaller systems, the uptake of HCB is eventually limited by the declining aqueous concentrations. It is not valid to assume that *C*_{aq} is constant, and an accurate estimate of the *BCF* would need to be based on measured values of *C*_{org} and *C*_{aq} once steady state is reached. This simple example demonstrates that the scale (spatial extent) of the laboratory test system can dramatically influence the toxicant exposure experienced by organisms, and our ability to extrapolate effects to the field.

### Dimensional analysis

Dimensional analysis focuses on the units associated with scaled quantities (Barenblatt, 1996; Legendre and Legendre, 1998). Dimensional analysis is employed to ensure the consistency of units associated with equations. Often the analysis proceeds to derive dimensionless quantities (ratios or products of variables in which the units cancel out). These dimensionless quantities can be useful for developing scaling relationships.

An example from fluid mechanics serves to illustrate the approach (Potter and Wiggert, 1991; Vogel, 1994). The Reynolds number (Re) is a dimensionless quantity that characterizes the degree of turbulence associated with fluid flow around an object:

where*U*is the fluid velocity (in units of length per time),

*L*is the length of the object, and ν is the kinematic viscosity (in units of length squared per time). To study turbulence in the wake of a submerged aquatic plant using a half-size scale model one might double the flow rate. At ordinary temperatures, the kinematic viscosity of water is about 15 times less than air, so similar Reynolds numbers are found at slower flows or for smaller objects in water than in air. The transition from laminar to turbulent flow displays a dynamical similarity across a wide range of aerodynamic or hydrodynamic conditions, a pattern that is made most apparent by translating the absolute scales of velocity, length and viscosity into the relative scaling provided by Reynolds numbers.

Dimensional analysis has a long history in physics and in engineering; however, recently more ecological applications have been explored. Legendre and Legendre (1998) provide a clear discussion of the theoretical foundations of dimensional analysis, and illustrate its application to simple equations or models with ecological relevance. Petersen and Hastings (2001) encourage the use of dimensional approaches for ecological problems in general, and to scaling issues in microcosms and mesocosms in particular.

Dimensional approaches provide a useful theoretical framework for attacking scaling problems, but they have their limitations. First, the similarity of processes across scale implied by maintaining dimensionless quantities cannot actually apply across all ranges of scales. The Reynolds number does not account for phenomena arising at molecular length scales, or at supersonic velocities. Second, maintaining one property across scales usually implies distortions in other properties. For instance, a constant Reynolds number may make the turbulence similar between a scale model and a full-scale airplane, but compressibility effects, which are important at high airspeeds, would differ. In ecotoxicological applications, dimensional approaches may provide a basis for scaling some properties, but such uses are at a fairly rudimentary level.

## Empirical relationships

We turn now from scaling relationships with a strong theoretical basis to those which are more phenomenological. These are opposite ends of a spectrum, rather than disjunct categories. Indeed, it is the normal process of science to take empirical generalizations and provide ever-stronger theoretical frameworks to explain or predict them. The ideal gas law started as a phenomenological description of empirical data until the development of molecular theory and statistical mechanics provided a theoretical basis for the mathematical form of the equation.

### Species-area scaling

One of the most studied empirical scaling relationships in ecology is the power law scaling of species richness with geographic area:

where*S*(

*A*) is the number of species found in a geographic region with area

*A*, and

*c*and

*z*are empirical constants. The values of

*c*and

*z*can vary, depending upon taxa and location. Rosenzweig (1995) provides a review of empirical data, as well as a description of various attempts at theoretical explanation. Recently, Harte et al. (1999) have emphasized the self-similarity implied by Equation (7). We defer a discussion of self-similarity until we consider fractals, but note that it has important implications for conservation, and can be used to predict species richness across biomes or to estimate extinction rates due to habitat loss (Harte, 2000).

Recent analyses of data on tropical tree species diversity have suggested a different mathematical form for the species-area relationship. Plotkin et al. (2000) observed that the data are more appropriately described by:

where*S*(

*A*) is the number of species found in a geographic region with area

*A*, and

*c*,

*z*and

*k*are empirical constants. Although Equation (8) is not appreciably different from a power law if

*A*varies over a limited range of scales (particularly if

*A*is small), divergence from the power law becomes apparent over a broader range of scales. Plotkin et al. (2000) argue that most species-area relationships have been derived from data spanning a relatively narrow range of scales, but that their data, which spans a wide range of spatial scales, allowed them to observe this previously unsuspected deviation from a power law.

### Allometric scaling

Allometry refers to the scaling of a biological variable as a function of body mass or some other index of organismal size. Allometric relationships can represent variations in biological variables between individuals within a species, or variations between species. A large number of allometric relationships have been empirically derived for both animals (Peters, 1983; Calder, 1984) and plants (Niklas, 1994). These relationships usually take the form of power laws:

where*X*is the biological or ecological variable of interest,

*M*is body mass, and

*a*and

*b*are empirical constants.

One of the most intensively studied and biologically fundamental allometric relationships is the scaling of basal metabolic rate with body mass. Beginning in the 1930s, Kleiber (1932) and Brody (1945) pioneered the investigation of the allometric scaling of metabolic rate. Based on their work and subsequent studies, the scaling exponent is, to a very good approximation, *b* = 3/4. This exponent holds for unicellular organisms, poikilotherms, and for homeotherms, although each of these groups requires a different value of *a* for a good fit (Peters, 1983).

The 3/4 exponent arose as an empirical generalization, but considerable effort has been expended in seeking a theoretical justification. If the metabolic requirements of an organism were proportional to the mass of biological tissue, then we would expect a scaling with *b* = 1. If metabolic rates were limited by the flux of materials across surfaces, then a scaling proportional to the surface-to-volume ratio might arise. Assuming that organisms are like simple Euclidean objects, we would expect the surface-to-volume ratio to scale with *b* = 2/3. None of these expectations are supported by the data. Recently, West et al. (1997) proposed a model which successfully predicts a scaling exponent of *b* = 3/4. They assume that metabolic rate is determined by the rate of supply of essential resources to biological tissues and that these materials are delivered by branching systems (e.g., bronchi, arteries, xylem) that are fractal, and that the energy efficiency of transport is optimized.

The empirical derivation of additional allometric relationships, and their theoretical justification, is an active area of current research. The edited volume by Brown and West (2000) compiles many recent examples. For the ecotoxicologist, it is relevant to note that toxicokinetic parameters, or physiological parameters that may be used in toxicokinetic models, often display allometric scaling (Cashman et al., 1996; Pastorok et al., 1996). Sample and Arenal (1999) have proposed an allometric model for interspecies extrapolation of toxicity values in wildlife ecology. Ecological variables that may be important in ecological risk assessments, such as home range size (Kelt and Van Vuren, 2001), dispersal distance (Sutherland et al., 2000), population density (Damuth, 1987), minimal viable population size (Calder, 2000) or population recovery time (Calder, 2000) may also scale allometrically with body mass.

### Fractals

Mandelbrot (1977) introduced the term fractal to describe geometric objects that are characterized by non-integer scaling exponents. These objects display the mathematical property of self-similarity (or, more generally, self-affinity). Precise mathematical definitions of these terms lie beyond the scope of this paper (the interested reader is directed to standard references on fractals (Mandelbrot, 1977; Hastings and Sugihara, 1993; Peitgen et al., 1992). Intuitively, however, self-similarity or self-affinity implies that as one changes scale (e.g., as one zooms in to examine a small piece of the geometric object), the object still looks more or less the same. Many natural objects are approximately fractal, at least over some range of length scales. Topography, for instance, is at least approximately fractal. If given a line drawing of a topographic surface, with no indication of scale, it is hard to distinguish an anthill from a mountain.

A pervasive feature of fractal geometry is the use of power laws to describe the scaling of geometric properties. A classic example is the fractal representation of a convoluted line, such as a coastline (Mandelbrot, 1977). Imagine estimating the length of a rugged coastline from a map. Assume that you have a pair of dividers, which you set using the scale bar on the map to a spacing representing a distance of 10 km. You then ‘walk’ the dividers over the relevant section of coastline, and count five steps. The apparent length of the coastline is the number of steps times the length of each step (= 5 steps × 10 km/step = 50 km). Now, reset the dividers spacing to represent a 2 km step length (changing the grain of the measurement). If the coastline were a straight line, it would now require 25 steps to traverse it. However, with a convoluted coastline, the smaller step size will allow the dividers to more closely follow the zigs and zags of the line, and more than 25 steps will be required, yielding an apparent length greater than 50 km. For a fractal line, length depends upon the scale of measurement. Specifically:

where*L*(

*s*) is the apparent length of the coastline measured with a step length

*s*, and

*D*is a non-integer quantity known as the fractal dimension, and ‘∼ ’ denotes proportionality. For the smooth lines of Euclidean geometry,

*D*= 1. For fractal lines, such as a coastline, embedded in a planar surface, 1 <

*D*< 2. Empirically, most real coastlines are in the range of

*D*= 1.1 to

*D*= 1.4.

Fractal geometry can be used to describe many natural patterns, including perimeter-area relationships of islands or lakes (Mandelbrot, 1977) or properties of river networks (Tarboton et al., 1988; Tarboton et al., 1989). Fractal patterns affect the dynamics of ecological processes, such as the diffusion of populations or genes resulting from movements of organisms in heterogeneous environments (Johnson et al., 1992; Johnson et al., 1995). Fractal descriptions can be applied to the temporal domain, and have been used to model fluctuations in river hydrology (Mandelbrot and Wallis, 1968) and population densities (Keitt and Stanley, 1998). Initially, power law relationships describing natural spatial or temporal patterns are usually empirical generalizations, even if motivated by a fractal model. However, if independent evidence supporting the fractal model is adduced, it may provide a theoretical explanation. As we have seen, fractal models play an important role in attempts to theoretically justify of the species-area relationship (Harte et al., 1999) and the allometric scaling of metabolism (West et al., 1997).

## Physical models

Ecotoxicologists have long relied on scaled physical models to predict the fate and effects of materials. These scaled physical models have been variously termed microcosms or mesocosms, depending upon size. In some respects, the use of ‘cosms’ by ecotoxicologists is not unlike the use of scale models by engineers. Depending upon the questions or hypotheses involved, a variety of approaches are available to extrapolate from scaled physical models to predict the dynamics of larger ecological systems. We must recognize, however, that we can conduct experiments in laboratory systems that have no bearing on or relationship to events outside the laboratory. This has led some researchers to question the relevance of such physical models in ecological research (Carpenter, 1996; Schindler, 1998). Rather than dismissing microcosm and mesocosm experiments, we advocate a systematic approach to the extrapolation problem. The challenge is to recognize those attributes that are amenable to accurate extrapolation. Results obtained may be predictive in a site-specific or a generic sense, depending upon the level of resolution desired.

### Considerations in microcosm/mesocosm design

We will focus on physical models of aquatic systems, although similar considerations apply to terrestrial systems. Crossland and LaPoint (1992) provide a thoughtful discussion regarding the rationale and design of mesocosm experiments. Considerations of spatial and temporal scale are of paramount importance in the design. The mesocosm must be large enough to capture the effects or phenomena of interest. For example, if a material impacts benthic invertebrates, it would be important that the mesocosm contain an appropriate assemblage of benthic organisms. Further, the mesocosm should be of sufficient size that sampling during the experiment does not itself constitute a major impact.

Petersen et al. (1999) reviewed the design of a variety of lentic microcosm and mesocosms. They discovered patterns in the scale-related choices made by researchers as inferred from the published literature. Studies showed a statistically significant trend toward experiments of longer duration in larger systems (*p* = 0.001), although the relationship displayed a wide amount of scatter (*R*^{2} = 0.05). Also, the number of replicates per experimental treatment declined as the volume of the experimental system increased. By plotting the reported physical dimensions of a wide variety of cylindrical microcosm and mesocosm studies, Petersen et al. (1999) uncovered an implicit scaling relationship. Changing the volume of a cylinder involves altering either the depth or the radius (or both). Petersen et al. (1999) found that, over about six orders of magnitude in volume, researchers tend to construct systems which preserve a relatively constant depth/radius (≈ 4.5). This relationship displays moderate scatter (*R*^{2} = 0.59) and is highly significant (*p* < 0.001). This relative constancy of shape may be the result of multiple interrelated factors, including human aesthetic sense, logistical or pragmatic concerns, and the dimensions of manufactured containers.

### Extrapolation

Microcosms or mesocosms could be studied as interesting systems in their own right. However, when used as physical models, the problem arises of extrapolating from the model to other ecosystems. To be specific, the extrapolation problem is this: how do we use the results of experiments with the model system to make accurate predictions about the ecosystem(s) it is designed to represent?

The extrapolation problem is multifaceted, but almost invariably one aspect of the problem is the difference in spatial extent between the model and the represented ecosystem. Sampling of the model and represented ecosystems may also involve differences in spatial grain. The duration of experiments in model systems may be shorter than the time frame for which we wish to predict effects in the natural ecosystem. Differences in sampling frequency may exist. Thus, the extrapolation problem involves, at a minimum, developing spatial and temporal scaling relationships spanning the scale differences between the model and represented ecosystems. We focus on those ecological variables for which either theoretical or empirical scaling relationships are available. We expect that as research on scaling proceeds, scaling relationships will be developed for an increasing number of variables of relevance to ecotoxicology. However, for the foreseeable future, ad hoc approaches may be required to deal with facets of the extrapolation problem for which scaling relationships per se are lacking.

Scaling relationships quantitatively describe how ecological attributes vary with changes in scale. Such scaling relationships can be represented by graphs, or by mathematical equations. Often these scaling relationships take the form of power laws, which are linear when both axes have logarithmic scales. However, in general, scaling relationships can take a variety of linear or nonlinear forms. Of particular importance for the extrapolation problem, scaling relationships may display discontinuities or abrupt changes in slope. For instance, foraging movements by animals can often be modeled, at some scale, by a random walk (Berg, 1983; Bovet and Benhamou, 1988). Theoretical arguments predict that the apparent diffusion rate of multiple random walkers can change with scale if the environment displays fractal heterogeneity (Gefen et al., 1983; Johnson et al., 1992). This is revealed in relatively abrupt changes in slope in spatial or temporal scaling relationships. Empirical studies of movements of flightless tenebrionid beetles (*Eleodes* spp.) across a range of spatial and temporal scales in a semi-arid landscape demonstrated the existence of changes in slope of scaling relationships that are qualitatively consistent with the theoretical expectations (Johnson et al., 1992).

In microcosm-mesocosm experiments addressing primary productivity in planktonic-benthic systems, Petersen et al. (1997) observed different scaling relationships under spring versus summer conditions. These researchers monitored gross primary production (*GPP*) in aquatic systems in cylindrical containers ranging in volume from 0.1 to 10 m^{3}. This included a constant shape series (depth/radius = 1.8), with radius *r* = 0.26, 0.57 or 1.22 m, and depth *d* = 0.46, 1.00 or 2.15 m. It was observed that, in the spring, *GPP* was proportional to the cross-sectional area of the cylinder (i.e., * GPP*∼

*r*

^{2}). However, in the summer,

*GPP*was proportional to the volume of the mesocosm (

*∼*

*GPP**r*

^{2}

*d*). Petersen et al. (1997) provide a plausible explanation for these different scaling relationships. In the spring, phytoplankton abundance is high and primary productivity is limited by light availability. The amount of light energy reaching phytoplankton in the water column depends upon the cross-sectional area of cylinder (assuming illumination from above). In the summer, primary production in the mesocosms became limited by nutrient availability. Dissolved nutrients are distributed throughout the water column, and so nutrient supply depends upon the volume of the cylinder. Based on the factors limiting primary production these systems, it is possible to derive simple and understandable scaling relationships. Note that a dimensionless quantity (the radius/depth ratio) was preserved in the experiments leading to these simple scaling relationships.

Perez et al. (1991) examined the fate and ecological effects of the pesticide Kepone in aquatic microcosms ranging in volume from 9.1 to 140 l. The depth of water was held constant, but the surface to volume ratio decreased with size, ranging from 395 to 106 cm^{2} l^{−1}. In the absence of toxicant, the microcosms exhibited size-dependent differences in the timing and magnitude of the phytoplankton bloom, which was later and larger as microcosm volume increased. Kepone reduced grazing pressure by zooplankton, resulting in an increase in phytoplankton densities. The time-course of Kepone in the water column was independent of microcosm size, but the concentrations found in surficial sediments were size-dependent. Clearly, extrapolation of these results to larger ecosystems requires the development of scaling relationships, aided by further elucidating the ecological mechanisms involved.

## Guidelines for scaling

The preceding review has demonstrated an increasing awareness of scale and scaling issues in ecological and ecotoxicological research. There is much yet to be learned, but several heuristic guidelines for future research can be offered:

### In scaling, the exposure (stimulus) requires as much attention as the responses measured

Without characterizing the stimulus, no response can be reliably attributed to the stimulus. The simple toxicokinetic example presented in this paper provides one example of how the scale of an experimental system can affect the exposure of organisms, and therefore modify ecotoxicological responses. For any given stimulus (e.g., chemical, physical or biological stressor), responses *may* be proportional to, or a function of concentration, magnitude, duration, intensity, frequency or periodicity of exposure. Each stimulus must be evaluated in its proper perspective or setting, which may depend upon the nature of the stimulus and of the receptor. Scale issues arise in terms of the spatial extent of the stressor versus the size of the experimental system, the response and recovery dynamics of the receptor versus the duration of the experiment, and the problem of extrapolating from models (physical, mathematical or computer) to other systems. To establish causation, it is necessary to verify the presence of the stimulus (e.g., analytical measurement of a chemical stressor) as well as documenting the response.

### Physical models, such as microcosms and mesocosms, are useful for a variety of purposes related to scaling

Physical models may be useful for a variety of purposes in ecotoxicology, including to 1) corroborate laboratory or modeling results, 2) provide comparative results for similar or homologous compounds, 3) provide integrated information on chemical stability, fate, and persistence, 4) examine the influence of environmental factors on chemical fate and effects, 5) determine mechanisms of action with different exposures of the same compound, 6) aid in design of field studies or monitoring programs, 7) produce ‘replicated’ systems for experimental designs providing strong inference and permitting assessment of margins of safety, and 8) empirically derive scaling relationships by systematically varying scale variables (grain and extent). Studies incorporating physical models can be used in an iterative process, coupled with fate and effects modeling, to enhance predictability and extrapolation. Microcosms and mesocosms are useful in scaling to understand field observations and responses that are appropriately attributed to a specific stimulus. Often these responses are masked or confounded in field studies where lack of control leads to mistaken implication of the wrong stimulus.

### A variety of theoretical and empirical approaches may be fruitfully applied in the pursuit of ecotoxicological scaling relationships

In this paper we have examined several approaches to scaling, including dimensional analysis, allometric relationships, fractal geometry, and physical models. All of these approaches have utility, and all have limitations. The current state of the art requires the researcher to be familiar with a suite of scaling approaches, and to employ each as the situation warrants. As science progresses, the theoretical basis for ecotoxicological scaling will undoubtedly grow stronger, but a mixture of theory and empiricism will probably always be required.

### Scaling relationships can be linear or nonlinear, continuous or discontinuous

We have presented examples of a variety of scaling relationships in this paper. Power laws, which are linear when the variables are log-transformed, are common, but by no means the only mathematical form observed. We have emphasized that scaling relationships may exhibit either gradual or abrupt changes as the range of scales examined is increased. Thus, a relationship that is linear over a narrow range of scales may be nonlinear, and perhaps exhibit discontinuities, over a broad range of scales. In experimental situations, if the stimulus is scaled correctly, then it is possible to observe responses that may be capable of being extrapolated. Inaccuracies in extrapolation often arise when dissimilar stimuli are compared, or when the necessity for nonlinear scaling is not recognized. Theoretical models may be useful for scaling of both stimulus and response.

## Research needs

Issues of scale and extrapolation across scales are important and pervasive in ecotoxicology. No single approach is likely to be completely successful in dealing with problems of this complexity. However, a sustained effort, attacking the problem from a number of different angles, will likely yield results. We have four suggestions to guide future research.

### A rigorous assessment is needed of the applications and limitations of allometric scaling in ecotoxicology

Existing data supports allometric scaling relationships for many physiological and ecological variables. The use of these relationships, particularly in addressing extrapolation issues, is still an area requiring substantial research. Allometric relationships may describe how the mean value of a physiological or ecological variable change over a large range of body mass, but there is often considerable variation around the trend in the means. In ecological risk assessment, it is often essential to characterize the variability (or uncertainty) as well as the mean response. Ecotoxicology will progress as we investigate the underlying causes for deviations from the mean. It is likely that more precise relationships can be derived if data are grouped on the basis of similarity of mode of action or other toxicologically relevant criteria. Combining allometric scaling with other ecotoxicological models holds promise.

### Greater focus is needed on identifying non-linearities and thresholds in empirical and theoretical scaling relationships

Most scaling relationships are described as simple continuous functions, often as linear functions of log-transformed variables. Such scaling relationships can be enormously useful. However, there is growing theory of complex, hierarchical systems which leads to the expectation that at least some ecological properties will display abruptly non-linear or threshold responses as a function of scale. Rather than simple linear functions of log-transformed variables, we should expect functions that are more appropriately described as piecewise linear, with possible discontinuities. The statistical analysis of such complex relationships is, of course, more difficult, but possible with modern computing technology. The key point is for researchers to be aware of the possibility of such phenomena, and attentive to the relevant empirical evidence.

### Attention should be directed toward integrating spatial and temporal aspects of scaling

Considerably greater attention is usually focused on the analysis of and extrapolation across spatial scale than on the corresponding issues in the temporal domain. However, in most natural systems, spatial and temporal scales are intimately connected. A comprehensive theory of scaling in ecotoxicology will have to address both space and time in an integrated fashion.

### There is a need for more experimental studies specifically designed to manipulate scale as a controlled treatment variable

Ecotoxicological experiments traditionally manipulate levels of toxicant exposure as a treatment variable. Other ecologically relevant factors may be experimentally manipulated, such as temperature, light regime, nutrient levels, food availability, and the presence or absence of predators. However, despite the recognition by ecotoxicologists that scale can affect the results of an experiment, scale per se is rarely treated as a treatment variable. Most of our knowledge of scale-dependent responses of ecological systems comes from post hoc examination of results from multiple experiments or empirical studies. Although this can be very illuminating, there are often accompanying differences in the composition of the systems under study, or the details of what responses were measured and how, which can obscure the effects of scale. The science of ecotoxicological scaling would be considerably advanced by a greater number of experimental studies in which scale is explicitly treated as an experimental variable.

## Acknowledgement

We thank Bernard Montuelle and CEMAGRAF for financial assistance in attending the 7th AEHMS Conference in Lyon, France, where this paper was presented.

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