Download Dimensional approaches to designing better experimental

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)
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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
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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
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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.). Travel funds were provided
by Umeå University, Department of Ecology and Environmental
Science (Umeå, Sweden). Many thanks to Allen Hastings, Michael
Kemp and John Lawton for stimulating discussion that contributed to this paper.
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