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HORIZONS
Planktonic ecosystem models: perplexing
parameterizations and a failure to fail
PETER J. S. FRANKS*
SCRIPPS INSTITUTION OF OCEANOGRAPHY, UNIVERSITY OF CALIFORNIA, SAN DIEGO, LA JOLLA, CA
92093-0218, USA
*CORRESPONDING AUTHOR: [email protected]
Received March 4, 2009; accepted in principle July 9, 2009; accepted for publication July 16, 2009; published online 12 August, 2009
Corresponding editor: Roger Harris
Planktonic ecosystem models have been used for many decades; modern models are only subtle variations on model structures established in the 1970s or earlier. Here I explore two problems that I
see with these models: (i) their formulation and parameterization and (ii) their use. Using nutrient
uptake by the phytoplankton as an example, I trace the history of why we use Michaelis –Menten
kinetics in our models, and show that this functional form may not be the best representation of
nutrient uptake by a diverse and changing phytoplankton community. I then discuss how models
are used—not as the hypotheses they are, but more like toasters. I make the point that treating
models as hypotheses could lead to much stronger and more robust insights into planktonic
dynamics. However, this requires better statistical comparisons of models and data, including the
“right” kinds of data. Finally I suggest some ways to move forward, to make planktonic ecosystem
models much more powerful tools in our investigations of ocean dynamics.
I N T RO D U C T I O N
Since the early models of Gordon Riley (e.g. Riley,
1946), planktonic ecosystem models have not changed a
great deal. One of the more sophisticated and wellparameterized models was that of Steele and Frost
(Steele and Frost, 1977); modern ecosystem models represent only subtle variations on this structure. I have
been constructing and using planktonic ecosystem
models for more than two decades (Franks et al., 1986).
During that time, I have become disillusioned by the
lack of innovation in planktonic ecosystem modeling,
even as the types of data we gather in the field has burgeoned. My issues with these models fall broadly into
two categories: how models are formulated and parameterized, and how they are used.
Planktonic ecosystem models typically follow a
“nutrient–phytoplankton–zooplankton” (NPZ) structure
in which phytoplankton take up dissolved nutrients,
zooplankton eat phytoplankton, and phytoplankton and
zooplankton both recycle nutrients back to the dissolved
pool (Franks, 2002). Variations on this structure include
separation of the planktonic state variables into size
classes or functional types, addition of state variables
such as bacteria and detritus, and the inclusion of
additional limiting nutrients (silicate, phosphate, iron,
etc.). The state variables are linked by “transfer functions” which have a particular functional form.
Typically, this is Michaelis– Menten for the phytoplankton uptake of nutrients, and a Holling-type grazing
response of the zooplankton to changing phytoplankton
concentrations (Franks, 2002).
The formulation of the model involves choosing the
state variables, and specifying the functional forms of
the transfer functions among state variables (linear
doi:10.1093/plankt/fbp069, available online at www.plankt.oxfordjournals.org
# The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: [email protected]
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response, saturating response, etc.). The parameterization of the model involves choosing particular values for
the parameters of the functional forms: maximum
uptake rates or half-saturation constants, for example
(see below). In my opinion, both these processes are
often performed somewhat mindlessly (noting that I am
as guilty of all these things as anyone else). As I detail
below, I believe that we are missing some opportunities
for significant improvements in our formulation and
parameterization of planktonic ecosystem models.
The second point I will address is the use of models,
and the comparison of models to data. The last decade
has seen significant advances in our ability to statistically compare models and data. Our ecosystem models
are hypotheses: concise mathematical statements
describing how we think the ecosystem operates.
However, unlike a hypothesis, I have yet to see an NPZ
model rejected. More typically, the parameters are
tweaked until the model gives an adequate description
of the data, and the paper is published. Again, I believe
we are missing significant opportunities for identifying
the most accurate and robust models, as well as missing
opportunities to improve the quality of data used to test
the models.
Here I detail my arguments—the perplexing parameterization of planktonic ecosystem models, and their
failure to fail. I briefly trace the history of how I believe
we got here, and offer some suggestions for moving
forward. I believe that we are poised to make some significant leaps in our ability to use models to understand
the planktonic ecosystems of the world’s ocean. It will
require, however, significant effort in shaking off the
shackles of several decades of stagnation. I should
emphasize that in this paper I am being purposely provocative in my presentation, and vague in my suggestions of ways forward. My hope is that you, the reader,
will be sufficiently annoyed at me, yourself or other
researchers that you will forge new paths that I cannot
even envision.
PERPLEXING
PA R A M E T E R I Z AT I O N S
One of my criteria for judging modeling papers is to
look at the sources for their parameter values. Authors
of “good” modeling papers have gone into the experimental and field literature and extracted parameters
based on our current understanding of the field.
Authors of “bad” modeling papers have taken their parameters from other models without any critical thought
or discussion. Tracing these parameters through the
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literature, one can often find that the original paper
that documented the parameter in question is either
many decades old (and may have been proven inaccurate in the meantime) or a modeling paper in which the
parameter was chosen with no supporting data. Many
parameters are chosen simply to make the model look
more like the data, with no regard to physiological or
ecological relevance.
To make my point about the curious history of the
formulation and parameterization of planktonic ecosystem models, I will trace the history of one transfer
function: nutrient uptake by phytoplankton. Although
I concentrate on this one process for the sake of brevity,
it is important to recognize that similar criticisms can
be made of any other transfer function in these models,
in particular grazing [often Holling type II (1959) or
Ivlev (1955) forms].
In 1913, Michaelis and Menten published a paper
describing the reaction rate of an enzyme (invertase)
with its substrate (Michaelis and Menten, 1913). The
rate of reaction—that is, the rate of formation of
product, V—is given by
V ¼
Vmax ½S
ks þ ½S
ð1Þ
where Vmax is the maximum rate of reaction, [S] the
concentration of the substrate and ks the half-saturation
constant. This equation suggests that the reaction rate
can be calculated from the substrate concentration,
knowing only two parameters: Vmax and ks. These parameters are presumed to be characteristic of a particular enzyme.
The Michaelis– Menten equation has several important assumptions: the equation describes the initial rate
of reaction (and the initial substrate concentration), the
reaction is at steady state, there is no change in the concentration of the enzyme or the substrate-enzyme
complex and the molecular dynamics are governed by
molecular diffusion of the enzyme and substrate.
In 1967, Richard Dugdale introduced Michaelis–
Menten kinetics to the planktonic modeling community
(Dugdale, 1967). Using four data points from Harvey
(Harvey, 1963), Dugdale made the case that the growth
(in this case, O2 production) of phytoplankton was well
described by a Michaelis– Menten curve. Dugdale
argued that the rate of nitrogen uptake by phytoplankton could be described by this same Michaelis– Menten
kinetics.
Dugdale included several interesting caveats in his
paper. At one point he states, “Wright and Hobbie
(1966) . . . designate a Kt, a coefficient for transport, to
distinguish it clearly from Ks, which has in the past
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always referred to a single species. The distinction is
important and will be observed here”. Dugdale thus
makes the point that an organism and a species have a
Ks, while a community has a Kt—a potentially different
half-saturation constant than the individual species.
Dugdale further states, “There is an urgent need to discover the important kinetic parameters for the uptake
of nutrients (if other than Michaelis – Menton [sic] kinetics as postulated in this paper) and to measure them
for phytoplankton algae characteristic of different productivity regimes”. He was clearly open to the suggestion that nitrogen uptake by phytoplankton might not
follow Michaelis – Menten kinetics, though there is the
implicit assumption that the uptake rates could be parameterized somehow.
Phytoplankton ecologists forged ahead, attempting to
determine the Vmax and ks of various species taking up
various nutrients. The pinnacle of such research might
have been the study by Titman (Titman, 1976) which
showed that the outcome of competition for two nutrients by two species of phytoplankton could be predicted
based on their relative Vmax and ks for a given nutrient.
In his experiments, Asterionella formosa outcompeted
Cyclotella meneghiniana under phosphate limitation, while
the opposite was true under silicate limitation. His
measurements of Vmax and ks for these species taking up
these nutrients predicted that this should be the
outcome.
From this point on, Michaelis– Menten kinetics
became firmly entrenched in phytoplankton modeling.
As an example, Table I shows the half-saturation constants ks from 10 recent well-cited planktonic ecosystem
models. These models are all based on the NPZ structure, and the ks values should be comparable among
them. However, the values vary over more than two
Table I: Michaelis– Menten half-saturation
constants for nitrate and ammonium uptake
from 10 recent ecosystem models
Model
Besiktepe et al. (2003)
Chai et al. (2002)
Christian et al. (2002)
Denman and Peña (1999)
Fujii et al. (2002)
Hood et al. (2001)
Kishi et al. (2007) small
phytoplankton
Kishi et al. (2007) large
phytoplankton
Laws et al. (2000)
Moore et al. (2004)
Wiggert et al. (2006)
NO3 Ks
(mmol N L21)
NH4 Ks
(mmol N L21)
0.5
0.5
0.25
0.1
3.0
0.5
1.0
0.2
0.05
0.05
3.0
0.3
0.015 –0.15
0.5 –2.5
0.4 –0.8
0.005–0.08
0.05
0.3
0.1
orders of magnitude for both nitrate and ammonium
uptake for model variables describing the same property in the ocean. There is clearly no consensus on the
value of this parameter. There is, however, an implicit
consensus that nutrient-limited uptake is controlled by
Michaelis– Menten kinetics, which can be described by
constant parameters.
In spite of the community consensus, I have fundamental issues with the assumption that nutrient uptake
by a community should necessarily be modeled as
Michaelis– Menten kinetics. First, recalling the foundations of the Michaelis– Menten formulation, it is
important to note that almost all experiments quantifying the Michaelis –Menten parameters of phytoplankton actually violate some or all the assumptions
inherent in the kinetic formulation. The substrate concentrations decrease markedly during experiments, and
the rates are not initial rates of reaction. Goldman and
Gilbert (Goldman and Glibert, 1982), for example,
showed that phytoplankton populations had markedly
enhanced uptake rates during the first minute or two of
an uptake experiment. These rates decreased by an
order of magnitude within about 15 min after labeled
nutrients were added. Flynn (Flynn, 1999) modeled
similar dynamics, relating changes in Vmax to the changing N:C ratio of the cell—and thus to the integrated
history of the cell’s growth and environment. This is not
consistent with Michaelis –Menten kinetics and constant
uptake parameters.
Second, there is no particular reason an individual
phytoplankton cell should behave (kinetically) like an
enzyme. Phytoplankton actively respond to their
environment, and can acclimate metabolically to changing conditions. For example, Dyhrman and Palenik
(Dyhrman and Palenik, 2001) showed that Prorocentrum
minimum expressed a great deal more alkaline phosphatase enzyme on its cell surface when under phosphate
stress than when in phosphate-replete conditions
(Fig. 1). With the hugely increased number of sites for
phosphate cleavage on the cell, it is entirely possible
that the net uptake rate of phosphate cleavage on the
cell would be the same under nutrient-limited and
nutrient-replete conditions. This is consistent with
analyses by Aksnes and Egge (Aksnes and Egge, 1991)
who showed that Vmax should increase linearly with the
number of transporters on the cell’s surface.
Third, it is not obvious that the nutrient uptake
dynamics of an individual or a population should represent the nutrient uptake dynamics of a diverse community. It is reasonable to argue that an individual
organism might have a saturating response to a
resource, and that the Michaelis– Menten formulation
is just a mathematically convenient way of representing
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Fig. 1. Immunofluorescence images of six Prorocentrum minimum cells from field samples of Narragansett Bay, Rhode Island. Labeling indicates
the presence of alkaline phosphatase on the cell surface of phosphate replete (þP) and phosphate stressed (2P) cells. Pre-immune treated cells
are shown in the left panels. Adapted from Dyhrman and Palenik (Dyhrman and Palenik, 2001).
this (see in particular Flynn, 2003, for a thoughtful discussion of this issue). The physiological basis for this
saturating response may have little to do with
Michaelis– Menten kinetics, but still fit that functional
form (e.g. Aksnes and Egge, 1991). Flynn (Flynn, 2008)
argues that modified quota models are more appropriate representations of population nutrient uptake
dynamics, and that simple Michaelis– Menten kinetics
will not allow an accurate fit of an uptake model to laboratory data. But a more fundamental problem exists in
modeling complex, dynamic communities: with many
different species of different sizes and relative abundances, and many different enzymes involved in uptake,
the emergent community uptake dynamics could be
quite different from those of any individual component
species. One example of this is shown by the model of
Fuchs and Franks (Fuchs and Franks, in preparation): a
near-continuum size-structured planktonic ecosystem
model. The 1024 size classes of phytoplankton all take
up nutrients following Michaelis– Menten kinetics; all
have the same ks, whereas the Vmax decreases allometrically with size. Running the model for a range of total
nutrients, the emergent functional form for community
phytoplankton nutrient uptake at steady state is basically
linear (Fig. 2).
Similar results were obtained for the aggregate zooplankton community grazing response in a model by
Fig. 2. Functional form of the community phytoplankton nutrient
uptake rate as a function of dissolved nutrient concentration from a
near-continuum size-structured planktonic ecosystem model (Fuchs
and Franks, in preparation). Note that the community uptake rate is
nearly linear, even though the underlying size classes each have a
Michaelis –Menten uptake response.
Leising et al. (Leising et al., 2003). They showed that a
zooplankton community made up of species with
different grazing parameters had an aggregate response
that was relatively linear over the range of prey
concentrations.
This is a significant result. Planktonic ecosystem
models are typically highly aggregated; a single
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phytoplankton variable is used to represent an extremely diverse—and changing—community. If the emergent community nutrient uptake rate as a function of
dissolved nutrient concentration is linear, then models
will be significantly easier to formulate and solve. The
non-linearities of the trophic transfer functions (nutrient
uptake, grazing, etc.) are one of the main obstacles to
obtaining analytical solutions to the model equations.
These non-linearities are also the source of oftenunrealistic oscillations in the model state variables.
The main problem with the parameterization and
formulation of planktonic ecosystem models is that
model state variables are parameterized as though they
were a single individual—or enzyme. The transfer functions of these models might be quite different if we
recognized that we are representing a diverse and changing community in each state variable. The parameter
values in models are usually fixed constants; these parameters are often either unmeasurable, or unrepresentative of real physiological processes. I believe that we
need to spend some effort exploring more rational, justifiable techniques to aggregate from a community to a
single model variable. A few models exist that have parameter values that vary with the magnitude of the state
variable, based on allometric relationships of biomass
and size structure of the community (e.g. Hurtt and
Armstrong, 1996). This is certainly a step in the right
direction, but considerably more work is required in
this field.
A FA I L U R E TO FA I L
The equations governing the movements of water in the
ocean—the Navier– Stokes equations—are generally
accepted as theory. In practice, the Navier–Stokes
equations are solved in a reduced form known as the
primitive equations. These equations describe changes
in a few state variables: momentum and density.
Although there is some art involved in representing physical dynamics such as turbulence, mixing and boundary
layer dynamics, it would be unusual for these equations
to fail to describe the fundamental dynamics. In contrast, the equations that we have formulated to describe
the biological dynamics in the ocean are not as well constrained as the Navier– Stokes or primitive equations.
Indeed, there is no generally accepted number or type of
state variable in these equations: they should be considered as hypotheses rather than theory.
A significant aspect of any hypothesis is that it should
be testable—and rejectable. But are we doing that with
our planktonic ecosystem models? I would argue that
we treat our models like toasters: we put in the bread,
push the lever and wait to see what color the toast
comes out. If it is the wrong color, we adjust a knob,
and try again. When the toast comes out the right
color, we say we have succeeded. This approach to
modeling (and I am as guilty as anyone) is common:
adjust the model parameters until the model “fits” the
data (if there are any data), and then conclude that we
have learned something; that the model is “right” or
“valid” (see Oreskes et al., 1994 for a discussion of
model “validation”). I do not believe I have ever seen a
planktonic ecosystem model rejected outright: this
appears to be a failure to fail.
The problem with this approach is that there might
be many other, distinct models that would fit the data
just as well. These are the alternate hypotheses. The
goal should be to discover which of these model(s) is
(are) “right”. Or more accurately, which one(s) cannot
be rejected. Strong scientific inference (Platt, 1964—a
paper we should all read every few years) is based on
the formulation of hypotheses and alternate hypotheses,
and the design of tests to reject them. With skillful
experimental design and testing, it should be possible to
whittle the suite of hypotheses (models) down to a
limited set, or sometimes a single hypothesis (model) to
explain the observations. Following this protocol leads
to hypotheses (models) that are robust, and whose
dynamics match the data because they are correct, not
because of a coincidence.
To reject hypotheses (models), we require data.
However, not all data have equal power in distinguishing among models. For example, it is relatively easy to
formulate a model that describes the seasonal change in
chlorophyll and nutrients in a region of the ocean.
Friedrichs et al. (Friedrichs et al., 2007) fit 12 extant
planktonic ecosystem models to data from the Arabian
Sea and the Equatorial Pacific, and showed that all the
models could be made to fit the data relatively well.
The degree to which they described the chlorophyll and
nitrate concentrations did not depend a great deal on
the inherent complexity of the model. What was particularly interesting was how the models balanced their
dynamics to fit these variables. Let me explain.
The net change of a property over time is determined
by the rate of production and the rate of loss of that
property. In the case of chlorophyll, to first order these
rates are primary productivity and grazing. For a given
change in chlorophyll, a higher rate of primary production must be balanced by a higher grazing rate to
produce that change in chlorophyll. There is an infinite
combination of primary productivity and grazing rates
that will give a particular net rate of change of chlorophyll. Although Friedrichs et al. (Friedrichs et al., 2007)
assimilated primary productivity (as well as chlorophyll
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Fig. 3. The relationship between mean integrated primary
production and grazing from 12 models (symbols) fit to data from the
Arabian Sea (AS) and the Equatorial Pacific (EP). Even though the
models all gave reasonable fits to the data, they did so by achieving
different balances of primary productivity and grazing. Adapted from
Friedrichs et al. (Friedrichs et al., 2007).
and nitrate) data into their models, these data were
sparse enough in space and time that the best model fits
gave considerable variation in total integrated primary
production among the models. The models fit the data
equally well, but they did it by achieving quite different
balances of primary productivity and grazing (Fig. 3). A
model with a high primary productivity necessarily had
a high grazing rate to achieve the same chlorophyll
changes as a model with low primary productivity.
This example indicates that not all data are equally
powerful for testing and constraining a model. In
general, measurements of state variables (chlorophyll,
nitrate, and zooplankton biomass) are much weaker
constraints than measurements of rates: growth rates,
grazing rates, etc. Unfortunately, rate measurements
tend to be the most difficult to perform in the field. I
would argue, however, that they are essential for strong
testing of our models.
A further issue (also inherent in the example above) is
the amount of data available: the required number of
observations increases with the complexity of the
model. If particular variables (such as diatoms or bacteria) are modeled but not measured, then they remain
constrained—fixed by their initial parameters—while
the remaining model dynamics change to accommodate
them. This issue was also identified in the Friedrichs
et al. (Friedrichs et al., 2007) study, and accounts for
much of the variation of integrated primary production
among models.
I maintain that we must use data to help distinguish
among models (hypotheses). Unfortunately, there is a
fundamental disconnect between what is measured and
what is modeled: we do not model what we measure,
and we do not measure what we model (see also Flynn,
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2005). In the field, we typically measure chlorophyll fluorescence. Careful researchers will calibrate these
measurements with frequent samples of extracted chlorophyll. Models, however, usually simulate the nitrogen
content of the phytoplankton. To compare the modeled
nitrogen content to the measured chlorophyll, we typically convert nitrogen to carbon using the Redfield
ratio, and then carbon to chlorophyll using some constant or algorithm. Unfortunately, none of these conversions is truly a constant—or even well known—giving
considerable room for errors in model-data comparison.
Several recent model-data comparisons have found that
including a variable carbon:chlorophyll ratio improved
the ability of the model to reproduce the data (e.g.
Hurtt and Armstrong, 1996, 1999; Spitz et al., 2001).
However, by adding a further variable, the number of
model parameters is increased, as is the ability of the
model to pass through all the data points since it now
has more scope for variability. It is not clear, however,
that the increased ability of the model to hind-cast the
data necessarily improves our understanding of the
system: we still do not know if the model is “right”.
I would advocate a more scientific approach to the
formulation and testing of models: let the data select
the model. A number of powerful statistical techniques
now exist for comparing models with data, and for
parameterizing models based on data [see special issue
of the Journal of Marine Systems, 2009, 26 (1 – 2)]. For
proper selection and testing of the models, separate,
independent data sets must be available. Numerous
models (hypotheses) should be formulated; these models
should contain combinations of reasonable functional
forms for nutrient uptake, grazing, etc., and various
levels of aggregation of the state variables (by size, functional type, etc.). The models should then be tested
against the data to identify those model structures that
cannot reproduce the data. Those models should be
rejected as reasonable hypotheses explaining the data.
This approach, though non-trivial in its implementation, has been used for some time in terrestrial
ecology (e.g. Pascual and Kareiva, 1996), though it has
not yet become widespread in oceanography.
Once the models have been winnowed to a few
“un-rejectable” candidates, further testing is then
required to attempt to distinguish among them. This
requires an iterative interaction between modeler and
experimentalist: the modeler can identify the dynamics
or conditions that would reveal the differences among
these models. The experimentalist can then design
experiments that would explicitly test those processes.
Following this protocol should lead to robust models for
simulating planktonic ecosystems. Fundamental to this
approach is the availability of data sets with sufficient
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resolution to act as a strong test of the models. As noted
above, this requires a significant number of rate
measurements. A significant outcome of this approach
should be a much more concrete understanding of what
we do and do not know about the dynamics of the particular ecosystem. If none of the models (hypotheses)
can be made to fit the data, then it is clear that our
understanding is flawed and must be revised.
I should be clear that I am not suggesting that there is
a single “correct” planktonic ecosystem model. I strongly
believe that the model should be designed around the
question being asked. A model simulating seasonal chlorophyll fluctuations in the north Pacific might be quite
different from a model exploring a harmful algal bloom
in the Gulf of Mexico. But within the constraints of our
research questions, we should always be asking ourselves
whether another, quite different, model might reproduce
the results just as well as the model we are using. Our
work is not done when one model fits the data—we need
to then try other models to discover how unique and
robust our conclusions are.
CONCLUSIONS
I have identified two areas of planktonic ecosystem
models that I believe require significant thought and
research. The first is the formulation and parameterization of the models. We have fallen into a historical rut
with our model formulations, not being critical of the
fundamental transfer functions that govern our models.
For example, almost every extant planktonic ecosystem
model uses Michaelis –Menten kinetics to describe nutrient uptake by phytoplankton. However, it is not clear
that this formulation should apply to an individual phytoplankter, let alone a diverse phytoplankton community.
I would urge researchers to investigate in more detail
how acclimating individual physiological responses to the
environment emerge as a community functional
response. It is the community response that we typically
model and measure, and it is possible that this community response is quite different from the functional
response of a single enzyme. We need more careful investigations of rational techniques for aggregating in ecosystem models: how many state variables are necessary,
what are they, how constant are their transfer functions?
Second, we need to treat models as the hypotheses
they are, and be more scientific in our formulation and
rejection of them. This requires the formulation of a set
of reasonable alternate hypotheses (models), and the
acquisition of data sets that are adequate to reject some
of the models. We must spend more time addressing the
issue of the types of data we gather in the field versus
the variables we model: they are seldom the same thing.
We must acquire more of the types of data that present
a strong test of models, in particular rate data. We also
need to formulate and test a greater variety of models.
Perhaps it is time to think beyond the traditional
NPZ model structure and look to different model architectures, modeling enzymes, for example, rather than
trophic groups.
There has been significant progress in the last decade
in improving the techniques for objectively comparing
models and data. These techniques should be applied
to improving our models as well as improving our
understanding. With new techniques being developed
for quantifying biological properties in the ocean, we
have a great opportunity to make significant strides in
our understanding of the dynamics of the planktonic
ecosystems in the world’s ocean.
AC K N OW L E D G E M E N T S
This paper was originally presented as the 2008 David
C. Chapman Lecture at the Woods Hole
Oceanographic Institution. The author gratefully
acknowledges the very useful insights of four reviewers.
FUNDING
The author was funded by NSF grants OCE 08-25154
and OCE 08-15025 during the writing of this paper.
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