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vol. 182, no. 6
the american naturalist
december 2013
A Three-Way Trade-Off Maintains Functional Diversity
under Variable Resource Supply
Kyle F. Edwards,1,* Christopher A. Klausmeier,1,2 and Elena Litchman1,3
1. Kellogg Biological Station, Michigan State University, Hickory Corners, Michigan 49060; 2. Department of Plant Biology, Michigan
State University, East Lansing, Michigan 48824; 3. Department of Zoology, Michigan State University, East Lansing, Michigan 48824
Submitted November 26, 2012; Accepted July 9, 2013; Electronically published October 31, 2013
Online enhancements: appendixes. Dryad data: http://dx.doi.org/10.5061/dryad.kf4rt.
abstract: The resources that organisms depend on often fluctuate
over time, and a variety of common traits are thought to be adaptations to variable resource supply. To understand the trait structure
of communities, it is necessary to understand the functional tradeoffs that determine what trait combinations are possible and which
species can persist and coexist in a given environment. We compare
traits across phytoplankton species in order to test for proposed
trade-offs between maximum growth rate, equilibrium competitive
ability for phosphorus (P), and ability to store P. We find evidence
for a three-way trade-off between these traits, and we use empirical
trait covariation to parameterize a mechanistic model of competition
under pulsed P supply. The model shows that different strategies are
favored under different conditions of nutrient supply regime, productivity, and mortality. Furthermore, multiple strategies typically
coexist, and the range of traits that persist in the model is similar
to the range of traits found in real species. These results suggest that
mechanistic models informed by empirical trait variation, in combination with data on the trait structure of natural communities, will
play an important role in uncovering the mechanisms that underlie
the diversity and structure of ecological communities.
Keywords: functional traits, coexistence, resource pulse, fluctuating
environment, phytoplankton.
Introduction
Many organisms live in a world where the resources they
depend on fluctuate greatly over time (Bode et al. 1997;
Polis et al. 1997; Yang et al. 2010). A variety of common
traits are thought to be adaptations to fluctuating resource
supply, such as resource storage, high maximum consumption rates, high growth or reproductive rates, high
resource use efficiency during periods of scarcity, and dormant life stages or seed banks (Noy-Meir 1973, 1974;
Sommer 1984; Grover 1991; Chesson et al. 2004; Evans
and Dennehy 2005). These traits are likely to be con* Corresponding author; e-mail: [email protected].
Am. Nat. 2013. Vol. 182, pp. 786–800. ! 2013 by The University of Chicago.
0003-0147/2013/18206-54293$15.00. All rights reserved.
DOI: 10.1086/673532
strained by trade-offs, such that an increase in one function causes a concomitant reduction in one or multiple
other functions (Stearns 1989; Tilman 1990). The particular phenotype(s) selected for by resource fluctuation will
depend on the structure of trade-offs, in combination with
environmental conditions. Therefore, in order to understand the distribution of traits that occur in nature, it is
necessary to understand the nature of the relevant tradeoffs.
Theory shows that trade-offs that constrain trait evolution may also promote temporal niche differentiation in
a fluctuating environment, leading to the coexistence of
multiple strategies (Ellner 1987; Grover 1991; Chesson et
al. 2004; Litchman et al. 2009). A better quantification of
empirical trade-off relationships can therefore enhance the
mechanistic understanding of both the maintenance of
species diversity and variation in the structure of communities across environmental gradients. This is especially
likely if a trade-off is quantified in terms of covariation
among functional traits that can be used to parameterize
models of competition; in this case, one can use a model
to ask whether different kinds of species are expected to
prevail under different conditions and whether empirically
supported forms of a trade-off can feasibly allow the coexistence of multiple species (Levine and Rees 2002; Litchman et al. 2009).
Here we investigate a potential trade-off inspired by
classic experiments by Sommer (1984), in which he found
that a greater diversity of phytoplankton species could be
maintained in culture when phosphorus was supplied in
pulses, compared to that under a constant rate of P supply.
He proposed that fluctuating nutrient supply promoted
diversity because natural phytoplankton communities possess a variety of strategies related to varying resource concentration: “velocity-adapted” species grow quickly when
nutrients are replete but decline steeply during nutrient
scarcity; “affinity-adapted” species are competitively superior under periods of nutrient scarcity but do not grow
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Three-Way Trade-Off Maintains Diversity 787
as quickly during nutrient excess; and “storage-adapted”
species take up and store nutrients when they are replete
and use these stores to grow when ambient nutrients are
depleted, but they do not exhibit the fast growth or persistence under prolonged nutrient limitation of species
using the other strategies. If these three types represent
extremes of adaptation, it is reasonable to posit a threeway trade-off between fast growth, competitive ability, and
ability to store nutrients. We refer to this proposed mutual
constraint as a velocity-affinity-storage trade-off.
Our goal in this study is twofold: first to test for potential
trade-offs among functional traits and then to use a model
parameterized by these trade-offs to see whether they promote coexistence and/or constrain community trait structure to empirically plausible values. We use a compilation
of phytoplankton functional traits to quantify variation
across species in traits related to velocity, affinity, and storage strategies. We then use the empirical structure of trait
covariation to parameterize a mechanistic model of competition under fluctuating nutrient supply. We use this
model to explore how species sorting leads to shifting
community trait structure and diversity across gradients
of environmental conditions.
This analysis expands on prior work by Grover (1991),
who used a model of competition under pulsed nutrient
supply to explore how nutrient storage and other traits
affect competition and coexistence of phytoplankton. He
examined several proposed bivariate trade-offs and found
that storage-adapted species could dominate under large
nutrient pulses, that affinity-adapted species could dominate under small pulses, and that uptake-adapted species
could dominate under small or large pulses, depending on
the particular trade-off. He also found that coexistence of
multiple strategies occurred only under a restricted bandwidth of pulse conditions. Our work builds on these results
in several ways to better resolve how fluctuating nutrient
supply operates in natural communities. We test for multivariate trade-offs, which may explain the lack of bivariate
correlations; we parameterize a model with empirical trait
correlations (rather than hypothesized trade-offs); and we
show how the likelihood of coexistence is affected by more
complex modes of nutrient supply.
Although proposed for phytoplankton, the velocityaffinity-storage trade-off has similarities to traits and
trade-offs proposed for many kinds of organisms. Fast
growth or high reproductive rates, efficient use of limiting
resources, and storage of temporally variable resources are
all considered important adaptations for both autotrophs
and heterotrophs (Noy-Meir 1973, 1974). A velocityaffinity trade-off is analogous to opportunist-gleaner
(Grover 1990; Litchman and Klausmeier 2001) or powerefficiency (Smith 1976) trade-offs, as well as some aspects
of r- versus K-selection (Reznick et al. 2002). Likewise, a
trade-off in allocation to growth or reproduction versus
that to energy storage is thought to be an important lifehistory constraint (Stearns 1989). Therefore, our analysis
may shed insight on the general problem of how organisms
persist and coexist in variable environments.
Methods
Functional Traits
In order to test for a velocity-affinity-storage trade-off, we
require lab-measured traits that quantify maximum
growth rate, nutrient competitive ability, and ability to
store nutrients. Maximum growth rate (mmax) is often estimated by measuring growth along a gradient of resource
input (light or a nutrient). In order to use estimates of
mmax that are comparable across species, we compiled data
from commonly performed experiments where exponential growth rate is measured as a function of irradiance.
In all of these experiments, species were grown in monoculture at low density, with nutrients supplied at nonlimiting levels and temperature at or near 20"C. Then, mmax
was estimated by fitting a curve to the growth-irradiance
relationship (fig. 1A; app. A; apps. A–C are available
online).
Because P is commonly limiting in freshwater systems
(Lampert and Sommer 2007) and to follow Sommer, we
focus on it as the nutrient. As a metric of competitive
ability for P, we use scaled uptake affinity for phosphate
(Psaff), which is a composite of three functional traits: maximum cell-specific phosphate uptake rate (Vmax), the halfsaturation constant for phosphate uptake (K), and the
minimum phosphorus quota (Qmin). The uptake measures
Vmax and K are the parameters of the Michaelis-Menten
curve typically fitted to measurements of uptake rate as a
function of nutrient concentration in the medium (fig.
1B). Uptake affinity (Vmax /K ) is the slope of the MichaelisMenten curve at the origin, which quantifies the cell-specific clearance rate for a nutrient as its concentration nears
0 (fig. 1B; Healey 1980). The value of Qmin is typically
estimated in the fitting of the Droop model of phytoplankton growth (Droop 1973), which relates growth rate
to internal nutrient content such that m p m " (1 !
Q min /Q), where m is specific growth rate, Q is cellular internal nutrient concentration or quota, Qmin is the minimum quota at which growth ceases, and m" is asymptotic
growth rate at an infinite quota (fig. 1C). Therefore, scaled
uptake affinity (Vmax /KQ min p Psaff) combines these three
traits in order to scale uptake ability (affinity) by the
amount of phosphorus required for growth. Furthermore,
in a model where Michaelis-Menten uptake is coupled to
the Droop growth equation as m r 0, R * r m/Psaff, where
m is the density-independent mortality rate and R * is the
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The American Naturalist
B
Uptake rate per cell
Specific growth rate
A
max
0
0
0
0
Irradiance
C
Vmax
Specific growth rate
788
inf
0
0 Qmin
K
Phosphate concentration
Phosphorus quota
Figure 1: Estimation of mmax and Psaff . A, mmax is estimated by fitting a curve to data measuring growth rate as a function of irradiance,
with nutrients supplied at nonlimiting levels. The equation for the curve is given in appendix A, available online. B, Uptake kinetics are
quantified with a Michaelis-Menten curve fitted to data measuring uptake rate as a function of phosphate concentration. The initial slope
of this curve, Vmax/K, is the uptake affinity (dashed line). C, Qmin is the cellular P content (quota) at which growth ceases, and it can be
estimated by fitting the Droop growth curve, which intersects the X-axis at Qmin and increases to the asymptote minf as P content increases
to infinity.
break-even nutrient concentration where growth equals
mortality (Tilman 1982; Litchman et al. 2007). Therefore,
as mortality goes to 0, for competitors with equal mortality, Psaff predicts the winner of competition at equilibrium, because higher Psaff results in a lower R * . In a compilation of chemostat studies, we found that scaled uptake
affinity is a good predictor of the winner in competition
at equilibrium, even at relatively high mortality rates (i.e.,
dilution rates; Edwards et al. 2012). We therefore consider
Psaff to be a good proxy of competitive ability for phosphate
under long periods of P scarcity. Unlike R *, it is independent of m", so it represents a pure expression of equilibrium
competitive ability. We compiled estimates of Vmax, K, and
Qmin from studies in which monocultures were grown under P limitation, with other nutrients and light at nonlimiting levels, with temperature at or near 20"C, and with
uptake kinetics measured using P-depleted cells.
As a metric of ability to store P, we compiled estimates
of the maximum measured P quota (Qmax), that is, the
maximum cellular P content, which approximates a cell’s
storage capacity for P. Reported estimates of Qmax were
measured in monoculture in one of two ways: P content
after a period of saturated phosphate uptake (3–24 hours)
or P content during resource-saturated growth under high
phosphate concentration. The Qmax measured with these
methods may underestimate the true maximal P content
(Elrifi and Turpin 1985; Flynn 2008), but estimates of Qmax
vary by 2 orders of magnitude across species (“Results”),
and therefore these values likely capture broad interspecific
differences in storage capacity.
We compiled estimates of mmax, Psaff, and Qmax for freshwater species; insufficient data were available for a similar
analysis with other nutrients, such as nitrate, or for marine
species. The data set includes 29 species for which at least
two of the focal traits were measured (app. B), including
10 chlorophytes, six cyanobacteria, six diatoms, four desmids, one prochlorophyte, one xanthophyte, and one euglenid. Cell volume for each species was also collected from
the literature. The data set is deposited in the Dryad Digital
Repository: http://dx.doi.org/10.5061/dryad.kf4rt (Edwards et al. 2013).
Statistical Analyses
In order to test for a trade-off among more than two traits,
it is necessary to use a technique, such as partial correlation, that quantifies the relationship between two variables while accounting for the effects of additional variables (van Noordwijk and de Jong 1986). For example, for
a trade-off among three traits, an increase in trait A may
result in a decrease in trait B, a decrease in trait C, or
both; a partial correlation will appropriately test whether
trait A is correlated with trait B while accounting for any
dependence of A and B on C. The trait data set has 18
species for which all functional traits have been measured
and 11 species for which one trait is missing. Thus, to
make full use of the information in the data, we used a
Bayesian approach that allowed us to quantify the multivariate structure of the trait data while accounting for
the fact that some data are missing. Details of methods
for estimating trait correlations are given in appendix A.
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Three-Way Trade-Off Maintains Diversity 789
dR
p s(R in ! R)
dt
Model of Competition under Pulsed P
As described in “Results,” our analysis of trait correlations
suggests a three-way trade-off among mmax, Psaff, and Qmax.
To explore the potential effects of this trade-off on phytoplankton communities, we use a model of nutrient competition parameterized with these traits. The model is derived from one analyzed by Morel (1987) and Grover
(1991, 2011) that has been shown to adequately predict
the outcome of competitive interactions under fluctuating
nutrient supply (Grover 1991; Ducobu et al. 1998; Chapelle et al. 2010), and it largely follows one used by Litchman et al. (2009) to model the evolution of diatom size.
We model phytoplankton dynamics under nutrient limitation in a well-mixed layer. For each species i, the growth
rate of cell density Ni (cells L!1) increases with P quota
Qi, following the Droop model (Droop 1973), with minimum P quota Qmin (mmol P cell!1) and theoretical growth
rate at infinite quota m" (day!1). Quota is increased by
uptake of phosphate R (mmol L!1), following MichaelisMenten kinetics, with maximum uptake rate Vmax (mmol
P cell!1 day!1) and half-saturation constant K (mmol P
L!1). We assume that increasing quota feeds back to reduce
nutrient uptake, such that Vmax declines linearly from
hi
lo
Vmax
, when quota is at its minimum Qmin, to Vmax
, when
quota is at its maximum Qmax (Gotham and Rhee 1981;
Grover 1991; Ducobu et al. 1998). Mortality occurs at a
density-independent rate m (day!1), and cell density also
declines because of turbulent mixing across the thermocline at rate s (day!1). Phosphate mixes across the thermocline at rate s (day!1), with a fixed deep-water concentration Rin, and is also supplied to the mixed layer in
periodic, instantaneous mixing events where a fraction a
of the mixed layer is replaced from water below the thermocline, with period T (days). This periodic mixing also
dilutes phytoplankton, which are assumed to be absent
from the deep waters. A fraction f of the phytoplankton
biomass loss to mortality within the mixed layer is instantaneously recycled to yield dissolved phosphate.
The model equations are
[
]
dQ i
Q i ! Q min , i
hi
hi
lo
p Vmax
, i ! (Vmax , i ! Vmax , i)
dt
Q max , i ! Q min , i
#
(
)
R
Q min , i
! m ", i 1 !
Qi,
R # Ki
Qi
(
)
dNi
Q
p m ", i 1 ! min , i Ni ! mNi ! sNi ,
dt
Qi
and
(1)
(2)
!
![
i
#
hi
hi
lo
Vmax
, i ! (Vmax , i ! Vmax , i)
R
N#
R # Ki i
!
i
]
Q i ! Q min , i
Q max , i ! Q min , i
fmNiQ i ,
(3)
with mixing events such that
R(jT #) p (1 ! a)R(jT !) # aR in ,
N(jT #) p (1 ! a)N(jT !)
for all positive integers j (which counts time in pulses),
where T ! indicates conditions immediately before the
pulse and T # indicates conditions immediately after the
pulse. The combination of slow turbulent nutrient input,
pulsed input due to mixing events, and nutrient recycling
within the mixed layer is intended as a minimal representation of the dynamics of phosphorus supply in lakes
during stratified, P-limited conditions. Stratification leads
to uptake of nearly all dissolved P, with low input from
across the thermocline or other sources and the majority
of production fueled by P remineralized within the mixed
layer (Caraco et al. 1992; Soranno et al. 1997; Kamarainen
et al. 2009). These conditions are interrupted by wind and/
or rainfall events that result in entrainment of nutrientrich water (Soranno et al. 1997; Robarts et al. 1998; Kamarainen et al. 2009) or resuspension of sediments in
shallow lakes (Istvánovics et al. 2004). These events cause
a large transient increase in dissolved P and can represent
a substantial fraction of total P supply during long stratified periods (Soranno et al. 1997; Kamarainen et al. 2009).
As we describe in “Results,” under relatively long pulse
periods, some species can persist by growing sufficiently
during the elevated nutrient conditions after a pulse occurs, while the population declines during the low-nutrient
interim between pulses. Under these conditions, Q r
Q min, production of new biomass approaches 0, and the
population declines at a per capita rate of m # s. Under
natural conditions of prolonged nutrient starvation, cells
likely adjust physiologically to enhance persistence (Dodson and Thomas 1977), but this process is poorly understood, and we do not model it here.
Parameterization
Variation across species in parameterization is derived from
empirical relationships between mmax, Psaff, and Qmax. A threeway constraint among these traits can be represented as a
surface in three-dimensional trait space. We used expectation maximization to obtain the maximum likelihood estimate for the variance-covariance matrix of the log-trans-
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790
The American Naturalist
formed traits, and we used that matrix to perform a
principal-components analysis. The first two principal components define a best-fit plane, in the sense that this plane
minimizes the sum of squared distances of the points from
the plane; 90% of trait variation is accounted for by this
plane. The trait values for all species in our simulations are
constrained to lie on this plane. We took this approach in
order to explore the effects of this trade-off when isolated
from other sources of trait variation. The real species in our
analysis presumably vary along many other trait dimensions,
leading to residual trait variation not explained by the tradeoff plane. For example, a grazer-resistant species may be
generally worse and a grazer-susceptible species may be generally better in terms of mmax, Psaff, or Qmax. Consistent with
this expectation, if we simulate competition using the trait
values for the real species, coexistence does not occur, and
a single species (Dictyosphaerium pulchellum) wins under
nearly all conditions (results not shown). This species has
the second-highest value of Psaff, the second-lowest value of
Qmax, and an intermediate value of mmax (table B1; tables B1,
C1 are available online). It appears that the combination of
having very high competitive ability for P without a correspondingly low maximum growth rate allows this species
to dominate under most conditions. Because this species
does not dominate all natural phytoplankton communities,
it likely possesses disadvantages not quantified in this analysis. By focusing on trait combinations that lie on the tradeoff plane, we can isolate the effects of this trade-off while
recognizing that additional trait dimensions are surely important for fully explaining phytoplankton diversity.
Scaled P affinity is a composite trait involving Vmax, K,
and Qmin, which are parameters in the model. Thus, in
order to parameterize the model using the empirical tradeoff, we must translate from Psaff to these component traits.
hi
We chose to hold Vmax
and Qmin constant at their median
values in the data set (3.89 # 10!7 mmol cell!1 day!1 and
1.74 # 10!9 mmol cell!1, respectively) and vary K across
species to obtain the appropriate values of Psaff, that is,
hi
K p Vmax
/PsaffQ min. This choice is justified because among
the three component traits, Psaff covaries most strongly with
K and is not significantly correlated with Vmax or Qmin (fig.
C1; figs. C1–C7 are available online).
Growth rate in the model is parameterized with m",
which can be calculated for each species with the constraint
derived in Morel (1987): m " p m maxQ max / (Q max ! Q min).
lo
lo
Likewise, Vmax
is constrained such that Vmax
p m maxQ max
(Morel 1987). For very high values of Qmax, this can result
lo
hi
in Vmax
; that is, the maximum uptake rate increases
1 Vmax
with increasing quota. When this occurred, we set
lo
hi
, but removing this constraint did not signifVmax
p Vmax
icantly alter the results (fig. C2). Other parameters in the
model (a, s, m, Rin, f ) were varied across simulations, as
described in the next section.
Simulation
Our simulations were designed to mimic species sorting
from a large regional pool that includes the range of feasible trait values, leading to varying community structure
as a function of environmental conditions. The trade-off
relationship that we use defines a plane in (log) trait space,
but the space of trait values that corresponds to potentially
persisting species is a bounded region of this plane, because
of several constraints. First, in order for a species to persist,
the nutrient concentration at which growth equals mortality (R *; Tilman 1982) must be lower than Rin, which is
the maximum possible value for dissolved nutrient concentration in the model. Second, mmax must exceed the
combined average loss rate, m # s ! log (1 ! a) /T. Finally,
in some regions of the plane Q max ! Q min, that is, storage
capacity is lower than the minimum subsistence quota,
and we define these species a priori as impossible. Given
these three constraints, the space of feasible species is
nearly triangular, with exact boundaries that depend on
parameter values and with extreme points corresponding
to maximum possible values of mmax, Psaff, and Qmax (fig.
C3). For all simulations we used a common 20 # 40 array
of trait values as the initial species pool (fig. C3); the
number of species in this array that were feasible depended
on parameter values. To speed up the simulations, some
areas of feasible trait space were not included, because
extensive preliminary simulation found that species persisting under competition never occurred outside of this
region of the trait surface for the parameter values used
here. All species were initiated at equal low densities relative to final abundances, and community dynamics were
simulated until the density of each species at the end of
a period changed by less than a factor of 10!4 between
successive periods; species were removed from the community if their mean density over one period became less
than 1 cell L!1. We performed invasibility analyses and
found that when the simulation reached an attractor of
coexisting species, this attractor was not invasible by any
species from the initial species pool (fig. C4).
We explored how the frequency of nutrient pulses determines community structure. In order to vary the frequency of nutrient pulses while holding integrated mortality and nutrient supply constant, we made the fraction
of mixed layer replaced with deep water proportional to
pulse period, such that a p 1 ! e!maT, where ma is the
instantaneous mortality rate that is equivalent to removing
a fraction a of biomass every T days. Twenty-five period
durations from 1 to 300 days, evenly spaced on a log scale,
were used. We used a baseline value of ma p 0.0255 day!1,
which results in replacement fractions that vary from
a p 0.025 for T p 1 to a p 0.3 for T p 14 and a p
0.9995 for T p 300. Baseline values of the other param-
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Three-Way Trade-Off Maintains Diversity 791
eters were m p 0.1, R in p 3, s p 0.1, and f p 0.7. To
compare these results to the limiting case of pulses that
are infinitely fast and small, we ran a simulation with no
pulses and m p 0.1255. We also used alternative parameter values to explore how community structure varied
with reduced lake productivity (R in p 0.1), reduced crossthermocline nutrient flux (s p 0.01), and increased mortality (m p 0.4). To evaluate whether our results were sensitive to uncertainty in the trait data, we resampled species
with replacement two times and used the trade-off surfaces
estimated from these bootstrap replicates to simulate communities under the baseline parameter values.
We performed several analyses to explore how the complexity of the nutrient supply regime affects community
structure. To create a scenario in which P comes solely from
pulsed input, we performed a set of simulations with crossthermocline P flux and recycling set to 0. To evaluate the
effects of stochastic variation in pulse period, we performed
simulations where the period between pulses was sampled
from an exponential distribution with mean T. Finally, to
evaluate the effects of multifrequency pulse regimes, we
performed simulations where high-frequency pulses occur
with period T and the corresponding replacement fraction
a f p 1 ! e!maT, and superimposed on this regime are slowfrequency pulses that occur with period 8T and corresponding replacement fraction a s p 1 ! e!ma8T.
In some simulations, we found that pairs of species that
coexisted in the same simulation had very similar values
for all traits. These species are adjacent on the species pool
array (fig. C5), and the coexistence of these similar species
likely implies that a species between them on the trait plane
would persist in their stead if it were present in the initial
species pool. In order to simplify visualization of the results
and focus on large differences in functional strategies, we
have averaged the trait values of these highly similar species
pairs and plotted them as single species. Plots showing all
persisting species for baseline parameter values are given in
figure C6.
Results
Trait Covariation
Raw bivariate relationships between mmax, Psaff, and Qmax
hint at negative relationships (fig. 2A–2C), but only mmax
versus Psaff is nearly significant (Pearson’s r p !0.41,
P p .056). In contrast, partial correlations among these
traits are all negative, with 95% credible intervals that
exclude 0 (table 1; fig. 2D–2F; three-dimensional scatterplot in fig. 3A); this is the case whether or not cell volume
is controlled for in calculating the partial correlation (table
1). Thus, consistent with a three-way constraint among
these traits, negative relationships between pairs of traits
are most evident when dependence on the third trait is
accounted for, and this constraint appears to be independent of cell size. Very similar patterns were found when
the analysis was limited to the 18 species for which all
traits were measured (table C1). There is some suggestion
of differences between taxa in average strategy; in particular, cyanobacteria tend to have high Psaff, desmids tend
to have a high Qmax, and nondesmid chlorophytes tend to
have high mmax (fig. 3B).
Species-Sorting Simulations
We first describe the effects of pulse frequency on community trait structure, with other parameters at their baseline values. With no pulses, a single species persists that
has high Psaff, low Qmax, and low mmax (fig. 4A). This species
is at the edge of feasible trait space and represents an
extreme affinity specialist (fig. 5). Under pulses with a 1day period, the high-affinity species coexists with a second
species that has higher Qmax, higher mmax, and lower Psaff.
With increases in pulse period, the second species continues to decline in Psaff while increasing in Qmax, and at the
same time mmax for the second species initially increases
and then decreases (fig. 4A). Thus, the high-affinity species
initially coexists with a second species that has an intermediate strategy, but increasingly this second species becomes a storage specialist, and at a period of ∼20 days it
reaches a peak value of Qmax that is ∼100 times greater
than Qmin. Beyond this point, a transition occurs where
the storage specialist is replaced by a fast-growing velocity
specialist that has low Psaff, low Qmax, and high mmax. Further
increases in pulse period result in the persistence of additional species that are intermediate on the Psaff-mmax axis
while having minimal Qmax (figs. 4A, 5).
Example dynamics under pulse periods of ∼10 and ∼100
days are shown in figure 6. Under a period of ∼10 days,
an affinity specialist and a storage specialist coexist (fig.
4A). Fluctuations in abundance are relatively small, with
the affinity specialist reaching peak abundance at the end
of the period and the storage specialist reaching peak abundance around day 6 (fig. 6A). Quota dynamics show that
the storage specialist rapidly takes up dissolved P until its
quota is near maximal and that its quota then declines
throughout the period as it continues to grow (fig. 6B).
In contrast, the affinity specialist has no variation in quota
(fig. 6B), because its Qmax is only slightly greater than Qmin
(fig. 5). Dissolved phosphate declines rapidly at first, followed by a slow decline through the rest of the period (fig.
6C). Under a pulse period of ∼100 days, the dynamics are
quite different. Three species coexist: one extreme affinity
specialist and two species with successively higher mmax and
lower Psaff (fig. 4A). All species show substantial fluctuation
in abundance, although the affinity specialist experiences
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A
B
100
100
P saff
1000
P saff
1000
10
C
1.5
µ max
792
1
0.75
10
0.5
0.5
0.75
1
1.5
4e-09
4e-08
µ max
4e-07
4e-09
Q max
D
E
F
-2
0.6
2
residual µ max
residual P saff
residual P saff
0
4e-07
Q max
2
-4
4e-08
0
0.2
-0.2
-2
-0.6
-0.1
0.4
-2
residual µ max
0
2
-2
residual Q max
0
2
residual Q max
Figure 2: Relationships between Psaff, mmax, and Qmax. A–C, Raw bivariate scatterplots of Psaff versus mmax, Psaff versus Qmax, and mmax versus
Qmax, respectively; D–F visualize partial correlations between these pairs of traits. In order to display partial correlations, the two focal traits
in each panel were each regressed against the remaining trait and cell volume, and the residuals of these regressions are plotted. For points
for which one of the controlled-for traits was not observed, the posterior mean value of that trait was used in the regression. For example,
in D, Psaff and mmax have each been regressed against Qmax and cell volume, and the residuals of these regressions are plotted. For points for
which Qmax was not observed, the posterior mean value from the Bayesian model was used.
less fluctuation than the two faster-growing species; in
addition, the faster-growing species have peak abundances
in succession at ∼10 and 20 days, after which they decline
greatly, while the affinity specialist continues to increase
to an asymptotic abundance throughout the period (fig.
6D). The quotas of all species do not vary, because they
are on the boundary of trait space where Qmax is only
slightly greater than Qmin (fig. 6E). Finally, the dynamics
of dissolved P have several phases. Initially, P declines very
slowly because of the low abundance of all species. As the
populations increase exponentially, they eventually cause
a decline in dissolved P, depleting P to successively lower
levels as species with greater Psaff increase in relative abundance (fig. 6F). By day 40, dissolved P is at a low level
that is maintained through the rest of the period.
Under alternative environmental conditions (higher
mortality, lower productivity, lower nutrient flux), the
same general pattern of shifting community structure with
Table 1: Partial correlations between Psaff, mmax, Qmax, and cell volume
Psaff
mmax
Qmax
Psaff
mmax
Qmax
Volume
...
!.59 (!.83, !.29)
!.45 (!.75, !.14)
!.63 (!.86, !.35)
...
!.45 (!.76, !.08)
!.52 (!.79, !.20)
!.56 (!.86, !.20)
...
.30 (!.07, .64)
.37 (!.03, .73)
.71 (.49, .90)
Note: Posterior means are listed, with the 95% highest posterior density credible interval in parentheses.
Above the diagonal are partial correlations that account for cell volume; below the diagonal are partial
correlations among the other three traits with volume not accounted for.
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Three-Way Trade-Off Maintains Diversity 793
A
B
-4
1
PC 2 (40%)
-8
0.5
1.0
1.5
2.0
2.5
-10
-0.4
-0.2
log10 µ max 0.0
0.2
3.0
log10 P
saff
log10 Q max
-6
µmax
0
P saff
Q max
-1
-2
-2
-1
0
1
2
PC 1 (51%)
Figure 3: Multivariate trait patterns. A, Three-dimensional scatterplot of Psaff, mmax, and Qmax. Open circles are for species with all traits
observed; filled circles are for species with two traits observed and the third trait estimated with the posterior mean from the Bayesian
model. The fitted plane is the plane derived from the first two principal-component (PC) axes fitted to these points. The equation for this
plane is Qmax p !5.77 ! 1.22 # Psaff ! 5.91 # mmax, where all traits are on a log10 scale. B, The same data plotted in PC space, with taxon
coded by symbol type; cyan p cyanobacteria; chlor p chlorophyte.
shifting pulse period occurs, but the trait values of the
most extreme specialist strategies are modulated by the
parameters (fig. 4). When mortality is increased, the minimum value of mmax needed to persist is increased, and this
in turn decreases the maximum possible value of Psaff. As
a result, the most extreme affinity specialist under higher
mortality has a relatively lower Psaff (fig. 4B). Higher mortality also causes the transition from a storage specialist
to a velocity specialist to occur at a shorter period (∼7
days), and the most extreme storage specialist has a relatively lower Qmax, compared to that under low-mortality
conditions. Finally, under the longest pulse periods, the
communities have a maximum of two coexisting species,
compared with three or four species under lower mortality.
When lake productivity is reduced (reduced Rin), there
is a moderate reduction in the highest achieved values of
both mmax and Qmax (fig. 4C). In other words, velocity and
storage specialists grow relatively slower and store less P
than under conditions of higher productivity. Reducing
the rate of mixing across the thermocline results in negligible changes from baseline parameter values (fig. 4D).
When communities are simulated with bootstrapped
trade-off surfaces and baseline parameter values, the primary patterns in trait structure remain, with some alteration of the most extreme persisting trait values (fig. C7).
We performed several analyses to explore how complexity of the nutrient supply regime affects community
structure. When cross-thermocline P flux and P recycling
are both set to 0, only a single species persists across all
conditions, and the affinity specialist is restricted to conditions of small pulses with a short period, that is, conditions of less variable nutrient supply (fig. 7A). This loss
of diversity requires that both recycling and turbulent P
influx are removed; with only one of these sources removed, the results are similar to those in figure 2A (results
not shown). In contrast, increasing the complexity of P
by including multiple pulse frequencies promotes functional diversity. Notably, when the high-frequency pulse
has a period ranging from ∼4 to 20 days, which corresponds to a period of ∼32–160 days for the low-frequency
pulse, distinct velocity, affinity, and storage strategies coexist simultaneously (fig. 7B). Finally, when pulses occur
randomly according to a Poisson process, patterns in trait
structure are very similar to those under periodic pulses
(results not shown).
Comparison with Observed Trait Distributions
Although our exploration of parameter space is not exhaustive, the different parameter values used in our simulations can be considered to represent a range of conditions
that may plausibly occur in natural lakes; we can then compare the trait variation that occurs across simulations with
the trait variation that occurs among real species. In general,
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794
The American Naturalist
P saff
1e+05
1e+03
1e-07
Q max
1e-07
1e-09
2.50
2.50
1.00
0.50
0.25
2
5
10
20
Period
50
120
1.00
0.50
0.25
300
0 1
P saff
1e+05
1e+03
1e-07
Q max
1e-07
1e-08
1e-09
2.50
2.50
5
10
20
Period
50
120
300
20
Period
50
120
300
50
120
300
Low diffusion
1e-08
1e-09
1.00
0.50
0.25
10
1e+03
1e+01
2
5
1e+05
1e+01
0 1
2
D
High mortality
µ max
P saff
1e-08
1e-09
B
Q max
1e+03
1e+01
1e-08
Low R in
1e+05
1e+01
0 1
µ max
C
Baseline
µ max
µ max
Q max
P saff
A
1.00
0.50
0.25
0 1
2
5
10
20
Period
Figure 4: Persisting species in community assembly simulations for a range of pulse periods, using baseline parameter values (A), low Rin
(B), high mortality (C), and low cross-thermocline P influx (D). Each panel contains three subpanels that show the persisting trait values
for Psaff, Qmax, and mmax. For each pulse period, coexisting species are denoted with distinct symbols. For example, in A at pulse period 45,
there are three coexisting species, coded with a filled circle, a triangle, and an open circle. The species coded with a filled circle has the
highest Psaff and the lowest mmax, while the species coded with the open circle has the lowest Psaff and the highest mmax. The species coded
with a triangle is intermediate for both of these traits. Units for Psaff are L day!1 (mmol P)!1; units for Qmax are mmol P cell!1; units for mmax
are day!1.
there is substantial overlap between the observed range of
traits and that predicted by the simulations (fig. 8). Both
real and modeled mmax range between ∼0.25 and ∼2.5 day!1,
while Qmax ranges between ∼10!9 and ∼10!7 mmol P cell!1
(fig. 8). There is more discrepancy for Psaff, which has a
minimum of ∼1 for real and modeled species but a maximum of ∼103 for real species and ∼106 for modeled species.
There is a large region of feasible trait space where Qmax is
very high and Psaff is low (fig. C3); species with these traits
never persisted in the simulations, and likewise there are
no real species with trait values in this region.
Discussion
Empirical Trait Patterns
Our analysis supports the existence of a three-way velocityaffinity-storage trade-off for phytoplankton. An increase
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Period = 1
Period = 10
-5
-6
-6
-6
log10 µ max
0.5
-9
-0.5
0.0
log10 µ max
Period = 21
-4
-2
0
2
4
6
0.5
-8
-9
-0.5
Period = 96
-6
-6
-6
0.0
log10 µ max
0.5
-8
-9
-0.5
0.0
log10 µ max
-4
-2
0
2
4
6
log10 P
saff
log10 P
saff
-9
-0.5
-4
-2
0
2
4
6
-7
0.5
log10 Q max
-5
log10 Q max
-5
-8
0.5
Period = 300
-5
-7
0.0
log10 µ max
-4
-2
0
2
4
6
log10 P
saff
0.0
-8
-7
-7
-8
-9
-0.5
0.0
log10 µ max
-4
-2
0
2
4
6
log10 P
saff
-0.5
-4
-2
0
2
4
6
log10 P
saff
-8
-7
log10 P
saff
-7
log10 Q max
-5
log10 Q max
-5
-9
log10 Q max
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log10 Q max
Period = 0
0.5
Figure 5: Coexisting species for six example pulse periods, under baseline parameter values. Small gray circles show the array of species that are initially present in the simulations.
Large black circles show the species that persist. The persisting species are the same as those plotted in figure 4A, replotted here to show their locations in three-dimensional trait space.
Figure C3, available online, compares the array of initial species to the full range of feasible species. “Period p 0” indicates the simulation with no pulsed nutrient supply.
796
The American Naturalist
Phosphate (
mol
L 1)
Quota (
molP
cell 1)
Abundance (cells L 1)
A
D
Abundance
Abundance
1e+09
1e+09
1e+07
1e+05
1e+08
1e+03
1e+01
0
10
B
Quota
20
0
E
2e-08
2e-08
5e-09
5e-09
2e-09
2e-09
0
10
C
Dissolved phosphate
20
3e+00
1e-02
1e-02
1e-04
1e-04
1e-06
1e-06
10
Day
20
180
Quota
0
3e+00
0
90
90
F
Dissolved phosphate
0
90
Day
180
180
Figure 6: Example dynamics for a ∼10-day pulse (A–C) and a ∼100-day pulse (D–F). Coexisting species are indicated by distinct line
patterns in A, B and D, E. A and D have Y-axes with different ranges, because of greater fluctuation in abundances for 100-day pulses.
Note that abundances are reduced after each simulated period by the mixing event, so that these represent stable population cycles.
in any one of mmax, Psaff, and Qmax tends to be correlated
with a decrease in both of the other traits (table 1). Within
this constraint, species exhibit a wide variety of strategies
(fig. 3). It is noteworthy that in this data set the trade-off
is independent of cell volume. In general, Qmax tends to
increase with increasing cell volume, while mmax and Psaff
decline with increasing cell volume (Grover 1989; Litchman et al. 2009; Edwards et al. 2012). Therefore, variation
in cell volume alone should tend to cause Qmax to be negatively correlated with the other two traits. However, in
this data set Qmax correlates with volume, while mmax and
Psaff do not (table 1). This is consistent with the fact that
allometric relationships for many traits tend to be noisy
(Edwards et al. 2012), and in this data set the variation
in cell volume is moderate (∼1,000-fold), compared to the
full range of variation among phytoplankton (∼108-fold).
We interpret these results to mean that the correlations
among the three focal traits in this analysis represent trait
variation that is orthogonal to variation in cell volume,
while changing cell volume tends to cause additional correlated change in these traits. An interesting avenue for
future research would be an analysis of the underlying
physiological basis for these trait correlations.
Our analysis of interspecific trait variation confirms the
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Three-Way Trade-Off Maintains Diversity 797
B
P saff
1e+05
1e+03
1e+05
1e+03
1e+01
1e-07
1e-07
Q max
1e+01
1e-08
1e-08
1e-09
1e-09
2.50
2.50
µ max
µ max
Q max
P saff
A
1.00
0.50
0.25
1
2
5
10
20
50
120
300
1.00
0.50
0.25
1
2
Period
5
10
20
Period
Figure 7: Effects of supply regime complexity on community structure. A, Community structure with no cross-thermocline nutrient flux
or recycling (s p 0, f p 0) and other parameters at baseline values. B, Community structure under multifrequency pulsing. The period
listed along the X-axis is the high-frequency pulse period T, and each community also receives a low-frequency pulse of period 8T.
importance of using a multivariate approach when attempting to decipher how traits covary. A prior analysis
of scaled phosphate affinity, scaled nitrate affinity, and cell
volume also found that some important patterns emerged
only when multiple trait dimensions were considered simultaneously (Edwards et al. 2011). In order to extend
these methods to a full consideration of phytoplankton
ecology and evolution, it will be important to measure
numerous key traits concurrently across a range of species
and to better incorporate traits reflecting interactions with
grazers and pathogens. It will also be important to extend
the use of missing-data methods that allow all of the available information to be used in quantification of multivariate patterns.
Across many kinds of organisms, traits analogous to
mmax, Psaff, and Qmax are considered to be important adaptations to fluctuating resource supply. For example, terrestrial plants in arid environments are often distinguished
by their ability to grow rapidly after a pulse of rainfall,
their ability to maintain metabolic activity during drought
conditions, and their ability to store water and/or energy
(Noy-Meir 1973; Chesson et al. 2004). An analysis of functional traits in a community of desert annuals found evidence for a trade-off between relative growth rate and
water use efficiency (Angert et al. 2009), analogous to a
mmax-versus-Psaff trade-off in phytoplankton. As research in
this area accumulates, it will be interesting to consider
whether such traits participate in multidimensional tradeoffs and whether the importance of such traits results in
common responses of community structure to resource
fluctuation across ecosystem types.
Species-Sorting Simulations
A primary result of the species-sorting simulations is that
a wide range of strategies can persist, depending on environmental conditions. Species with high maximum
growth rates tend to occur under large pulses of low frequency (period longer than ∼20 days), presumably because
these conditions result in long periods of P saturation that
allow for sustained growth at maximal rates. This is consistent with the dominance of fast-growing species during
the early stages of annual successional cycles (Sommer
1996), cycles that have some similarity to a large nutrient
pulse with annual frequency. Relatively high P concentration below the mixed layer is also necessary to favor the
greatest mmax, because this increases the postpulse duration
of prolonged growth at saturated rates.
Species with high storage capacity are favored by pulses
of intermediate frequency (∼5–20 days), presumably be-
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The American Naturalist
A
B
12
Frequency
8
Frequency
C
4
0
Frequency
798
8
4
0
1e-25
1e-10
P saff
1e+05
6
4
2
0
0.2
2
µ max
15
1e-09
1
Q max
1e+09
Figure 8: Observed and modeled ranges for Psaff (A), mmax (B), and Qmax (C). Observed trait distributions are shown as histograms in gray.
Modeled trait distributions are demarcated with dashed lines. The X-axes are scaled to include the range of feasible trait values in figure
C3, available online. Units for Psaff are L day!1 (mmol P)!1; units for mmax are day!1; units for Qmax are mmol P cell!1.
cause these conditions allow for rapid uptake of pulsed P,
followed by growth using stored P, followed by the occurrence of a new pulse around the time that stores have
been depleted. In addition, high nutrient concentration
below the mixed layer also benefits storage specialists, presumably by increasing the amount of nutrient that can be
stored after a pulse. These results are consistent with prior
models showing that a high storage capacity is favored
under pulses of intermediate frequency with high nutrient
concentration (Grover 1991, 2011; Litchman et al. 2009).
We also find that high mortality rates reduce the magnitude of storage capacity that is favored (fig. 4B); this is
likely because cells experiencing a high mortality rate are
dying before they can fully utilize their stored resources.
Species with high scaled affinity are always present in
simulations that include cross-thermocline P influx or recycling of P from dead phytoplankton within the mixed
layer. This usually results in the coexistence of multiple
strategies (fig. 4). This result contrasts with some prior
models showing that a single species typically persists under pulsed nutrient supply, with the strategy of this species
depending on conditions (Grover 1991, 2011). The difference in our results appears to be due to the fact that
our model contains additional sources of continuous nutrient supply, that is, mixing across the thermocline and
nutrient recycling. Thus, it appears that multiple modes
of nutrient supply can promote coexistence of multiple
strategies, with a high-affinity species possessing a relative
advantage during times of low nutrient supply, after pulsed
nutrients have been consumed and transformed into biomass. Furthermore, inclusion of multiple pulse frequencies
promotes additional diversity and allows velocity, affinity,
and storage strategies to coexist simultaneously under a
range of conditions (fig. 7B). This contrasts with singlefrequency pulses, which rarely allow strategies of rapid
growth and high storage to coexist (fig. 2). Multiple
sources of nutrient loading are a common feature of pelagic ecosystems (Dugdale and Goering 1967; Caraco et
al. 1992; Hudson et al. 1999; Kamarainen et al. 2009), and
we suggest that incorporating this complexity will be important for understanding how the nutrient supply regime
determines the structure of planktonic communities.
We find a reasonable fit between the real and modeled
ranges of trait values (fig. 8). Although the real trait values
were used to fit the trade-off plane that parameterizes the
model, our simulations were initialized with the full spectrum of trait values from the plane that could feasibly
persist. Therefore, the simulations included a much
broader range of trait values than is observed in our data
set (fig. 8). Because the range of trait values that persist
in the model is nonetheless similar to the range of observed
trait values, it is plausible that the trade-off we have identified plays an important role in generating trait diversity
in natural communities, as well as in constraining which
trait values can exist in nature. Variation across lakes in
productivity, mortality, and the temporal regime of nutrient supply may lead to spatial variation in dominant
trait values, while temporal variation within lakes may
maintain local functional diversity. A more direct empirical
test of our simulation results will require the synthesis of
spatial and temporal community survey data with functional trait data for the surveyed species. In general, we
think that combining data on interspecific trait correlations with mechanistic models will be crucial for advancing
trait-based approaches to community structure and better
synthesizing these approaches with an understanding of
the maintenance of species and trait diversity.
Acknowledgments
This research was supported by National Science Foundation grants DEB-0845932 (to E.L.), DEB-0845825 (to
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Three-Way Trade-Off Maintains Diversity 799
C.A.K.), and OCE-0928819 (to E.L. and C.A.K.). We thank
F. Adler and two anonymous reviewers for helpful comments on a prior version of this manuscript. This is Kellogg
Biological Station publication number 1724.
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Associate Editor: Frederick R. Adler
Editor: Mark A. McPeek
“Like Van Diemans Land, now changed to Tasmania, New Zealand is unfortunate in its name, as it is in every way in contrast with the
Zealand of the Netherlands; while the latter is nearly as level and uniform as the sea, the former possesses some of the wildest and grandest
scenery in the world.” From “A Sketch of New Zealand with Pen and Pencil” by I. C. Russell (The American Naturalist, 1879, 13:65–77).
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