Journal of
Plankton Research
plankt.oxfordjournals.org
J. Plankton Res. (2016) 38(4): 977–992. First published online June 4, 2015 doi:10.1093/plankt/fbv038
Contribution to the Themed Section: Advances in Plankton
Modelling and Biodiversity Evaluation
Flexible phytoplankton functional type
(FlexPFT) model: size-scaling of traits
and optimal growth
S. LAN SMITH1,2*, MARKUS PAHLOW3, AGOSTINO MERICO4,5, ESTEBAN ACEVEDO-TREJOS4, YOSHIKAZU SASAI1,
CHISATO YOSHIKAWA6, KOSEI SASAOKA7, TETSUICHI FUJIKI1, KAZUHIKO MATSUMOTO8 AND MAKIO C. HONDA8
1
ECOSYSTEM DYNAMICS RESEARCH GROUP, RESEARCH AND DEVELOPMENT CENTER FOR GLOBAL CHANGE, JAMSTEC, 3173-25 SHOWA-MACHI, KANAZAWA-KU,
2
3
YOKOHAMA, JAPAN, CREST, JAPAN SCIENCE AND TECHNOLOGY AGENCY, TOKYO, JAPAN, GEOMAR, HELMHOLTZ CENTRE FOR OCEAN RESEARCH KIEL, KIEL,
4
5
GERMANY, SYSTEMS ECOLOGY, ZMT (LEIBNIZ CENTER FOR TROPICAL MARINE ECOLOGY), BREMEN, GERMANY, JACOBS UNIVERSITY, BREMEN, GERMANY,
6
7
INSTITUTE OF BIOGEOSCIENCES, JAMSTEC, YOKOSUKA, JAPAN, GLOBAL CHEMICAL AND PHYSICAL OCEANOGRAPHY GROUP, JAMSTEC, YOKOSUKA, JAPAN AND
8
DEVELOPMENT OF ENVIRONMENTAL GEOCHEMICAL CYCLE RESEARCH, YOKOSUKA, JAPAN
*CORRESPONDING AUTHOR: [email protected]
Received December 4, 2014; accepted May 1, 2015
Corresponding editor: Zoe Finkel
Recent studies have analysed valuable compilations of data for the size-scaling of phytoplankton traits, but these
cannot be employed directly in most large-scale modelling studies, which typically do not explicitly resolve the relevant trait values. Although some recent large-scale modelling studies resolve species composition and sorting within
communities, most do not account for the observed flexible response of phytoplankton communities, such as the
dynamic acclimation often observed in laboratory experiments. In order to derive a simple yet flexible model of phytoplankton growth that can be useful for a wide variety of ocean modelling applications, we combine two trade-offs, one
for growth and the other for nutrient uptake, under the optimality assumption, i.e. that intracellular resources are dynamically allocated to maximize growth rate. This yields an explicit equation for growth as a function of nutrient concentration and daily averaged irradiance. We furthermore show how with this model effective Monod parameter
values depend on both the underlying trait values and environmental conditions. We apply this new model to two contrasting time-series observation sites, including idealized simulations of size diversity. The flexible model responds differently compared with an inflexible control, suggesting that acclimation by individual species could impact models of
plankton diversity.
KEYWORDS: phytoplankton; ecosystem model; trait; acclimation; size-scaling
available online at www.plankt.oxfordjournals.org
# The Author 2015. Published by Oxford University Press. All rights reserved. For permissions, please email: [email protected]
JOURNAL OF PLANKTON RESEARCH
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I N T RO D U C T I O N
Trait-based modelling is now widely applied to study
plankton ecosystems (Verdy et al., 2009; Follows and
Dutkiewicz, 2011; Acevedo-Trejos et al., 2014), and size is
widely considered a meta- or master trait for phytoplankton ecology (Litchman et al., 2007; Edwards et al., 2012).
However, whereas most observation-based size-scalings
are reported for traits such as nutrient uptake and subsistence cell quotas (Litchman et al., 2007; Edwards et al.,
2012; Marañón et al., 2013), most models of plankton
ecosystems, particularly those applied at large scales, are
formulated in terms of the Monod equation for growth,
without explicitly resolving the dynamics of nutrient
uptake or cell quotas. Although the Monod equation for
growth and the Michaelis– Menten (MM) equation for
nutrient uptake are of exactly the same shape (Healey,
1980), the half-saturation value for growth as applied in
the Monod equation must be systematically less than the
half-saturation value for nutrient uptake as applied in the
MM equation (Morel, 1987). Furthermore, no consistent
theoretical relationship has yet been derived for the
size-scaling of Monod growth parameters, nor of overall
growth response, in terms of commonly reported allometry relations for underlying trait values. This means that
the reported allometric scaling relationships for traits, no
matter how informative they may be, cannot be applied
directly in most large-scale models.
Culture experiments with single phytoplankton species
(Flynn, 2003) and predator– prey interactions (Yoshida
et al., 2003; Fussmann et al., 2005) clearly show flexible response to changing environmental conditions. However,
simple models based on Monod kinetics (Flynn, 2003) do
not capture this flexibility, and even those based on the
more realistic Droop growth model (Caperon, 1968;
Droop, 1968), do not account for optimal allocation of
intracellular resources. Optimality-based formulations,
by combining traits and trade-offs, provide one means of
representing such flexible responses without greatly increasing model complexity (Smith et al., 2011). The combination of traits and trade-offs holds further promise
(Smith et al., 2014b). For example, while allometric
scaling of nutrient uptake parameters based on laboratory studies (Fiksen et al., 2013) does not account for the
full range of responses observed by ship-board experiments in the ocean, a model incorporating both
size-scaling of traits and the trade-off of optimal uptake
(OU) kinetics (Pahlow, 2005; Smith et al., 2009) captured
the wide observed range of half-saturation values (Smith
et al., 2014a). Moreover, Pahlow and Oschlies (Pahlow
and Oschlies, 2013) recently developed an optimalitybased model of phytoplankton growth and used it to give
the first theoretical derivation of the Droop equation
NUMBER 
j
PAGES
 –  
j 
(Caperon, 1968; Droop, 1968), which had been long
established purely on an empirical basis.
Here, we derive a relatively simple and flexible model
of phytoplankton growth and show that it can be
expressed as an equation of Monod form, with effective
Monod parameter values depending on both underlying
trait values and ambient environmental conditions. This
new flexible phytoplankton functional type (FlexPFT)
model accounts for the adaptive response to light and nutrient levels in terms of two trade-offs (Fig. 1) for allocation of intracellular resources: (i) carbon versus nitrogen
assimilation (Pahlow and Oschlies, 2013), and (ii) affinity
for nutrient versus maximum uptake rate (Pahlow, 2005;
Smith et al., 2009). This provides a framework for modelling the growth response based on the commonly
reported size-scalings for the parameters of the Droop
and MM equations. In order to evaluate how the flexible
response impacts model performance, we apply the
FlexPFT, and an inflexible control model (hereafter
“control”), to contrasting time-series from two observation sites in the North Pacific.
M O D E L E Q UAT I O N S
Balanced growth assumption
Burmaster (Burmaster, 1979) showed that, at steady state,
the widely applied Monod kinetics can be derived by
combining (i) the Droop quota model (Caperon, 1968;
Droop, 1968) for growth as a function of intracellular nutrient content, (ii) the MM kinetics (Michaelis and
Menten, 1913) applied for nutrient uptake at the cellular
level (Dugdale, 1967) as a function of external (ambient)
nutrient concentration and (iii) the assumption that the
rates of growth, m (day21) and nutrient uptake, V (mol N
(mol C)21 day21), are balanced so that
V ¼ mQ
ð1Þ
where Q (mol N:mol C) is intracellular nutrient content
per unit C biomass. Doing so results in an equation of
Monod form, in which the parameters of the Droop and
MM equations are combined to give the effective Monod
parameters for growth. The recent global scale study of
Ward et al. (Ward et al., 2013) applied this relationship to
model the growth response of a size-based phytoplankton community in terms of the underlying size-scalings
for the parameters of the empirically based Droop and
MM equations. Here, we apply the balanced growth assumption to derive a simple model of phytoplankton
growth by combining an optimality-based model for
growth, which has recently been used to derive the
Droop quota model (Pahlow and Oschlies, 2013), with
978
S. L. SMITH ET AL.
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FLEXIBLE PHYTOPLANKTON FUNCTIONAL TYPE (FlexPFT) MODEL
Fig. 1. Schematic of the FlexPFT model structure, which combines two trade-offs, from optimal uptake kinetics (Pahlow, 2005; Smith et al., 2009)
and the optimal growth model (Pahlow and Oschlies, 2013), respectively, under the assumptions of balanced growth (Burmaster, 1979) and
instantaneous optimal resource allocation.
OU kinetics (Pahlow, 2005; Smith et al., 2009) for nutrient uptake. Then we show that this provides a novel
trait-based framework for modelling the flexible adaptive response of phytoplankton, based on the optimality
principle that intracellular resources are dynamically
allocated in order to maximize growth rate (Smith et al.,
2011, 2014b), and that empirical size-scalings of traits
(Supplementary Material online, Appendix SB) can
easily be incorporated into this framework to model size
diversity.
Growth as a function of nutrient quota
The Droop-type quota model equation for growth rate,
m (day21), is
Q0
ð2Þ
m ¼ m1 1 Q
where m1 is the asymptotic growth rate at infinite cell
quota, Q is the nitrogen cell quota, and Q 0 is the subsistence, i.e. absolute minimum, cell quota at which growth
rate becomes zero. Based on a recent optimality-based
model (Pahlow and Oschlies, 2013), which also relates
net-specific growth rate to nutrient quota, we formulate
our model in terms of the structural cell quota, Q s, so
that
2Q s
ð3Þ
m¼m
^I 1 Q
where m
^ I is the potential maximum growth rate at
ambient light ( just as m1 above implicitly accounts for
light limitation). Equation (3) is equivalent to the Droop
model with Q 0 ¼ 2Q s. The first trade-off relates the
actual growth and nutrient uptake rates to their respective potential maximum values via the fractional allocation of resources towards nutrient uptake, fV
Qs
I
m ¼ 1
ð4Þ
^I
fV m
Q
V N ¼ fV V^ N
ð5Þ
where V^ N is the potential maximum nutrient uptake rate,
specified further below. Thus increasing fV will increase
the rate of nutrient uptake [Equation (5)], at the expense
of reducing the carbon-based growth rate [Equation (4)].
We define the dependence on irradiance, I, and temperature, T, as
^ I ðI; T Þ ¼ m
m
^ 0 SðI; T ÞF ðT Þ
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VOLUME 
where m
^ 0 is the potential maximum growth rate. S(I,T)
specifies the dependence on light
(
)
^
aI QI
SðI; T Þ ¼ 1 exp
ð7Þ
^ 0 F ðT Þ
m
as in Pahlow et al. (Pahlow et al., 2013), where aI is the
chl-specific initial slope of growth versus light intensity
^ is the
(here assumed constant and independent of size), Q
chl:C ratio of the chloroplast and I is the intensity of
photosynthetically active radiation (PAR). The optimal
^ Q
^ o ; which maximizes the potential lightvalue of Q;
limited growth rate [Equation (6)], is calculated by balancing the costs versus benefits of chlorophyll (chl) synthesis
on a daily averaged basis (Pahlow et al., 2013):
NUMBER 
zchl Rchl
M
:
I0 ¼
Ld aI
The total chl content, Q (g chl (mol C)21), is then
Qs
^
Q¼ 1
fV Q
Q
ð9Þ
 –  
fVo ¼
j 
Qs
zN ðQ 2Q s Þ
Q
ð12Þ
or alternatively, in terms of m
^ I and V^ N :
fVo ¼
m
^
Rchl
aI I
þ 0 1 W0 1 þ M exp 1 þ chl
Ld m
^0
aI I
z m
^0
where z chl (mol C (g chl)21) is the respiratory cost of
photosynthesis, RM chl is the loss rate of chl (day21), Ld is
the fractional day length (fraction of 24 h), I is the daily
averaged irradiance and W0, the zero-branch of
Lambert’s W function, which can be calculated numerically. Equation (8) satisfies the optimality condition and is
valid only for I . I0, the threshold irradiance below
which the respiratory costs outweigh the benefits of pro^ ¼ 0). This threshold irradiance
ducing chl (so that Q
level is
PAGES
temperature (8C) and Tref is the reference temperature
(taken as 208C).
Independent of the specific functions assumed for
light, temperature and nutrient dependence, the optimal
value of fV is calculated (Pahlow and Oschlies, 2013) by
maximizing growth rate subject to the trade-off specified
by Equations (4) for growth and (5) for nutrient uptake
^o ¼ 1
Q
zchl
ð8Þ
j
m
^ I ðI; T ÞQ s
N
V^ ðN ; T Þ
2
3
v"
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!#1
u
I
u
m
^ ðI; T Þ
6
7
t Q
þ zN
þ 15
41 þ
s
N
V^ ðN ; T Þ
ð13Þ
where z N is the energetic respiratory cost of assimilating
inorganic N, estimated as 0.6 mol C (mol N)21 by
Pahlow and Oschlies (Pahlow and Oschlies, 2013). The
cell quota under the assumption of optimal resource allocation, Q o, depends on N and I:
2
3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"
!#1
u
I
u
m
^ ðI; T Þ
6
7
Q o ¼ Q s 41 þ t 1 þ Q s
þ zN
5
N
^
V ðN ; T Þ
ð14Þ
where the superscript “o” denotes that this value of the
cell quota assumes optimal resource allocation.
Nutrient uptake as a function of ambient
concentration
ð10Þ
Thus, the model calculates the chl content within the
chloroplast, and hence of the whole cell, assuming instantaneous adjustment to the daily averaged irradiance.
Arrhenius-type temperature dependence is assumed
Ea
1
1
F ðT Þ ¼ exp
ð11Þ
R T þ 298 Tref þ 298
where Ea is the activation energy (taken here as 4.8 104
J mol21, to approximate the widely applied “Q 10 ¼ 2,”
i.e. doubling of rate for a temperature increase from 10 to
208C), R is the gas constant (8.3145 J (mol K)21), T is
The affinity-based equation for nutrient uptake rate, V
(mol N (mol C)21 day21), is
V ¼
Vmax AN
Vmax þ AN
ð15Þ
where Vmax is the maximum uptake rate (mol N (mol C)21
day21), A is the affinity (m3 (mmol C)21 day21) and
N is the dissolved inorganic nitrogen (DIN) concentration
(mmol N m23). The second trade-off is between nutrient
affinity and maximum uptake rate, as specified by OU
kinetics (Pahlow, 2005; Smith et al., 2009). This trade-off
is defined in terms of the fractional allocation, fA, of nitrogenous resources for nutrient uptake, such that increasing
fA increases nutrient affinity, A ¼ fAA0, at the expense of a
980
S. L. SMITH ET AL.
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FLEXIBLE PHYTOPLANKTON FUNCTIONAL TYPE (FlexPFT) MODEL
decrease in maximum uptake rate, Vmax ¼ (1 2 fA)V0,
where A0 and V0 are the potential maximum values of affinity and maximum uptake rate, respectively.
Incorporating this trade-off, the potential maximum
uptake rate is
are optimized instantaneously, gives a single equation
for growth rate explicitly in terms of the nutrient
concentration, N:
"
I
ð1 fA ÞV0 fA A0 N
V ð fA ; N Þ ¼
ð1 fA ÞV0 þ fA A0 N
^N
mðI ; N ; T Þ ¼ m
^ ðI; T Þ 1 þ Q 0
ð16Þ
0
13
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!1ffi
u
I
u
2
m
^ ðI; T Þ
B
C7
@1 t1 þ
þ zN
A5
Q 0 V^ N ðN ; T Þ
This will be multiplied by the allocation factor fV
described above to obtain the actual rate. Independent of
fV, the optimal allocation of resources for maximizing V^ N
is then (Pahlow, 2005)
fAo
rffiffiffiffiffiffiffiffiffi 1
A0 N
¼ 1þ
V0
ð17Þ
which can be substituted back into Equation (16) to give
the nutrient-limited uptake rate assuming instantaneous
optimization of fA (Pahlow, 2005; Smith et al., 2009). The
actual uptake rate depends upon the allocation factor fV
as per Equation (5), and the actual values of uptake parameters are
^0
A0 ¼ fV A
ð18Þ
V0 ¼ fV F ðT ÞV^0
ð19Þ
where affinity is assumed not to depend on temperature.
Assuming instantaneous optimization of fA, the potential
nutrient uptake rate is
V^ N ðN ; T Þ ¼
V^ N
q0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^ 0 Þ þ 2 ðV^ 0 N =A
^0 Þ þ N
ðV^ 0 =A
ð22Þ
The cell quota as a function of light intensity and nutrient concentration, Equation (14), can then be expressed
as:
2
3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"
!#1
u
I
u
Q06
Q0 m
^ ðI; T Þ
7
t
þ zN
Qo¼
41 þ 1 þ
5
2
2 V^ N ðN ; T Þ
ð23Þ
For the control model, the rates of C-based growth and
nutrient uptake are calculated independently (based on
fixed values of fA and fV), and the cell quota is calculated dynamically from the balanced growth assumption, Equation (1).
The rate of change of phytoplankton biomass, BP
(mmol C m23), is then
dBP
¼ ðmnet ðI; N ; T Þ MBP ÞBP
dt
ð20Þ
mðI; N ; T Þ ¼
^0 N
m
^ I ðI; T ÞfVo ð1 fAo ÞV^ 0 fAo A
I
o ^
m
^ I ðI; T ÞQ 0
^ ðI; T ÞQ 0 ð1 fA ÞV 0 þ ðm
o
o ^
o^
þ fV ð1 fA ÞV 0 Þ fA A0 N
ð21Þ
Note that we have employed the equality Q 0 ¼ 2Q s.
Substituting Equations (13) and (17), under the assumption that the fractional resource allocations, fA and fV,
ð24Þ
where m net is the net-specific growth rate, and M is the
specific mortality rate, which implicitly represents losses
to grazing.
Combining growth and uptake
Combining Equations (3) and (16) under the balanced
growth assumption, Equation (1), and substituting
Equations (18) and (19) gives the growth rate as a function of external nutrient concentration and light level, in
an equation of the same form as Equation (15) for nutrient uptake
!
m
^ I ðI; T Þ
N
þz :
N
V^ ðN ; T Þ
mnet ðI; N ; T Þ ¼ mðI; N ; T Þ Rchl
ð25Þ
Here R chl is the biomass-specific respiratory cost of maintaining chl (Pahlow et al., 2013):
chl ^
Rchl ðI; N ; T Þ ¼ ðm
^ I ðI; T Þ þ Rchl
M Þz Q
Qs
fv
1
Q ðI; N ; T Þ
ð26Þ
Effective values of Monod parameters
Here we examine how our optimality-based model relates
to the well-known Monod growth kinetics, which are
widely applied in existing phytoplankton models as well
as for interpreting observations. The FlexPFT constitutes
a novel framework for expressing effective values of
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parameters for the Monod growth equation in terms of
more commonly reported trait values (e.g. nutrient uptake
parameters and subsistence cell quotas), which cannot be
utilized directly in the Monod equation. Equation (21) can
be re-arranged into Monod form:
meff N
m ¼ effmax
Km þ N
ð27Þ
where the effective Monod parameter values are:
meff
max ¼
m
^ I fVo ð1 fAo ÞV^ 0
m
^ Q0 þ fVo ð1 fAo ÞV^ 0
I
Kmeff ¼
Q0
meff
^ 0 max
fVo fAo A
ð28Þ
eff
Vmax
ð1 fAo ÞV^ 0
¼
^0
Aeff
fAo A
ð30Þ
Then, the half-saturation values for growth and uptake
are related as follows:
Kmeff ¼
m
^ I Q0
KVeff
m
^ I Q0 þ fVo ð1 fAo ÞV^ 0
ð31Þ
which shows that Kmeff , KVeff . Morel (Morel, 1987)
showed this inequality via a different relationship in terms
of the ratio of the maximum and minimum cell quotas
multiplied with the ratio of short- to long-term maximum
uptake rates. Equation (31) furthermore provides a basis
for expressing the size-scaling of Kmeff in terms of the
size-scalings of MM parameters. It is also informative to
consider the effective growth-based affinity (the initial
slope of growth rate vs. nutrient concentration):
Aeff
m
^0
Aeff
f of oA
¼
¼ V A
Q0
Q0
ð32Þ
which is a better metric of competitive ability for nutrient
than the half-saturation constant of MM/Monod kinetics
(Button, 1978; Healey, 1980; Smith et al., 2014b). With this
optimality-based formulation the growth-based affinity
j
PAGES
 –  
j 
(Aeff
m ) varies with environmental conditions, as resources
are dynamically allocated so as to maximize growth rate.
Mass balance for nutrient and detritus
The mass balance for DIN must account for the
balanced growth assumption, which implies that the cell
quota instantaneously adjusts to changing nutrient concentration and (daily averaged) light in the ocean environment. This results (Supplementary Material online,
Appendix SA) in the following equation for nutrient concentration N:
dN
¼
dt
ð29Þ
both of which vary with the flexible response through the
two fractional allocations, fA and fV .
For comparison with the widely reported halfsaturation constants based on the MM equation for nutrient uptake, the expression for Kmeff can be re-written in
terms of KVeff ; the half-saturation constant for nutrient
uptake, which can be expressed in terms of the OU parameters as:
KVeff ¼
NUMBER 
kd DN þ km ðNb N Þ þ EN ðmnet ðI; N ; T ÞQ
þð@Q =@IÞðdI=dtÞÞBP
1 þ ð@Q =@N ÞBP
ð33Þ
where kdDN is the source from regeneration of detritus,
km(Nb 2 N) accounts for mixing across the bottom of the
mixed layer, below which concentration Nb is prescribed
as dynamic forcing, EN is entrainment of nutrients as the
mixed layer deepens (Supplementary Material online,
Appendix SA). The term in the denominator accounts
for the fact that as nutrient concentration changes, the
cell quota changes instantaneously in the same direction
(balanced growth), because of the assumption of an instantaneous balance between nutrient uptake and
growth. The derivative of the cell quota with respect to
the nutrient concentration is positive, so that the denominator is greater than unity and, therefore, this model will
produce slower changes in concentration of nutrients,
compared with models that resolve explicitly the dynamics of the cell quota (i.e. unbalanced growth). The mass
balance for detrital nitrogen, DN, is:
hv
i
dDN
s
¼ QMmax F ðT ÞBP2 þ kd F ðT Þ DN
dt
H
ð34Þ
where vs (m day21) is the sinking rate of detritus, H is the
mixed layer depth ( prescribed as a time-varying forcing),
and kd (day21) is the specific degradation rate of detritus.
S I M U L AT I O N S
This simple set-up was chosen not to realistically reproduce the time-series observations in their entirety, but
rather to test the performance of the FlexPFT model
versus the control model. In the latter, fV [Equations (4)
^ [Equation (7)] and fA [Equation (16)] were
and (5)], Q
each fixed at a constant value. Each version of the ecosystem model included only three compartments: phytoplankton C biomass for a single PFT (BP), DIN (N) and
982
S. L. SMITH ET AL.
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FLEXIBLE PHYTOPLANKTON FUNCTIONAL TYPE (FlexPFT) MODEL
detrital N (DN). Zooplankton was simulated only implicitly as a mortality term (quadratic in BP).
Comparison to time-series sites
A zero-dimensional (box) model of the oceanic mixed
layer, based on the model of Spitz et al. (Spitz et al., 2001),
was applied with variable mixed layer depth prescribed
as forcing for two contrasting time-series observation
sites (Fig. 2). Entrainment of nutrients by deepening of
the mixed layer was simulated here based on a variable
nutrient concentration at the bottom of the mixed layer,
which was prescribed as forcing based on the World
Ocean Atlas (http://www.nodc.noaa.gov/OC5/WOA09/
pr_woa09.html).
The Adaptive Metropolis algorithm (Smith, 2011) was
used to fit values of selected model parameters to the
observations (averaged within the mixed layer) at both
sites simultaneously and for each model version. Thus for
each model, an ensemble of simulations was conducted
and the Akaike Information Criterion, as calculated
based on the ensemble mean log likelihood, was the
metric for goodness of fit. This results in an ensemble of
parameter values and a corresponding ensemble of simulated values, which can be used to assess the overall
agreement of each model with the observations.
Common values for both observation sites were fitted
for five key phytoplankton traits (i.e. model parameters):
^ 0 and V^0 ; aI, Q 0, A
^ 0 and V^0 ; as well as the respiration
A
rate for chl maintenance, Rchl
M ; and mortality rate conK2
S1
and Mmax
; separately for each station, restants, Mmax
^
spectively. For the control model, a constant value for Q
was also fitted (common to both sites). Initial estimates
for the trait values were obtained from the size-scalings in
Table I, assuming a cell size of 1 mm.
Size-based multi-species model
The model set-up above was applied with 200 PFTs
having size-scaled traits (Table I and Supplementary
Material online, Appendix SB), but without fitting the
models to the data. Size classes were evenly distributed in
log space over the range 0.2– 50 mm ESD, for both the
FlexPFT and the control model. Again, effects of zooplankton were included only implicitly through the mortality term, which was modified to reproduce competitive
exclusion (Record et al., 2013) by making the mortality
rate for each PFT dependent on the mean concentration
P ; and its own concentration, BP,i. Thus,
(of all PFTs), B
the mortality rate of the ith PFT becomes:
1f 1þf
BPi
mi ¼ Mmax B
P
ð35Þ
where f is a parameter (0, 1) which determines the
degree of competitive exclusion. With f ¼ 1, this expression reduces to an independent quadratic mortality term
for each PFT. With f ¼ 0, competitive exclusion is strong
because of the dependence on the mean concentration,
P : In order to compare the results with and without
B
competitive exclusion, we conducted simulations with
f ¼ 0 and f ¼ 1, with both models. We quantified the
simulated size diversity using a continuous diversity
index, here denoted h, which was estimated for the ensemble of discrete PFTs as described by Quintana et al.
(Quintana et al., 2008). They denoted this quantity m, but
we have changed the notation to avoid confusion with
the growth rate.
Time-series observation sites
Fig. 2. Locations of time-series observation sites, subarctic station K2
and subtropical station S1, in the North Pacific.
We applied the models to subarctic station K2 (478N,
1608E, water depth 5300 m) and subtropical station S1
(308N, 1458E, water depth 5800 m), both of which are
maintained by the Japan Agency for Marine-Earth
Science and Technology (JAMSTEC). Between 2010
and 2013, observations were conducted of plankton and
biogeochemistry in order to allow comparative studies of
the lower trophic ecosystems and biological pump, and
thus to inform modelling studies of how biological activity and material cycles may change in the future. At both
stations, parameters such as nutrients, chl, primary
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Table I: Values of model parameters and size-scaling factors
Parameters
Value
Units
Description (reference)
aC
Q C
am
m
^ 0
aQ
Q 0
aA
A0
aV
V0
aI
aI
chl
RM
z chl
zN
Ea
K2
Mmax
S1
Mmax
vs
kd
km
2.8
0.0180
0.0
5.0
20.18
0.039
20.80
0.15
0.2
5.0
0.0
1.0
0.1
0.8
0.6
4.8 104
1.0
1.0
10
0.50
0.1
–
pmol C cell21
Size-scaling exponent for carbon content (Menden-Deuer and Lessard, 2000)
Carbon content per cell (Menden-Deuer and Lessard, 2000)
Size-scaling exponent for maximum growth rate (null hypothesis)
Potential maximum growth rate (Pahlow et al., 2013)
Size-scaling exponent for cell quota (Edwards et al., 2012)
Minimum (subsistence) cell quota (Edwards et al., 2012)
Size-scaling exponent for affinity (heuristic assumption)
Potential maximum nutrient affinity (Pahlow et al., 2013)
Size-scaling exponent for maximum uptake rate (Marañón et al., 2013)
Potential maximum uptake rate (Pahlow et al., 2013)
Size-scaling exponent for chl-specific initial slope
Chl-specific initial slope of growth versus irradiance (Pahlow et al., 2013)
Loss rate of chlorophyll (Pahlow et al., 2013)
Cost of chlorophyll synthesis (Pahlow et al., 2013)
Cost of nitrogen assimilation (Pahlow et al., 2013)
Energy of activation
Mortality rate coefficient at station K2
Mortality rate coefficient at station S1
Sinking velocity of detritus
Degradation (remineralization) rate of detritus
Mixing rate at bottom of mixed layer (Spitz et al., 2001)
day21
–
mol N (mol C)21
–
m3 (mmol C)21 day21
–
mol N (mol C)21 day21
–
m2 E21 mol (g chl)21
day21
mol C (g chl)21
mol C (mol N)21
J mol21
m3 (mmol C)21 day21
m3 (mmol C)21 day21
m day21
day21
day21
Pre-exponential factors, denoted by asterisks, apply at l ¼ 0 (ESD ¼ 1 mm), from reported size-scaling relationships (Edwards et al., 2012) or from Wirtz
(Wirtz, 2013). Other rates and parameters are based on the values of Pahlow et al. (Pahlow et al., 2013) and Wirtz (Wirtz, 2013). Quantities reported on a
per cell basis were converted to per mol C basis using reported values of C content per cell (Menden-Deuer and Lessard, 2000). Distinct values of the
mortality rate for phytoplankton were fitted (Table II) for each station, respectively.
Table II: Fitted values of model parameters
Ensemble mean [+SD]
Parameter
Initial estimate
FlexPFT
Control model
Units
^ 0
m
aI
A0
V0
Q 0
chl
RM
K2
Mmax
S1
Mmax
5.0
1.0
0.15
5.0
0.039
0.10
1.0
1.0
0.6
5.2 [+0.6]
0.24 [+0.05]
0.11 [+0.02]
5.0 [+0.6]
0.037 [+0.0008]
0.10 [+0.01]
0.50 [+0.08]
0.11 [+0.02]
NA
5.6 [+0.5]
1.1 [+0.1]
0.085 [+0.02]
1.6 [+0.17]
0.057 [+0.0006]
0.10 [+0.01]
0.67 [+0.1]
0.092 [+0.01]
0.59 [+0.04]
day21
m2 E21 mol (g chl)21
m3 (mmol C)21 day21
mol N (mol C)21 day21
mol N (mol C)21
day21
m3 (mmol C)21 day21
m3 (mmol C)21 day21
g chl (mol C)21
^
Q
For each model, the Adaptive Metropolis algorithm was run for an ensemble of 2 105 simulations and convergence was verified. Statistics of the fits
overwhelmingly favoured the FlexPFT. Ensemble mean log likelihoods were 653 for the FlexPFT model and 589 for the control model, giving Akaike
weights (Anderson et al., 2000) of .0.999 for the FlexPFT and ,0.001 for the control model.
productivity and settling particles were observed seasonally by research vessels and mooring systems (Honda
et al. submitted to Journal of Oceanography and references
therein). Data used herein from observations of nutrients,
chl and primary production (PP) are available online
at http://ebcrpa.jamstec.go.jp/k2s1/en/. Forcing for
the model consisted of daily averaged PAR from the
MODIS satellite product. Temperatures and mixed layer
depths were based on temperature and salinity observed
by autonomous ARGO floats (http://apdrc.soest.hawaii
.edu/argo/). The mixed layer was calculated as the
depth at which the gradient of potential density reached
0.03 (kg m23) m21.
R E S U LT S
Flexible response
The FlexPFT and control were fitted simultaneously to
the data from both stations, resulting in a single parameter set for each model (Table II). By dynamically allocating resources to maximize growth rate, the FlexPFT
maintains a faster growth rate as either light or nutrient
becomes limiting, compared with the control (Fig. 3).
Resource allocation in the FlexPFT model depends on
the ambient light and nutrient environment, shifting
more toward nutrient uptake rather than carbon assimilation at subtropical station S1, and vice versa at subarctic
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Fig. 3. Growth rate (A) versus nutrient concentration, for different light levels (denoted by line thickness), and (B) versus light level for different
nutrient concentrations (denoted by line thickness), for the flexible model, FlexPFT (orange lines) in which the allocation of internal resources is
calculated dynamically to optimize growth rate, and for the control model (grey lines), in which the allocation of resources is fixed, giving lower
growth rates as either resource becomes limiting. Values of model parameters are for a cell size of 1 mm (Table I), and for the control model
^ ¼ 0.7 g chl (mol C)21.
fA ¼ 0.5, fV ¼ 0.25 and Q
Fig. 4. Modelled values of optimal fractional resource allocation for the optimal uptake trade-off (left column) and the optimal growth trade-off
(right column) as calculated for the FlexPFT model at subarctic station K2 (A and B, top row) and subtropical station S1 (C and D, bottom row), for
the 4 years modelled. Values on the horizontal axes increase to the left. Dynamic resource allocation (orange lines) shifts to higher (lower) values of
both allocation factors during summer (winter) at each station, and their means are higher at nutrient-poor, high-light station S1, compared with the
relatively high-nutrient, lower light station K2. Constant fractional allocations (grey vertical lines) were assumed for the control model.
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Fig. 5. Model results for subarctic station K2 with the FlexPFT (orange lines), the control (grey lines) and data (dots) from the time-series
observations averaged over the mixed layer depth (A through C only). (A) Primary production, (B) dissolved inorganic nitrogen (DIN), (C)
chlorophyll, (D) chl:C ratio, (E) phytoplankton biomass and (F) N-cell quota.
station K2 (Fig. 4). Furthermore, the FlexPFT responds
dynamically to seasonal changes, whereas resource allocation is fixed in the control. At subarctic station K2
(Fig. 5), the FlexPFT model reproduces better the magnitude and timing of PP and chl, compared with the
control. chl is also much more variable in the FlexPFT
model, compared with the control, because of the active
regulation of chl content, Equation (8). The simulated
patterns are more similar at subtropical station S1
(Fig. 6). Simulated values over the ensemble range for the
FlexPFT agree better with the observations, compared
with the control (Fig. 7).
The FlexPFT model reproduces better the seasonality
and magnitude of phytoplankton production at these two
contrasting locations, even though one more degree of
freedom was allowed in fitting the control (nine parameters fitted) versus the FlexPFT (eight parameters
fitted). Whereas the FlexPFT model calculates the chl:C
^ which
ratio dynamically, the control assumes constant Q;
was fitted as a free parameter. Thus, even in the control
model, the whole-cell chl:C ratio varies with the cell
quota, Q , via Equation (10). This allows some variation
of the overall chl content in the control model, although
much less than for the FlexPFT model.
Size-scalings for effective Monod
parameters
Size-scalings of effective Monod parameters are obtained
by substituting the size-scalings of trait values into
Equations (27), (28), (29) and for the growth-based affinity, Equation (32). Figure 8 shows the resulting curves,
based on the FlexPFT model, for growth rate versus DIN
concentration and the effective values of Monod
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Fig. 6. Model results for subtropical station S1 (same notation as Fig. 4).
parameters for different cell sizes, at different levels of
light limitation. The effective maximum growth rate is
greatest at some intermediate size, and larger sizes are
favoured at higher ambient nutrient concentrations
(Fig. 8A and B). Moreover, compared with reported halfsaturation values for uptake, half-saturation values for
growth are consistently lower and tend to increase less
steeply with cell size (Fig. 8C and D). This same relationship is also clear in terms of the growth-based affinity,
which decreases with both ambient nutrient concentration and cell size (Fig. 8E and F).
Comparison of modelled size diversity
For the simulations without competitive exclusion ( f ¼ 1),
model output (not shown) was similar to that from the
single PFT models, after increasing the mortality rate coefficient, Mmax by a factor of 200 (to account for the fact
that the biomass was initially evenly divided among the
200 PFTs). With competitive exclusion, results were also
similar (not shown) to the single PFT models, using the
same value of Mmax (Table II). Although the overall simulated response was similar to the single PFT models,
either with or without competitive exclusion, the simulated log-mean size and size diversity did depend strongly
on whether or not competitive exclusion was modelled
(Fig. 9). Both models reached a greater log-mean size at
Station K2 than at Station S1, as expected given the advantage of large (vs. small) cells under high (vs. low) nutrient conditions.
The FlexPFT maintained greater size diversity overall in
the simulations without competitive exclusion (Fig. 9C and
D), particularly during summer through autumn, when the
control lost more size diversity under low-nutrient conditions. The control model tended to lose diversity faster over
the 3-year period, both with (Fig. 9C and D) and without
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Fig. 7. Comparison of the ensemble mean simulated values (circles) on the vertical (FlexPFT in orange, control in grey) versus the corresponding
observed values on the horizontal for dissolved inorganic nitrogen, DIN (A and B), chlorophyll (C and D) and primary production, PP (E and F) at
stns. K2 (left column) and S1 (right column). Observed values are averages within the mixed layer, the same as shown in Figs 3 and 4. Solid diagonal
lines show the 1:1 relationship. RMSEs of the ensemble means versus observations are also shown in each panel. Vertical lines (with slight
horizontal offset for the two models) centred on each circle show the 90% quantile range from the adaptive metropolis ensemble of simulations,
which reveals that the FlexPFT comes closer to covering the observations.
(Fig. 9G and H) competitive exclusion, and its size diversity
was more sensitive to seasonal cycles at nutrient-rich station
K2. With competitive exclusion, both models lost size diversity over time as expected (Fig. 9G and H) although at both
stations, the flexible model maintained somewhat greater
diversity compared with the control. At nutrient-rich station
K2, the median size was smaller by approximately a factor
of 2 for the FlexPFT compared with the control.
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Fig. 8. Size-scalings of effective Monod parameters for growth, as a function of cell size (ESD for equivalent spherical diameter) as simulated for
different ambient nutrient concentrations (N), under light-limited (left) and light replete (right) conditions. Effective maximum growth rate (A and
B), half-saturation values for growth (Km) and uptake (KV), respectively, with data for the latter from laboratory experiments (open circles)
as compiled by Edwards et al. (Edwards et al., 2012) (C and D), and modelled growth-based affinity and affinity for nutrient uptake, respectively
(E and F).
DISCUSSION
Although trait-based models and size-scalings of traits
are currently much discussed in plankton ecology, few
modelling studies have actually applied such scalings and
tested model output against oceanographic observations.
Among these, Verdy et al. (Verdy et al., 2009) applied empirical allometric scalings within their model, which is a
valuable step for assessing the implications of the
size-scalings based on laboratory studies. However, their
empirical model did not account for either intra- or interspecific flexible responses. Ward et al. (Ward et al., 2013)
went further by applying empirical allometry relations in
a global scale model, thus accounting for variations in
cell quotas but not for other flexible resource allocations.
Wirtz (Wirtz, 2013) developed a different model, which
allows for flexibility via multiple trade-offs, to investigate
niche formation in plankton communities. Most recently,
Acevedo-Trejos et al. (Acevedo-Trejos et al., 2014) applied a
size-based model to capture a glimpse of the future composition of phytoplankton communities in two contrasting
regions of the Atlantic Ocean under different climate
change scenarios, and Terseleer et al. (Terseleer et al., 2014)
tested a size-based model of diatoms against observations
from a eutrophic coastal area of the North Sea.
Our new FlexPFT model accounts for the flexible response of phytoplankton subject to two eco-physiological
trade-offs and can easily be combined with empirically
based allometric relations for traits. Its optimality-based
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Fig. 9. Median size, exp(l ), where l is the mean loge(ESD) and size diversity, h (Quintana et al., 2008) for simulations with 200 phytoplankton
PFTs with size-scaling of traits (Table I and Supplementary Material online, Appendix SB) for stns. K2 (left) and S1 (right). Initial biomass was
evenly divided among size classes. Without competitive exclusion (A–D), median size is similar for the FlexPFT (black lines) has greater median
size at stn. S1 compared to the inflexible control model (dashed grey lines), and the FlexPFT maintains greater size diversity at both sites. With
competitive exclusion (E–H), the control has greater median size at stn. K2, the FlexPFT maintains greater size diversity than the inflexible control at both sites, and size diversity declines over time in all cases.
formulation reproduces better the pattern of blooming at
the two contrasting time-series sites (Figs 5 and 6) compared with the control model. Previous modelling studies
(Pahlow, 2005; Wirtz and Pahlow, 2010; Pahlow et al.,
2013) have found that accounting for optimal intracellular resource allocation results in better agreement with laboratory experiments. The FlexPFT model is simple
enough to be applied in large-scale models of the ocean,
and would provide a means of accounting for the flexible
response of phytoplankton under changing environmental conditions. This model also considers the energetic
costs of synthesizing and maintaining chl and associated
biomolecules. By coupling this FlexPFT model with
physical models of the ocean, future studies can examine
how the flexible response itself, as well as different
assumptions about size-scalings of key traits, contribute
to determining biogeographical patterns of phytoplankton growth and community composition.
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Idealized simulations applying size-scalings for key traits
have revealed differences in median size and size diversity
for the FlexPFT compared with the control. This suggests
that including the flexible response of individual PFTs in
multispecies models may indeed produce different results
in terms of size diversity, compared with existing models in
which the individual PFTs do not acclimate to changing
environmental conditions (e.g. Ward et al., 2013). We compared results with and without competitive exclusion using
an implicit formulation in terms of mortality (Record et al.,
2013) as a simplistic way to represent the maintenance of
diversity by the “Kill-the-Winner” (KTW) grazing response (Vallina et al., 2014b). Accounting for the flexible
response of both predator and prey in a theoretical model,
Mougi (Mougi, 2012) found quite different dynamics
compared with inflexible models. Future studies should
therefore test the effects of the flexible response of phytoplankton in combination with an explicit model of the
KTW grazing response by zooplankton, which will likely
yield different dynamics and greater diversity (Vallina
et al., 2014a).
Our study only accounts for environmental variability
by imposing time-dependent forcing for mixed layer
depth, light, temperature and nutrient concentration.
Future studies using spatially explicit physical –ecological
models will be able to better examine the effects of environmental variability and transport on size diversity.
Furthermore, Mandal et al. (Mandal et al., 2014) have recently shown that accounting for realistic spatial variability at the mm scale leads to quite different response of a
typical inflexible phytoplankton PFT, compared with the
usual assumption of uniform concentrations of nutrient
and phytoplankton within each grid cell. Future studies
should also consider eco-physiological flexibility and environmental variability (at various scales) in combination,
given that the former is the result of natural selection
subject to the latter.
Our new and relatively simple FlexPFT model reproduces better the magnitude and seasonality of production
and chl observed at two contrasting time-series stations in
the North Pacific, compared with the control. This new
model can be applied as a single PFT without size-scaling
of traits (as in our application to the two time-series sites)
or to model size-structured communities. The latter can
be done either by assigning different trait values to PFTs
representing discretely resolved cell sizes (Ward et al.,
2013) or in an “adaptive dynamics” framework representing a continuous distribution of cell sizes and associated trait values (Merico et al., 2009). The new model
presented herein provides a framework for testing the
overall effect of allometric trait scalings based on laboratory experiments on phytoplankton growth and size diversity. This modelling framework, coupled with physical
ocean models, can help to resolve some of the considerable uncertainty that remains regarding size-scalings of
traits and their combined net effects on growth (Wirtz,
2013; Marañón et al., 2013), by providing testable predictions for comparison with oceanic observations and
further laboratory studies.
S U P P L E M E N TA RY DATA
Supplementary data can be found online at http://plankt.
oxfordjournals.org.
AC K N OW L E D G E M E N T S
We thank Sergio Vallina for helpful suggestions concerning the description of the model and results, and also the
two anonymous reviewers whose comments helped us to
substantially improve the manuscript. We gratefully acknowledge funding from a CREST Project (PI: SLS) provided by the Japan Science and Technology Agency.
FUNDING
This work was supported by funding from a CREST
Project (PI: SLS) provided by the Japan Science and
Technology Agency. The work of MP was supported by
Deutshe Forschungsgemeinschaft (DFG) project SFB754
(Sonderforschungsbereich 754 ‘Climate-Biogeochemistry
Interactions in the Tropical Ocean’, www.sfb754.de),
and that of AM and EA-T was supported by DFG priority program DynaTrait, (Schwerpunktprogramm 1704,
subproject 19).
REFERENCES
Acevedo-Trejos, E., Brandt, G., Steinacher, M. and Merico, A. (2014) A
glimpse into the future composition of marine phytoplankton communities. Front. Mar. Sci., 1, doi:10.3389/fmars.2014.00015.
Anderson, D. R., Burnham, K. P. and Thompson, W. L. (2000) Null hypothesis testing: problems, prevalence, and an alternative. J. Wildl.
Manage., 64, 912– 923.
Burmaster, D. E. (1979) The continuous culture of phytoplankton:
mathematical equivalence among three steady-state models. Am. Nat.,
113, 123– 134.
Button, D. K. (1978) On the theory of control of microbial growth
kinetics by limiting nutrient concentration. Deep Sea Res., 25,
1163–1177.
Caperon, J. (1968) Population growth response of Iscochrysis galbana to
nitrate variation at limiting concentrations. Ecology, 49, 866–872.
Droop, M. R. (1968) Vitamin B12 and marine ecology. 4. The kinetics
of uptake, growth and inhibition of Monochrysis lutheri. J. Mar. Biol.
Assoc. UK, 48, 689 –733.
Dugdale, R. C. (1967) Nutrient limitation in the sea: dynamics, identification, and significance. Limnol. Oceanogr., 12, 685 –695.
991
JOURNAL OF PLANKTON RESEARCH
j
VOLUME 
j
NUMBER 
Edwards, K. F., Thomas, M. K., Klausmeier, C. A. and Litchman, E.
(2012) Allometric scaling and taxonomic variation in nutrient utilization traits and maximum growth rate of phytoplankton. Limnol.
Oceanogr., 57, 554– 566.
Fiksen, Ø., Follows, M. J. and Aksnes, D. L. (2013) Trait-based models
of nutrient uptake in microbes extend the Michaelis– Menten framework. Limnol. Oceanogr., 58, 193– 202.
Flynn, K. J. (2003) Modeling multi-nutrient interactions in phytoplankton: balancing simplicity and realism. Progr. Oceanogr., 56, 249–279.
Follows, M. J. and Dutkiewicz, S. (2011) Modeling diverse communities
of marine microbes. Ann. Rev. Mar. Sci., 3, 427–451.
Fussmann, G. F., Ellner, S. P., Nelson, G., Hairston, J., Jones, L. E.,
Shertzer, K. W. and Yoshida, T. (2005) Ecological and evolutionary
dynamics of experimental plankton communities. Adv. Ecol. Res., 37,
221–243.
j
PAGES
 –  
j 
method for the measurement of size diversity with emphasis on data
standardization. Limnol. Oceanogr. Methods, 6, 75–86.
Record, N. R., Pershing, A. J. and Maps, F. (2013) Food for thought: the
paradox of the paradox of the plankton. ICES J. Mar. Sci.,
doi:10.1093/icesjms/fst049.
Smith, S. L. (2011) Consistently modeling the combined effects of temperature and concentration on nitrate uptake in the ocean. J. Geophys.
Res., 116, G04020.
Smith, S. L., Merico, A., Hohn, S. and Brandt, G. (2014a) Sizing-up
nutrient uptake kinetics: combining a physiological trade-off with size
scaling of phytoplankton traits. Mar. Ecol. Prog. Ser., 511, 33– 39.
Smith, S. L., Merico, A., Wirtz, K. W. and Pahlow, M. (2014b) Leaving
misleading legacies behind in plankton ecosystem modelling.
J. Plankton Res., 36, 613–620.
Healey, F. P. (1980) Slope of the Monod equation as an indicator of advantage in nutrient competition. Microb. Ecol., 5, 281–286.
Smith, S. L., Pahlow, M., Merico, A. and Wirtz, K. W. (2011)
Optimality-based modeling of planktonic organisms. Limnol.
Oceanogr., 56, 2080– 2094.
Litchman, E., Klausmeier, C. A., Schofield, O. M. and Falkowski, P. G.
(2007) The role of functional traits and trade-offs in structuring
phytoplankton communities: scaling from cellular to ecosystem level.
Ecol. Lett., 10, 1170–1181.
Smith, S. L., Yamanaka, Y., Pahlow, M. and Oschlies, A. (2009)
Optimal uptake kinetics: physiological acclimation explains the
pattern of nitrate uptake by phytoplankton in the ocean. Mar. Ecol.
Prog. Ser., 384, 1 –12.
Mandal, S., Locke, C., Tanaka, M. and Yamazaki, H. (2014)
Observations and models of highly intermittent phytoplankton distributions. PLoS One, 9, e94797.
Spitz, Y. H., Moisan, J. R. and Abbott, M. R. (2001) Configuring an
ecosystem model using data from the Bermuda Atlantic Time Series
(BATS). Deep Sea Res. II, 48, 1733– 1768.
Marañón, E., Cermeño, P., López-Sandoval, D. C., Rodriguez-Ramos,
T., Sobrino, C., Huete-Ortega, M., Blanco, J. M. and Rodriguez, J.
(2013) Unimodal size scaling of phytoplankton growth and the size
dependence of nutrient uptake and use. Ecol. Lett., 16, 371–379.
Terseleer, N., Bruggeman, J., Lancelot, C. and Gypens, N. (2014)
Trait-based representation of diatom functional diversity in a plankton functional type model of the eutrophied Southern North Sea.
Limnol. Oceanogr., 59, doi:10.4319/lo.2014.59.6.0000.
Menden-Deuer, S. and Lessard, E. J. (2000) Carbon to volume relationships for dinoflagellates, diatoms, and other protist plankton. Limnol.
Oceanogr., 45, 569– 579.
Vallina, S. M., Follows, M. J., Dutkiewicz, S., Montoya, J. M.,
Cermeno, P. and Loreau, M. (2014a) Global relationship between
phytoplankton diversity and productivity in the ocean. Nat. Commun.,
5, 4299.
Merico, A., Bruggeman, J. and Wirtz, K. (2009) A trait-based approach
for downscaling complexity in plankton ecosystem models. Ecol.
Model., 220, 3001–3010.
Michaelis, L. and Menten, M. M. (1913) Die kinetik der invertinwirkung. Biochem. Zeitung, 49, 333–369.
Morel, F. M. M. (1987) Kinetics of nutrient uptake and growth in phytoplankton. J. Phycol., 23, 137 –150.
Mougi, A. (2012) Unusual predator-prey dynamics under reciprocal
phenotypic plasticity. J. Theoret. Biol., 305, 96–102.
Pahlow, M. (2005) Linking chlorophyll-nutrient dynamic to the Redfield
N:C ratio with a model of optimal phytoplankton growth. Mar. Ecol.
Prog. Ser., 287, 33– 43.
Pahlow, M., Dietz, H. and Oschlies, A. (2013) Optimality-based model
of phytoplankton growth and diazotrophy. Mar. Ecol. Prog. Ser., 489,
1–16.
Pahlow, M. and Oschlies, A. (2013) Optimal allocation backs droop s
cell-quota model. Mar. Ecol. Prog. Ser., 473, 1– 5.
Quintana, X. D., Brucet, S., Boix, D., Lopez-Flores, R., Gascon, S.,
Badosa, A., Sala, J., Moreno-Amich, R. et al. (2008) A nonparametric
992
Vallina, S. M., Ward, B. A., Dutkiewicz, S. and Follows, M. J.
(2014b) Maximal feeding with active prey switching: a kill-the-winner
functional response and its effect on global diversity and biogeography. Prog. Oceanogr., 120, doi:10.1016/j.pocean.2013.08.001.
Verdy, A., Follows, M. and Flierl, G. (2009) Optimal phytoplankton cell
size in an allometric model. Mar. Ecol. Prog. Ser., 379, 1– 12.
Ward, B. A., Dutkiewicz, S. and Follows, M. J. (2013) Modelling spatial
and temporal patters in size-structured marine plankton communities: top-down and bottom-up controls. J. Plankton Res., 36, 31– 47.
Wirtz, K. W. (2013) Mechanistic origins of variability in phytoplankton
dynamics: part i: niche formation revealed by a size-based model.
Mar. Biol., doi:10.1007/s00227-012-2163-7.
Wirtz, K. W. and Pahlow, M. (2010) Dynamic chlorophyll and nitrogen:carbon regulation in algae optimizes instantaneous growth rate.
Mar. Ecol. Prog. Ser., 402, 81– 96.
Yoshida, T., Jones, L. E., Ellner, S. P., Fussmann, G. F. and Hairston, N.
G. (2003) Rapid evolution drives ecological dynamics in a predatorprey system. Nature, 424, 303–306.