GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-9-4071-2016PhytoSFDM version 1.0.0: Phytoplankton Size and Functional Diversity ModelAcevedo-TrejosEstebanesteban.acevedo@leibniz-zmt.dehttps://orcid.org/0000-0003-4222-7062BrandtGunnarSmithS. LanMericoAgostinohttps://orcid.org/0000-0001-8095-8056Systems Ecology Group, Leibniz Center for Tropical Marine Ecology, Fahrenheitstrasse 6, 28359 Bremen, GermanyMarine Ecosystem Dynamics Research Group, Research and Development Center for Global Change, Japan Agency for Marine-Earth Science and Technology, Yokohama, JapanFaculty of Physics & Earth Sciences, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germanycurrent address: Brockmann Consult, Max-Planck-Str. 7, 21052 Geesthacht, GermanyEsteban Acevedo-Trejos (esteban.acevedo@leibniz-zmt.de)14November20169114071408528April201630May201630September201621October2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/9/4071/2016/gmd-9-4071-2016.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/9/4071/2016/gmd-9-4071-2016.pdf
Biodiversity is one of the key mechanisms that facilitate the adaptive
response of planktonic communities to a fluctuating environment. How to allow
for such a flexible response in marine ecosystem models is, however, not
entirely clear. One particular way is to resolve the natural complexity of
phytoplankton communities by explicitly incorporating a large number of
species or plankton functional types. Alternatively, models of aggregate
community properties focus on macroecological quantities such as total
biomass, mean trait, and trait variance (or functional trait diversity), thus
reducing the observed natural complexity to a few mathematical expressions.
We developed the PhytoSFDM modelling tool, which can resolve species discretely and can capture aggregate
community properties. The tool also provides a set of methods for treating
diversity under realistic oceanographic settings. This model is coded in
Python and is distributed as open-source software. PhytoSFDM is implemented
in a zero-dimensional physical scheme and can be applied to any location of
the global ocean. We show that aggregate community models reduce computational
complexity while preserving relevant macroecological features of
phytoplankton communities. Compared to species-explicit models, aggregate
models are more manageable in terms of number of equations and have faster
computational times. Further developments of this tool should address the
caveats associated with the assumptions of aggregate community models and
about implementations into spatially resolved physical settings
(one-dimensional and three-dimensional). With PhytoSFDM we embrace the idea
of promoting open-source software and encourage scientists to build on this
modelling tool to further improve our understanding of the role that
biodiversity plays in shaping marine ecosystems.
Introduction
Numerical models are simplified abstractions of complex phenomena. They are
engineered for the problem at hand and cannot be designed to maximize
simultaneously the three key requirements of generality, precision, and
realism, because one of these must be sacrificed in favour of the other two
. Marine ecosystem models are no exceptions, and the
scientific community has questioned the trend towards increasing model
complexity in terms of large numbers of state variables and parameters
. Alternatives such as
trait-based models have been put forward as a way to simplify overly
parameterized ecosystem models .
In the past 2 decades, trait-based models of planktonic ecosystems have
become important tools for elucidating the fundamental mechanisms behind
emergent patterns of community structure and diversity. Most of these models
describe the phytoplankton community by a discrete representation of
many species or functional groups . Alternatively, models
have been developed that treat the whole phytoplankton species assemblage as
a single entity (; ;
; ; ;
; ; ).
These models use aggregate community properties such as total
biomass, mean trait, and trait variance to describe changes in phytoplankton
community composition. Hence, by approximating the full spectrum of species
or functional types with just a few macroecological properties, these models
present a way of reducing the complexity of natural communities .
The simplification of both types of trait-based models (i.e. discrete and
aggregate) relies on the use of a key trait, for which relationships with
other traits can be formulated. Cell size is recognized as one of the most
important traits for characterizing phytoplankton communities
, and it has been
commonly used in plankton ecosystem models
.
This morphological trait affects trophic organization of foodwebs and the
sequestration of CO2 into the ocean interior .
Phytoplankton size also impacts on many ecological and physiological
functions and is linked to other relevant traits via trade-off relationships
see reviews by. Therefore,
studies on how cell size is associated with ecological and physiological
processes and on the impact that these associations have on the structure and
functioning of planktonic communities are of fundamental importance
.
Here we present the new Phytoplankton Size and Functional Diversity Model
(called PhytoSFDM) that allows for five different ways of describing the size
composition of phytoplankton communities in the upper mixed layers of the
world oceans. In the first variant, the phytoplankton community is described
according to the classical approach that resolves the discrete assemblage of
many different species and then we present four alternative ways of
expressing aggregate community properties of phytoplankton based on four
different ways of treating size diversity. We provide this model as
open-source so that it can be used, modified, and redistributed freely with
the aims of fostering reproducibility and encouraging investigations about
the impact of environmental conditions on properties of phytoplankton
community structure and diversity.
Model description
PhytoSFDM is developed from the study of , which
used a size-based model of aggregate community properties to investigate the
phytoplankton size structure and size diversity in two environmentally
contrasting regions of the Atlantic Ocean. In this model, the phytoplankton
community self-assembles according to a trade-off emerging from relationships
between cell size and (1) nitrogen uptake, (2) zooplankton grazing, and
(3) phytoplankton sinking. In PhytoSFDM we have extended this work by
providing four ways of treating size diversity using a moment-based
approximation seeand Sect. 2.1.3 in this
study. In addition, we include a discrete version
of the model (hereafter referred to as the full model) to better illustrate
the potential of using aggregate models as compared to the equivalent
discrete version. In the following, we present the mathematical equations, a
description of the code structure, and easy-to-follow examples of how to use
the model.
Mathematical formulationsMixed-layer scheme
The zero-dimensional physical set-up consists of two vertical layers, the
upper mixed layer containing the pelagic ecosystem and the abiotic bottom
layer with nitrogen concentration as forcing. Following and
, we describe material exchange between the two layers
(K) as a function of the mixed-layer depth (M),
K=κ+h+(t)M(t),
where κ is a constant that parameterizes diffusive mixing across the
thermocline and h+(t) is a function that describes entrainment and
detrainment of material. The latter is given by h+(t)=max[h(t),0],
with h(t)=dM(t)/dt.
Zooplankton are considered capable of maintaining themselves within the upper
mixed layer; thus, their mixing term simplifies to KZ=h(t).
Dynamics of the full phytoplankton community
The description of the phytoplankton community is a trait-based variant of
the classical nutrient–phytoplankton–zooplankton–detritus (NPZD) model
. We consider only one nutrient, nitrogen, which
constitutes the currency of our model, one zooplankton population (composed
of individuals assumed to be identical), and a single detritus pool. We
define n morphologically distinct phytoplankton types (hereafter referred
to as morphotypes), and we consider n equal to either 10 or 100. Each
morphotype is characterized by a
biomass Pi and a cell size Si, in units of µm equivalent
spherical diameter (ESD). The distribution of biomass along the size
dimension is known to be positively skewed (i.e. an asymmetrical size
distribution with a pronounced right tail compared to its left tail), due to
physiological, morphological, and ecological constraints that limit
phytoplankton from a minimum size of around 0.15 µm ESD to a
maximum size of about 575 µm ESD .
Consequently, we assume a log-normal distribution of size to represent the
size of each morphotype, thus transforming the cell size Si as follows:
Li=ln(Si). The net growth rate of the whole phytoplankton community (P)
is then given by
dPdt=∑i=1nfi(Li,E)⋅Pi,
where fi(Li,E) is the net growth rate of size class i, which we assume
to be a proxy for fitness . Hence fi accounts for the
gains and losses of each morphotype as a function of cell size Li and
environment E. The latter includes changes in nitrogen, irradiance,
temperature, and grazing. The equation describing the fitness functions of
each size class i is thus given by
fi=μP⋅F(T)⋅H(I)⋅U(Li,N)-μZ⋅G(Li,Pi)⋅Z-V(Li,M)-mP-K,
where μP indicates the maximum growth rate and F(T)=e0.063⋅T is Eppley's formulation for temperature-dependent growth
. The light-limiting term, H(I), represents the total
light I available in the upper mixed layer. According to Steele's
formulation ,
H(I)=1M(t)∫0MI(z)Is⋅e1-I(z)Isdz,
where Is is the light level at which photosynthesis saturates and I(z)
is the irradiance at depth z. The exponential decay of light with depth is
computed according to the Beer–Lambert law with a generic extinction
coefficient kw:
I(z)=I0⋅e-kw⋅z.
The current version of our model does not specify any size dependence for
light absorption, although we provided suggestions on how this could be done
(Sects. 4 and 6).
The nutrient-limiting term U in Eq. () is determined by a Monod
function with a half-saturation constant KN, which scales
allometrically with phytoplankton cell size L,
U(Li,N)=NN+KN=NN+(βU⋅eLi⋅αU),
with βU and αU, respectively, intercept and slope of the
KN allometric function (i.e. the power law βU⋅SαU). This empirical relationship is based on observations of
different phytoplankton groups see Fig. 3b in, with
the regression parameters rescaled from cell volume to ESD.
The loss term G(Li,Pi) in Eq. () represents zooplankton
grazing. As mentioned above, here we consider a single zooplankton
population, which is assumed to be an assemblage of identical individuals
with a size-selective feeding preference given by
G(Li,Pi)=eLi⋅αG∑i=1nPi⋅eLi⋅αG+KP,
where αG is the slope for size-dependent grazing (or the power law
1⋅SαG) and KP is the half-saturation
constant. This formulation is inspired by meta-analyses of laboratory data
and reflects a grazing preference of
zooplankton for smaller phytoplankton cells. For demonstration purposes, we
use here a simple formulation for zooplankton grazing; however, other
functional relationships can be implemented and tested in future versions of
PhytoSFDM (see also Sects. 4 and 6).
The loss term V(Li,M) in Eq. () represents the sinking of
phytoplankton as a function of size and depth of the mixed layer,
V(Li,M)=βV⋅eLi⋅αVM(t),
where the constants αV and βV are the parameters of the
function relating phytoplankton cell size to sinking velocity according to
Stokes' law or the power law βV⋅SαV. These parameters are expressed here in units of metres per
day.
Our model formulation does not specify an explicit size dependence for the
phytoplankton maximum growth rate (μP). Various compilations of
data from laboratory experiments reveal different size scalings for
μP, either as a power law of cell volume
or as a unimodal distribution in terms of
cell size . Therefore, we adopted an
approach similar to that of , who reproduced the unimodal
distribution of realized growth rate over size using two physiological
trade-offs. We specified our trade-off in terms of three allometric
relationships, and this results in an indirect size dependence of
phytoplankton growth rate.
Parameter definitions, their units, and their default values as
provided in PhytoSFDM.
DefinitionSymbol (units)ValueDiffusive mixing across the thermoclineκ (m day-1)0.1Light attenuation constantkw (m-1)0.1Optimum irradianceIs (E m-2 day-1)30P max growth rateμP (day-1)1.5P mortality ratemP (day-1)0.05Z grazing rateμZ (day-1)1.35Z mortality ratemZ (day-1)0.3P half-saturationKP (mmol N m-3)0.1P assimilation coefficientδZ (–)0.31Mineralization rateδD (day-1)0.1Immigration rateδI (mmol N day-1)0.008Trait diffusivity parameterν (–)0.008Slope for allometric grazer preferenceαG ([µm ESD]-1)-0.75Intercept of the KN allometric functionβU (mmol N m-3)0.14257Slope of the KN allometric functionαU (mmol N m-3 [µm ESD]-1)0.81Intercept of the V allometric functionβV (m day-1)0.01989Slope of the V allometric functionαV (m day-1 [µm ESD]-1)1.17Size variance of immigrating PV0 (Ln [µm ESD]2)0.58Number of morphotypesn (–)10 or 100
The loss term mP in Eq. () accounts for all phytoplankton
losses other than those from grazing and mixing.
Differential equations for the nutrient (N), zooplankton (Z), and
detritus (D) complete the model system:
dNdt=-∑i=1nμP⋅F(T)⋅H(I)⋅U(Li,N)⋅Pi+δD⋅D+K⋅(N0-N),dPdt=∑i=1n(μP⋅F(T)⋅H(I)⋅U(Li,N)-μZ⋅G(Li,Pi)⋅Z-V(Li,M)-mP-K)Pi,
dZdt=∑i=1nδZ⋅μZ⋅Z⋅G(Li,Pi)⋅Pi-mZ⋅Z2-KZ⋅Z,dDdt=∑i=1n(1-δZ)⋅μZ⋅Z⋅G(Li,Pi)⋅Pi+∑i=1nmP⋅Pi+mZ⋅Z2-δD⋅D-K⋅D,
where δD is the mineralization rate and N0 is the
nitrogen concentration below the upper mixed layer. μZ,
δZ, and mZ are, respectively, maximum growth
rate, prey assimilation coefficient, and mortality rate of zooplankton. All
parameter values and their units are reported in Table 1.
Dynamics of the aggregate phytoplankton community
The phytoplankton community comprising many distinct morphotypes
(Eqs. to ) can be approximated with the so-called
moment-based approach
.
, , and used a Taylor
expansion together with a moment closure technique to approximate the whole
community with three macroscopic properties, which correspond to the first
three order moments of the approximated biomass distribution. These
properties are total biomass, mean trait, and trait variance. These works
were inspired by earlier
applications in quantitative genetics and have been
reviewed by and more recently by .
Here the whole phytoplankton community is characterized by the morphological
trait cell size and by a trade-off that emerges from three allometric
relationships described by Eqs. ()–(). The equations of the
respective macroscopic properties are
dPdt≈P⋅(f+12⋅f(2)⋅V),dL‾dt≈f(1)⋅V,dVdt≈f(2)⋅V2,
where f is the net growth rate (or the fitness function; see
Eq. ) and f(n) is the nth derivative of the net growth with
respect to the trait. Due to competitive exclusion, however, the
phytoplankton community loses functional diversity over time, i.e. the
variance declines to zero with time, in both full and aggregate model
formulations . We name this standard formulation
“unsustained variance”.
Alternatively, one can use the approximated model to focus only on changes in
the mean trait, thus ignoring changes in the variance by fixing it to an
arbitrary constant value:
dPdt≈P⋅(f+12⋅f(2)⋅V),dL‾dt≈f(1)⋅V,dVdt=0.
While using these two formulations (i.e. unsustained and fixed variance) can
be acceptable in some special cases (e.g. in experiments that lead to
competitive exclusion or where diversity is being manipulated), it is clear
that they fail to account for changes in the adaptive capacity of the
community, which requires allowing the size variance, and thereby functional
diversity, to vary over time .
Within our modelling tool we also provide two alternative ways of treating
the size variance: immigration following and trait
diffusion following. The treatment with immigration
considers the introduction of biomass and new trait values from hypothetical
adjacent communities into the resident community. The addition of incoming
amount of biomass per day is named immigration I,
dPdt≈P[f+12⋅f(2)⋅V]+I,dL‾dt≈f(1)⋅V+IP(LI-L‾),dVdt≈f(2)⋅V2+IP[(VI-V)+(LI-L‾)2],
where LI and VI are, respectively, the mean size and
the size variance of the immigrating community. As implemented by
, we treat I as a density-dependent process (i.e.
I=δI⋅P), and set LI equal to the
mean size of the resident community (i.e. LI=L‾). Thus,
we assume that phytoplankton immigrating from adjacent areas are
characterized by sizes similar to the simulated community, implying that the
immigrating community has been exposed to the same selection pressures as the
simulated community . We also assume that the rate of
immigration increases proportionally to the concentration of phytoplankton,
consistent with observations of diversity patterns along the Atlantic Ocean
.
The treatment of the size variance based on trait diffusion
gives
dPdt≈P⋅f+12⋅f(2)⋅V+12⋅ν(r4⋅V-3⋅r2),dL‾dt≈f(1)⋅V+ν(r3⋅V-3⋅r1),dVdt≈f(2)⋅V2+ν(r4⋅V2-5⋅V⋅r2+2⋅r),
where ν is the trait diffusivity parameter, r is the reproduction rate
(or gross growth), and rn is the nth derivative of gross growth with
respect to the trait. Note that the process of trait diffusion (last term in
Eq. ) depends on the gross growth r, via the trait diffusivity
constant ν; thus, an increase in phytoplankton gross growth causes an
increase in trait variance .
The system of differential equations for all variance treatments is completed
by equations describing gains and losses in nitrogen (N), zooplankton
(Z), and detritus (D):
dNdt=-μP⋅F(T)⋅H(I)⋅U(L‾,N)⋅P+δD⋅D+K⋅(N0-N),dZdt=δZ⋅μZ⋅Z⋅G(L‾,P)⋅P-mZ⋅Z2-KZ⋅Z,dDdt=(1-δZ)⋅μZ⋅Z⋅G(L‾,P)⋅P+mP⋅P+mZ⋅Z2-δD⋅D-K⋅D.
The first term in Eq. () represents a reduction of the nitrogen
pool due to phytoplankton growth, which is a function of temperature, light,
nitrogen, and mean size (see the description of Eq. in the
previous section). The last two terms in Eq. () represent sources
of nitrogen due to remineralization and mixing. The first term in
Eq. () describes size-dependent grazing, while the last two terms
describe losses of zooplankton due to mortality and mixing. The first term in
Eq. () represents a fraction of phytoplankton biomass that is not
assimilated by zooplankton and the following two terms represent the
mortality of phytoplankton and zooplankton, respectively. The detritus pool
is reduced by remineralization and mixing. Parameter values and their units
are reported in Table 1.
Environmental forcing
We compiled monthly climatological forcing data for mixed-layer depth (MLD),
photosynthetic active radiation (PAR), sea surface temperature (SST), and
concentration of nitrogen immediately below the upper mixed layer (N0).
The MLD data were obtained from using the variable
density criterion and are openly accessible from
https://www.nodc.noaa.gov/OC5/WOA94/mix.html. The PAR data were
obtained from the Moderate Resolution Imaging Spectroradiometer (MODIS), for
the time period 2002–2011. This dataset is managed and distributed by the
NASA's Ocean Biology Processing Group
(http://oceancolor.gsfc.nasa.gov/cms/). SST and N0 were obtained
from the World Ocean Atlas 2009 (WOA09), which is maintained and distributed
by NOAA (https://www.nodc.noaa.gov/OC5/WOA09/pr_woa09.html). For
consistency and efficiency, all data were transformed from their original
formats (e.g. TXT and HDF) to NetCDF. All monthly forcings were spatially
averaged over the selected location (square boxes in Fig. )
and then interpolated to obtain daily values (Fig. 2).
Environmental forcing variables considered in PhytoSFDM. The data
shown are the annual average of mixed-layer depth (MLD), photosynthetic
active radiation (PAR), sea surface temperature (SST), and nitrogen
concentration below the mixed layer (N0). The square boxes mark the
location of the test-case simulation.
Test-case simulation
A test-case model configuration is provided for a location of the North
Atlantic Ocean at 47.5∘ N 15.5∘ W (Fig. ), a
region where seasonal changes in mean size and size diversity are well known
. This region presents the typical oceanographic
conditions of a temperate environment (Fig. ). The
environmental conditions produce a pronounced phytoplankton bloom in spring,
which stimulates secondary production and almost the full depletion of
nitrogen (Fig. ). Overturning of the water column in autumn
restocks the pool of nitrogen and light limitation together with lower
temperatures halt primary production (Figs. and
).
Comparison of full and aggregate models
Within PhytoSFDM, we provide a practical example of how to implement and
compare phytoplankton community models that aim to describe (a) a full
assemblage of species or morphotypes (see Sect. 2.1.2), and (b) an aggregate
community (see Sect. 2.1.3). The aggregate community model is an
approximation of the full assemblage of species or morphotypes
.
Figures and show the results of,
respectively, the full model and the aggregate model for the unsustained
variance case. N, P, Z, and D are unaffected by the type of model considered.
As expected, the dynamics of P, L‾, and V produced by the
aggregate model are good approximations of those produced by the full model.
Both models exhibit competitive exclusion, as indicated by the reduction in
the number of morphotypes and consequently in the loss of size variance over
time (Fig. ). The phytoplankton community evolves towards the
optimal trait value, which is expressed by the fittest few morphotypes for
the chosen parameterization and the prevailing environmental conditions.
Although competitive exclusion is well established theoretically
, natural communities of phytoplankton are typically very
diverse; hence, we will explore in the following the effects of different
ways of sustaining the variance.
Temporal variation of the environmental variables. The monthly
climatology data (red dots) are spatially averaged over the test location
(square boxes in Fig. 1). The interpolation (continuous line) is obtained
with a third- (MLD and PAR) and a fifth-order (SST and N0) polynomial.
Computation time in seconds for the full model with 10 and
100 morphotypes and the four variants of the aggregate model.
NPZD dynamics of the full model (Sect. 2.1.2) and of its equivalent
aggregate model (Unsustained variance, Sect. 2.1.3) for the last year of the
simulations. The total phytoplankton in the full model corresponds to the sum
of all Pn. The red dots are observations of nitrogen concentrations
(monthly data obtained from the World Ocean Atlas) and the green dots are
remotely sensed Chl-a data (8-day composite obtained from MODIS).
Number of morphotypes and size variance over the first year of the
simulation. Here we included the morphotypes with a biomass greater than
0.01 mmol N m-3. Models that do not consider a mechanism to sustain
variance exhibit competitive exclusion of morphotypes and a rapid decline of
size diversity.
Nutrient, phytoplankton, zooplankton, and detritus dynamics over a
seasonal cycle for the four variants of the aggregate model (see
Sect. 2.1.3), named unsustained and fixed variance, trait diffusion, and
immigration.
Comparison of variance treatments
The key aspect of trait-based models is their ability to describe the
phytoplankton community in terms of mean trait and trait variance.
Figures and show the results of 1-year
simulation after an initial spin-up phase of 4 years. While the four
treatments produce very similar, if not identical, dynamics for N, P, Z, and
D (Fig. ), the results for the mean size and the size
variance differ considerably among treatments (Fig. ).
As already discussed, the system loses diversity over time when variance is
unsustained. The loss of diversity reduces the capacity of the community to
adapt to changing environmental conditions via shifts in species composition,
as a flat year-round mean trait shows (Fig. , grey lines).
Under fixed variance, size diversity is locked at an arbitrary value. If this
value is high enough, the mean size can adapt in response to changes in
nutrient availability and grazing regimes (Fig. ). This
treatment can be useful for studies focusing only on the size structure of
the community, but it is otherwise based on an arbitrarily fixed level of
diversity and cannot offer meaningful insights, for example about
biodiversity and ecosystem functioning relationships.
Dynamics of the size-structured phytoplankton community and its
functional size diversity for the four variance treatments (see Sect. 2.1.3),
named unsustained and fixed variance, trait diffusion, and immigration.
Sensitivity of four variance treatments to an increase and a
decrease by 25 % in the default parameter values. The values and
definitions of all parameters are given in Table 1.
Trait diffusion and immigration show similar results for the mean size but
not for the size variance (Fig. ). Since the mechanism of
trait diffusion depends on reproduction, i.e. gross growth (see
Eq. ), the highest diversity of the community is reached in spring
under high growth rates and declines when moving towards winter. Size
diversity also peaks in spring for the case of immigration because this
mechanism is density-dependent (see Eq. ), but the variances
predicted in autumn and winter are, respectively, lower and higher than those
obtained with trait diffusion (Fig. ). As mentioned above,
this originates from the different assumptions underlying the trait diffusion
and immigration treatments, which consider, respectively, an internal or
external source of phytoplankton biomass, mean trait, and trait variance. In
the case of trait diffusion, such an internal source is gross growth because
the size variance of the phytoplankton community is proportional to it via
the diffusivity constant ν (last terms in Eqs. , , and
). In contrast, immigration represents a source of biomass (I) and
size variance (I/P[VI-V]) external to the phytoplankton community
being simulated (e.g. from an adjacent patch). Hence, during the
autumn–winter transition, the size variance tends to decline in the trait
diffusion case as phytoplankton gross growth is reduced by growth-limiting
processes. Instead, the trait variance keeps building up to values similar to
the variance of the immigrating community in the case of immigration.
Sensitivity to changes in parameter values
We tested the sensitivity of the annual mean in P, L‾, and V
to variations of ±25 % in parameter values. To quantify this
sensitivity, we formulated an index S that accounts for relative changes in
model results:
S=X(p)-X(p′)X(p)⋅100,
where X(p) is the result of the state variable X obtained with the
standard parameter p and X(p′) is the result of the state
variable X obtained with the modified parameter p′=p±25 %.
The four treatments of size variance respond similarly to changes in
parameter values (Fig. ). The annual means of all three state
variables (P, L‾, and V) are sensitive to changes in the
parameters controlling zooplankton grazing (i.e. μZ,
mZ, KP, δZ). However, P also shows a
sensitive response to parameters affecting phytoplankton gross growth, such
as kw, Is, μP, and mP. Mean size is the most
robust variable, with less than 10 % relative change compared to the
standard run. The size variance treatments for immigration and trait
diffusion are affected by the parameters controlling the input of exogenous
(i.e. δI for immigration) or endogenous variance (i.e. ν
for trait diffusion). The results of the unsustained variance model are very
sensitive to changes in μZ, and the case of fixed variance
shows a sensitivity that is similar to the other cases, except for the
variance itself.
Computational efficiency
Trait-based models that aim at resolving the complexity of natural
communities by incorporating many different species or functional types can
be expensive in terms of computational time
.
Alternatively, trait-based models that focus on aggregate community
properties such as total biomass, mean trait, and trait variance can be more
computationally efficient. In Table 2 we report a quantification of the
computation time required for calculating the full and aggregate models
presented here. We obtained a more than 10-fold longer computation time for
the full model than for the aggregate model. In addition, when we increase
the resolution of the full model from 10 to 100 morphotypes, the difference
in computation time increases by more than 20-fold. Thus, increasing the
realism in terms of number of species or morphotypes comes at a significant
computational cost.
Components of the size variance (V), where f(2) is the second
derivative of the fitness function with respect to the trait; f(2)U(L‾,N), f(2)G(L‾,P), and
f(2)V(L‾,M) are, respectively, nitrogen uptake, zooplankton
grazing, and phytoplankton sinking components of f(2).
Strength and weakness of moment-based approximations
Models are simplifications of reality and, as such, are based on assumptions.
For example, the simple exponential growth model is based on a number of
assumptions that do not hold in all circumstances (many factors affect the
intrinsic growth rate, which is often not time-invariant, not all individuals
within a population are identical, nothing can grow indefinitely, etc.).
However, this model is widely used within its range of validity. Likewise,
the approximation of full models with moment-based approaches requires an
assumption about the shape of the phytoplankton trait distribution
. Typically, unimodal distributions,
e.g. normal or log-normal, are assumed. However, depending on how the fitness
function (i.e. the net growth rate of the phytoplankton community) is
constructed and parameterized, the value of f(2), that is the rate of
change of the variance (Eqs. , , , and ),
can be positive, implying a disruptive selection or branching. This
represents an indication that the unimodality assumption does not hold
. Alternatively, f(2) can remain negative over
time, implying that the community continually loses variance, thus
constituting a strong indication against the occurrence of disruptive
selection. Therefore, models based on moment approximations require careful
checks about the validity of the unimodality assumption throughout the time
of the simulations. Figure shows, for our test case, the
predicted variance V, f(2), and its components for the four variance
treatments. In our test case, f(2) is negative for all treatments and
its changes are jointly driven by bottom-up, f(2)U(L‾,N), and top-down processes, f(2)G(L‾,P), i.e. the second derivatives with respect to the
trait for nitrogen uptake (Eq. ) and grazing (Eq. ) terms.
Sinking plays a role mainly during spring, but its influence is minor
compared to the effects of nitrogen uptake and grazing.
It is unclear whether unimodality in size distributions is a robust feature
in the oceans. Observational evidence from recent work
suggests that at large temporal scales, from 5 to 20 years, unimodality of
size distributions is a consistent feature of phytoplankton communities of
the North Sea. By contrast, multimodality is typically observed on temporal
scales of less than 1 year . We consider that the
observational evidence available remains insufficient to identify general
patterns. However, the ocean is a highly variable environment and we
considered it more likely that multimodality, for example because of
size-selective grazing events, is a short-term, transient situation rather
than the norm, because mixing would continuously reshuffle plankton
assemblages and restore homogeneous conditions.
An aspect that our model does not include in its current version is the
dependency of light acquisition on phytoplankton cell size. Given that the
effect of cell size on light harvesting is well understood
, it could be implemented in the
model. Future versions of PhytoSFDM could address this gap by considering the
vertical attenuation of light as a function of both phytoplankton biomass and
cell size, following the approach proposed by .
Uncertainty remains about how to describe the zooplankton population, which
we simplified as an assemblage of identical individuals. This has been the
standard approach in plankton ecosystem modelling for decades and we based
the first version of PhytoSFDM on this simple and classical formulation. In
recent years, however, significant efforts have been made to increase the
level of detail of the zooplankton component in ecosystem models. Approaches
are numerous and include the consideration of different zooplankton
functional types, different size classes, and different feeding preferences
and strategies . A trait-based description of zooplankton can help
in reducing model complexity while maintaining an adequate representation of
diversity. The selection of traits to consider for ecosystem models will
depend on the questions under scrutiny. For example, traits that could
characterize zooplankton-related processes in ecosystem models that focus on
nutrient cycling are maximum growth rates, stoichiometric requirements, and
the size distribution of food particles . Since many
zooplankton traits scale allometrically with body size, scaling laws should
be considered because they are effective ways to generalize the relationships
among different traits and thus to reduce model complexity (i.e. add
size-related functionality without the need for discretely parameterized
zooplankton classes). Implementing such a diversity of grazing mechanisms and
processes is a natural step forward in the development of ecosystem models.
However, a consistent representation of different grazing strategies remains
an aspect under development . PhytoSFDM
constitutes a starting model platform for gradually building model complexity
at different trophic levels.
Concluding remarks
Biological communities are complex adaptive systems
characterized by many components and interconnections that lead to emergent
properties and non-linear responses. Models help us to formalize and simplify
the complexity we observe in nature. This simplification allows us to render
natural phenomena treatable and testable
. Over time, however,
phytoplankton models have grown more complex, computationally more
complicated, and often inaccessible to the wider scientific community,
aspects that can all hamper advancements in the field. To help reverse this
trend we developed PhytoSFDM as a tool to promote the use of trait-based
models (whether species-explicit or aggregate models) of marine ecosystems.
A key decision in modelling is choosing an appropriate level of detail for
the problem at hand. For example, a species-explicit model offers obvious
advantages, which aggregate models cannot offer, when the interest lies in
understanding the relative importance of particular species in providing
certain ecological services or in quantifying the effect of disruptive
selection. Aggregate models, instead, can be more useful at a higher level of
abstraction, when the interest lies in macroecological properties. In
addition, as we have shown, aggregate models present an advantage with
respect to computation time when compared to full models. The advantages in
terms of reducing complexity and computation time remain unproven in
spatially explicit settings (e.g. in 1-D and 3-D), although preliminary
applications have shown promising results .
PhytoSFDM provides a set of methods, under the open-source concept, to
quantify macroecological properties of phytoplankton communities, as an
alternative to the traditional discrete, species-explicit approach. This
effort, we hope, will foster our understanding about the role that
biodiversity plays in shaping marine ecosystems.
Code availability
PhytoSFDM is written in Python (version 2.7.x) as a lightweight and
user-friendly package to facilitate use and re-distribution. We provide
PhytoSFDM as free software under GNU General Public License version 2. The
python package is hosted in (a) GitHUB
(https://github.com/SEGGroup/PhytoSFDM), a software repository that
allows for version control, (b) Zenodo
(https://zenodo.org/record/49849), an open scientific repository, and
(c) PyPI (https://pypi.python.org/pypi/PhytoSFDM), one of the most
popular Python package repositories. To be able to install and operate the
package, the user should be familiar with the Python language and should have
a running Python distribution (preferably version 2.7.x) that includes the
latest versions of the pip and setuptools libraries.
Additional required libraries are matplotlib, numpy,
scipy, and sympy.
PhytoSFDM can then be conveniently installed by typing the following command
from a terminal window:
or by downloading the tarball from the GitHub repository. This is installed
using the source file setup.py
contained in the PhytoSFDM folder by typing
The package consists of three main modules: Example,
SizeModels, and EnvForcing. Example is the entry
point: it computes and compares full and aggregate models with the four
treatments of variance (unsustained, fixed, trait diffusion, and immigration)
at the testing location in the North Atlantic Ocean (centred at
47.5∘ N and 15.5∘ W). The example is run from a terminal by
typing
or from an interactive python shell by typing
>> import phytosfdm.Example.example
as exmp
>>> exmp.main()]]>
The module SizeModels contains the model variants. Here the user can
(a) modify the default parameters, (b) symbolically solve the derivatives
with respect to the trait, and (c) log-transform mean trait and trait
variance. To run the model at a specific location in an interactive Python
shell, one should type
>> from phytosfdm.SizeModels.sizemodels
import SM
>>> Lat=47.5
>>> Lon=344.5
>>> RBB=2.5
>>> SM1= SM(Lat,Lon,RBB,"Imm")]]>
In the above example, the model is executed at a location in the North
Atlantic Ocean centred at 47.5∘ N and 15.5∘ W (here
transformed to a scale of 0 to 360∘). RBB (range of the bounding box) specify the range of the bounding box (in degrees)
for averaging the environmental forcing variables. The fourth argument SM1 is
an object that contains the call of the function SM, which runs the size
model at the specified location and with the desired treatment for the size
variance, in this case immigration.
The last module, EnvForcing, consists of a class containing
spatially averaged forcing data. The climatological data have monthly
resolution, but we include a method to interpolate the data to a daily time
step. Spatially averaged and temporally interpolated forcing at a specific
location can be extracted by typing
>> MLD=ExtractEnvFor(Lat,Lon,RBB,'mld')]]>
Additional information on the usage of the package is contained in the Readme
file and in the repository webpage in GitHUB. The source code of our model is
fully and freely accessible. Users can modify or add new model variants. This
can be done by manipulating the SizeModels module, which contains
model variants as separated methods within the class SM. By using a version
control system such as GitHUB, users can fork our repository, i.e. create a
copy, which allows one to freely change and experiment without affecting the
original code. Users can also modify the original code and submit a new
version by pulling a request. More details can be found in our GitHUB
repository (https://github.com/SEGGroup/PhytoSFDM; ).
Acknowledgements
We would like to thank Jorn Bruggeman for his support on earlier versions of
the model and for his suggestions while we were preparing the draft of this
paper. Esteban Acevedo-Trejos and Agostino Merico are supported by the German
Research Foundation (DFG) through priority programme DynaTrait
(DFG-Schwerpunktprogramm 1704, subproject 19). S. Lan Smith received support
from the Japan Science and Technology Agency (JST) through a CREST project.
We are also grateful to Andrew Yool, Mark Baird, and an anonymous reviewer
whose constructive suggestions helped improve our manuscript. Edited by: A. Yool Reviewed by: M. Baird and
one anonymous referee
References
Abrams, P., Matsuda, H., and Harada, Y.: Evolutionarily unstable fitness
maxima and stable fitness minima of continuous traits, Evol. Ecol., 7,
465–487, 1993.Acevedo-Trejos, E., Brandt, G., Bruggeman, J., and Merico, A.: Mechanisms
shaping phytoplankton community structure and diversity in the ocean, Sci.
Rep., 5, 8918, 10.1038/srep08918, 2015.Acevedo-Trejos, E., Brandt, G., Smith, S. L., and Merico, A.: PhytoSFDM: Phytoplankton Size and Functional Diversity Model, available at:
https://github.com/SEGGroup/PhytoSFDM (last access: 10 November 2016),
2016.
Andersen, K. H., Berge, T., Gonçalves, R. J., Hartvig, M., Heuschele, J.,
Hylander, S., Jacobsen, N. S., Lindemann, C., Martens, E. A., Neuheimer, A. B.,
Olson, K., Palacz, A., Prowe, F., Sainmont, J., Traving, S. J., Visser, A. W.,
Wadhwa, N., and Kiørboe, T.: Characteristic
sizes of life in the oceans, from bacteria to whales, Annu. Rev. Mar. Sci.,
8, 1–25, 2015.
Anderson, T. R.: Plankton functional type modelling: running before we can
walk?, J. Plankton Res., 27, 1073–1081, 2005.
Anderson, T. R.: Progress in marine ecosystem modelling and the unreasonable
effectiveness of mathematics, J. Mar. Syst., 81, 4–11, 2010.
Agustí, S.: Allometric scaling of light absorption and scattering by
phytoplankton cells, Can. J. Fish. Aquat. Sci., 48, 763–767, 1991.
Baird, M. E. and Suthers, I. M.: A size-resolved pelagic ecosystem model,
Ecol. Modell., 203, 185–203, 2007.
Banas, N. S.: Adding complex trophic interactions to a size-spectral plankton
model: Emergent diversity patterns and limits on predictability, Ecol.
Model., 222, 2663–2675, 2011.
Barton, A. D., Dutkiewicz, S., Flierl, G., Bragg, J. G., and Follows, M. J.:
Patterns of diversity in marine phytoplankton, Science, 327, 1509–1511,
2010.
Bonachela, J. A., Klausmeier, C. A., Edwards, K. F., Litchman, E., and Levin,
S. A.: The role of phytoplankton diversity in the emergent oceanic
stoichiometry, J. Plankton Res., 38, 1021–1035, 2015.
Bruggeman, J.: Succession in plankton communities: A trait-based perspective,
PhD thesis, Department of Theoretical Biology, Vrije Universiteit Amsterdam, the Netherlands,
2009.
Bruggeman, J. and Kooijman, S. A. L. M.: A biodiversity-inspired approach to
aquatic ecosystem modeling, Limnol. Oceanogr., 52, 1533–1544, 2007.
Chisholm, S. W.: Phytoplankton Size in Primary productivity and
biogeochemical cycles in the sea, edited by: Falkowski, P. G. and Woodhead,
A. D., Plenum Press, 213–237, 1992.
Chust, G., Irigoien, X., Chave, J., and Harris, R. P.: Latitudinal
phytoplankton distribution and the neutral theory of biodiversity, Glob.
Ecol. Biogeogr., 22, 531–543, 2013.
Downing, A. S., Hajdu, S., Hjerne, O., Otto, S. A., Blenckner, T., Larsson, U., and Winder, M.: Zooming in
on size distribution patterns underlying species coexistence in Baltic Sea
phytoplankton, Ecol. Lett., 17, 1219–1227, 2014.
Edwards, K. F., Thomas, M. K., Klausmeier, C. A., and Litchman, E.:
Allometric scaling and taxonomic variation in nutrient utilization traits and
maximum growth rate of phytoplankton, Limnol. Oceanogr., 57, 554–566, 2012.
Eppley, R.: Temperature and phytoplankton growth in the sea, Fish. Bull., 70,
1063–1085, 1972.
Evans, G. and Parslow, J.: A model of annual plankton cycles, Biol.
Oceanogr., 3, 327–347, 1985.
Fasham, M., Ducklow, H. W., and Mckelvie, S. M.: A nitrogen-based model of
plankton dynamics in the oceanic mixed layer, J. Mar. Res., 48, 591–639,
1990.
Finkel, Z. V. and Irwin, A. J.: Modeling size-dependent photosynthesis: Light
absorption and the allometric rule, J. Theor. Biol., 204, 361–369, 2000.
Finkel, Z. V.: Light absorption and size scaling of light-limited metabolism
in marine diatoms, Limnol. Oceanogr., 46, 86–94, 2001.
Finkel, Z. V., Beardall, J., Flynn, K., Quigg, A., Rees, T. A. V., and Raven,
J. A.: Phytoplankton in a changing world: cell size and elemental
stoichiometry, J. Plankton Res., 32, 119–137, 2010.
Follows, M. J. and Dutkiewicz, S.: Modeling diverse communities of marine
microbes, Annu. Rev. Mar. Sci., 3, 427–451, 2011.
Follows, M. J., Dutkiewicz, S., Grant, S., and Chisholm, S. W.: Emergent
biogeography of microbial communities in a model ocean, Science, 315,
1843–1846, 2007.
Fulton, E. A., Smith, A. D. M., and Johnson, C. R.: Effect of complexity on
marine ecosystem models, Mar. Ecol. Prog. Ser., 253, 1–16, 2003.
Hansen, B., Bjørnsen, P. K., and Hansen, P. J.: The size ratio between
planktonic predators and their prey, Limnol. Oceanogr., 39, 395–403, 1994.Hansen, P. J., Bjørnsen, P. K., and Hansen, B. W.: Zooplankton grazing and
growth: Scaling within the 2–2,000-µm body size range, Limnol.
Oceanogr., 42, 687–704, 1997.
Hardin, G.: The competitive exclusion principle, Science 131, 1292–1297,
1960.
Hood, R. R., Laws, E. A., Armstrong, R. A., Bates, N. R., Brown, C. W., Carlson,
C. A., Chai, F., Doney, S. C., Falkowski, P. G., Feely, R. A., Friedrichs, M. A. M.,
Landry, M. R., Moore, J. K., Nelson, D. M., Richardson, T. L., Salihoglu, B.,
Schartau, M., Toole, D. A., and Wiggert, J. D.: Pelagic
functional group modeling: Progress, challenges and prospects, Deep-Sea Res.
Pt. II, 53, 459–512, 2006.
Kiørboe, T.: Turbulence, phytoplankton cell size, and the structure of
pelagic food webs, Adv. Mar. Biol., 29, 1–72, 1993.
Levin, S. A.: Ecosystems and the biosphere as complex adaptive systems,
Ecosystems, 1, 431–436, 1998.
Levins, R.: The strategy of model building in population biology, Am. Sci.,
54, 421–431, 1966.
Litchman, E. and Klausmeier, C. A.: Trait-based community ecology of
phytoplankton, Annu. Rev. Ecol. Evol. Syst. 39, 615–639, 2008.
Litchman, E., Klausmeier, C. A., Schofield, O., and Falkowski, P. G.: The
role of functional traits and trade-offs in structuring phytoplankton
communities: scaling from cellular to ecosystem level, Ecol. Lett., 10,
1170–1181, 2007.
Litchman, E., de Tezanos Pinto, P., Klausmeier, C. A., Thomas, M. K., and
Yoshiyama, K.: Linking traits to species diversity and community structure in
phytoplankton, Hydrobiologia, 653, 15–28, 2010.
Litchman, E., Ohman, M. D., and Kiørboe, T.: Trait-based approaches to
zooplankton communities, J. Plankton Res., 35, 473–484, 2013.
Norberg, J., Swaney, D. P., Dushoff, J., Lin, J., Casagrandi, R., and Levin,
S. A.: Phenotypic diversity and ecosystem functioning in changing
environments: a theoretical framework, P. Natl. Acad. Sci. USA, 98,
11376–11381, 2001.
Marañón, E.: Cell Size as a Key Determinant of Phytoplankton
Metabolism and Community Structure, Annu. Rev. Mar. Sci., 7, 1–24, 2015.
Marañón, E., Cermeño, P., López-Sandoval, D. C., Rodríguez-Ramos,
T., Sobrino, C., Huete-Ortega, M., Blanco, J. M., and Rodríguez, J.:
Unimodal size scaling of phytoplankton growth and the size dependence of
nutrient uptake and use, Ecol. Lett., 16, 371–379, 2013.
Mariani, P., Andersen, K. H., Visser, A. W., Barton, A. D., and Kiørboe,
T.: Control of plankton seasonal succession by adaptive grazing, Limnol.
Oceanogr., 58, 173–184, 2013.
Merico, A., Bruggeman, J., and Wirtz, K.: A trait-based approach for
downscaling complexity in plankton ecosystem models, Ecol. Model., 220,
3001–3010, 2009.
Merico, A., Brandt, G., Smith, S. L., and Oliver, M.: Sustaining diversity in
trait-based models of phytoplankton communities, Front. Ecol. Evol., 2, 1–8,
2014.
Monterey, G. I. and Levitus, S.: Seasonal variability of the mixed layer
depth for the world ocean, US Gov. Printing Office, Washington, DC, USA, 1997.
Prowe, A. E. F., Pahlow, M., Dutkiewicz, S., Follows, M. J., and Oschlies,
A.: Top-down control of marine phytoplankton diversity in a global ecosystem
model, Prog. Oceanogr., 101, 1–13, 2012.
Ryabov, A. B., Morozov, A., and Blasius, B.: Imperfect prey selectivity of
predators promotes biodiversity and irregularity in food webs, Ecol. Lett.,
18, 1262–1269, 2015.
Smith, S. L., Pahlow, M., Merico, A., and Wirtz, K. W.: Optimality-based
modeling of planktonic organisms, Limnol. Oceanogr., 56, 2080–2094, 2011.
Smith, S. L., Merico, A., Wirtz, K. W., and Pahlow, M.: Leaving misleading
legacies behind in plankton ecosystem modelling, J. Plankton Res., 36,
613–620, 2014.
Smith, S. L., Pahlow, M., Merico, A., Acevedo-Trejos, E., Sasai, Y., Yoshikawa, C., Sasaoka, K., Fujiki, T., Matsumoto, K., and Honda, M.: Flexible
phytoplankton functional type (FlexPFT) model: size-scaling of traits and
optimal growth, J. Plankton Res., 38, 977–992, 2015.
Steele, J.: Environmental control of photosynthesis in the sea, Limnol.
Oceanogr., 7, 137–150, 1962.
Terseleer, N., Bruggeman, J., Lancelot, C., and Gypens, N.: Trait-based
representation of diatom functional diversity in a plankton functional type
model of the eutrophied Southern North Sea, Limnol. Oceanogr., 59, 1–16,
2014.
Vallina, S. M., Ward, B. A., Dutkiewicz, S., and Follows, M. J.: Maximal
feeding with active prey-switching: A kill-the-winner functional response and
its effect on global diversity and biogeography, Prog. Oceanogr., 120,
93–109, 2014.
Ward, B. A., Dutkiewicz, S., Jahn, O., and Follows, M. J.: A size-structured
food-web model for the global ocean, Limnol. Oceanogr., 57, 1877–1891, 2012.Wirtz, K.: Non-uniform scaling in phytoplankton growth rate due to
intracellular light and CO2 decline, J. Plankton Res., 33, 1325–1341,
2011.
Wirtz, K.: Who is eating whom?: Morphology and feeding type determine the
size relation between planktonic predators and their ideal prey, Mar. Ecol.
Prog. Ser., 445, 1–12, 2012.
Wirtz, K. W.: Mechanistic origins of variability in phytoplankton dynamics:
Part I: niche formation revealed by a size-based model, Mar. Biol., 160,
2319–2335, 2013.Wirtz, K. W. and Eckhardt, B.: Effective variables in ecosystem models with
an application to phytoplankton succession, Ecol. Model., 92, 33–53, 1996.
Wirtz, K. W. and Sommer, U.: Mechanistic origins of variability in
phytoplankton dynamics. Part II: analysis of mesocosm blooms under climate
change scenarios, Mar. Biol., 160, 2503–2516, 2013.