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SPEAD 1.0 -- Simulating Plankton Evolution with Adaptive Dynamics in a two-trait continuous fitness landscape applied to the Sargasso Sea


Abstract and Figures

Diversity plays a key role in the adaptive capacities of marine ecosystems to environmental changes. However, modeling phytoplankton trait diversity remains challenging due to the strength of the competitive exclusion of sub-optimal phenotypes. Trait diffusion (TD) is a recently developed approach to sustain diversity in plankton models by allowing the evolution of functional traits at ecological timescales. In this study, we present a model for Simulating Plankton Evolution with Adaptive Dynamics (SPEAD), where phytoplankton phenotypes characterized by two traits, nitrogen half-saturation constant and optimal temperature, can mutate at each generation using the TD mechanism. SPEAD does not resolve the different phenotypes as discrete entities, computing instead six aggregate properties: total phytoplankton biomass, mean value of each trait, trait variances, and inter-trait covariance of a single population in a continuous trait space. Therefore SPEAD resolves the dynamics of the population’s continuous trait distribution by solving its statistical moments, where the variances of trait values represent the diversity of ecotypes. The ecological model is coupled to a vertically-resolved (1D) physical environment, and therefore the adaptive dynamics of the simulated phytoplankton population are driven by seasonal variations in vertical mixing, nutrient concentration, water temperature, and solar irradiance. The simulated bulk properties are validated by observations from BATS in the Sargasso Sea. We find that moderate mutation rates sustain trait diversity at decadal timescales and soften the almost total inter-trait correlation induced by the environment alone, without reducing the annual primary production or promoting permanently maladapted phenotypes, as occur with high mutation rates. As a way to evaluate the performance of the continuous-trait approximation, we also compare the solutions of SPEAD to the solutions of a classical discrete entities approach, both approaches including TD as a mechanism to sustain trait variance. We only find minor discrepancies between the continuous model SPEAD and the discrete model, the computational cost of SPEAD being lower by two orders of magnitude. Therefore SPEAD should be an ideal eco-evolutionary plankton model to be coupled to a general circulation model (GCM) at the global ocean.
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SPEAD 1.0 – A model for Simulating Plankton Evolution with
Adaptive Dynamics in a two-trait continuous fitness landscape
applied to the Sargasso Sea
Guillaume Le Gland1, Sergio M. Vallina2, S. Lan Smith3, and Pedro Cermeño1
1Institut de Ciències del Mar (CSIC), Passeig Marítim de la Barceloneta 37-49, 08003 Barcelona, Spain
2Spanish Institute of Oceanography (IEO), Ave Principe de Asturias 70 bis, 33212 Gijon, Spain
3Earth SURFACE Research Center, Research Institute for Global Change, JAMSTEC, Yokosuka, Japan
Correspondence to: Guillaume Le Gland (
Diversity plays a key role in ecosystem adaptive capacities. However, modeling phytoplankton trait diversity remains chal-
lenging due to the strength of the competitive exclusion of sub-optimal phenotypes. A recent approach to sustain diversity,
called "trait diffusion", consists in allowing evolution to occur at contemporary timescales.
In this study, we present a model for Simulating Plankton Evolution with Adaptive Dynamics (SPEAD), where phenotypes5
characterized by two traits, nitrogen half-saturation constant and optimal temperature, can mutate at each generation. SPEAD
does not resolve all phenotypes, computing instead six aggregate properties: biomass, mean traits, trait variances and inter-trait
covariance. The adaptive dynamics are driven by vertical and seasonal variations in water temperature, irradiance, vertical
mixing, and nutrient concentration. The bulk properties are validated by observations from station BATS in the Sargasso Sea.
We find only minor discrepancies between SPEAD and a similar model that represents the full phenotype distribution, but10
SPEAD has a lower computational cost by two orders of magnitude. Moderate mutation rates are shown to sustain trait diversity
at decadal timescales and to soften the almost total inter-trait covariance induced by the environment alone, without reducing
the annual primary production or promoting permanently maladapted phenotypes, as occurs with high mutation rates. The
response to environmental changes is faster than in single-trait models.
Future axes of improvement include increasing the number of traits, beginning with optimal irradiance, refining the descrip-15
tion of the phytoplankton community by resolving several functional groups, and coupling SPEAD with a general circulation
1 Introduction
Phytoplankton are a polyphyletic group of single-cell primary producers widespread in aquatic environments. Despite account-
ing for only 1% of the global photosynthetic biomass, they perform more than 45% of Earth’s net primary production (Field20
et al., 1998; Falkowski et al., 2004). They are the basis of all oceanic food webs and play key roles in biogeochemical cycles
(Falkowski, 2012). In particular, they have a large impact on global climate through the export of detritic carbon from the
surface to the ocean interior, sequestrating carbon in the deep ocean for timescales from a few years to more than a millenium,
depending on the depth they reach (DeVries and Primeau, 2011; DeVries et al., 2012). This process, called the "biological
carbon pump", regulates the concentration of carbon dioxide in the atmosphere (Volk and Hoffert, 1985; Falkowski et al.,
Phytoplankton are highly diverse and live in many different environments. They differ in their ecological interactions and5
the processes through which they mediate biogeochemical cycles. For instance, some species can fix atmospheric nitrogen
and enrich oligotrophic regions, some produce ballast minerals (mainly silica and calcium carbonate) and sink faster to the
deep ocean, some promote the formation of clouds by producing dimethylsulfide, and others are mixotrophic, being able
to both photosynthesize and feed on organic sources (Le Quéré et al., 2005). Most species are denser than seawater and
eventually sink but some are buoyant (Lännergren, 1979; Villareal, 1988). Phytoplankton size ranges from less than 1 µm for10
cyanobacteria like Prochlorococcus (Chisholm et al., 1988) to more than 1 mm for the giant diatom Ethmodiscus rex (Swift,
1973; Villareal and Carpenter, 1994). Their half-saturation constants for the main limiting nutrients range over three orders
of magnitude (Edwards et al., 2012), reflecting adaptation to different nutrient supply levels. They are also adapted to very
different temperatures: some diatoms can grow within sea ice (Ackley and Sullivan, 1994), whereas some hyperthermophilic
cyanobacteria can grow at up to 75C in hot springs (Castenholz, 1969). Most oceanic species have an optimal temperature for15
growth between 0 and 35C (Thomas et al., 2012). Even within the same species or genus, wide variability has been observed
for key traits such as iron requirement (Strzepek and Harrison, 2004), light requirement (Biller et al., 2015) and resistance to
predation (Yoshida et al., 2004). Given that any change in the abundance and composition of phytoplankton has far-reaching
consequences for other organisms and for the Earth’s climate, it is important to understand the factors driving the dynamics of
such communities.20
Numerical modelling studies can address this issue by finding the mechanistic equations and parameters that most correctly
account for the observations, and thereby provide invaluable insights into the general rules controlling ecosystems. Models
can also be used to make predictions of how phytoplankton will impact or be impacted by future environmental changes
(Norberg et al., 2012; Irwin et al., 2015). Mathematical models of phytoplankton growth as a function of nutrient concentration,
temperature and radiation have been developed since the 1940’s (Riley, 1946; Steele, 1958; Riley, 1965), leading to the25
now common NPZD ("Nutrient, Phytoplankton, Zooplankton, Detritus") models representing one or several compartments of
nutrients, phytoplankton, zooplankton and detritus (Fasham et al., 1990). Since the early 1990’s (Maier-Reimer, 1993), the
increase in computational power allowed biogeochemical models to be fully coupled with ocean circulation (Aumont et al.,
2003; Follows et al., 2007). However, representing all the phytoplankton diversity in models is neither feasible nor desirable.
The computational cost would be high, and even if computationally feasible, the existing observations would not suffice to30
constrain the many free parameters.
Instead, models account for biodiversity through a few key traits representing physiological characteristics or adaptation to
different environments. The most widely investigated phytoplankton traits are cell size, nutrient niche, optimal temperature,
optimal irradiance and resistance to predation. Some trait-based models divide the phytoplankton community into discrete
entities or "boxes" with different traits. The boxes can be as simple as diatoms and small phytoplankton groups (Aumont35
et al., 2015), with diatoms having higher nutrient concentration niches, or include more complex divisions into functional
groups (Baretta et al., 1995; Le Quéré et al., 2005; Follows et al., 2007). The other approach, which further reduces the
number of equations while still allowing communities to adapt to changes in their environments, is to consider one or several
continuously distributed traits and to compute only the dynamics of aggregate properties, such as community biomass, mean
trait values and trait variances (Wirtz and Eckhardt, 1996; Norberg et al., 2001; Bruggeman and Kooijman, 2007; Merico et al.,5
2009; Acevedo-Trejos et al., 2016; Smith et al., 2016; Chen and Smith, 2018). In this method trait variance can be used as a
quantitative index of biodiversity. A community with a higher trait variance is considered to be more diverse because it has a
wider spread and more even distribution of trait values (Li, 1997), although it does not necessarily have a higher number of
taxonomic species ("richness") (Vallina et al., 2017).
One weakness induced by the simplification of phytoplankton communities in both the aggregate and discrete models,10
however, is that competitive exclusion (Hardin, 1960; Hutchinson, 1961) often leads to a collapse of the modeled diversity
(Merico et al., 2009) unless trait variance is imposed (Norberg et al., 2012; Wirtz, 2013) or a mechanism is added explicitly
to sustain it. One way to sustain biodiversity is through immigration from a distant community (Norberg et al., 2001; Savage
et al., 2007). Yet, immigration does not explain the diversity observed in closed laboratory experiments, including continuous
cultures (Fussmann et al., 2007; Kinnison and Hairston, 2007; Beardmore et al., 2011). Biodiversity can also be sustained15
by viruses (Thingstad and Lignell, 1997) or predators (Murdoch, 1969; Kiørboe et al., 1996) if they specialize on a narrow
range of preys or switch their preference to the most common phytoplankton species. This is the idea behind the "Kill The
Winner" theory (Thingstad, 2000; Vallina et al., 2014b), where predation concentrating more so on the most dominant species
maintains diversity, because then each prey species persists at the abundance where the predation rate equals its growth rate.
An alternative approach recently introduced to sustain diversity in models is to allow the simulated phytoplankton to mutate20
their physiological traits (Kremer and Klausmeier, 2013; Merico et al., 2014). Due to their short generation times of around
1 day (Marañon et al., 2013), phytoplankton are known to evolve at the timescale of a few years (Schlüter et al., 2016). For
phytoplankton, the "ecological" timescales, featuring successions of dominant species in reaction to changes in the environ-
ment and selection of the fittest, overlap with the "evolutionary" timescales, where species can also evolve genetically to adapt
to their new environment (Irwin et al., 2015). As far as we know, the first aggregate phytoplankton model allowing a phyto-25
plankton trait to randomly mutate through subsequent generations, before being selected by the environment, was developed
by Merico et al. (2014). They called their scheme "Trait diffusion" (TD), where "diffusion" is a mathematical term referring
to the spreading of a property, in this case the trait value, not to physical transport. Trait diffusion of a single physiological
trait was recently introduced in a model coupled with oceanic circulation (Chen and Smith, 2018). Upgrading the trait dif-
fusion framework to several traits requires more complex equations and the introduction of a new class of state variables: the30
covariances between traits. However, multi-trait models are more realistic and conceptual modeling studies have shown that
the dynamics of correlated traits sometimes differ from those of single-trait models (Savage et al., 2007).
Here we present a new aggregate phytoplankton model called SPEAD (Simulating Plankton Evolution with Adaptive
Dynamics), an eco-evolutionary model using the trait diffusion framework for two key phytoplankton traits: nitrogen half-
saturation constant and optimal temperature for growth. Our model is based on a NPZD model (Vallina et al., 2014a, 2017),35
where the phytoplankton compartment is represented by the community biomass, mean trait values, trait variances and covari-
ance. SPEAD is embedded in a 1D (water column) physical setting simulating the Sargasso Sea using data from the Bermuda
Atlantic Time-series Studies (BATS). We chose the 1D rather than a simpler 0D setting because vertical turbulent diffusion
(not to be confused with trait diffusion) is the main source of covariance by mixing communities from different depths. Since
the trait diffusion equations can easily be adapted to a discrete model, we have also built a discrete version of SPEAD where5
the phytoplankton community consists of 625 different phenotypes (i.e. 25 half-saturation constants and 25 optimal temper-
atures), each characterized by its own fixed set of trait values. The discrete version is more intuitive and easier to program,
and provides a useful control experiment. SPEAD is intended as a prototype to be coupled later with 3D general circulation
models. Its equations for mean trait, trait variance and covariance can be used as a starting point to build more comprehensive
trait-based models, with or without mutations. In particular, we plan to add optimal irradiance as a third trait in the near future10
for a more realistic description of phytoplankton distributions over depth.
In the following sections, we first describe our ecological model, the differential equations controlling the growth of phyto-
plankton and the adaptive evolution of their trait distribution, as well as the physical model setting. Then, we present the model
outputs. In order to validate SPEAD and to highlight its novelties, we will focus on answering the following four questions: 1)
How well does SPEAD represent the bulk properties of phytoplankton communities observed in the Sargasso Sea? 2) Do the ag-15
gregate and discrete approaches agree? 3) How are phytoplankton dynamics changed by the value of the mutation rate? 4) Can
the mean value and variance of each trait be represented independently by a 1-trait model where only nitrogen half-saturation
or optimal temperature varies between phenotypes? Finally, we discuss the reach of our modeling framework, focusing on three
aspects: the performance of aggregate models, the choice of phytoplankton traits, and the relationship between trait diffusion
and evolution.20
2 Methods
2.1 A phytoplankton community model with 2 traits
Our phytoplankton community model SPEAD extends an existing nitrogen-based NPZD model (Vallina et al., 2017). Nitrogen
is partitioned into four pools, all expressed in millimoles of nitrogen per cubic meter (mmolN m3): phytoplankton (Pin the
equations), zooplankton (Z), Dissolved Inorganic Nitrogen or "DIN" (N) and Particulate Organic Nitrogen or "PON" (D25
as in "detritus"). Phytoplankton increases its biomass by taking up DIN (Up). Zooplankton increases its biomass by grazing
phytoplankton (Gz). The non-predation mortalities of phytoplankton (Mp) and zooplankton (Mz) and the nitrogen exudation
by zooplankton (Ez) are divided between DIN and PON. Given that nitrogen is the limiting nutrient for phytoplankton growth,
we do not consider nitrogen exudation by phytoplankton. ωp,ωzand zare constants representing the respective proportions
of Mp,Mzand Ezgoing to DIN. PON is remineralized to DIN (Md). The fluxes from one pool to another are controlled by30
the pool concentrations and by two environmental forcings: temperature (T, in C) and Photosynthetically Available Radiation
or "PAR" (I, in W m2). The main state variables of the model and their relationships are shown in Table 1 and Fig. 1a and the
Phytoplankton (P)
Zooplankton (Z)
Dissolved Inorganic
Nitrogen (N)
Particulate Organic
Nitrogen (D)
Photosynthetically Available
Radiation (I)
Temperature (T)
b. Multi-phenotype (discrete) phytoplankton model
Vertical diffusion
Half-saturation ()
c. Aggregate (continuous) phytoplankton model
Bivariate normal trait distribution
6 state variables:
Total phytoplankton P
Volume below the curve
Mean trait values
Peak location
Trait variances
Distribution width
Inter-trait covariance 
Direction of the major axis
Vertical diffusion
Particle sinking
Box n (<<)
a. NPZD (Nutrient, Phytoplankton, Zooplankton, Detritus) model
Figure 1. NPZD (Nutrient, Phytoplankton, Zooplankton, Detritus) model within its physical setting (a). The phytoplankton pool is repre-
sented by a discrete set of species with different traits (b) or by moments of the trait distributions, assuming a bivariate normal distribution
(c). Colors in b and c represent phytoplankton concentration.
expressions of the fluxes are given by the following equations, with their dependencies:
dt =Up(P, N, T,I )Mp(P, T )Gz(P, Z, T )(1)
dt =Gz(P, Z, T )Ez(P, Z, T )Mz(Z, T )(2)
dt =zEz(P, Z, T ) + ωpMp(P, T ) + ωzMz(Z, T ) + Md(D, T )Up(P, N, T, I)(3)
dt = (1 z)Ez(P ,Z, T ) + (1 ωp)Mp(P, T ) + (1 ωz)Mz(Z, T )Md(D, T )(4)5
Zooplankton, DIN and PON are generic pools, characterized by a single variable: their concentration. Phytoplankton and
zooplankton mortality, zooplankton exudation, grazing and the particle remineralization rate have simple expressions as a
Table 1. State variables of the ecosystem model
Symbol Description Unit
Prognostic variables of the aggregate model
PPhytoplankton concentration mmolN m3
xMean nitrogen half-saturation logarithm (trait 1)
yMean optimal temperature (trait 2) C
VxHalf-saturation logarithm variance
VyOptimal temperature variance C2
Cxy Inter-trait covariance C
Prognostic variables of the multi-phenotype model
Pij Concentration of phytoplankton phenotype i,j mmolN m3
Prognostic variables common to both models
ZZooplankton concentration mmolN m3
NDissolved Inorganic Nitrogen (DIN) concentration mmolN m3
DParticulate Organic Nitrogen (PON) concentration mmolN m3
Diagnostic variables related to trait
σxStandard deviation of half-saturation logarithm
σyStandard deviation of optimal temperature C
Rxy Inter-trait correlation
Other diagnostic variables
Chl Chlorophyll a concentration mgCHL m3
P P Primary production mgC m3d1
Environment variables
TTemperature C
IPhotosynthetically available radiation (PAR) W m2
KzVertical diffusivity m2.d1
function of the nitrogen pool concentrations and temperature:
Gz(P, Z, T ) = g0eαh(TT0)P2
Ez(P, Z, T ) = (1 βz)Gz(P,Z ,T )(6)
Mz(Z, T ) = ψzeαh(TT0)Z2(7)
Mp(P, T ) = ψpeαh(TT0)P(8)5
Md(T) = ψdeαh(TT0)D(9)
The constants appearing in this equation (αh,T0,Kp,βz,ψz,ψpand ψd) are described in Table 2. Zooplankton mortality
depends on the square of zooplankton concentration in order to prevent an explosion of zooplankton concentration. This
stabilizing quadratic mortality term represents consumption by animals higher on the trophic chain, which is expected to
increase faster than a linear function of Z biomass. Grazing is formulated as a Holling type III function (Holling, 1959) of10
phytoplankton concentration, with a niche at low concentrations to prevent the whole phytoplankton community from going
extinct, even when they have very low growth rates. Grazing, mortality and remineralization are considered as heterotrophic
processes and as such increase exponentially with temperature. The exponential factor is αh= 0.092 C1. This is equivalent
to multiplying the speed of all these processes by 2.5 when the temperature increases by 10 C, as in a "Q10 = 2.5" formulation.
This value of Q10 is close to measured values for zooplankton grazing (Hansen et al., 1997) and to the theoretical predictions15
of the metabolic theory of ecology for respiration (Gillooly et al., 2001; Allen et al., 2005).
On the contrary, the phytoplankton pool is composed of diverse organisms responding to environmental conditions in differ-
ent ways. The diversity of phytoplankton is represented by variations in the values of two traits: the logarithm of half-saturation
constant for nitrogen uptake (x) and the optimal temperature for growth (y). From now on, we will refer to each set (x,y) of trait
values as a "phenotype". Nutrient uptake by phytoplankton depends on the trait distribution. The bivariate trait distribution is20
represented by a density p(x,y)(in mmolN m3C1) so that the biomass of phytoplankton (in mmolN m3) with trait values
between x1and x2and between y1and y2is equal to Rx2
y1p(x,y)dxdy and by extension the total phytoplankton biomass
P is equal to the density integrated over the whole trait domain. Any phenotype has its own uptake rate up(x, y). The uptake
rate is the product of a constant (u0
p) and three growth factors: a nutrient factor (γn(N, x)), a temperature factor (γT(T, y))
and an irradiance factor (γi(I)). Two of these factors, γn(N,x)and γi(I), represent limitations by resources. The third factor,25
γT(T,y), represents the kinetic effect of temperature on growth. In this study, we use the Monod approach (Monod, 1949), so
that cells do not store nutrients and the uptake rate is equal to the reproduction or growth rate. Since phytoplankton are unicel-
lular and we do not consider changes in their cell volumes, we will use the words "growth" and "reproduction" interchangeably.
All phenotypes share the same rates of mortality and grazing, respectively.
The last term in the equation of trait density (Eq. 10) is trait diffusion, as defined by Merico et al. (2014). Trait diffusion30
represents the fact that offspring can exhibit different trait values than their parents, due to mutations or otherwise heritable
plasticity. In our numerical model, we assume only that these mutations are heritable and random. They can represent genetic
mutations as well as other, e.g. epigenetic, phenotypic plasticity. We assume that mutations on xand yare independent of each
Table 2. Parameters of the ecosystem model
Symbol Description Value Unit
Phytoplankton parameters
T0Reference temperature 20 C
pPhytoplankton maximum uptake rate for x= 0 and y=T01.1 d1
TDifference between optimal and maximal temperature 5 C
pPhytoplankton exudation fraction going to DIN 1/3
ψpPhytoplankton mortality rate at 20 C 0.05 d1
ωpPhytoplankton mortality fraction going to DIN 1/4
I0Phytoplankton optimal irradiance 25 W m2
χPhytoplankton photoinhibition factor 12
αaTemperature dependence factor for autotrophic processes 0.056 C1
Speed multiplied by 1.75 ("Q10") with a temperature increase of 10 C
νxHalf-saturation diffusivity parameter 105– 0.1
νyOptimal temperature diffusivity parameter 104– 1 C2
Other ecological parameters
g0Zooplankton maximum grazing rate at 20 C 1.5 d1
KpHalf-saturation for grazing 0.4 mmolN m3
βzZooplankton assimilation efficiency 0.4
zZooplankton exudation fraction going to DIN 1/3
ψzZooplankton (quadratic) mortality rate at 20 C 0.25 (mmolN m3)1d1
ωzZooplankton mortality fraction going to DIN 1/4
ψdPON remineralization rate at 20 C 0.1 d1
wPON sinking speed 1.2 m d1
kwPAR vertical attenuation 0.04 m1
αhTemperature dependence factor for heterotrophic processes 0.092 C1
Speed multiplied by 2.5 ("Q10") with a temperature increase of 10 C
Numerical parameters
nxNumber of half-saturation values (discrete model) 25
nyNumber of optimal temperature values (discrete model) 25
xmin Minimum half-saturation logarithm (discrete model) -2.5
xmax Maximum half-saturation logarithm (discrete model) +1.5
ymin Minimum optimal temperature (discrete model) 18 C
ymax Maximum optimal temperature (discrete model) 30 C
zmax Maximum model depth 200 m
other. In the limit of small but frequent mutations, stochasticity can be neglected (Dieckmann and Law, 1996; Champagnat
et al., 2006) and this process can be represented as a deterministic diffusion, depending on diffusivity parameters νxand νy
and on the second derivatives of the growth rate (up(x, y)) relative to each trait respectively. Note that in TD, "diffusion" is a
mathematical term referring to the spreading of a property, in our case trait values, not to a physical mixing process. It should
therefore not be confused with vertical turbulent diffusion, which is also present in our model (see 2.3.). To avoid ambiguity,5
from now on, we will refer to the trait diffusivity parameters as "mutation rates". νxand νyare mutation rates per generation,
not per unit of time, therefore time does not appear in their units. They have the same units as trait variances. The derivation of
the trait diffusion term is explained in Appendix A. The differential equations followed by a given phenotype (x,y) are:
∂p(x, y, t)
∂t =up(N,T , I, x, y)Mp(P, T )
PGz(P, Z, T )
Pp(x,y,t) + νx
up(N, T,I ,x,y) = u0
pγn(N, x)γT(T,y)γi(I)(11)10
Like all biodiversity models, SPEAD must not allow a phenotype to outcompete all other phenotypes in all environments,
because any such Darwinian demon would drive all its sub-optimal competitors to extinction and trait variance would collapse
to zero. In order to make competition for nutrients possible, we have defined two uptake traits so that either low or high values
are advantageous in certain environments and disadvantageous in others. The shape of the two trade-offs and the three growth
factors are presented in Fig. 2.15
The first trait allowed to mutate in SPEAD is the half-saturation constant that controls the nutrient limitation factor γn(N, x).
The half-saturation constant can be linked to the well-known trade-off between the affinity for a nutrient and the maximum
uptake rate, also known as the "gleaner-opportunist" trade-off (Frederickson and Stephanopoulos, 1981). The biomass-specific
nitrogen uptake rate up(N, T,I ,x,y)of a given phenotype is proportional to its affinity for nitrogen fp(in d1mmol1m3)
at low nitrogen concentration and reaches the maximum uptake rate u
p(in d1) in nutrient-replete environments. The uptake20
rate follows a Michaelis-Menten function of nutrient concentration:
up(N, T,I ,x,y) = u
p(T,I ,x,y)N
p(T,I ,x,y)
fp(T,I ,x,y)+N
p(T,I ,x,y) = lim
up(N, T,I ,x,y) = u0
pγT(T,y)γi(I) lim
γn(N, x)(13)
fp(T,I ,x,y) = lim
up(N, T,I ,x,y)
pγT(T,y)γi(I) lim
γn(N, x)
Phenotypes that specialize in taking up the few available nutrients at low concentrations (high fp) are called "gleaners", whereas25
those that specialize in taking up nutrients quickly at saturating nutrient concentrations (high u
p) are called "opportunists".
We assume that the product fpu
pis independent of x(Meyer et al., 2015; Vallina et al., 2017), a relation that defines
a gleaner-opportunist trade-off. The half-saturation constant Kn=u
fp(in mmolN m3) is the DIN concentration at which
nitrogen uptake rate is equal to one half of the maximum uptake rate for the same temperature and solar irradiance. The
half-saturation constant is assumed to be independent of the temperature and irradiance of the environment, as well as of the30
phytoplankton optimal temperature. Half-saturation makes the analysis of our results more straightforward, because it has
Figure 2. Phytoplankton growth factors γn(nutrient-dependent), γt(temperature-dependent) and γi(PAR-dependent). a) and b) represent
the growth factor as a function of nutrient concentration and temperature respectively, for different phenotypes. c) and d) represent the growth
factor as a function of the corresponding trait for different values of the environmental parameter. The maximum of each curve corresponds
to the phenotype most adapted to a given environment. e) and f) are normalized versions of c) and d), respectively, so that their maximum is
always 1. g) is the PAR-dependent growth factor, which is common to all phenotypes in this version of SPEAD.
units of concentration and can therefore be compared directly to the ambient DIN concentration. Because the concentrations
are always positive and span several orders of magnitude, we use a natural logarithmic scale and define x=log Kn
nas our
first trait axis, with K0
n= 1 mmolN m3as a reference value for Kn. Thus, the nutrient limitation factor is:
γn(N, x) = γ
Kn(x) + N=γ
n(x) = Kn
For any nutrient concentration N, we note that the phenotype corresponding to the largest growth rates is x=log(N). This is
why, under the assumption of the gleaner-opportunist trade-off defined above, the Kndefines the optimal nutrient concentration
of each xphenotype where they are competitively superior (Vallina et al., 2017). This result, however, is dependent on the
specific model assumption that fpu
pis a constant.
The second phytoplankton trait that is allowed to mutate in SPEAD is the optimal temperature. Temperature affects microbes
in two ways. One is generic and applies to the whole plankton community. An increase in temperature increases the speed of
both primary production and heterotrophic processes for thermodynamic reasons. This effect is often assumed to be exponen-
tial. In our model, the exponential factor for autotrophic primary production is αa= 0.056 C1, which corresponds to a Q10
of 1.75, slightly lower than the classical value of 1.88 from Eppley (1972) but higher than the values based on the metabolic5
theory of ecology for photosynthesis (López-Urrutia et al., 2006). The second effect of temperature is phenotype-specific. Each
phenotype has an optimal temperature for growth, which is the second trait axis and is denoted by y. The effect of temperature
on a given phenotype (x,y) is asymmetric: at temperatures more than 5C above ygrowth ceases but temperatures below y
merely slows growth. We defined our temperature multiplicative growth factor to be as close as possible to the species-specific
curves of Eppley (1972):10
γT(T,y) = e(Ty)
Ty+ ∆TT
The temperature tolerance Tis set to 5C. T0is a reference temperature with no ecological meaning. For a fixed value of
y,γT(T,y)has a maximum at T=ywith a value of eαa(yT0). At T=y+ ∆Tand warmer, growth is impossible. For a
given value of the environment temperature T, the phenotypes with the largest growth rates have an optimal temperature y
around 2C larger than T. This apparent mismatch, where the dominant phenotype at temperature Tcan grow even faster15
at temperatures a few degrees warmer, is both coherent with other models (Beckmann et al., 2019) and observed in nature
(Thomas et al., 2012; Irwin et al., 2012).
In this study, the PAR limitation factor γi(I)is the same for all phenotypes. It includes an optimal PAR (Iopt ) of 25 Wm2
and photoinhibition above this level. Our value for Iopt is in the middle of the range considered by Follows et al. (2007) and
our expression for γi(I)is equivalent to theirs:20
Iopt eln(1+χ)
Iopt (18)
i=χ+ 1
χ+1 )(19)
In the above equation, Γ0
iis a normalization factor (to ensure that γi(I)cannot exceed 1) and χis an inhibition factor. The
higher the inhibition factor, the less photoinhibition there is at irradiances larger than Iopt. In this study, we use χ= 12, which
is the average value in Follows et al. (2007) for large phytoplankton and corresponds well to published photoinhibition curves25
(Platt et al., 1980; Whitelam and Codd, 1983; Walsh et al., 2001).
For comparison with data, two additional variables can be estimated from the model: primary production and chlorophyll a
concentration. Primary production (P P ) is expressed in mgC m3d1. Our model-based estimate is calculated by multiplying
the phytoplankton concentration and the uptake rate, normalizing from nitrogen to carbon with the 106:16 Redfield molar
ratio (Redfield, 1934) and then converting from amount of substance to mass using the molar mass of carbon (12 g.mol1).30
Chlorophyll a concentration (Chl, in mgCHL m3) is obtained by dividing the phytoplankton concentration in mass of carbon
by a variable carbon to chlorophyll mass ratio (C:Chl). The C:Chl ratio is estimated as in Vallina et al. (2008) using a function
of depth and time developed by Lefèvre et al. (2002), with parameter values calibrated with the observations of Goericke and
Welschmeyer (1998). At the surface, C:Chl is a sinusoidal function of the day of year, varying between a maximum of 160
mgC mgCHL1at the summer solstice and a minimum of 80 mgC mgCHL1at the winter solstice. From the depth where
I(z,t) = 25 W.m2to the bottom, C:Chl decreases linearly with I(z,t) down to a value of 40 mgC mgCHL1when light is
2.2 Aggregate and multi-phenotype models5
Traits xand yhave an infinity of possible values. In order to solve the equations numerically, the problem needs to be simpli-
fied. Two approaches are considered here. In the "multi-phenotype" or "discrete" model approach (Fig. 1b), the trait-space is
discretized and only a finite number of phenotypes, with fixed trait values, are simulated. Phenotypes with intermediate trait
values are neglected. In the "aggregate" or "continuous" model approach (Fig. 1c), the state variables are total phytoplankton
concentration, the mean trait values, the trait variances and the inter-trait covariance. In the continuous-trait model, a specific10
shape of the trait distribution must be assumed a priori (Wirtz and Eckhardt, 1996; Bruggeman and Kooijman, 2007). In the
discrete-trait model, the trait distribution is an emergent property and thus it does not need to be assumed beforehand.
In the multi-phenotype model, only nxvalues of xand nyvalues of yare allowed. The phytoplankton community is divided
into nx×nyphenotypes. The values of both traits are explicitly bounded by xmin,xmax,ymin and ymax . Each phenotype
is separated from its immediate neighbors by a trait interval x=xmaxxmin
nx1on xor a trait interval y=ymaxymin
ny1on y.15
Mutation fluxes at the boundaries (i.e. mutations of the phenotypes with the highest or lowest trait values leading out of the
domain) are set to zero. In the interior of our trait domain, the concentration of the phenotype with the jth value of xand the
kth value of y, noted Pjk, is controlled by the following equation, where aj,k =uj,k gj,k mj,k is the net growth rate:
dt =aj,k(N ,T, I)Pj,k +νx
(∆x)2(2Pj,kuj,k Pj1,k uj1,k Pj+1,kuj+1,k )
(∆y)2(2Pj,kuj,k Pj,k1uj,k1Pj,k+1uj,k+1)(20)20
In all our discrete simulations, we impose xmin =2.5(Kn=0.082 mmolN m3), xmax = +1.5(Kn=4.48 mmolN m3),
ymin = 18C and ymax = 30C. All model values of temperature and DIN concentrations are within these boundaries. We
set nx= 25 and ny= 25 in order to ensure that in most cases xand yare less than the standard deviations of xand y,
respectively. Thus, the total number of discrete phenotypes (x,y) is 25 ×25 = 625.
In the aggregate model, the trait distribution is assumed to be continuous. In this case, and contrary to the multi-phenotype25
case, the trait axes are formally unbounded, although phenotypes with extreme trait values always have low net growth rates,
making them extremely rare. The prognostic variables are six statistical moments of the trait distribution: the total phytoplank-
ton concentration P(t), the mean trait values x(t)and y(t), the trait variances Vx(t)and Vy(t)and the inter-trait covariance
Cxy(t). They are defined as follows:
P(t) = Z Z p(x,y,t)·dxdy (21)
x(t) = 1
P(t)Z Z x·p(x,y,t)·dxdy (22)
y(t) = 1
P(t)Z Z y·p(x,y,t)·dxdy (23)
Vx(t) = 1
P(t)Z Z (xx(t))2p(x,y,t)·dxdy (24)5
Vy(t) = 1
P(t)Z Z (yy(t))2p(x,y,t)·dxdy (25)
Cxy(t) = 1
P(t)Z Z (xx(t))(yy(t))p(x, y, t)·dxdy (26)
The second order moments (Vx,Vyand C) are difficult to interpret directly due to their dimensions. In the analyses, we thus
transform variances into standard deviations (σxand σy) and covariance into correlation (Rxy ) as follows:
σx(t) = pVx(t)(27)10
σy(t) = qVy(t)(28)
Rxy(t) = Cxy (t)
These three diagnostic variables, along with P,xand y, are also computed for the multi-phenotype model for comparison.
The standard deviations have the same dimensions as the mean traits and can thus be compared to them. Ecologically, they
represent trait diversity. Inter-trait correlation is a dimensionless number between -1 and +1, which is easier to interpret than15
the covariance. A correlation of -1 means above-average values of xalways coincide with below-average values of yand vice-
versa. A correlation of +1 means above-average values of xalways coincide with above-average values of y. A correlation of
0 means all combinations are equally possible (i.e. the two traits are independent).
We follow the method developed by Norberg et al. (2001), based on Taylor expansions of the uptake and net growth rates,
to derive the differential equations for the moments of the trait distribution. We assume a bivariate normal distribution of traits,20
which is a generalization of the 1D Gaussian function. Normal distributions are observed in nature for the logarithm of size
(Cermeño and Figueiras, 2008; Quintana et al., 2008; Downing et al., 2014) and are convenient assumptions for models because
they produce the simplest forms for the equations (Wirtz and Eckhardt, 1996; Merico et al., 2009). The derivation is explained
in detail in Appendix B. In the absence of trait diffusion, our equations are a particular case of the general equations derived
by Bruggeman (2009) for multivariate normal trait distributions. In the single trait case, they are simpler than the original25
equations of Merico et al. (2014) and identical to the more recent formulation of Coutinho et al. (2016). The differential
equations followed by the prognostic variables are:
∂t =Pa+1
∂x∂ y (30)
∂t =Vx
∂x +Cxy
∂y (31)
∂t =Vy
∂y +Cxy
∂x (32)
∂t =V2
∂x2+ 2VxCxy
∂x∂y +C2
∂y2+ 2νxu+1
∂x∂ y (33)5
∂t =V2
∂y2+ 2VyCxy
∂x∂y +C2
∂x2+ 2νyu+1
∂x∂ y (34)
∂t =VxCxy
∂x2+ (VxVy+C2
∂x∂y +VyCxy
The net growth rate aand its derivatives with respect to traits are in all cases taken near the mean trait values (xand y) and
for the values of N,Tand Iat time t. The growth rate of the whole phytoplankton community depends first on the growth
rate of the most abundant phenotype ("winner" of the competition), a(x,y,t), with correction terms for the less abundant, and10
generally less fit, phenotypes ("losers"). Mean traits increase when larger trait values are associated with larger net growth rates,
and decrease in the opposite case. The change is faster when trait variances are high. As a consequence, the overall effect of trait
diversity on primary production depends on the environmental conditions. In a stable environment, high trait variances diminish
the primary production, because phenotypes with low growth rates are present. Under frequent disturbances, however, high trait
variances increase the short-term adaptive capacity, allowing the community to maintain mean traits close to the optimum and15
thereby increasing primary production (Smith et al., 2016). We note that due to covariance, the change in each trait depends
on both environmental factors (Nand T). Variances decrease due to competition when mean trait values are close to the values
that maximize the net growth rate (2a
∂x2<0and 2a
∂y2<0). This is most often the case, since phenotypes that are not optimal
tend to be outcompeted. Trait diversity must therefore be maintained by some other process: this is the role of trait diffusion. In
these equations, trait diffusion is a source of variance but does not affect the equations for phytoplankton concentration, mean20
traits, nor covariance. This is coherent with the fact that mutations are symmetrical (no effect on xand y) and neither create nor
remove biomass (no effect on P). Trait diffusion does not affect covariance because mutations of the two traits are independent
of each other. There is no mechanistic relationship between optimal temperature and half-saturation. Mutations can create all
combinations: cold-water gleaners, warm-water gleaners, cold-water opportunists and warm-water opportunists. However, by
increasing variances, trait diffusion decreases the absolute value of correlation. Only the environment can correlate the traits25
by favoring some combinations over others. Although correlation is defined as a local quantity, for a given depth and time, it is
expected to be influenced by the spatio-temporal patterns of environment variations, because local communities always contain
remnants of past communities and migrants from other locations.
2.3 Physical setting
SPEAD 1.0 has one spatial dimension: the vertical. A depth-resolved simulation is the minimal physical setting in the ocean
to resolve the different temperature and nutrient niches and the decisive effect of the vertical mixing on the variances and the
covariance. The model is divided into 20 vertical levels, from surface to 200 m deep, with a uniform vertical step of 10 m.
Two processes can transport matter from one vertical level to another, and thus need to be added to the differential equations.5
First, PON sinks at a speed of w= 1.2 m.d1. Sinking is solved by a first order upwind scheme. Second, tracers are verti-
cally mixed by turbulent diffusion. Vertical turbulent diffusion (called "vertical diffusion" from now on, and unrelated to trait
diffusion) tends to homogenize the spatial distribution of each tracer. It is controlled by the vertical diffusivity parameter κz,
expressed in m2s1. The diffusion of a tracer A is
∂z κz(z,t) A(z,t)
∂z , where z is the vertical dimension. Diffusion operates
on concentrations N, P, Z and D, but not on the phytoplankton trait distribution moments, as these are not material quantities10
and thus are not conserved during mixing. For instance, the mixing of two phytoplankton communities with different mean
traits creates additional variance. However, vertical mixing conserves the sum of phytoplankton trait values (P x and P y ) and
the sum of squared trait values (P(Vxx2),P(Vyy2)and P(Cxy xy)). Therefore we need to convert the trait distribution
moments to P x,P y,P(Vxx2),P(Vyy2)and P(Cxy xy)before applying vertical mixing to them, and then we con-
vert them back to their original values in the next numerical time step to compute growth and loss terms. Vertical diffusion is15
represented by an implicit scheme in order to avoid numerical instability. The depth-resolved model is solved in time with a
fourth-order Runge-Kutta numerical scheme.
Three environmental forcings are necessary to run the model: temperature, PAR and vertical diffusivity. All three depend
on depth and time and have been set to values from the Sargasso Sea. The forcings are seasonal. Interannual variations and
the day/night cycle are neglected. Temperature and surface PAR (I0(t)) directly affect the rates of plankton growth and death.20
They are set for each day using observations collected during the Bermuda Atlantic Time-series Study (BATS) (Steinberg
et al., 2001). PAR availability is assumed to decrease exponentially with depth (I(z, t) = I0(t)ekwz), with a PAR vertical
attenuation coefficient (kw) of 0.04 m1. Self-shading by phytoplankton is neglected. The vertical diffusivity κzis the third
forcing. Contrary to temperature and PAR, it has not directly been observed. Therefore, the turbulent diffusivity comes from
the physical model GOTM for the Sargasso Sea (Bruggeman and Bolding, 2014). All three forcings as functions of time and25
depth are shown in Fig. 3.
2.4 List of simulations
The simulation of the aggregate 2-trait model with mutation rates νx= 0.001 and νy= 0.01 C2is our standard simulation for
this study. νxis expressed without unit because the trait axis xis in logarithmic scale, but like νyit is a variance increase per
generation. Most of the results presented in Figures 4, 5, 6, 7 and 9 come from this standard simulation. The bulk properties30
of SPEAD 1.0 (total primary production, total phytoplankton biomass, nutrient concentration) are validated using observations
from the BATS station in the Sargasso Sea (Steinberg et al., 2001; Vallina et al., 2008; Vallina, 2008). The multi-phenotype
discrete version of SPEAD is used to validate i) the assumption made in the aggregate continuous model that traits are normally
Figure 3. Distribution in depth and time of three environmental variables: a) Temperature, b) and d) Photosynthetically Available Radiation
(PAR) and c) Vertical turbulent diffusivity. The black curve in d) represents the lower limit of the mixed layer.
distributed and ii) the simulated values and tendencies of the moments of the continuous-trait distribution. In order to better
understand the behavior of the model, the standard simulation is also compared to simulations with different mutation rates
and to simulations with adaptive dynamics for only 1 trait, keeping the other trait unable to mutate but at its optimal value.
Trait diffusion is a relatively recent concept and the values of the mutation rates are not yet well calibrated by observations.
To obtain a qualitative idea of the ecosystem model behavior, we tried a wide range of values for νx(from 0.00001 to 0.1).5
The largest value was chosen for its similarity to the trait diffusivity parameter chosen by Merico et al. (2014) and Chen and
Smith (2018) to account for the observed trait variance. However, νx= 0.1allows the phytoplankton to reach a variance of
Vx= 4 in only 20 generations, since 2νxis added to the phytoplankton population variance at each generation. This variance
is the maximum allowed in the discrete model and corresponds to having half the community at each extreme of the trait
axis (x=2.5and x= +1.5). However, laboratory experiments based on single clones show significant evolution only on10
timescales of hundreds to thousands of generations (Schlüter et al., 2016). For this reason we also conducted simulations with
mutation rates as low as 0.00001, and a control simulation without trait diffusion at all (νx= 0). As the mutation rate has the
dimension of trait squared and as the range of temperature is around three times larger than the range of nutrient concentration
logarithms, we fixed the same ratio of mutation rates, νy
νx=10 C2, for all simulations. We checked that departing from this
ratio did not qualitatively affect our results. In total, we conducted simulations for 10 sets of mutation rates, including the15
control case.
A 2-trait model is not simply the superposition of two 1-trait models, for at least two reasons. First, when two environmental
factors limit biomass growth, but only one is included in the model, the simulation is likely to overestimate the phytoplankton
growth rate. Second, when there is a strong inter-trait correlation, each environmental factor impacts both traits. For instance, if
the ambient DIN concentration (N) is below the (geometric) mean half-saturation constant (ex), the competition for nutrients
will select for phenotypes with a lower half-saturation constant. If at the same time the half-saturation is negatively correlated5
with optimal temperature (i.e. if phenotypes with low half-saturation constants tend to also have a high optimal temperatures),
the competition for nutrients will also increase the amount of phenotypes with high optimal temperatures, in addition to the
effect of environment temperature. In the conceptual model of Savage et al. (2007), the inter-trait correlation in a 2-trait model
led to higher variances and to a considerable improvement in the ability of the mean phytoplankton traits to track optimal
values controlled by environmental conditions compared with 1-trait models. In order to know whether these results also apply10
to our model, we compare the dynamics of traits xand yin SPEAD to the dynamics of simplified 1-trait models where either
xor yvary between phenotypes and the other trait is optimized instantaneously (i.e. set to the optimal value at each location
and time given the environmental conditions).
The time step for our simulations is 6 hours. At the first time step and at all vertical levels, DIN concentration is initial-
ized to 1.8 mmolN m3, phytoplankton and zooplankton concentrations to 0.1 mmolN m3, and PON concentration to 0.015
mmolN m3. The total amount of nitrogen in the water column is conserved, and every loss below 200 m due to PON sinking
is compensated by an equivalent gain of DIN, also at 200 m. Mean logarithm of half-saturation and optimal temperature are
initialized at -0.5 (corresponding to Kn=0.61 mmolN m3) and 24C respectively, with initial standard deviations of 0.1 and
0.3C. Each simulation is run for at least 3 years and until convergence is reached. Our convergence criterion is that, for every
day of year and every depth level, the difference between the two last years should be less than 0.1% for P, Vxand Vy, less than20
0.1% of the modeled range for xand yand less than 0.001 for Rxy. In other words, convergence is achieved when the seasonal
cycle of the model state variables is repeated from year to year. The results shown are in all cases from the last simulated year.
We checked that total nitrogen was the only feature in the initial conditions that affected the results.
3 Results
3.1 Bulk modeled properties and comparison with observations25
The first step to validate the SPEAD model is to compare some bulk properties with observations from the Sargasso Sea.
In Fig. 4, the primary production, chlorophyll, DIN and PON concentrations of the aggregate model are compared to a 10-
year climatology of monthly observations of primary production, chlorophyll, nitrate and PON concentrations (Vallina et al.,
2008). With carefully chosen values for the parameters of the ecosystem model (Table 2), the model and observations agree
well, although some minor discrepancies exist, due to the simplicity of our parameterizations.30
Primary production is the state variable best reproduced by the model, with a maximum around 10 mgC m3d1in the first
50 m in February and March in both the model and the observations. From May to September, primary production spreads
slightly more in depth but is overall around half its maximal value. Primary production is negligible deeper then 80 m.
Figure 4. Distribution in depth and time of a) model primary production, c) chlorophyll concentration, e) dissolved inorganic nitrogen
concentration and g) particulate organic nitrogen concentration. Each variable is compared with equivalent observations in the Sargasso Sea
(b, d, f and h).
Chlorophyll concentration is reproduced with the right order of magnitude, an absolute maximum correctly located in Febru-
ary between 50 m and 80 m deep, at about 0.25 mgCHL.m3, and a deep chlorophyll maximum around 100 m in Summer.
However, the model chlorophyll concentration is lower than the observations in Spring and higher in late Autumn. Chloro-
phyll concentration begins to increase in December in the model but only in February in the observations. Both chlorophyll
concentration and primary production are proportional to phytoplankton concentration. The reason for the temporal mismatch5
between SPEAD and the observations in chlorophyll concentration, but not in primary production, must then be related to the
temporal variability of the other factors affecting these two quantities: the nitrogen uptake rate and the C:CHL ratio. The rela-
tively high primary production and very low chlorophyll concentration observed in December might be accounted for better if
the uptake rate were faster in December than in February, despite the lower availability of nutrients and light, or if turbulence
were included in the estimation of the C:CHL ratio so that it reaches its lowest value in February, when the waters are best10
mixed and phytoplankton cannot stay close to the surface (Taylor et al., 1997), rather than in December, when the surface light
intensity is minimum (Lefèvre et al., 2002; Jakobsen and Markager, 2016).
Dissolved Inorganic Nitrogen is compared to the observed nitrate concentration, knowing that this form of DIN dominates at
depth but co-occurs with nitrite and ammonium, which are also components of DIN. The modeled DIN and the observed nitrate
concentrations share the same range, with a maximum of 2.8 mmol m3in the observations and 3.3 mmol m3in the model.
Another common point between the model and the observations is that both reach a maximum at the bottom of our setting,
at 200 m, with concentrations between 2.1 and 2.8 mmol m3during most of the year. Because of the strong vertical mixing,5
from February to April, the concentrations are lower, between 1.5 and 2.0 mmolm3, but still a maximum. However, from
June to January, the modeled DIN concentration exhibits a second maximum that is absent from the observations. This second
maximum is located just below the euphotic layer, between 100 m and 150 m deep, with values as high as 3.3 mmol m3
in late November. Both modeled and observated concentrations are minimal at the surface, due to the nitrogen uptake by
phytoplankton. However, their values diverge by more than one order of magnitude. Modeled DIN concentrations at the surface10
vary between 0.18 and 1.34 mmol m3, with a mean of 0.67 mmol m3, whereas observed nitrate concentrations vary between
0 and 0.11 mmol m3, with a mean of 0.03 mmol m3. We assume that these discrepancies are due to the contribution of
ammonium, and possibly nitrite, since the few studies reporting measured concentrations of ammonium in the Sargasso Sea
(Menzel and Spaeth, 1962; Brzezinski, 1988) showed that ammonium was more homogeneously distributed in the upper 200
m than nitrate and was the dominant form of dissolved nitrogen from surface to 100 m deep.15
Particulate organic nitrogen distributions from the model and observations are relatively similar, with a maximum around 0.5
mmol m3(0.45 mmol m3in the observations, 0.51 mmol m3in the model) in April and May at depths between 30 m and
80 m. However, the seasonality and vertical gradients are much larger in the model, where particles are very rare in Autumn
and nearly absent at depths greater than 100 m, whereas observed PON concentrations are never below 0.08 mmol m3. The
observations might be better explained if a minority of particle production went to a slowly remineralizing refractory pool,20
enabling them to stay during the whole year and to reach greater depths (Aumont et al., 2017), but we did not increase the
complexity of the particle parameterization because this is not the focus of our study.
3.2 Trait distribution of SPEAD and comparison with a multi-phenotype model
The second step to validate the aggregate SPEAD model and the only validation of its bivariate trait distribution is done by
comparing it to the multi-phenotype model. Although both are models and thus simplify reality in similar ways, the multi-25
phenotype model is used as a reference for two reasons. First, it is more intuitive than the aggregate model, with birth and
death processes and mutations to the nearest neighbors as the only terms in the equations. Therefore, the moments of the trait
distribution in the multi-phenotype model can be used as a control to confirm that the equations of the aggregate model are
correct. Second, the multi-phenotype model does not assume any particular trait distribution shape and can be used to validate
the a priori assumption of the aggregate model that the trait distribution is a bivariate normal distribution.30
The spatial and temporal patterns of phytoplankton concentration, mean traits, trait standard deviations and inter-trait cor-
relation for the standard simulation with mutations rates of νx= 0.001 and νy= 0.01 C2are shown in Fig. 5. The value of x
varies between -0.83 (Kn=0.44 mmolN m3) and +0.6 (Kn=1.82 mmolN m3), with standard deviations between 0.31 and
0.77. The value of yvaries between 22.0 C and 26.1 C, with standard deviations between 0.81 C and 1.92 C. By compari-
son, the modeled DIN concentration varies between 0.18 and 3.31 mmolN m3, and the water temperature varies between 18.5
and 27.8 C. As expected, the mean trait values remain consistently within the range of the environmental drivers to which they
adapt. Because the best competitor at a given time and depth needs tens of generations to become dominant after having been
a rare phenotype, the mean traits react with a delay of 1 to 2 months and with a lower amplitude than their drivers. Cold-water
opportunists (high x-trait, low y-trait) dominate in Winter and Spring throughout the water column. In Summer, they are slowly5
replaced by warm-water gleaners (low x-trait, high y-trait) in the upper 70 m but retain dominance at greater depths, where
their half-saturation constants continue increasing and their optimal temperatures continue decreasing. The coexistence of two
very distinct communities in Summer and early Autumn is made possible by the intense stratification, which creates a physical
barrier between the different depth levels. In late Autumn, the two communities are rapidly mixed by the vertical turbulent
diffusion, producing a peak in the standard deviation of each trait, in other words a peak in the local (alpha) diversity. Then,10
as the water column becomes more homogeneous, competition selects for a single dominant phenotype, reducing the trait
diversity until the next Autumn. Inter-trait correlation is negative at all times and depths, due to the negative correlation of the
environmental drivers. High DIN concentrations generally coincide with low temperatures, favoring cold-water opportunists.
This happens during Winter because turbulent mixing brings nutrient-rich cold waters from the deep layers up to the surface.
During Summer, the consumption of nutrients by primary producers leads to a coincidence of warm temperatures with low DIN15
concentrations at the surface. In late Autumn, the negative correlation reaches its maximum absolute value when the two main
communities are suddenly mixed. During the rest of the year, trait diffusion progressively reduces the inter-trait correlation.
In Fig. 6, the state variables of the aggregate model are compared to the trait distribution moments of the discrete model.
The discrete model is considered as a "truth" and the difference between the two models as an "error", positive if the value is
higher in the aggregate model. The aggregate model reproduces Pand xvery precisely, with linear determination coefficients20
(R2) of 0.998 and 0.988 respectively. The biases (mean error) are very low: +0.0005 mmolN m3on Pand -0.04 on x. The
bias of yis larger, at -0.50 C, but the coefficient of determination is still very high, at R2= 0.862. The error is largest in
the deep community in Summer and early Autumn, reaching a maximum of -1.51 C in early September around 100 m. The
most likely reason why the aggregate model underestimates y, but not x, is that the response of phytoplankton to temperature
is asymmetrical. Increases in the environment temperature put more selective pressure on the phytoplankton community than25
decreases. This feature is poorly taken into account by the aggregate SPEAD model because of its assumption that traits are
normally distributed. There is a mismatch between the asymmetrical shape of temperature niches and the imposed symmetrical
shape of the distribution of optimal temperatures (y-trait) in the aggregate model. This mismatch does not happen for the x-
trait. As is typically the case in aggregate models, there are more errors in the higher order moments, in our case the standard
deviations and the inter-trait correlation. The coefficients of determination for σx,σyand Rxy are R2= 0.813, R2= 0.462 and30
R2= 0.896 respectively. Their biases are -0.04, +0.09 C and -0.001 respectively, which is around 10% of the mean value for
σyand σxand negligible for Rxy. All three variables decrease much faster in early Winter in the aggregate model than in the
multi-phenotype model. Additionally, in Summer, there is a strong discrepancy for σy. In the discrete model, σycan reach as
low as 0.57 C but in the aggregate model it is never less than 0.81 C and very rarely less than 1.0 C.
Figure 5. Distribution in depth and time of trait distribution moments for νx= 0.001: a) phytoplankton concentration, b) (geometric) mean
half-saturation, c) optimal temperature, d) inter-trait correlation, e) half-saturation logarithm standard deviation and f) optimal temperature
standard deviation. For readability, the mean value of trait xis transformed into a nitrogen concentration in b). To speak properly and contrary
to other means present in this study, the "mean half-saturation" is a geometric mean, not an arithmetic mean.
The main errors on σx,σyand Rxy are caused the aggregate model’s assumption of a multi-variate normal distribution,
which is not strictly correct based on the results of the discretely resolved model. In Fig. 7, we show in a 2D color plot
how the two traits are distributed in the discrete model at three different depths (surface, 50 m and 100 m) at the end of
each season (March, June, September, December). This distribution is compared with the bivariate normal distribution of the
aggregate model, represented by ellipses. In March, when the waters are well-mixed, and in June, when the stratification has5
just begun, the traits are normally distributed and the two models agree. There is only a small error on the distribution of
optimal temperature. In March at all depths and in June at 100 m, the normal distribution of the aggregate model contains
more phenotypes with low optimal temperatures than the distribution of the discrete model. In Summer, the traits are also
normally distributed near the surface. However, the distribution of optimal temperature is markedly right-skewed deeper in
the water column. Optimal temperatures below that of the most common phenotypes are extremely rare whereas those larger10
than this level are more common. The aggregate model has its ybelow the peak of the multi-phenotype model and has a
much larger σy. This is the largest error for both a mean trait value and a trait standard deviation in this study. Since nothing
similar occurs with half-saturation, this error must be linked with the right skew in the temperature-dependent growth factor
when expressed as a function of optimal temperature (Fig. 2d). In stable environments with little change in temperature with
time and little vertical mixing, the distribution of optimal temperature tends to become naturally right-skewed. However, our15
Figure 6. Comparison at all depths and time of the aggregate and multi-phenotype model state variables for νx= 0.001: a) total phytoplank-
ton concentration, b) (geometric) mean half-saturation, c) mean optimal temperature, d) inter-trait correlation, e) half-saturation logarithm
standard deviation and f) optimal temperature standard deviation. Blue is Winter, green is Spring, red is Summer and black is Autumn.
results show that re-mixing (in late Autumn), fast environmental change (near the surface) and trait diffusion can reduce or
eliminate this skew, so that the trait distribution is often close to normality. In December during the re-mixing phase, the
trait distribution completely deviates from normality and becomes bimodal, with a community of warm-water gleaners and a
community of cold-water opportunists co-occurring throughout the water column. At this time the standard deviations and the
inter-trait correlation are at their annual maxima. The moments of the trait distribution at that time are very well captured by the5
aggregate model. However, assuming a normal trait distribution is not only wrong in terms of ecological description but also
leads to incorrect dynamics during Winter. In Winter, the ecological selection in the now mixed waters reduces trait diversity,
and trait diffusion reduces the inter-trait correlation. These processes occur faster in the aggregate model than in the multi-
phenotype model (see Fig. 6) because the selective pressure is larger for a normal distribution than for a bimodal one. From
a mathematical point of view, this can be shown in a simplified 1-trait model. Selection through competition reduces the trait10
variance at a speed equal to 1
∂x2(this is a 1-trait unskewed version of equation B7), where M4is the fourth order
moment or "kurtosis". In a Gaussian distribution, M4= 3V2
x. Bimodal distributions have a lower kurtosis, therefore they are
affected more slowly by ecological selection. By design, the aggregate model cannot account for this effect because it assumes
a unimodal Gaussian distribution. From a more ecological point of view, it can be noted that in order to replace a bimodal
distribution by a unimodal one with a smaller variance, a previously rare intermediate phenotype must rise to prominence15
Figure 7. Concentrations of each phenotype of the multi-phenotype model (color) compared with lines of equal density of the aggregate
model. Subplots correspond to days 71, 161, 251 and 341, and depths of 0, 50 and 100 m. Dashed lines indicate the optimal competitor.
and previously dominant phenotypes must become rare, which is a dramatic change. By comparison, in an already unimodal
Gaussian distribution, reducing the variance only means making rare and extreme phenotypes even rarer.
3.3 Trait dynamics with different mutation rates
In this section, we compare the results of simulations conducted with 9 different sets of mutation rates, from νx= 0.00001 and
νy= 0.0001 C2to νx= 0.1and νy= 1.0C2. A control simulation with no trait diffusion is also included. That amounts to5
a total of 10 simulations. The ratio of mutation rates, νy
νx=10 C2, is the same in all simulations. This comparison highlights
the unique role played by trait diffusion in SPEAD, even at very low mutation rates.
For each simulation, Fig. 8 shows the values of depth-integrated primary production per year and the yearly averaged values
and ranges of x,y,σx,σyand Rxy. Additional diagnostics are presented in Table 3. The number of years to converge to
a steady state, beyond being just a numerical issue, can also serve as an ecological indicator of the time needed to damp a10
perturbation or to adapt to a new physical setting, although, by design, our convergence times cannot be less than 3 years. The
Control simulation does not fully converge even after decades and we decided to run it 38 years, which is the convergence
time of the simulation with the lowest non-zero mutation rates. However, we find that the trait diversity of the Control scenario
Figure 8. Primary production and trait distribution moments for different mutation rates. The ratio νy
νxis kept constant and equal to 10C2.
Moment ranges are represented by error bars, and their mean by dots. The mean traits are compared to the extreme values of their environ-
mental drivers (dissolved inorganic nitrogen concentration and temperature). The trait standard deviations are compared to their values in a
uniform distribution within the boundaries of the discrete model.
does not collapse to zero as we expected because the standard deviations of its xand ytraits ends up being higher than their
initial values and continue slowly increasing. Yet, the standard deviations of the Control scenario is significantly smaller than
for the scenarios with non-zero trait diffusion. For either trait, the maximum value of V
2νis the number of generations required
to reach the highest trait variances of the simulation in the absence of ecological selection and with trait diffusion as the only
source of variance. Although highly idealized, this number is a proxy for the timescale of evolutionary processes. Table 3 also5
assesses whether bimodality occurs at some point in the year in the discrete model and whether the mean traits come within
one standard deviation of the discrete model boundaries.
Primary production is around 146.9 gC m2yr1for all mutation rates between νx= 0.00001 and νx= 0.001, then de-
creases at higher trait diffusivities to finally reach 130.8 gCm2yr1for νx= 0.1. The primary production of the control
simulation is 146.5 gC m2yr1. This result agrees with the model of Chen et al. (2019) as applied to the North Pacific.10
Their phytoplankton community was characterized by one trait, cell size, which is somehow related to our half-saturation trait
x. They found that primary production was diminished when νxincreased, but they only considered relatively high mutation
rates between 0.01 and 0.1, as well as a control simulation. Under relatively stable conditions, we find that fast mutation
rates (νx>0.01) are a drawback for primary production because they promote large trait variances, allowing non-competitive
Table 3. Convergence time and various properties of SPEAD 1.0 simulations with different mutation rates
νxνyConvergence time maxV
2νOut of range Bimodality Adapts faster
– [C2] [years] [generations] (discrete model) with 2 traits
0 0 No Yes Yes
0.00001 0.0001 38 12512 No Yes Yes
0.00003 0.0003 14 4213 No Yes Yes
0.0001 0.001 10 1363 No Yes Yes
0.0003 0.003 7 631 No Yes Yes
0.001 0.01 5 297 No Yes No
0.003 0.03 4 139 No Yes No
0.01 0.1 4 53 Yes No No
0.03 0.3 4 21 Yes No No
0.1 1 3 7 Yes No No
phenotypes (i.e. under-performers) to proliferate. However, phytoplankton mutating very fast could be invaded by phenotypes
mutating more slowly. Therefore we do not expect them to be common in nature.
In the simulations with νx= 0.01,νx= 0.03 and νx= 0.1, the mean trait values remain close to their environmental drivers
and their range over the year is as wide as that of DIN concentration and temperature, respectively. On average, the community
adapts nearly instantaneously to its environment. However, the cost for this apparent success in fast-tracking the environmental5
conditions is that the standard deviations of the trait distribution are very high and close to that of a uniform distribution
between the trait boundaries of the discrete model. Given that the trait domain of the discrete model is already wider than the
ranges of DIN concentration and in-situ temperature, this result suggests that either 1) phenotypes that are maladapted at all
depths and throughout the year are common (explaining the low primary production) or 2) that we have reached the limit of
validity of the aggregate approach. These simulations do not have skewed or bimodal distributions, and their correlations are10
negligible, even in December when the stratification is broken, because trait diffusion is a symmetrical process that constantly
replenishes all rare phenotypes, including warm-water opportunists and cold-water gleaners that are maladapted at all depths
and during all the year. The simulations with large mutation rates converge in 3 or 4 years and can sustain their variances in
less than 60 generations, that is, in less than a year. They use mutations to follow the seasonal cycle of their environment faster
than the usual timescales of evolution, even for phytoplankton (Schlüter et al., 2016).15
When the mutation rates are lower, the mean traits still vary during the year but not as much nor as fast as the physical
environment, and no phenotypes are found outside of the trait domain of the discrete model. With νxat 0.001 or lower, several
years are required to sustain the variance and to converge to a seasonally stable state. In this case, the mutations create variance
over the long term, facilitating the ecological successions of phenotypes seasonally and the adaptive evolution inter-annually.
However, low mutation rates do not allow the community to evolve seasonally. Bimodality is present in the discrete version,20
at least during the late Autumn mixing, and lasts longer as the mutation rates decrease. The variances increase when the trait
diffusivity parameters increase, which is what trait diffusion was designed for. We note that, contrary to chemostat models
(Merico et al., 2009), SPEAD 1.0 does not require trait diffusion to sustain a positive trait diversity: the trait variances do not
collapse to zero even in the absence of trait diffusion. The late Autumn mixing is a source of variance in its own right, avoiding
the collapse of trait diversity even in the Control simulation. However, the trait standard deviations in the Control case are very5
low, between 0.12 and 0.41 for x. Trait diversity appears even lower when accounting for the fact that correlation between x
and yis blocked at -1. The xand the ytraits totally determine each other, as if there were only one trait and no extra degree of
freedom. The only active phenotypes are located on a straight line. Trait diffusion is not necessary to sustain variance, but it is
necessary to allow the model to explore the entire trait space and to adapt to entirely new sets of environmental conditions.
Increasing trait diffusion to νx= 0.00001 does not lead to any significant increase in trait variance, which keeps being10
extremely low. The variance is still overwhelmingly controlled by the December mixing, producing very large correlations.
Just above this level, the cases from νx= 0.00003 to νx= 0.0003 share features that are coherent with our expectations on the
effect of inter-generational mutations: a high primary production, a moderately high but never total correlation and timescales
of a few years ( 100s to 1000s of generations) to adapt to their environment.
3.4 Trait dynamics compared with 1-trait models15
In Figs. 9 and 10, the trait distribution moments of SPEAD at the surface are compared with the environmental drivers (DIN
concentration and water temperature) and with the outputs of two single-trait aggregate models, where only the half-saturation
constant or only optimal temperature is allowed to vary between phenotypes, subject to trait diffusion. Figure 9 shows the
comparison for the standard values of the mutation rates: νx= 0.001 and νy= 0.01 C2. However, the differences between
2-trait and 1-trait models are likely to be larger when correlations between xand yare large. Therefore, in Fig. 10, we compare20
the 2-trait and 1-trait model distributions obtained with the lowest non-zero mutation rates: νx= 0.00001 and νy= 0.0001 C2,
which lead to inter-trait correlations between -0.8 and -1 in the 2-trait model.
With standard mutation rates, the trait dynamics are very similar in all three models. The 2-trait model has slightly lower
standard deviations than the 1-trait models during some parts of the year, but the difference is always within 10%. The seasonal
patterns are very similar in both timing and amplitude. The greatest differences are found in Summer, from mid-June to mid-25
August, when the 1-trait model with adaptive dynamics for half-saturation has a greater phytoplankton concentration by as
much as 29% and a lower nutrient concentration by as much as 24% compared to the other two models, which concentrations
are very similar to each other. This result means that at the onset of Summer, the most important factor decreasing the ability
of phytoplankton to grow is not the lack of nutrients, but temperature itself. In other words, the phenotypes that dominated
in Spring decline, not because they are not adapted to oligotrophic conditions, but because they are not adapted to the high30
temperatures of Summer and the growth rate of a phenotype declines sharply when temperature exceeds its optimal value. This
effect is negligible at the highest mutation rates, because in this case the community is able to evolve and adapt very quickly
to the Summer warming, but becomes more important as the mutation rates, and hence the optimal temperature variances,
Figure 9. SPEAD state variables at surface for νx= 0.001 (standard) compared with the state variables of 1-trait models: a) total phytoplank-
ton concentration, b) (geometric) mean half-saturation, c) mean optimal temperature, d) inter-trait correlation, e) half-saturation logarithm
standard deviation and f) optimal temperature standard deviation. The dashed lines represent the environmental drivers.
The differences between models are larger at low mutation rates. With the smallest non-zero mutation rates, the Summer
difference in phytoplankton biomass increases. The 1-trait half-saturation model has now a phytoplankton biomass as much as
57% greater and a DIN concentration as much as 30% smaller than in the other models. Trait variances are again lower in the
2-trait model during most of the year but sometimes exceed the 1-trait variances during the Autumn mixing. However, the most
notable change is that the seasonal amplitude of mean half-saturation (x) is now 56% higher in the 2-trait model than in the5
1-trait half-saturation model. Having a second trait allows the ecosystem to adapt faster to environmental changes. This effect
is even more notable when considering that both the half-saturation variance and the nutrient-mediated selective pressure are
lower in the 2-trait model. This effect does not extend to the other trait, although the mean optimal temperature of the 2-trait
model and that of the 1-trait optimal temperature model sometimes show slight departure from each other, in a seasonally
dependent way.10
The effects described above are related to inter-trait correlation, which is driven by correlated environmental conditions and
becomes very large in the case of low mutation rates. Equations 30 to 35 can help understand the effect of trait correlation on the
seasonality of mean traits and trait variances. In the mean-trait equations (31 and 32), correlation implies that both temperature
and DIN concentration drive changes in both mean trait values. The covariance term can either accelerate or slow down the
response of each mean trait, but generally the sign of covariance is such that the change is accelerated. This is what occurs from15
Figure 10. SPEAD state variables at surface for νx= 0.00001 (low mutation rate) compared with the state variables of 1-trait models: a) total
phytoplankton concentration, b) (geometric) mean half-saturation, c) mean optimal temperature, d) inter-trait correlation, e) half-saturation
logarithm standard deviation and f) optimal temperature standard deviation. The dashed lines represent the environmental drivers.
December to March, when the environment selects for higher half-saturation constants and lower optimal temperatures, and the
environmentally induced negative correlation between traits further accelerates this adaptation. From June to October, the same
effect occurs but is significant only for half-saturation. During these months, the 2-trait model actually experiences a slower
increase in optimal temperature then the 1-trait model because it has a smaller variance and because the selective pressure
of high temperatures is much sharper than the nutrient-mediated pressure conveyed by the correlative term. In November,5
correlation has the opposite effect: as the temperature decreases while the DIN concentration remains low, the environment
at that time selects for both low optimal temperature and low half-saturation, and the negative correlation prevents optimal
temperature from decreasing.
The effect of correlation on variance is even more convoluted. In Equations 33 and 34, inter-trait correlation adds a second
variance-reducing competition term (C2
xy 2a
∂y2and C2
xy 2a
∂x2, respectively, both very likely to be negative). This is why variances10
are smaller in the 2-trait model during most of the year. However, one source of variance is not accounted for in these equations:
vertical mixing. Trait variance is not a conservative tracer. Indeed, mixing two communities with different mean trait values
"creates" additional variance. As phytoplankton adapts better to their environment in the 2-trait model than in the 1-trait models,
the difference between surface and sub-surface communities when the water column is stratified is larger in the 2-trait model,
and therefore the late Autumn mixing event adds more trait variance in the 2-trait model. This is why the variances are higher
in the 2-trait model than in the 1-trait models in December.
4 Discussion
4.1 Strengths and weaknesses of aggregate models
SPEAD is an aggregate phytoplankton model. Aggregate models, used as far as we know since Wirtz and Eckhardt (1996),5
do not compute the abundance of each phytoplankton species as discrete entities, but represent the phytoplankton community
by its total biomass together with the mean values, variances and covariances of a few key traits controlling its competitive
ability along different environmental gradients. Aggregate models are known to reduce the computational cost of ecosystem
models by at least one order of magnitude. In a 0D physical setting (i.e. not spatially resolved), the single-trait aggregate model
of Acevedo-Trejos et al. (2016) was found to be 18 times faster to run than its alternative discrete model with as few as 1010
phenotypes. SPEAD 1.0 is 70 times faster in its aggregate version than in its alternative discrete version with 25 ×25 = 625
phenotypes, despite the two model versions having to compute the same number of non-phytoplankton variables. The exact
factor of cost reduction depends on the number of phenotypes used in the discrete model, and it is likely to be even larger if the
number of traits increases. The computational cost of a multi-phenotype model with phenotypes covering the entire trait domain
is an exponential function of the number of traits. For instance, allowing 25 values of optimal irradiance would multiply the cost15
of our discrete model by 25. By contrast, the cost of an aggregate model is a quadratic function of the number of traits. In our
case, adding a third trait would simply increase the number of phytoplankton state variables from 6 to 10 (total phytoplankton
concentration, 3 mean traits, 3 trait variances, 3 inter-trait covariances) and add a few new terms in each equation, which is
far less computationally demanding. This makes aggregate models promising tools to explore high-dimensional trait spaces of
the ecology and evolution of microbial ecotypes (?). We note that some discrete models use an approach different from the20
one described in our study: they run several simulations with a relatively low number of randomly sampled phenotypes, and
then make an ensemble mean of all their simulations. For instance, in Follows et al. (2007), each member of the ensemble
contains 78 random phenotypes, which is not a particularly large number given that they have 4 functional types and that their
traits include half-saturation constants for several nutrients, optimal temperature, and optimal irradiance. Still, running 10 such
simulations to compute an ensemble mean is computationally expensive. Aggregate models do not need ensemble means for25
sampling the trait space since they are continuous by design: their phytoplankton communities fill the trait space completely,
without the need of any arbitrary sampling.
Aggregate models are very efficient because their state variables are the quantities that make most ecological sense. In ther-
modynamics, computing the trajectory of each atom or molecule is not only unfeasible, but also of little use. The collection of
trajectories does not provide more information on the macroscopic behavior of a thermodynamic system than aggregate prop-30
erties such as temperature, pressure and density. Equally, modeling the dynamics of thousands of species would be incredibly
costly, and sufficient observational data would not be available to validate the models. Furthermore, the results would also be
extremely difficult to interpret (Levins, 1966). Given that the most important quantities for understanding a community of
species with similar niches are biomass, followed by mean trait values and trait diversity, the aggregate model focuses com-
putational power where it is most needed, without much loss of information. Aggregate models also explicitly quantify the
factors controlling biodiversity, such as the second derivatives of the net growth rate and the trait diffusivity parameters (Chen
et al., 2019).
However, the aggregate approach has one major weakness: a specific shape for the trait distribution must be assumed a priori,5
with only as many degrees of freedom as there are free parameters (Wirtz and Eckhardt, 1996; Bruggeman and Kooijman,
2007). There is no universal distribution shape for phytoplankton traits, which is why the equations describing their dynamics
are not as precise as the equations of thermodynamics. In this study, we assumed that optimal temperature was normally
distributed and that half-saturation constant was lognormally distributed. This Gaussian closure was chosen because of its
simple moment equations and low number of free or arbitrary parameters. Normally distributed temperature and irradiance10
niches have been observed by Irwin et al. (2015). Half-saturation constant is strongly correlated to cell size (Litchman et al.,
2007; Edwards et al., 2012), and lognormal distributions have been observed in nature for size (Cermeño and Figueiras, 2008;
Quintana et al., 2008; Schartau et al., 2010; Downing et al., 2014; Marañon, 2015), although not in all cases. Another very
common distribution is the power (or "log-linear") law (Rodríguez, 1994; Cermeño and Figueiras, 2008; Huete-Ortega et al.,
2012), although the power law must be truncated on at least one side. We note that a power law distribution with a cutoff on the15
left might be better able to represent the right-skewed size distribution (even in logarithmic scale) of oligotrophic environments
where Prochlorococcus, the smallest known phytoplankton, dominates and coexists only with larger species (Marañon, 2015).
What neither a unimodal nor a power law distribution can capture is bimodality, which is known to occur at least in lakes, due
to common herbivores, in particular daphnids, feeding optimally on preys of intermediate size (Gaedke and Klauschies, 2017).
Normal distributions are symmetrical, unimodal and unbounded. If the real trait distribution deviates from these three prop-20
erties, errors will arise in aggregate models based on a normal distribution. Not only is information lacking by not including
higher-order moments such as skewness and kurtosis, but the dynamics of mean traits and trait variances could be signifi-
cantly altered. If the trait distribution is skewed, the community will respond faster to a certain type of perturbation than to
the opposite perturbation. For instance, if optimal temperature is right-skewed, the phytoplankton community will adapt faster
to warming than to cooling environmental conditions. Phytoplankton with larger optimal temperature will need less time to25
become dominant in case of warming than cold-water phenotypes in case of cooling because they will start from a higher
concentration. To express the above in terms of moment equations, the variance of a right-skewed distribution increases when
the environment favors larger trait values, thus facilitating the adaptation, but decreases when the environment favors smaller
trait values (see Appendix B and the neglected term M30 in equation B7). If the trait distribution is multimodal, the reduction
in trait diversity induced by competitive exclusion (Hardin, 1960) will be slower. This is because replacing all pre-existing30
communities by intermediate and previously rare phenotypes takes more time than making the most abundant phenotype even
more abundant and the rarest even rarer, as in a unimodal distribution. (see Appendix B and the neglected kurtosis term M40
in equation B7, knowing that multimodal distributions have low kurtosis).
Normal distributions are unbounded, with the assumption that extreme values are rare and ecologically meaningless. The
consequence of this apparently reasonable assumption is that model phytoplankton can adapt to any environmental change if35
they are given enough time, irrespective of the intensity of that change. This contrasts with expectations on the behavior of
real phytoplankton communities, as explained in the following example. If a closed (i.e. without immigration) phytoplankton
community experiences temperatures between 15 C and 25 C, the local phenotypes should be adapted to temperatures
between 15 C and 25 C and not a single individual should be optimized for temperatures out of the boundaries, since it
would be outcompeted at all places and times. If the environment suddenly warms, the phenotypes with an optimal temperature5
closer to 25C should come to dominate. However, if temperature reaches 30C, no phenotype with an optimal temperature of
30C can rapidly come to dominate, since no such phenotype pre-exist in the system. Adaptation to temperatures larger than
25C can only occur through mutations or immigration. In an aggregate model, however, none of these processes are required.
Phytoplankton will be able to adapt to any warming because an extremely small but non-zero biomass of phenotypes adapted
to very high temperatures is always present by model design and can become dominant if the environment selects them.10
In the present study, the aggregate (continuous-trait) model agrees very well with a multi-phenotype (discrete-trait) model,
where no distribution shape is imposed but trait distribution is spontaneously close to normality during most of the year.
Skewness and kurtosis occur during some times of the year, only to be removed later, and do not strongly impact our estimates
of the lower order moments. The assumed normal trait distribution is symmetrical and unimodal, therefore some errors occur
when the trait distribution is skewed or bimodal. The mean optimal temperature is slightly underestimated and its variance is15
overestimated because SPEAD does not account for the slight right skew of optimal temperatures distributions. The other main
error is that variance tends to decrease too fast in Winter, after the remixing of the previously stratified water column, because
SPEAD cannot account for bimodality. The seasonal cycle and the orders of magnitude, however, are accurate. Our results
are similar to that of Acevedo-Trejos et al. (2016). However, other studies show much larger errors (Coutinho et al., 2016;
Klauschies et al., 2018) and consider Gaussian-based aggregate models to be inaccurate.20
Whether the trait distribution of a model ecosystem is normal or not depends on the ecological processes included by the
modeler. At least two factors in SPEAD play in favor of a normal distribution. The first factor is trait diffusion. In a fluid, the
diffusive movement of a tracer follows a Gaussian law, provided that the diffusivity coefficient is constant (Einstein, 1905).
Trait diffusion plays the same role here for traits and tends to erase skewness and bimodality. The second factor is the simplicity
of our ecological model. All our phytoplankton phenotypes compete for the same resource. In a given environment defined25
by nutrient concentration and temperature, there is a single most competitive phenotype and the phytoplankton net growth
rate decreases continuously when moving away from this optimum. Grazing and mortality rates do not act against this trend
because we impose them to be identical for all phenotypes. Conversely, two factors in SPEAD play against normality, but their
reach is relatively minor. The asymmetry of the temperature response curve promotes right-skewed distributions of optimal
temperature. However, and despite the naive expectation, this right-skew is generally not large in the discrete model because30
the standard deviation of optimal temperature is always smaller than the temperature tolerance (T= 5 C). The second effect
is the alternation of stratification and mixing during the year. In Summer, stratification leads to the formation of two distinct
communities in surface and in subsurface. When the vertical mixing strengthen again in late Autumn, the two communities
mix into a temporary bimodal distribution.
Other ecological settings yield more widespread multimodality. Multimodality can be induced by immigration (Norberg
et al., 2001), resting stages (Beckmann et al., 2019), fast environmental oscillations (Beckmann et al., 2019), spatially hetero-
geneous environments (Wickman et al., 2019), "convex" trade-offs favoring extreme phenotypes (Coutinho et al., 2016), and
zooplankton prey selectivity (Wirtz, 2013; Klauschies et al., 2018). In particular, evolutionary branching of both phytoplankton
and zooplankton into tens of size clusters can occur when each zooplankton grazes only on a small size range (Sauterey et al.,5
2017). However, it is important to point out that only fixed zooplankton preferences cause disruptive selection. By contrast,
active switching by grazers ("Kill The Winner") is insufficient to promote evolutionary branching as by design it promotes
uniform distributions, flattening the peaks and filling the gaps in trait distributions. Using a Gaussian-based aggregate model in
a setting promoting branching would of course be inappropriate: SPEAD would be unable to simulate evolutionary branching
of phenotypes in these kind of ecological scenarios10
Alternatives to Gaussian closures have been proposed since early in the development of aggregate models. Norberg et al.
(2001) and Norberg (2004) used more complex closures to estimate skewness and kurtosis. However, these closures had
free parameters, varying from ecosystem to ecosystem, and a discrete model was required to compute them, canceling the
advantage of aggregate models in terms of computational cost. Klauschies et al. (2018) replaced the normal distribution
by a beta distribution. The beta distribution is bounded, allows bimodality, and was proven to increase the realism of trait-15
based aggregate models in a bounded trait scenario. However, applying this method requires defining fixed boundaries for
phytoplankton traits. Phytoplankton has a minimum size (and half-saturation) at 0.5 µm, which is the size of Prochlorococcus,
but it does not have a well-defined maximum. Also, optimal temperature at local scales does not have clear boundaries either.
Therefore, any set of boundaries would be arbitrary and might prevent further adaptation to changing environments beyond
those limits.20
A more practical approach to account for non-Gaussian distributions would be to divide the community into several func-
tional groups, each one having a normal trait distribution of its own (Terseleer et al., 2014; Chen and Laws, 2017). The sum of
these communities can have a skewed or multimodal trait distribution. Trait variances (in particular size variance) could be high
within phytoplankton as a whole, without bolstering the adaptive capacity of each functional group (see 4.3.). All phenotypes
within a given functional groups must feed on the same nutrients, be subject to the same trade-offs and should not be subject25
to processes promoting evolutionary branching. Ideally, and in order to prevent the convergence of all functional groups on the
same trait values, each group should have distinct qualitative properties or trade-offs. Functional groups could include diatoms,
mixotrophs, diazotrophs or Prochlorococcus, among others. Two communities defined by the same parameters could coexist
and avoid merging if an intermediate trait range is permanently disadvantageous, due for instance to a convex trade-off or to
a size-specific grazer. The multi-Gaussian approach would combine the moderate computational cost of Gaussian aggregate30
models with the more thorough description of planktonic ecosystems allowed by discrete models.
4.2 Traits in phytoplankton community models
In nature, many different traits define phytoplankton niches: nitrogen, phosphorus or iron uptake abilities, requirements in
other nutrients (for instance silica or calcite), stoichiometry, optimal temperature, optimal irradiance, mixotrophy, diazotrophy,
motility, buoyancy, resistance to predation, toxicity, and many others. In many trait-based models, this complexity is reduced to
one trait. The most common trait is cell size (Terseleer et al., 2014; Acevedo-Trejos et al., 2016; Smith et al., 2016; Chen and
Smith, 2018). Cell size is used as a master trait because it is the most observable trait and correlates strongly with many other
phytoplankton traits, such as light requirements (Taguchi, 1976; Edwards et al., 2015; Álvarez et al., 2017) and resistance
to predation (Kiørboe, 1993; Thingstad et al., 2005). Some other commonly modeled traits include resistance to predation5
(Norberg et al., 2001; Merico et al., 2009), optimal temperature (Norberg et al., 2012; Beckmann et al., 2019) and optimal
irradiance (Follows and Dutkiewicz, 2011).
The first trait included in SPEAD, half-saturation for nutrients, is known to be strongly correlated to cell size (Litchman
et al., 2007; Edwards et al., 2012). Small species, such as cyanobacteria, have low half-saturation constants and can thrive in
oligotrophic waters. They correspond to the "gleaners" of our model. Large species are more likely to be involved in blooms10
but require more nutrients. They correspond to the "opportunists" of our model. In this study, we chose not to use size but to
impose only a simple gleaner-opportunist trade-off. This choice was made for two reasons: to maintain compatibility with the
Darwin model (Follows et al., 2007; Vallina et al., 2014a) and to facilitate the analysis of the outputs of our otherwise complex
model, as in our setting the best competitors in a given environment have a half-saturation equal to the dissolved inorganic
nitrogen concentration. Our modeling framework can be, however, easily adapted to use size instead of half-saturation as the15
first trait (Smith et al., 2016).
The trait dynamics of models with 2 traits differ from those of simpler and less realistic single-trait models. Savage et al.
(2007) obtain larger trait variances and much larger adaptive capacities when two traits are modeled together rather than in
separate models. In our study, we also find a larger adaptive capacity, although it is conveyed by inter-trait correlation only.
We actually find decreased variances, caused by stronger competition, during most of the year, except during the late Autumn20
re-mixing of the water column, following the Summer stratification. This discrepancy might have been caused by the presence
of an immigration term to sustain variance in Savage et al. (2007) but not in SPEAD, since our re-mixing of the water column
is most analogous to a dispersal or migration process and we simulate it explicitly. SPEAD simulations coupled with a realistic
3D circulation model where phytoplankton is explicitly allowed to migrate in all directions (ideally in a patchy environment)
would finally tell us if variance is increased or decreased by including more traits.25
The low computational costs of aggregate models allows increasing the number of modeled traits, provided that sufficient
observational data are available to constrain the corresponding trade-offs. Since the environmental drivers, such as nutrient
concentrations, temperature, and light are correlated with each other, the traits are likely to be correlated, unless some processes
erasing the correlations are introduced. Regardless of the effect of interactions between traits on variance, multi-trait models
will be able to adapt to their environments faster without the need for large and unrealistic mutation rates or other terms30
sustaining large variances, such as immigration or Kill The Winner grazing.
4.3 Trait diffusion, variance and evolution
Trait diffusion is a key process in SPEAD. Indeed, SPEAD is the first model to include diffusion of multiple traits, providing
insights into how both mutations and selection can impact phytoplankton communities. For each modeled trait, a diffusivity
parameter, or "mutation rate", has to be set. The chosen values of these parameters decisively affect trait dynamics. However,
the mutation rates remain poorly constrained. The most appropriate rate depends on what the modeler intends to represent by
trait diffusion. To understand why, we will need to discuss the notions of "ecological" and "evolutionary" timescales, as well
as the notions of "adaptation" and "species".
A first interpretation of trait diffusion is that it is the most conservative way to add variance when the exact processes sustain-5
ing trait variance are unknown or too complex to be implemented in models. Indeed, trait diffusion simply adds new variance
(aggregate approach) or disperse phytoplankton in a trait space (discrete approach), leaving little room to arbitrary parameters.
By contrast, immigration (Norberg et al., 2001) requires assumptions about both immigration rates and the composition of im-
migrant communities? Likewise, Kill The Winner (Vallina et al., 2014b) requires assumption about whether it is mediated by
viruses or zooplankton, whether consumers are specialized or can actively switch their preference between preys, and whether10
or not there is a time lag in their response.
This intepretation of trait diffusion as a generic source of variance is implicitly followed when the diffusivity parameter is set
by an optimization algorithm in order to account for the observed trait variance and no other mechanism sustaining variance is
included. This way, Chen and Smith (2018) found a diffusivity for the logarithm of size of 0.1, equal to our largest diffusivity
for the logarithm of half-saturation. Even in the homogeneous environments of mesocosms, Wirtz (2013) reports a logarithmic15
size variance of 0.2 to 0.5, which corresponds to standard deviations between 0.45 and 0.7. In SPEAD, reaching this high
values of trait variance is only possible with diffusivities superior or equal to 0.003, despite the fact that our physical setting
creates its own variance by mixing phenotypes adapted to the environmental conditions of different depths. This interpretation
of trait diffusion is also coherent with the use of trait diffusion as a "variance treatment" (Chen et al., 2019) to study the effect
of diversity on primary production and with the original goal of trait diffusion, which was to sustain trait variance in 0D settings20
(Merico et al., 2014; Acevedo-Trejos et al., 2016). In these models, trait diffusion is never run combined with other variance-
sustaining mechanisms. In real ecosystems, however, mutations are expected to occur at the same time as migration, Kill The
Winner grazing, and many other mechanisms that may promote diversity, including multiple convex trade-offs (Beardmore
et al., 2011) and mixotrophy (Ward and Follows, 2016).
The way trait diffusion is derived opens a second interpretation of what it represents. Trait diffusion is symmetrical: mu-25
tations occur at the same rate toward higher and lower trait values. Trait diffusion is also heritable: mutants transfer their
mutations to their offspring. These properties correspond to the evolutionary process of random mutations and selection of
the fittest by the environment. It does not correspond to environmentally induced non-heritable variations such as phenotypic
plasticity (Ghalambor et al., 2007) or to any selective ecological process driven by the environment, even if some promote
According to Fussmann et al. (2007), "Evolution is the change of genotype frequencies within populations or species,
whereas community dynamics represent the change of abundances of different species". This definition depends on the no-
tion of "species". Like many phytoplankton models, SPEAD lacks the notion of species and does not distinguish between
intraspecific and interspecific trait variance. Our trait space is continuous by design. By mutating, phytoplankton can cross
the boundaries between phenotypes, as if they were all of the same species. The only distinction in SPEAD is between mu-35
tations, an evolutionary process represented by trait diffusion, and selection, necessary to both ecological (interspecific) and
evolutionary (intraspecific) processes and represented by an adaptive change in the mean traits and a decrease in trait variance.
Some cases of adaptive evolution to environmental changes in only a few generations have been reported (Fussmann et al.,
2007; Kinnison and Hairston, 2007) but are not necessarily caused by mutations occurring at these timescales. They can also
be driven by ecological selection on a previously existing intraspecific diversity. In order to choose correct mutation rates these5
two evolutionary processes must be distinguished.
An alternative interpretation is that ecological processes are particular cases of eco-evolutionary processes where the phe-
notypes of the offspring are identical to that of their parents (Doebeli et al., 2017). This definition emphasizes the lack of
fundamental difference between the two types of processes from a mechanistic point of view, since both are rooted on the same
birth-death dynamics. Under this alternative interpretation, the "ecological" timescales are simply the timescales at which the10
effect of mutations is small and the "evolutionary" timescales are those at which the effect of mutations is large. Therefore,
the ecological selection timescales overlap with the adaptive evolution timescales and the difference between the two is diffuse
at intermediate timescales. Schlüter et al. (2016) showed that an algal culture starting with a single clone of the abundant
coccolithophore Emiliania huxleyi could evolve new traits in response to ocean acidification in a time measurable in the lab-
oratory. Their experiment lasted for 2100 generations and the changes after only 100 generations were small. These numbers15
agree with the seminal studies on Escherichia coli where bacteria were shown to adapt to temperature increases or to changes
in nutrient availability in 100 to a few thousand generations (Bennett et al., 1990; Lenski et al., 1991; Travisano et al., 1995).
These are the timescales a "mutation rate" should reflect. In the case of phytoplankton this means a few years, which falls
under the category of "contemporary evolution" but does not allow each species to adapt easily to a seasonal cycle or to faster
perturbations. The corresponding trait diffusivity parameters in our study are in the middle or our range, between 3 ×105
and 103for half-saturation and between 3 ×104and 102C2for optimal temperature.
The trait distributions in SPEAD provide additional insights. In absence of trait diffusion, the two traits become almost
totally correlated: one cannot vary without the other. This is a soft version of the diversity collapse observed in 0D models of a
1-trait fitness lanscape. In our 1D model of a 2-trait fitness landscape, trait variances do not collapse to zero but a bidimensional
trait space becomes unidimensional and phytoplankton lose their ability to adapt in other directions. Nutrient and temperature25
niches are known to be correlated in nature, but their correlation is never perfect (Irwin et al., 2012). A small trait diffusivity
is sufficient to avoid the collapse of the dimensional trait space into a unidimensional one and likely limits such trait correla-
tions in natural ecosystems, given that mutations affecting half-saturation and optimal temperature are likely independent and
hence able to freely fill the full trait space. With very fast trait diffusion, the mean phenotypes adapt instantaneously to their
environment but at the cost of keeping a large pool of maladapted phenotypes, which regularly represent more than 15% of the30
community even if they have very low fitness, because mutations are continuously creating them. These maladapted pheno-
types explain the decrease, by up to 10%, of the modeled annual primary production. In nature, the optimal niches of species
do not cover all the variability of nutrient concentration and water temperature (Irwin et al., 2012). Furthermore, most real
mutations having an effect on the phenotype are deleterious (Timofeeff-Ressovsky, 1940). A mutation rate able to permanently
sustain maladapted phenotypes despite strong selection against them would imply large amount of deleterious mutations not
represented in our model, and hence massive mortality. Moderate mutation rates are therefore more likely.
Modeling several communities with their own trait distributions and their own mutations might relax the contradiction
between the use of trait diffusion to explain trait variance and the use of trait diffusion to represent evolutionary processes.
The variance within a species or a group is lower than the total community variance, and can be sustained with lower mutation5
rates. This approach can also be used to separate the adaptive evolution of each species from the ecological successions (i.e.
inter-group competition) in response to environmental change (Norberg et al., 2012), finally disentangling all components of
eco-evolutionary processes.
4.4 Future directions
SPEAD 1.0 is the first step of the SPEAD project, whose aim is to simulate plankton evolution with adaptive dynamics10
in the ocean. In this first version of the model, we kept the complexity manageable, with only one spatial dimension (the
vertical) and two physiological traits, in order to facilitate the validation of our aggregate approach and to diagnose the effect
of trait diffusion. Three axes of potential future improvement have already been identified: 1) coupling SPEAD with a general
circulation model, 2) increasing the number of traits and 3) dividing the community into several functional groups, which
implies combining the continuous trait distribution approach with the discrete ecotypes approach.15
More concretely, our goal for the near future is to include optimal solar irradiance as a third physiological trait and implement
the aggregate approach with trait diffusion in a 3D trait space into the Darwin model (Follows et al., 2007; Dutkiewicz et al.,
2009; Barton et al., 2010; Follows and Dutkiewicz, 2011; Ward et al., 2012; Dutkiewicz et al., 2013). Darwin is a versatile
model that allows many discrete ecotypes to be resolved along several environmental axes and can be coupled with the MIT
general circulation model (Marshall et al., 1997). Optimal irradiance has been present as a trait in the Darwin model since its20
origin. Indeed, light can be a limiting resource for phytoplankton in the ocean, both in mixed water columns, where plankton
cannot stay close to the surface, and near the deep chlorophyll maximum of stratified water columns. Optimal irradiances are
known to cover more than two orders of magnitude (Edwards et al., 2015) and to determine the niches of many ecotypes, even
within the same species (Biller et al., 2015). The inclusion of optimal solar irradiance as a mutating trait in SPEAD will be a
key step to better capture ecological successions. One challenge will be to allow phenotypes to adapt to their environment while25
accounting at the same time for the mechanistic correlation between nutrient and irradiance niches, since both are related to cell
size. The phytoplankton in Darwin were originally divided into four functional groups: Prochlorococcus analogs, other small
phytoplankton, diatoms and other large phytoplankton. Representing each functional group by its own normal distribution is a
possible starting point to develop the multi-Gaussian approach.
Improved versions of SPEAD should be able to address various ecological issues related to community assembly and re-30
sponses to climate change. By including both trait diffusion and ecological selection of the fittest phenotypes competing in a
given environment, SPEAD can potentially be used to disentangle the role of ecological and evolutionary processes in shaping
diversity patterns in phytoplankton. In particular, it can be used to determine the conditions under which species or functional
groups may survive climate change by evolving new traits or may be replaced by other species or functional groups from other
regions. The effect of environmental changes, such as warming and increased stratification, on plankton size structure, and the
effect of biodiversity – controlled by trait diffusion among other processes – on primary production and ecosystem functioning,
are other examples of contemporary ecological questions that SPEAD might contribute towards answering.
5 Conclusions
In this article, we present an aggregate model of phytoplankton community called SPEAD (Simulating Plankton Evolution5
with Adaptive Dynamics), where different phenotypes competing for dissolved inorganic nitrogen are characterized by two
traits: their half-saturation constants for nitrogen uptake (in logarithmic scale) and their optimal temperature for growth. The
phytoplankton community is represented by the six lowest order moments of its trait distribution: total concentration, the mean
value of each trait, the variance of each trait, and the inter-trait covariance. The dynamics of these state variables are driven
by three environmental factors: nutrient concentration, temperature, and solar irradiance. The physical setting represents a10
water column down to 200 m. The seasonal alternation of stratification and vertical mixing also has a strong effect on the
trait distribution. Trait diffusion through subsequent generations is included to represent heritable mutations and hence sustain
trait diversity. To our knowledge, SPEAD is the first aggregate model to include at the same time two traits (with a proper
representation of inter-trait correlation) and trait diffusion.
The ecological parameters of SPEAD were set to reproduce the observed primary production, chlorophyll, nitrate and par-15
ticulate organic nitrogen concentrations observed by the BATS time series in the Sargasso Sea. Despite its strong assumption
that traits are normally distributed, SPEAD was shown to agree precisely with a discrete model explicitly representing all phe-
notypes, with only minor deviations at depth in Summer, where optimal temperature is underestimated, and in early Winter,
where trait variances decrease too fast. This good agreement is made possible by trait diffusion and by the simplicity of our
ecological setting and might not be extendable to all ecosystem models. The trait dynamics depend strongly on the imposed20
trait diffusivity parameters. With very high diffusivities, primary production is low, variances are high and the two traits are
independent, filling the entire trait space. With very low diffusivities, variances are low (albeit non-zero) and the two traits are
very strictly correlated: only warm-water gleaners and cold-water opportunists can survive. We think that intermediate values
are more realistic, but the precise value depends on whether trait diffusion is meant to sustain the trait diversity of a whole
community or to represent the mutations occurring within a given species.25
SPEAD has a computational cost two orders of magnitudes lower than a full discrete model and its variables are readily
interpretable in ecological terms. This effectiveness makes it possible to increase the number of traits. As optimal irradiance
is key to explain phytoplankton distribution in the water column and is already present in the Darwin model, the next step
of the SPEAD project will be to include it as a third dynamic trait. In agreement with Savage et al. (2007), we showed that
adding traits accelerated the response of preexisting traits to environmental changes. Other venues of future improvement30
include representing various functional groups, each with their own distinct normal distributions, and coupling SPEAD with a
general circulation model. Future versions of our multi-trait framework may address ecological questions related to the impact
of selection, mutations, and biodiversity on community dynamics and to the response of phytoplankton to climate change.
6 Code and Data availability
The code and data of SPEAD 1.0 are freely available on GitHub ( The code
for SPEAD 1.0 is written in MATLAB. To be able to install and operate SPEAD, the user should be familiar with MATLAB
and have the version R2010b or a more recent one. The executions has been tested on Windows with a 2.5 GHz Intel i5-3210M
processor. The main code modules are:5
jamstecrest_gaussecomodel1D_main is the main script to launch SPEAD, calling all functions.
jamstecrest_gaussecomodel1D_keys is the function where the different options are declared.
jamstecrest_gausssecomodel1D_parameters where the values of the model parameters are assigned.
jamstecrest_gaussecomodel1D_ode45eqs is a function called at each time step to solve the ordinary differential equations
of the aggregate (continuous) model.10
jamstecrest_discretemodel1D_ode45eqs is a function called at each time step to solve the ordinary differential equations
of the multi-phenotype (discrete) model.
SPEAD 1.0 also contains numerous other functions to plot figures and to represent each physical or ecological process (vertical
mixing, aggregate trait diffusion, discrete trait diffusion ...). The 4 observations files (for primary production, chlorophyll
concentration, nitrate concentration and particulate organic nitrogen concentration) and the 4 external forcing files (for water15
temperature, surface PAR, vertical mixing and mixed layer depth) are located in the INPUTS folder. Once all files are loaded,
SPEAD is run simply by calling jamstecrest_gaussecomodel1D_main.
Appendix A: Why mutations can be represented as "Trait Diffusion"
In this study, we represented phytoplankton mutations as a "trait diffusion" term, following the work of Merico et al. (2014).
In this appendix we show how the expression for trait diffusion is derived and discuss its conditions of validity.20
Let us consider the dynamics of a phytoplankton community, where each individual is characterized by the values of two
traits, called "x" (in trait unit x or "tux") and "y" (in tuy) . Trait is distributed with a density p(x,y,t)(in mmolN.m3.tux1.tuy1).
The mass concentration of phytoplankton cells with values of the first trait between xand x+dx and values of the second trait
between yand y+dy is p(x,y,t)·dxdy if dx and dy are small.
If traits are strictly inherited, the equation governing p(x,y,t)for a given phenotype (x, y)depends on the reproduction25
(u(x,y,t), in d1) and death (d(x,y,t), in d1) rates:
∂t (x, y, t)=(u(x,y,t)d(x, y, t))p(x, y, t)
∂t (x, y, t) = a(x, y, t)p(x,y,t)
In the above equation, a(x,y,t) = u(x,y,t)d(x, y, t)is the net growth rate. In our study, the reproduction rate is identical
to the nitrogen uptake rate because nitrogen, the limiting nutrient, is not exuded, cell size is considered independent of time5
and all nitrogen taken up is used for reproduction. However, genetic mutations or phenotypic plasticity can produce offspring
with trait values different from that of their parents. For simplicity, we will consider that mutations increasing or decreasing
the traits are equally probable. We assume that the offspring of a parent with trait value xwill have trait value xδxwith a
probability ax=νx(δx)2and trait value xδxwith the same probability, where νxis a diffusivity parameter, expressed in
tux2, considered independent of trait value, mutation step (δx) and time. Mutations also occur on trait y, with a mutation step10
δy and a y-diffusivity parameter νy. Mutations on both traits are assumed independent. The probability of having a mutation
on both xand yis just the product of the probabilities of each mutation. Hence the time derivative of p(x, y, t)with mutations
∂t (x, y, t) = [(1 2ax)(1 2ay)u(x, y, t)d(x,y,t)]p(x, y, t)
+ (1 2ay)ax[u(xδx, y, t)p(xδx, y, t) + u(x+δx, y, t)p(x+δx, y, t)]15
+ (1 2ax)ay[u(x,y δy, t)p(x,y δy, t) + u(x,y +δy, t)p(x,y +δy, t)]
+axay[u(xδx, y δy, t)p(xδx, y δy,t) + u(x+δx,y δy, t)p(x+δx, y δy, t)
+u(xδx, y +δy, t)p(xδx, y +δy,t) + u(x+δx,y +δy, t)p(x+δx, y +δy, t)]
In the limit of small but frequent mutations, this equation can be simplified by making a second-order approximation of u·p.
∂t (x, y, t) = [(1 2ax)(1 2ay)u(x, y, t)d(x,y,t)]p(x, y, t)20
+ (1 2ay)axu·pδx
∂x +1
∂x +1
+ (1 2ax)ayu·pδy
∂y +1
∂y +1
∂x δy
∂y +1
∂x δy
∂y +1
∂x +δy
∂y +1
∂x +δy
∂y +1
The sum of terms in (u·p)
∂x ,(u·p)
∂y and (u·p)
∂x∂y is zero, and the sum of factors before u·pis 1. hence the above equation
simplifies to:
∂t (x, y, t) =[u(x, y, t)d(x,y,t)]p(x,y,t) + axδ2
=a(x,y,t)p(x,y,t) + νx
This second-order approximation is valid in the limit as mutations become small (δxand δytend to zero) and frequent
(ax=νx(δx)2and ay=νy(δy)2tend to infinity), so that higher-order terms can be neglected. The mathematical expression5
of the mutation term νx2(u·p)
∂y2is somewhat analogous to that representing the diffusion of a tracer in physical
space. In this analogy, the trait space replaces the physical space, νx·uand νy·ureplace the diffusivity, and the density p(x,y,t)
is the diffused tracer. This analogy is the origin of the phrase "trait diffusion". We note that in a multi-trait space, whatever the
number of traits, the diffusion of each trait has the same expression as in the one-trait space.
Appendix B: Differential equations of a multi-trait aggregate model with trait diffusion10
Phytoplankton community models can be discrete or aggregate. In a discrete model, the phytoplankton community is divided
into a finite number of phenotypes, each characterized by a different set of trait values. Mutations are discrete with steps equal
to the difference between a phenotype and its nearest neighbors. The differential equation for a discrete phenotype is intuitive
and depends only on its net growth rate and a trait diffusion term.
The variables of aggregate models and the differential equations they follow are less intuitive. In an aggregate model, a15
general shape must be assumed for the trait distribution, with some degrees of freedom, and the prognostic variables are the
moments of the trait distribution that are free to vary. In a single-trait model, the most commonly assumed shape is the normal
(or "Gaussian") distribution (Wirtz and Eckhardt, 1996; Bruggeman and Kooijman, 2007), where the phytoplankton density
p(x,t)(in mmolN m3tux1, with "tux" the "trait unit x") is equal to:
p(x,t) = P(t)
In this distribution, the three free parameters are the phytoplankton concentration P(t), the mean trait value x(t)and the trait
variance Vx(t). They are respectively equal to Rp(x,t)·dx,Rx·p(x,t)·dx and R(xx)2p(x, t)·dx. In a multi-trait space,
we will assume that the traits follow a multivariate normal distribution, which is a generalization of a normal distribution. If N
is the number of dimensions:
p(x,t) = P(t)
2(xx(t))TΣ(t)1(xx(t)) (B2)25
In this case, xis the vector containing all traits, x(t)is the vector of mean trait values and Σ(t)is the matrix of variances
and covariances. There are (N+1)(N+2)
2free parameters in total: 1 phytoplankton concentration, Nmean trait values, Ntrait
variances and N(N1)
2covariances. In the following, we will show how to derive the differential equations for each type of
variable. To this end, we use the method developed by Norberg et al. (2001), based on Taylor expansions of the rates of
reproduction and net growth. The assumption of normal trait distribution is only required to compute the time derivatives of30
variance and covariance, but not for the equations of total biomass and mean trait values.
The trait space is considered to be unbounded, with the implicit assumption that extreme values are extremely rare and
ecologically meaningless. This is expressed in the fact that pand all products including por any of its derivatives tend to
0 when a trait tends to (plus or minus) infinity. For simplicity, in the following part of this section, we will not show the
dependencies on the environmental factors and will limit ourselves to two traits, but our method can be extended to derive
the equations for any given number of traits. The reproduction and net growth rates of the phenotype defined by trait values5
(x,y) are denoted u(x,y)and a(x,y)respectively. Integrals are over the whole bi-dimensional domain. We will first derive the
equations in the absence of trait diffusion and then discuss what terms are added by the trait diffusion scheme. The net growth
of a given phenotype is:
dt =a(x,y,t)p(x,y,t)(B3)
As a consequence, the equation controlling P(t)is:10
dt =Z Z a(x,y,t)p(x,y,t)·dxdy
We will use the notations aj k (in d1.tuxj.tuyk) for a(x,y,t)derivated jtimes with respect to xand ktimes with respect
to y, normalized by the factorials of jand k, and Mjk (in tuxj.tuyk) for the central moment of order jwith respect to xand k
with respect to y:
ajk (t) = 1
Mjk (t) = 1
P(t)Z Z (xx(t))j·(yy(t))k·p(x, y, t)·dxdy
We note that M00 = 1 (by definition of the total concentration), M10 = 0,M01 = 0 (by definition of the mean traits), M20 (t) =
Vx(t),M02(t) = Vy(t)(by definition of the variance) and M11(t) = Cxy (t).
A Taylor expansion of the net growth rate on xand yaround (x(t), y(t)) yields:20
a(x,y,t) =
ajk (t)·(xx(t))j·(yy(t))k
The time derivative of P(t)depends on the time derivative of p(x, t). Unless explicitly indicated otherwise, all derivatives
with respect to traits are taken at the current mean trait values.
dt =Z Z p
∂t (x, y, t)·dxdy25
=Z Z a(x,y,t)p(x,y,t)·dxdy
=Z Z
ajk (t)·(xx(t))j·(yy(t))k·p(x,y,t)·dxdy
ajk (t)Z Z (xx(t))j·(yy(t))k·p(x,y,t)·dxdy
ajk (t)Mjk(t)
The first and largest term of this sum is the net growth rate at the mean trait values, denoted a(x(t),y(t),t). This is the
expected growth rate of a community without trait variance. Since M10 = 0 and M01 = 0, there is no term depending on a
∂x or5
∂y . As we want to estimate the effect of trait diversity on the community, we consider the second order terms, proportional to
Vx(t),Vy(t)or Cxy(t). Higher order terms, which vanish when variance is small, are neglected, so that:
dt =P(t)a(x(t),y(t),t) + 1
∂x∂ y (B4)
Second order derivatives are expected to be negative if (x(t),y(t)) is in the neighborhood of the optimal trait value, and
Vx(t)and Vy(t)are always positive. Therefore, the second order terms are generally negative. This means that having large10
trait variances, or in other words having a large proportion of cells with non-optimal trait values, has a negative effect on the
net community growth.
In the equation for mean trait however, having a large variance is an advantage. Let us define an intermediate variable Sx(t)
(in mmolN.m3.tux) as P(t)x(t)or, equivalently, as R R x·p(x,y,t)·dxdy. The time derivative of Sx(t)is:
dt =Z Z x∂p
∂t (x, y, t)·dxdy15
=Z Z x·p(x,y,t)a(x,y,t)·dxdy
=Z Z (xx(t)) ·p(x,y,t)a(x,y,t)·dxdy +x(t)Z Z p(x, y, t)a(x, y, t)·dxdy
=Z Z
ajk (t)·(xx(t))j+1 ·(yy(t))k·p(x,y,t)·dxdy +dP
dt x(t)
ajk (t)Z Z (xx(t))j+1 ·(yy(t))k·p(x,y,t)·dxdy +dP
dt x(t)
ajk (t)Mj+1,k(t) + dP
dt x(t)20
As Sx(t)is a product, its derivative can also be written as:
dt =dx
dt P(t) + dP
dt x(t)
By equating the two previous expressions, we get:
dt =
ajk (t)Mj+1,k(t)
In this equation, we only consider the highest order terms:25
dt =Vx(t)∂a
∂x +Cxy (t)∂a
∂y (B5)
This equation represents the adaptation of the community to its environment. If the mean trait is not optimal, it will increase
or decrease in order to maximize the net specific growth rate. The speed of this selection process is proportional to variance
: biodiversity is required to track the environmental conditions. The covariance term means that if traits are correlated, the
optimization of trait y will also affect trait x.5
The mean value of trait y follows a similar equation:
dt =Vy(t)∂a
∂y +Cxy (t)∂a
∂x (B6)
The equations describing time changes in variances and covariance require more assumptions. As previously, we define an
intermediate variable Zx(t)(in mmolN.m3.tux2) as P(t)Vx(t)or, equivalently, as R R (xx(t))2·p(x,y,t)·dxdy. In this
integral, two terms depend on time: x(t)and p(x,y,t). Hence, the time derivative of Zx(t)is:10
dt =Z Z (xx(t))2· p
∂t (x, y, t)·dxdy +Z Z (xx(t))2
∂t p(x, y, t)·dxdy
=Z Z (xx(t))2·p(x,y,t)a(x,y,t)·dxdy 2dx
dt Z(xx)·p(x,y,t)·dxdy
By definition of x(t), we have R R (xx)·p(x,y,t)·dxdy = 0. Thus:
dt =P(t)
ajk (t)Mj+2,k(t)
As Zx(t)is a product, its derivative can also be written as:15
dt =dVx
dt P(t) + dP
dt Vx(t)
By equating the two previous expressions, we get:
dt =1
dt dP
dt Vx(t)
ajk (t)Mj+2,k(t)Vx(t)
ajk (t)Mjk(t)
In this equation, the two terms proportional to a(x(t),y(t),t)compensate and it is no longer possible to neglect the third and20
fourth orders of the trait distribution. With only the two lowest order non-zero terms retained, the time derivative of variance
dt =M30(t)a
∂x +M21 (t)∂a
∂y +1
x(t)) 2a
2(M22 VxVy)2a
∂y2+ (M31 VxCxy)2a
∂x∂y (B7)
The moments M30(t)and M40 (t)are called the skewness and kurtosis of x respectively. They represent the shape of the
trait distribution. These moments could be described by their own equations but their time derivatives depend on moments of25
even higher orders, and so on. In order to limit mathematical complexity and computation time, we do not explicitly compute
moments of higher order than variance. Instead, we close our system by giving these moments the same value as in a bivari-
ate normal distribution. In a bivariate normal distribution, odd order moments (where j+kis odd) are zero (for reasons of
symmetry) and even order moments can be expressed as a function of variances (Isserlis, 1916):
M22(t) = VxVy+ 2C2
The equation for Vx(t)then simplifies to:10
dt =V2
∂x2+ 2VxCxy