Diversity plays a key role in the adaptive capacity of marine ecosystems to environmental changes. However, modelling the adaptive dynamics of phytoplankton traits remains challenging due to the competitive exclusion of sub-optimal phenotypes and the complexity of evolutionary processes leading to optimal phenotypes. Trait diffusion (TD) is a recently developed approach to sustain diversity in plankton models by introducing mutations, therefore allowing the adaptive evolution of functional traits to occur at ecological timescales. In this study, we present a model called Simulating Plankton Evolution with Adaptive Dynamics (SPEAD) that resolves the eco-evolutionary processes of a multi-trait plankton community. The SPEAD model can be used to evaluate plankton adaptation to environmental changes at different timescales or address ecological issues affected by adaptive evolution. Phytoplankton phenotypes in SPEAD are characterized by two traits, the nitrogen half-saturation constant and optimal temperature, which can mutate at each generation using the TD mechanism. SPEAD does not resolve the different phenotypes as discrete entities, instead computing six aggregate properties: total phytoplankton biomass, the mean value of each trait, trait variances, and the 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, wherein 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 Bermuda Atlantic Time-series Studies (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, with 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, with the computational cost of SPEAD being lower by 2 orders of magnitude. Therefore, SPEAD should be an ideal eco-evolutionary plankton model to be coupled to a general circulation model (GCM) of the global ocean.

Phytoplankton are a polyphyletic group of microscopic primary producers widespread in aquatic environments. They are mainly single-celled, although colonial or multicellular species exist in most phytoplankton phyla

Phytoplankton are highly diverse and live in many different environments.
They differ in their ecological interactions and 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, and others are mixotrophic, being able to both photosynthesize and feed on organic sources

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

Instead, ecological 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

One weakness induced by the simplification of phytoplankton communities in both aggregate and discrete models, however, is that competitive exclusion

An alternative approach recently introduced to sustain diversity in models is to allow the simulated phytoplankton to mutate their functional traits

There are several other types of plankton ecological (without mutations) or eco-evolutionary (with mutations) models, such as individual-based models, resident–mutant models, and models with stochastic mutations. Each represents and sustains diversity in its own unique way. All the modelling approaches mentioned in this paper and others are reviewed with their assumptions, costs, and benefits by

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: the nitrogen half-saturation constant and optimal temperature for growth.
The SPEAD model is based on an NPZD model

In the following sections, we first describe our ecological model, the differential equations controlling the growth of phytoplankton, 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.

How well does SPEAD represent the bulk properties of phytoplankton communities observed in the Sargasso Sea?

Do the aggregate and discrete approaches agree?

How are phytoplankton dynamics changed by the value of the mutation rates?

Can the mean value and variance of each trait be represented independently by a one-trait model wherein only nitrogen half-saturation or optimal temperature varies between phenotypes?

Finally, we discuss the reach of our modelling framework, focusing on three aspects: the performance of aggregate models, the choice of phytoplankton traits, and the relationship between trait diffusion and evolution.

Our phytoplankton community model SPEAD extends an existing nitrogen-based NPZD model

State variables of the ecosystem model.

NPZD (nutrient, phytoplankton, zooplankton, detritus) model within its physical setting

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 function of the nitrogen pool concentrations and temperature.

Parameters of the ecosystem model.

In contrast, the phytoplankton pool is composed of diverse organisms responding to environmental conditions in different ways.
The diversity of phytoplankton is represented by variations in the values of two traits: the logarithm of the half-saturation constant for nitrogen uptake (

The last term in the equation of trait density (Eq. 10) is trait diffusion (TD), as defined by

Phytoplankton growth factors

The first trait allowed to mutate in SPEAD,

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 exponential.
In our model, the exponential factor for autotrophic primary production is

In this study, the PAR limitation factor

In the above equation,

For comparison with data, two additional variables can be estimated from the model: primary production and chlorophyll

Traits

In the multi-phenotype model, only

In the aggregate model, the trait distribution is assumed to be continuous. In this case, and contrary to the multi-phenotype 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 phytoplankton concentration

The second-order moments (

We follow the method developed by

The net growth rate

SPEAD 1.0 has one spatial dimension: the vertical. A depth-resolved simulation is the minimum 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 the 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 of the 0D model presented in Sect. 2.1 and 2.2. First, PON sinks at a speed of

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. Inter-annual variations and the day–night cycle are neglected. Temperature and surface PAR

Distribution in depth and time of three environmental variables:

The simulation of the aggregate two-trait model with mutation rates

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 behaviour, we tried a wide range of values for

A two-trait model is not simply the superposition of two one-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 (

The time step for our simulations is 6 h. At the first time step and at all vertical levels, the DIN concentration is initialized to 1.8 mmol N m

The first step to validate the SPEAD model is to compare some bulk properties with observations from the Sargasso Sea in order to assess the realism of the trait-independent biogeochemical parameters (Table 2).
In Fig. 4, the primary production, chlorophyll, and 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

Distribution in depth and time of

Primary production is the state variable best reproduced by the model, with a maximum around 10 mg C m

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 modelled DIN and the observed nitrate concentrations share the same range, with a maximum of 2.8 mmol m

Particulate organic nitrogen distributions from the model and observations are relatively similar, with a maximum around 0.5 mmol m

The second step to validate the aggregate SPEAD model and the only validation of its bivariate trait distribution is comparing it to the multi-phenotype model. Although both are models and thus simplify reality in similar ways, the multi-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 neighbours 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.

The spatial and temporal patterns of the phytoplankton concentration, mean traits, trait standard deviations, and inter-trait correlation for the standard simulation with mutations rates of

Distribution in depth and time of trait distribution moments for

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”, which is positive if the value is higher in the aggregate model.
The aggregate model reproduces

Comparison at all depths and times of the aggregate and multi-phenotype model state variables for

The main errors on

Concentrations of each phenotype of the multi-phenotype model (colour) compared with lines of equal density of the aggregate model (

In this section, we compare the results of simulations conducted with nine different sets of mutation rates, from

For each simulation, Fig. 8 shows the values of depth-integrated primary production per year and the yearly averaged values and ranges of

Primary production and trait distribution moments for different mutation rates. The ratio

Convergence time and various properties of SPEAD 1.0 simulations with different mutation rates.

Primary production lies between 146.8 and 147.0 g C m

In the simulations with

When the mutation rates are lower, the mean traits still vary during the year but not as much or as fast as the physical environment, and no phenotypes are found outside the trait domain of the discrete model. With

Increasing trait diffusivity to

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, wherein 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:

SPEAD state variables at the surface for

SPEAD state variables at the surface for

With standard mutation rates, the trait dynamics are very similar in all three models. The two-trait model has slightly lower standard deviations than the one-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-August, when the one-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, with concentrations that 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 high 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 ti becomes more important as the mutation rates and hence the optimal temperature variances decrease.

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 one-trait half-saturation model has 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 two-trait model during most of the year but sometimes exceed the one-trait variances during the autumn mixing. However, the most notable change is that the seasonal amplitude of mean half-saturation (

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 (29) to (34) can help us understand the effect of trait correlation on the seasonality of mean traits and trait variances. In the mean trait equations (30 and 31), correlation implies that both temperature and the 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 from 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 two-trait model actually experiences a slower increase in optimal temperature than the one-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, 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 the optimal temperature from decreasing.

The effect of correlation on variance is even more convoluted. In Eqs. (32) and (33), inter-trait correlation adds a second variance-reducing competition term (

SPEAD is an aggregate phytoplankton model. Aggregate models, used as far as we know since

Aggregate models are very efficient because their state variables are the quantities that make the most ecological sense.
In thermodynamics, computing the trajectory of each atom or molecule is not only infeasible, but also of little use. The collection of trajectories does not provide more information on the macroscopic behaviour of a thermodynamic system than aggregate properties such as temperature, pressure, and density. Equally, modelling 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

However, the aggregate approach has one major weakness: a specific shape for the trait distribution must be assumed a priori, with only as many degrees of freedom as there are free parameters

Normal distributions are symmetrical, unimodal, and unbounded. If the real trait distribution deviates from these three properties, 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 significantly 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 to become dominant in the case of warming than cold-water phenotypes in the 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 favours larger trait values, thus facilitating the adaptation, but it decreases when the environment favours smaller trait values (see Appendix B and the neglected term

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 if they are given enough time, irrespective of the intensity of that change.
This contrasts with expectations of the behaviour of real phytoplankton communities, as explained in the following example. If a closed (i.e. without immigration) phytoplankton community experiences temperatures between 15 and 25

In the present study, the aggregate (continuous trait) model agrees very well with a multi-phenotype (discrete trait) model, wherein no distribution shape is imposed but the 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 is 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 re-mixing 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

Whether the trait distribution of a model ecosystem is normal or not depends on the ecological processes included by the modeller.
At least two factors in SPEAD play in favour 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

Other ecological settings yield more widespread multimodality. Multimodality can be induced by immigration

Alternatives to Gaussian closures have been proposed since early in the development of aggregate models.

A more practical approach to account for non-Gaussian distributions would be to divide the community into several functional groups, each one having a normal trait distribution of its own

In nature, many different traits define phytoplankton niches: nitrogen, phosphorus or iron uptake abilities, requirements for 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

The first trait included in SPEAD, half-saturation for nutrients, is known to be strongly correlated with cell size

The trait dynamics of models with two traits differ from those of simpler and less realistic single-trait models.

The low computational cost of aggregate models allows increasing the number of modelled 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 terms sustaining large variances, such as immigration or “kill the winner” grazing.

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 modelled 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 modeller 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 sustaining trait variance are unknown or too complex to be implemented in models. Indeed, trait diffusion simply adds new variance (aggregate approach) or disperses phytoplankton in a trait space (discrete approach), leaving little room to arbitrary parameters. By contrast, immigration

This interpretation 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,

According to

An alternative interpretation is that ecological processes are particular cases of eco-evolutionary processes whereby the phenotypes of the offspring are identical to those of their parents

The trait distributions in SPEAD provide additional insights.
In the 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 one-trait fitness landscape. In our 1D model of a two-trait fitness landscape, trait variances do not collapse to zero, but a bi-dimensional trait space becomes unidimensional and phytoplankton lose their ability to adapt in other directions. Nutrient and temperature niches are known to be correlated in nature, but their correlation is never perfect

Modelling 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 mutation 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

SPEAD 1.0 is the first step of the SPEAD project, whose aim is to simulate plankton evolution with adaptive dynamics in the ocean. In this first version of the model, we kept the complexity manageable, with only one spatial dimension (the vertical) and two functional traits, in order to facilitate the validation of our aggregate approach and to diagnose the effects of trait diffusion. The equations of SPEAD 1.0 for the mean trait, trait variance, and covariance resolving two functional traits can be used as a starting point to build more comprehensive trait-based models in multi-dimensional continuous trait spaces, with or without mutations. 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 ecotype approach.

More concretely, our goal for the near future is to include optimal solar irradiance as a third functional trait and implement the aggregate approach with trait diffusion in a 3D trait space into the Darwin model

Improved versions of SPEAD should be able to address various ecological issues related to community assembly and responses to climate change that current models cannot address. The response of the phytoplankton community to environmental changes in a simple NPZD model can only be an increase or a decrease in primary production. Estimates of changes in primary production from NPZD models are likely to be inaccurate because the parameters of NPZD models are validated against observations of present environments. Under environmental changes, the composition of the plankton community is likely to change and the model parameters have no reason to remain valid. Models based on plankton functional types are more accurate, as they can account for ecological selection: a plankton type (e.g. diatoms) can be replaced by another type (e.g. cyanobacteria) in response to a perturbation (e.g. increased stratification). However, they do not represent adaptive evolution through which groups or species can change their traits and maintain local dominance. Therefore, they might overestimate the extinction rate and the shift in community composition. Eco-evolutionary models like SPEAD do not explain why groups that diverged millions of years ago exist, but a multi-Gaussian version of SPEAD, wherein each group follows its own adaptive dynamics, could account for their contemporary evolution in new environments. 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. Predicting changes in phytoplankton composition is particularly important as species perform different functions and have different impacts on their environment. For instance, they contribute differently to the carbon pump (e.g. by sinking more or less fast) and to the nitrogen cycle (e.g. by fixing atmospheric nitrogen or not). Therefore, changes in community composition might dramatically impact global climate and should be included in climate prediction models. 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 issues that SPEAD might contribute to addressing.

In this article, we present a new aggregate model of a phytoplankton community called SPEAD (Simulating Plankton Evolution with Adaptive Dynamics), wherein 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 a 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 simultaneously include 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 as well as the chlorophyll, nitrate, and particulate organic nitrogen concentrations observed by BATS 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 phenotypes, with only minor deviations at depth in summer when optimal temperature is underestimated and in early winter when trait variances decrease too fast. This good agreement is made possible by trait diffusion and by the simplicity of our ecological setting, and it might not be extendable to all ecosystem models. The trait dynamics depend strongly on the imposed 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 of mutation rates 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.

SPEAD has a computational cost 2 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 explaining 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

In this study, we represented phytoplankton mutations as a trait diffusion term, following the work of

Let us consider the dynamics of an isolated phytoplankton community wherein each individual is characterized by the values of two traits:

If traits are strictly inherited, the equation governing

In the above equation,

In the limit of small but frequent mutations, this equation can be simplified by making a second-order approximation of

The sum of terms in

This second-order approximation is valid in the limit as mutations become small (

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 neighbours. 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, a 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

In this distribution, the three free parameters are the phytoplankton concentration

In this case,

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

As a consequence, the equation controlling

We will use the notation

A Taylor expansion of the net growth rate on

The time derivative of

The first and largest term of this sum is the net growth rate at the mean trait values, denoted

Second-order derivatives are expected to be negative if

In the equation for the mean trait, however, having a large variance is an advantage. Let us define an intermediate variable

As

By equating the two previous expressions, we get

In this equation, we only consider the highest-order terms:

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

The mean value of trait

The equations describing time changes in variances and covariance require more assumptions. As previously, we define an intermediate variable

By definition of

As

By equating the two previous expressions, we get

In this equation, the two terms proportional to

The moments

The equation for

The equation for

These expressions represent the effect of competition on trait variance. If the mean trait is near an optimum, then

The equation for covariance is derived in a similar way. We define

The last two terms are zero since

Since

By equating the two previous expressions, we get

In the case of a bivariate normal distribution, the only terms remaining produce the following equation.

Replacing

Competition tends to reduce variance, which must therefore be sustained by another process. In our model, this is the role of trait diffusion, as described in Appendix A and originally derived by

The additional term for the total biomass time derivative due to the diffusion of

This result comes from the fact that

The effects of

This integral is similar to

Thus, the new equations of

Because

The effect on

Thus, trait diffusion does not add nor remove covariance to the phytoplankton community. However, by increasing the variances, trait diffusion decreases the correlation. In other words, trait diffusion decorrelates the traits by making rare trait combinations more likely.

We note that, in the absence of trait diffusion, our equations are a particular case of the general equations derived by

The code and data for SPEAD 1.0 are freely available on GitHub (

SPEAD_1D is the main script to launch SPEAD, calling all functions.

SPEAD_1D_keys is the function with which the different options are declared.

SPEAD_1D_parameters assigns the values of the model parameters.

SPEAD_gaussecomodel1D_ode45eqs is a function called at each time step to solve the ordinary differential equations of the aggregate (continuous) model.

SPEAD_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 four observation files (for primary production, chlorophyll concentration, nitrate concentration, and particulate organic nitrogen concentration) and the four external forcing files (for water 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 SPEAD_1D.

The supplement related to this article is available online at:

GLG, SMV, SLS, and PC conceived and designed the study. SMV wrote the initial code for the one-trait model and acquired the observational data. GLG derived the equations for the two-trait model and the trait diffusion scheme, built the final code, and wrote the first draft. All authors contributed to and reviewed the paper.

The authors declare that they have no conflict of interest.

The authors thank all the scientists who produced the data and developed all the concepts used in this article. In particular, we thank Agostino Merico, Esteban Acevedo-Trejos, and Marcel Oliver for the discussions we had on trait diffusion. The Institute of Marine Sciences (ICM – CSIC) is supported by a “Severo Ochoa” Centre of Excellence grant (CEX2019-000928-S) from the Spanish government.

This work was funded by national research grant CTM2017-87227-P (SPEAD) from the Spanish government. We acknowledge support for the publication fee by the CSIC Open Access Publication Support Initiative through its Unit of Information Resources for Research (URICI).

This paper was edited by Andrew Yool and reviewed by Fanny Monteiro and one anonymous referee.