Mesocosm experiments on phytoplankton dynamics under high

Oceans are a sink for about

In some experiments, the different treatment levels, i.e., different

Mesocosms typically enclose natural plankton communities, which is a more
realistic experimental setup compared to batch or chemostat experiments with
monocultures

To account for all possible factors that determine all differences in
plankton dynamics is practically infeasible, which also impedes a
retrospective statistical analysis of the experimental data. However, since
unresolved ecological details might propagate over the course of the
experiment, it is meaningful to consider a dynamical model approach to
upgrade the data analysis. From a modeling perspective, some important
unresolved factors translate into (i) uncertainties in specifying initial
conditions (of the state variables), and (ii) uncertainties in identifying
model parameter values. Here, we apply a dynamical model to estimate the
effects of ecophysiological uncertainties on the variability in POC
concentration of two mesocosm experiments. Our model describes plankton
growth in conjunction with a dependency between

Differences among replicates of the same sample can be interpreted as unresolved random variations (named uncertainties hereafter).

Uncertainties can amplify during the experiment and generate considerable variability in the response to a given treatment level.

Which uncertainties are more relevant can be estimated by the decomposition of the variability in the experimental data.

States variables and their dynamics.

Parameter values used for the reference run,

For our data-supported model analysis of variability decomposition we
consider the propagation of distributions

Potential sources of variability are estimated following a procedure already
applied in system dynamics, experimental physics, and engineering

Variability decomposition method based on uncertainty propagation
(summary of the basic principles given in Sect. 5.1.1 and 5.6.2. and Annex B
in

In this section, we describe the biological state that was used as reference
dynamics. Our model resolves a minimal set of state variables insofar
monitored during experiments that are assumed to be key agents of the
biological dynamics. Model equations are shown in Table

Field data of aquatic

The mean cell size in the community, represented as the logarithm of the mean
equivalent spherical diameter (ESD), was used as a model parameter. It
determines specific ecophysiological features by using allometric relations
that are relevant for the computation of subsistence quota, as well as
nutrient and carbon uptake rates. Regarding the latter, to resolve
sensitivities to different DIC conditions, we used a relatively accurate
description of carbon acquisition as a function of DIC and size. It has been
suggested by previous observations and models that ambient DIC concentration
increases primary production

We considered that uncertainties were only present in the initial setup of
the system; this allowed us to perform a deterministic non-intrusive forward
propagation of uncertainty, which neglects the possible coupling between
uncertainties and temporal dynamics unlike in intrusive methods

Our approach is based on a Monte Carlo method for the propagation of
distributions. It is based on the repeated sampling from the distribution for
possible inputs and the evaluation of the model output in each case

We considered model factors,

Solid lines show reference runs for POC, PON, and DIN simulating the
mean of the replicates per treatment level, with different colors for the
three experimental

This expression is based on the assumption that changes in

Hereafter, the standard deviation of any given factor, i.e., factor
uncertainty, will be given as percentage of the reference values and will be
called

To numerically calculate the ensemble of

Environmental data showed low variability among similar treatment replicates,
(see Fig.

Notably, our analysis suggested sufficient (but not necessary;

Our model reproduces the means of PON, POC, and DIN experimental data per
treatment level, i.e., for the future, present, and past

Tolerance of mesocosms experiments to differences among replicates,
given as a percentage of the reference factor value listed in Tables

As in Fig.

Reference simulations of POC for high

The estimation of the tolerance thresholds of the dynamics to uncertainty
propagation for the two test-case experiments, per acidification levels and
per factor uncertainty, are listed in Table

POC variability decomposition per factor,

As Fig.

Sensitivity coefficients (

Our method allows for decomposition of POC variability in factor-specific
components

POC variability during the prebloom phase can be explained mainly by the
differences of factors related to subsistence quota, i.e.,

The exploration of the sources of variability in an experiment with
a multi-way repeated measures ANOVA design with 3 acidification levels
requires a multi-factorial high-dimensional setup (left panel). Alternately,
we numerically simulate the biomass dynamics with

Environmental data from PeECE II and III are taken as model inputs. Error bars denote the standard deviation of the same treatment replicates.

Differences in the initial nutrient concentration,

Amplified variability in the postbloom phase (third column in Figs.

Perturbations of the initial detritus concentration,

POC variability throughout the bloom phases (right column in Figs.

We used the uncertainty quantification method to decompose POC variability by
using a low-complexity model that describes the major features of
phytoplankton growth dynamics. The model fits the mean of mesocosm
experimental PeECE II and III data with high accuracy for all

The results of our analyses are conditioned by the dynamical model equations
imposed. Deliberately, the model's complexity is kept low, mainly to limit
the generation of structural errors with respect to model design. At the same
time, the level of complexity resolved by the model suffices to explain POC
measurements of two independent mesocosm experiments with identical parameter
values (see Table 2), which highlights model skill. The used equations comply
with theories of phytoplankton growth

Differences among replicates in the initial nutrient concentration
substantially contribute to POC variability, a sensitivity that is,
interestingly, not well expressed when varying the initial cellular carbon or
nitrogen content of the algae,

For PeECE II, experimentally measured DIN concentration at day 0 was

We found a limited tolerance to variations in the mean cell size of the
community,

Another major contributor to POC variability during the bloom phase is
phytoplankton biomass loss,

To test the hypotheses outlined in the Introduction entails two important
aspects. First, heuristic exploration of variability would require
experiments designed to quantify the sensitivity of mesocosms to variations
in potentially relevant factors that specify uncertainties in environmental
conditions, cell physiology, and community structure. However, this would
require high-dimensional multi-factorial setups (see Appendix

In many cases, the mean and the variance of the sample are taken as a fair
statistical representation of the effect of the treatment level and its
variability. However, summary statistics such as the mean and the variance
might fail to describe distributions that do not cluster around a central
value, i.e., when the data are not normally distributed in the sample. This is
because a feature of normally distributed ensembles is that the mean
represents the most typical value and deviations from that main trend (caused
by unresolved factors not directly related to the treatment) might cancel out
in the calculation of the ensemble average. Actually, this cancellation is
the reason for using replicates

In our simulations, the

We provided an estimation for the uncertainty thresholds that can be used for
improving future sampling strategies with a low number of replicates, i.e.,

Strategies to reduce

In addition, our analysis results help interpreting non-conclusive results
and provide plausible explanations for the negative results for the detection
of potential acidification effects

Finally, we found the same main contributors to POC variability for all the
treatment levels, even if experimental variability is about

Our model projections indicated that phytoplankton responses to OA were mainly expected to occur during the bloom phase, presenting a higher and earlier bloom under acidification conditions. Moreover, we found that amplified POC variability during the bloom that potentially reduces the low signal-to-noise ratio can be explained by small variations in the initial DIN concentration, mean cell size, and phytoplankton loss rate.

The results of the model-based analysis can be used for refinements of experimental design and sampling strategies. We identified specific ecophysiologial factors that need to be confined in order to ensure that acidification responses do not become masked by variability in POC.

With our approach we reverse the question of how experimental data can
constrain model parameter estimates and instead determine the range of
variability in experimental data that can be explained by modeling with
variational ranges bounding uncertainties of specific control factors. We
tested the hypothesis of whether small differences among replicates have the
potential to generate higher variability in biomass time series than the
response that can be attributed to the effect of

In this study, we established a foundation for further model-based analysis
for uncertainty propagation that can be generalized to any kind of
experiments in biogeosciences. Extensions comprising time-varying
uncertainties by introducing a new random value for parameters at every time
step or including covariance matrices, showing the simultaneous interaction
of variations in two factors, can be straightforward implemented

Experimental data are available via the data portal Pangaea

Relative growth rate

Primary production rate reflects the limiting effects of light, dissolved
inorganic carbon (DIC), temperature, and nutrient internal quota as follows:

The allometric factor

The last term in Eq. (

Efforts related to nutrient uptake

To describe the loss rate of phytoplankton biomass, we used a density-dependent term

We used measured aquatic

The applied model equations attribute phytoplankton, detritus, and
herbivorous heterotrophs to particulate organic matter. Measurements of
particulate organic carbon also include some fractions of large
bacterioplankton, carnivorous zooplankton, as well as extracellular gel
particles such as transparent exopolymer particles. These additional organic
contributions to POC measurements are not explicitly resolved in our model.
Therefore, for comparisons between simulation results and observations, we
redefine the raw data from PANGAEA, named POC

Heuristic exploration of the potential origins of the observed variability
uses statistical inference tools, such as a multi-way repeated measures
ANOVA,
exploring which independent factors are contributing the most to the standard
deviations. Such approaches have the advantage of accounting for interacting
effects between combinations of factors (and not only for the synergistic
effects of each factor and acidification, as in our model-based approach; see
Sect.

For the model–data fit shown in Figs.

Cumulative residuals for PeECE III.

Kai Wirtz, Markus Schartau, and Maria Moreno de Castro developed the model code; Maria Moreno de Castro performed the simulations and prepared the manuscript, which was revised by Kai Wirtz and Markus Schartau.

The authors declare that they have no conflict of interest.

We thank Sabine Mathesius for the PAR and temperature data for both the PeECE II and III experiments and Kaela Slavik for the English edition of the preliminary version of the manuscript. We acknowledge our two anonymous reviewers for their helpful comments and suggestions. This work is a contribution to the National German project Biological Impacts of Ocean Acidification (BIOACID) and it is also supported by the Helmholtz society via the program PACES.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: M. Grégoire Reviewed by: two anonymous referees