The global carbon cycle is strongly controlled by the source/sink strength of vegetation as well as the capacity of terrestrial ecosystems to retain this carbon. These dynamics, as well as processes such as the mixing of old and newly fixed carbon, have been studied using ecosystem models, but different assumptions regarding the carbon allocation strategies and other model structures may result in highly divergent model predictions. We assessed the influence of three different carbon allocation schemes on the C cycling in vegetation. First, we described each model with a set of ordinary differential equations. Second, we used published measurements of ecosystem C compartments from the Harvard Forest Environmental Measurement Site to find suitable parameters for the different model structures. And third, we calculated C stocks, release fluxes, radiocarbon values (based on the bomb spike), ages, and transit times. We obtained model simulations in accordance with the available data, but the time series of C in foliage and wood need to be complemented with other ecosystem compartments in order to reduce the high parameter collinearity that we observed, and reduce model equifinality. Although the simulated C stocks in ecosystem compartments were similar, the different model structures resulted in very different predictions of age and transit time distributions. In particular, the inclusion of two storage compartments resulted in the prediction of a system mean age that was 12–20 years older than in the models with one or no storage compartments. The age of carbon in the wood compartment of this model was also distributed towards older ages, whereas fast cycling compartments had an age distribution that did not exceed 5 years. As expected, models with C distributed towards older ages also had longer transit times. These results suggest that ages and transit times, which can be indirectly measured using isotope tracers, serve as important diagnostics of model structure and could largely help to reduce uncertainties in model predictions. Furthermore, by considering age and transit times of C in vegetation compartments as distributions, not only their mean values, we obtain additional insights into the temporal dynamics of carbon use, storage, and allocation to plant parts, which not only depends on the rate at which this C is transferred in and out of the compartments but also on the stochastic nature of the process itself.

The global carbon cycle is strongly controlled by the
source/sink strength of terrestrial ecosystems. Vegetation in particular, is
one of the major controls of global C sources and sinks with respect to the
atmosphere

The C storage capacity of an ecosystem is determined by the collective
behavior of vegetation compartments such as foliage, wood, and roots,
which may also act as C sources and sinks among each other

It is indeed possible that the carbon stored in vegetation compartments,
including NSCs, has been fixed at different times, resulting in a mix of ages

Previous studies have focused mostly on determining ages of NSCs using
radiocarbon-derived mean residence times, but this approach has
limitations. One limitation is the ambiguity in the term “mean
residence time”, which has been defined in different ways across
studies; in some cases it implies the mean age of C in an ecosystem or
ecosystem compartment and in other cases it implies the time it takes
C molecules to leave the system of compartments

The study of C age distribution in vegetation requires challenging
empirical methods, but can also be approached using ecosystem C cycle
models. However, not all of the models perform equally well because
the assumptions behind their structures may result in highly divergent
predictions

Graphical representation of the concepts of age and transit
time distributions in a vegetation model. Carbon particles are
represented here as clocks that measure the time they have been in
the system. “System age” can be defined as the age of all
particles in the system at a given time, while “transit
time” as the age of particles in the output flux. Adapted from

At each time step, a particle may stay where it is, given by a certain
probability, or flow to the next compartment with a rate or probability given
by the transfer coefficients (also know as cycling rates). This means that
the age of carbon in a system results from stochastic and deterministic
processes, which are illustrated in Fig.

Lets describe a system of well-mixed C (distributed in multiple
compartments) with the system of ordinary differential equations

Given that each particle in the system has its own age and transit time, the
age and transit time of the whole system can be considered as random
variables. Additionally, the age and transit time of particles in a system's
compartment is exponentially distributed. Then, the age and transit time
distributions of the entire system would be the sum of those exponential
distributions, i.e., a phase-type distribution

The calculation of how many C particles have a certain age, or

The mean age is given by the expected value (

Likewise, the transit time density distribution (

The mean transit time is defined as

In this case, the definition of mean transit time coincides with the
commonly used

From these equations it is evident that age and transit time
calculations mainly depend on the schemes of C partitioning
(

In this contribution, we address the following question: how do different
C allocation schemes affect the ages and transit times of carbon in
vegetation models? In particular, we are interested in understanding
whether different carbon allocation strategies would lead to different
patterns of mixing of ages for the NSC compartment. For this work, we
implemented three carbon allocation schemes based on

Each model was written as a set of ordinary differential equations (based on
Eq.

As means to assess whether the carbon allocation strategies had an impact on
the mixing of C age in vegetation compartments, we implemented three models
whose carbon allocation strategies varied depending on the number of storage
compartments (0, 1, or 2; Fig.

Three carbon allocation strategies in vegetation models. These
strategies differ in the number of storage compartments, Storage: 0, Storage: 1, and
Storage: 2. Adapted from

Fixed photosynthetic input enters the system through the “Photoassimilates”
compartment, and part of the carbon is released back to the atmosphere at
each time step, in a flux proportional to the size of Photoassimilates and
the constant rate

To obtain results that can be related to a particular ecosystem, we performed
a parameter estimation procedure using published measurements of two
ecosystem C compartments from the Harvard Forest Environmental Measurement
Site (see footnote
links

We calculated C stocks in wood and foliage from the abovementioned
aboveground biomass, LAI, and leaf mass per area (LMA), using allometric equations.
Although in previous studies, performed at the same site, the use of woody
biomass increment and LAI reduced the uncertainties in the predictions of net
C sequestration and foliage dynamics, respectively

We were also interested in observing the uncertainty in the model
simulations (see section below), so we performed a Bayesian
optimization, which gave us alternative parameter sets after exploring
the parameter space. This optimization procedure was started using the
result of a classical optimization method using the R package

As means to evaluate whether the parameters could be estimated from
the given data sets, i.e., parameter identifiability, we performed
a local sensitivity analysis and estimated the collinearity of the
parameter sets with the package

Given the high correlation between some of the parameters, we decided
to run all model simulations using the parameter set that was most
frequently chosen by the Bayesian optimization method. We then
calculated C stocks, release fluxes, radiocarbon values based on the
bomb spike, ages, and transit times, using functions implemented in

In order to explore model predictions that could result from different parameter sets that were possible and likely, we extracted a random sample of 1000 posterior parameter sets from the Bayesian optimization that used Markov chain Monte Carlo. We ran the models with the unique sets, and calculated the weighted mean and standard deviation of the C stocks, the released C from each compartment, and the system's mean age and transit time. The weights corresponded to the number of times that each parameter set was repeated in the sample.

The C stock simulations obtained from the three models were within
the uncertainty range of the available data (Figs.

Carbon stocks estimated for each model, comparing the observed data
and the model predictions of C stocks in Wood

Parameter values obtained from the optimization procedures (

Final: parameter set that was most frequently chosen by the Bayesian optimization method and was used for all of the simulations, unless otherwise noted. Best1: parameter set obtained from the classical optimization procedure.Best2: the remaining columns were the result of the Bayesian optimization.

Carbon stocks estimated for each compartment and their
uncertainties. Carbon in the

All of these predictions were obtained using the parameter set that was most
frequently chosen by the Bayesian optimization method (Table

Interestingly, some of the parameters were strongly correlated with
each other. For the three model structures and the available empirical
data, the number of parameters that can be simultaneously estimated
with a collinearity index

To assess the impact of different carbon allocation strategies on the ecosystem C cycling, we used the following metrics: (1) C release fluxes, (2) dynamics of radiocarbon for individual compartments, (3) transit time distribution of C through the system, and (4) age distribution of C in the system and in each compartment. The calculations required for these metrics were performed using the parameter set that was most frequently chosen by the Bayesian optimization method for each model, unless otherwise noted.

The three models predicted different mean fluxes of C released from
each compartment at steady state (Fig.

C release fluxes from the compartments at steady state, with uncertainty ranges obtained from the set of posterior parameters obtained by Bayesian optimization.

The simulated radiocarbon content of fast cycling compartments
(e.g., Photoassimilates, Str. Foliage, and Storage fast) had
a stronger resemblance to the atmospheric

Radiocarbon simulations for the three model structures. The
black curve corresponds to the

Differences in radiocarbon values for the different compartments hint to different levels of mixing of carbon fixed at different times. For the fast cycling compartments such as the Photoassimilates, the degree of mixing is relatively low because most of the radiocarbon reflects the values in the atmosphere. For other compartments that cycle at slower rates, the mix of recent and old radiocarbon results in important divergences from the atmosphere. Mixing of carbon of different ages can be further studied with ages and transit time distributions.

The age and transit time distributions were calculated assuming that
the system was in steady state. These distributions had a wide range,
expanding from zero to several decades old carbon, and their shape varied
according to the model structure (Fig.

System ages and transit times.

At the compartment level, the abovementioned age-dependent ranking of
the models only holds true for Wood (Fig.

The only compartment that had an age maximum at 0 years was the
Photoassimilates. Hence, it had a unique distribution curve reflecting
the fact that all new carbon (

Age densities simulated for the compartments:

The age densities of the storage compartments, just as the ones for
Foliage, Wood, and Roots, consisted of curves with peaks at young ages
and long tails (Fig.

Age densities simulated for the models with storage
compartments.

Our simulation results showed that C cycling in ecosystems can be largely influenced by different carbon allocation strategies, which may result in diverging carbon cycling predictions for specific simulations. However, not all of the different prediction metrics were impacted with the same strength by the assumed number of compartments and values of cycling rates, so results here need to be interpreted within the context of predicted uncertainties.

The simulated ecosystem properties that were more strongly impacted by the assumptions behind model structure were (i) the fluxes of C released from each compartment and (ii) the ages and transit time distributions of carbon in the system and in each compartment. This sensitivity to different carbon allocation strategies makes them good candidates for diagnosing model performance.

Given that the C release fluxes from Photoassimilates, Roots, and Wood were
highly sensitive to the three model structures, empirical measurements of
these compartments could be used as constraints during the parameter
estimation procedure. Radiocarbon accumulation was less sensitive, but can be useful to diagnose
the models' performance according to the cycling speed of their compartments.
As an example, we could identify the short delay of the

Overall, the age and transit distributions were the best candidates to diagnose model performance and potentially constrain parameter estimations, because they were the most sensitive to differences in model structure and parameter values. So, what had the highest impact on these distributions, the differences in cycling rates or the inclusion of storage compartments? The differences in cycling rates and the inclusion of storage compartments had respectively direct and indirect effects on the predictions of the abovementioned distributions.

On the one hand, as we initially inferred from
Eqs. (

On the other hand, model structure had an indirect effect on the
predictions of age and transit time, most likely because the addition
of storage compartments impacted the outcome of the parameter
estimation and these different parameter values then lead to different
age and transit time predictions. An illustration of this is the fit
of the models to the data (Fig.

As expected, systems with ages distributed towards older values also
have older transit time distributions. In fact, their correlation can
be confirmed by observing the formulas once again. The calculation of
these two properties depends on the matrix of transfer coefficients
(

We also diagnosed the model performance by comparing the predicted
ages for the storage compartments (Fig.

Another important observation regarding the mean age predictions is the fact that these calculations were performed under the assumption that the system, in this case the forest, was in steady state. Since the C stocks in Harvard Forest continue growing, the calculated mean ages and transit times should be interpreted as predictions of the mean age that the carbon may have in this forest once it is in steady state. Based on this, the time that this forest will take to reach the steady state is highly divergent among the three models. As an example, the model with two storage compartments would predict 20 years more of growth to reach a steady-state close to that of the model with no compartments.

In the case of systems that are driven by environmental variables and
result in time dependencies of inputs (GPP) and process rates, the
mean ages and transit time distributions would also change over
time. To calculate the means of these time-dependent distributions one
would need to know the complete history of inputs and cycling rates
for the duration of the simulation

It is also noteworthy that what we assume to be a compartment,
e.g., Wood, does not necessarily meet the well mixed assumption, so its
particles may not have the same probability to leave the compartment
at all times.

Although there are still knowledge gaps regarding plant physiology,
and current C-dating methods only measure mean age of C rather than
age distributions

Model equifinality

Model equifinality as well as the impossibility to uniquely identify certain
parameters (parameter non-identifiability) is expressed as high correlations
between the parameter sets. Positive parameter correlations may indicate

Since this study was limited by the availability of relevant empirical
data, the parameter values that we used are only one of many possible
outcomes of parameter estimations using the same data sets. Therefore,
it is possible that none of these models accurately depict the C cycle
in the Harvard Forest. However, these problems experienced with
parameter non-identifiability are not an isolated case; the process of
finding unknown rates of C sequestration by fitting biosphere models
to empirical data

We obtained age and transit time distributions of carbon for simple vegetation models with contrasting carbon allocation schemes. Our results show that mixing of carbon in different vegetation compartments results in C age distributions not explored in previous studies. The shape of these distributions depends largely on model structure, and in particular on how carbon allocation is represented in models.

Models with none or one storage compartment may fail to explain the mixing of ages found in different vegetation compartments, but they are more parsimonious than the model with two storage compartments. Nonetheless, parameter collinearity and model equifinality were persistent problems that might be solved if more constraints are added, since the time series of C in foliage and wood are not enough to parameterize a full vegetation model.

Although all models predicted similar C stocks in vegetation compartments, the inclusion of a carbon storage compartment resulted in very different predictions of age, transit time distributions, C release, and isotopic composition. Thus C ages and transit times, which can be indirectly measured using isotope tracers, can be used to improve biosphere models via examination of their structure and estimation of parameter values, which then can be used to assess the strength of C sources or sinks from vegetation.

Finally, it is advantageous to consider age and transit times as distributions, rather than only mean values; with their distributions we obtain additional insights into the temporal dynamics of carbon use, storage, and allocation, which not only depends on the rate at which C flows into and out of the compartments but also on the stochastic nature of the process itself.

All of the simulations and figures for this work can be reproduced using the code and data provided in the Supplement.

Pairwise plots of sensitivity functions for the model Storage: 0.

Pairwise plots of sensitivity functions for the model Storage: 1.

Pairwise plots of sensitivity functions for the model Storage: 2.

Carbon stocks estimated for each model, comparing the observed data
and the model predictions of C stocks in Wood

Radiocarbon simulations for the three models. The three
models were run using the same parameter set. The black curve
corresponds to the

System ages and transit times.

Age densities simulated for the compartments:

Age densities simulated for the models with storage
compartments; the models were run using the same parameter
set.

Scatter plot of mean age vs. mean transit times on a log
scale. The three models have distributions below the

Number of positive and negative correlations between parameters. Only

The authors declare that they have no conflict of interest.

Research at Harvard Forest is supported by the National Science Foundation's LTER program (DEB-1237491). This material is based upon work supported by the US Department of Energy, Office of Science, Office of Biological and Environmental Research. Part of this work was the result of a research visit to the Terrestrial Ecosystems and Global Change group, Department of Organismic and Evolutionary Biology, Harvard University. It was funded by the Max Planck Society and the German Research Foundation through its Emmy Noether Program (SI 1953/2–1). The article processing charges for this open-access publication were covered by the Max Planck Society. Edited by: Akihiko Ito Reviewed by: four anonymous referees