CO2 molecules are fixed continuously by photosynthesis during the growing season; so, C particles enter the vegetation (from here on called “system”) at different times during the year. After fixation, the photosynthetic products transit through the vegetation compartments until they eventually leave the system, either as CO2 back to the atmosphere or as litter and exudates to the soil. This means that at a given time (t) each particle in the system has a different age, which is the time that it has remained within the system since its fixation from the atmosphere. The time that each particle spends transiting through the system, from arrival until exit, is called transit time .

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. ; given two organic molecules in a compartment, one that has remained in the system for longer time than the other, they both have the same probability to either leave the system or move on to another compartment. Thus, if by chance older molecules remain for longer times, then the system's age gets older. But the pace at which these molecules are transiting is moderated by the cycling rates. This is why slow cycling compartments have older C.

Lets describe a system of well-mixed C (distributed in multiple compartments) with the system of ordinary differential equations x˙(t)=B×x(t)+β⋅u,x(0)=x0, where x˙(t) is how much the quantity of carbon in vegetation compartment x changes with respect to time, B is a matrix of carbon transfer coefficients between the plant compartments, x(t) is a vector of states for vegetation (state variables), β is a vector containing the partitioning coefficients of photosynthetic input, and u is a scalar that represents that input. This linear system does not include environmental variables, or any other variables that depend on time; thus, it is an autonomous linear system with multiple interconnected compartments.

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 age density distribution of a system (fA(y)), is determined by the probability of entering the system through a given compartment and the rates at which C is transferred from one compartment to another until it leaves the system. Consistent with the symbols from the previous equation fA(y)=zT⋅ey⋅B⋅x∗||x∗||. fA(y) is a function of (i) how fast the carbon is leaving the system: the row vector of release rates, which is the column-wise sum of the elements of the B matrix (zT=-1TB); (ii) the transition probability matrix (eyB); and (iii) the relative amount of C stock at steady state with respect to the total (x∗||x∗||). Notice that we use here the symbol ||⋅|| to represent the vector norm, which is the sum of all entries of the vector.

The mean age is given by the expected value (E[A]) E[A]=||B-1×x∗||||x∗||.

Likewise, the transit time density distribution (fFTT(t)) is also a function of zT and the transition probability matrix (etB), as well as the vector of input distributions (β). fFTT(t)=zT⋅et⋅B⋅β.

The mean transit time is defined as E[FTT]: E[FTT]=||B-1×β||=||x∗||||u||.

In this case, the definition of mean transit time coincides with the commonly used stock over flux approach (turnover time), but note that the definitions presented here can only be applied to autonomous systems at steady state. For non-autonomous systems, i.e., models in which inputs and process rates change over time, formulas for the estimation of mean age and transit time can be found in .

From these equations it is evident that age and transit time calculations mainly depend on the schemes of C partitioning (β) and cycling (B) within a vegetation model. Therefore, if we want to understand processes such as the mixing of old and newly fixed NSC using ecosystem models, it is critical to model proper carbon allocation (CA) strategies. Unfortunately, it is still uncertain what assumptions and simplifications should be done. How many carbon compartments are necessary to describe carbon cycling in vegetation? How are these compartments interconnected and how fast are they transferring C among each other? These are important questions that need to be addressed to improve our understanding of vegetation dynamics, and predict consequences of environmental change on vegetation.

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 ; the models have either no storage, one storage compartment, or two storage compartments (fast and slow C cycling). Our approach is mostly theoretical, and we are mainly interested in introducing the concepts of age and transit time distributions as useful model diagnostics as well as an approach to explain mixes of C age in NSC compartments. For this purpose, we used published measurements on ecosystem C compartments from the Harvard Forest Environmental Measurement Site to find suitable parameter values for the different model structures. We also diagnosed the performance of these models using as metrics (1) C release fluxes (respiration and other carbon losses such as litterfall), (2) the dynamics of radiocarbon (based on the bomb spike) for individual compartments, (3) the transit time distribution of the system, and (4) the age distribution of C in the system and in each compartment.

Each model was written as a set of ordinary differential equations (based on Eq. ) within the environment of the R package SoilR . All models met the requirements of Eq. (); they are autonomous (their dynamics do not depend on variables that change with time) linear systems with multiple interconnected compartments. The initial carbon stocks, and some of the parameter values needed to solve those equations, were obtained from the literature, from a deciduous–evergreen model with similar carbon allocation schemes . Other parameter values were obtained from an optimization procedure (see below).

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. ), following the hypotheses proposed by . Given that we aimed at a theoretical comparison of the abovementioned strategies, we eliminated other potential sources of variation that may act as confounding factors by assuming that all C transfers between the compartments depended on constant rates. We also assumed that there was a constant photosynthetic input – gross primary production (GPP): u – of 1400 gCm-2 year-1 . Environmental variability, which operate mostly on hourly to daily timescales, was not considered here because we ran the models at an annual timescale; i.e., without diurnal cycles or phenology.

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 Ra (Fig. ). In the model without a storage compartment, the C stored in the Photoassimilates is partitioned into “Structural Foliage” (from here on Str. Foliage), Wood (including branches and coarse roots), and Fine Roots (from here on Roots), with the constant rates Af, Aw, and Ar; part of the C stored in these three compartments also leaves the system with constant rates Lf, Lw, and Lr, which comprise all the carbon released through respiration and other losses (e.g., litterfall). For the models with storage, the C is transferred from the Photoassimilates to the fast cycling storage, from which it is then partitioned to the rest of the compartments. In addition to the fast cycling storage, the model with two storage compartments also has a slow cycling compartment.

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

http://atmos.seas.harvard.edu/lab/hf/index.html, http://ameriflux.lbl.gov/doi/AmeriFlux/US-Ha1, http://ameriflux.lbl.gov/sites/siteinfo/US-Ha1

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.  and ). The simulations of Wood and Foliage C (Photoassimilates + Structural) stocks were in accordance with the stocks estimated from the aboveground biomass inventory data and the LAI, respectively.

All of these predictions were obtained using the parameter set that was most frequently chosen by the Bayesian optimization method (Table ). This table also shows two quantiles of the distribution of each parameter value after exploring the parameter space.

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 <20 for the models Storage: 0, 1, and 2 was 3, 4, and 5, respectively. The correlations can be seen in the pairwise plots of sensitivity functions (Figs. –). Table  summarizes the number of parameter correlations that we observed under the diagonal of the pairwise plots of sensitivity functions. For this reason, the results presented here need to be interpreted within the context of predicted uncertainties.

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 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 Δ14C signature of the Str. Foliage compartment with respect to the current year's atmosphere, in the models with storage (Fig. ). Although such a delay could indicate that this compartment had a slightly slower C cycling than what is expected for a deciduous forest , it could also indicate the flux of old C from the slow cycling compartments into the foliage. This delay in radiocarbon accumulation was also observed as shifts to older ages in the age distributions (Fig. ), with a resolution of months. In general, radiocarbon measurements can also help to constrain model parameters with respect to how fast or slow different compartments cycle C , but with less resolution than the age and transit time distributions.

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. ()–(), the calculation of age and transit time distributions depends on the C partitioning schemes (β) and the transfer – cycling – rates between plant compartments (B). Therefore, different parameter values that compose the vector β and matrix B directly result in the different calculations of ages and transit time distributions. This was tested by running the three models using the same parameter set; although they still differed in the number of storage compartments, there was almost no difference between the age and transit time distributions in the whole system and in the compartments (Figs. , , and ). These similarities were also observed for the radiocarbon accumulation (Fig. ). The only exceptions to the above were the foliage compartments of the model Storage: 0, which were faster than in the other two models.

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. ): Running the three models with the parameter set that gave the best fit for the model with two storage compartments hampered the fit of the model with no storage compartments to the woody and foliage C measurements. Thus, although the system had an external C input that could never be depleted because it was assumed to be a constant flux, the parameter estimation accounted for extra compartments by modifying the cycling rates to optimize the fit of the models to the data.

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 (B), but there are other factors driving these two distributions. The additional factor driving the mean age calculation is the relative amount of C stock at steady state; whereas the mean transit times depend on the C partitioning schemes (β; ). So, the mean transit time calculation is only limited by the rates of C transfer, while the mean age of C in the system also depends on its mass. This is why the mean age of C in vegetation is determined by the age of the compartment where the majority of the mass is stored, which in this case is Wood. We further explored this relation in the scatter plot in (Fig. ), where the distribution of the points below the 1:1 line indicate that the three models have mean ages greater than their mean transit times. This means that they have large masses of old carbon, but they also have highly dynamic compartments through which carbon transits very fast.

We also diagnosed the model performance by comparing the predicted ages for the storage compartments (Fig. ) with the mean age of NSC measured in previous empirical studies . The mean age of NSCs from red maple cores obtained at Harvard Forest was 7.2±7.7 years; for the fast cycling storage compartments of the models Storage: 1 and Storage: 2, it was 1.25 and 1.55 years, respectively, and 13.25 years for the slow cycling storage. Clearly, the mean ages of fast storage compartments are smaller than the mean value calculated empirically, but given the uncertainty in the measurements, they are still within the observed range. Furthermore, the density distributions of these storage compartments show that even though they are well mixed, their C do not have the same age. Thus, it is also likely to find C between 0–5 and 0–50 years in the fast and slow cycling storage compartments, respectively. This resonates with the fact that although stemwood NSC is highly dynamic on seasonal timescales, it can also be surprisingly old . Then, the two hypotheses regarding C mixing – inward mixing of younger and older C in one compartment and two compartments (young and old) that mix – converge in the concept of age distributions because C is simultaneously been fixed and removed from the compartments at different times, a process that results in C age distributions. We can think about these dynamics in the context of a stochastic process. The total amount of C that enters and leaves each compartment is fixed and given by the deterministic model, but the time that each C particle stays in a compartment varies stochastically within them. So, the age distribution of C particles in each compartment is a mix of new and old carbon, with distribution functions emerging from the deterministic model (Eq. ).

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 , information that is not available for Harvard Forest. Nonetheless, we can predict that if there were external factors influencing the simulations, there would be a different prediction of mean ages and transit times at each time step, but the model structure (C allocation scheme in particular) would play a major role determining the shape of these distributions.

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. found a low concentration of old NSC in old rings of stemwood, and a high concentration of old NSC in coarse roots and fine roots of pine. Additionally, they found young and old C in roots. These dynamics were interpreted as poor C mixing and reserves that cycle on different timescales , but they may also obey physiological constraints. For example, the parenchyma in heartwood is thought to be dead, so NSC trapped in there may no longer be accessible to the plant . So, to study the physiological significance of these findings with models, we might have to include such details regarding tree physiology. However, the important point we want to emphasize is that mixing of carbon in different vegetation compartments results in C age distributions that have been little studied previously. Our results are a first attempt to obtain these distributions using a number of assumptions, but in the future other analysis with more complex models and explicit formulas for time-dependent age distributions would help to obtain better predictions of ages and transit times as affected by specific physiological processes.

Although there are still knowledge gaps regarding plant physiology, and current C-dating methods only measure mean age of C rather than age distributions , we expect our results to motivate future work, particularly in the use of isotope tracers and their time evolution to approximate age distributions. Biosphere models can be enhanced with structural adjustments and the uncertainties in the parameter values can be reduced by constraining them with age and transit time distributions. These improved models could then be used to test hypotheses regarding physiological questions, and assess the sustainability of current terrestrial C sinks, given changes in environmental forcing.

Model equifinality was evident from the fact that despite having a different number of compartments and values of cycling rates, all three of the models had similar simulations of C stocks (Figs.  and ). Along with model equifinality, we obtained a high collinearity between some parameters, implying that they are non-identifiable, i.e., they cannot be uniquely estimated from the given data sets . Thus, for these particular models, the time series measurements from only 2 out of 4–6 vegetation compartments is not sufficient to estimate the values of 7–10 parameters.

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 practical non-identifiability, where the insufficiency or poor quality of data is not a good constraint for the parameters. In addition, a negative parameter correlation can be a symptom of structural non-identifiability, which is the result of a redundant parameterization . Thus, the three models had practical and structural non-identifiabilities, which means that they need to be constrained with more and better data, and they need to be restructured in order to avoid compensation of fluxes into and out of the compartments.

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 is often hampered by parameter non-identifiability . This is a real problem because parameters such as those that correspond to carbon turnover explain most of the variation in the response of terrestrial vegetation to future climate and CO2 and are highly important in determining C age and transit times.