The fate and transit time of carbon in a tropical forest

Tropical forests fix large quantities of carbon from the atmosphere every year; however, the fate of this carbon as it travels through ecosystem compartments is poorly understood. In particular, there is a large degree of uncertainty regarding the time carbon spends in an ecosystem before it is respired and returns to the atmosphere as CO2 . We estimated the fate of carbon (trajectory of photosynthetically fixed carbon through a network of compartments) and its transit time (time it takes carbon to pass through the entire ecosystem, from fixation to respiration) for an old‐growth tropical forest located in the foothills of the Andes of Colombia. We show that on average, 50% of the carbon fixed at any given time is respired in <0.5 years, and 95% is respired in <69 years. The transit time distribution shows that carbon in ecosystems is respired on a range of time‐scales that span decades, but fast metabolic processes in vegetation dominate the return of carbon to the atmosphere. Synthesis. The transit time distribution integrates multiple ecosystem processes occurring at a wide range of time‐scales. It reconciles measurements of the age of respired CO2 with estimates of mean residence time in woody biomass, and provides a new approach to interpret other ecosystem level metrics such as the ratio of net primary production to gross primary production.

organic compounds, it does not contribute to the greenhouse effect in the atmosphere (Neubauer & Megonigal, 2015;Noble et al., 2000;Sierra et al., 2020). Therefore, whether respired carbon from ecosystem is young or old, gives an idea of the time photosynthetically fixed carbon remains stored. This lapse of time when carbon is removed from the atmosphere is particularly relevant for tropical ecosystems given their dominance in the global GPP flux.
Studies with tropical trees have shown that healthy mature trees respire mostly recent carbon assimilates (< 2 years-old carbon), but can respire decades-old carbon under stress (Muhr et al., 2013(Muhr et al., , 2018Vargas et al., 2009). In fact, observational studies with temperate trees as well as modelling studies have shown that trees can respire carbon of a wide range of ages, from days-to decades-old carbon (Carbone et al., 2013;Ceballos-Núñez et al., 2018;Herrera-Ramírez et al., 2020;Trumbore et al., 2015). Therefore, one would expect that respiration in tropical ecosystems is composed by a mixture of carbon of different ages (Trumbore, 2006;Trumbore & Barbosa De Camargo, 2013), but such a mixture is difficult to quantify. Isotopic labelling experiments in temperate ecosystems have shown that respired carbon is mostly young, but with a high degree of mixing difficult to characterize from the isotopic data alone (Hopkins et al., 2012;Keel et al., 2006).
In contrast to isotopic labelling studies, data from permanent plots across the tropics suggest that carbon stays in the woody biomass pool, on average, by about 50 years or more (Galbraith et al., 2013;Malhi et al., 2013). Plot-level estimates of the time carbon stays in the woody biomass of tropical forests are commonly obtained by dividing wood biomass carbon stocks over stem growth.
This approach relies on three main assumptions: (1) the forests are in a dynamic equilibrium in which inputs of carbon are balanced by losses from mortality and respiration, (2) the obtained mean value characterizes an unknown underlying distribution of the time carbon spends in an ecosystem and (3) the woody biomass pool is representative of the dynamics of the entire ecosystem, so dynamics in detritus and soil carbon pools can be ignored. Assumption 1 is reasonable for old-growth tropical forests because it is expected that over the long-term, climate variability, disturbances and internal forest dynamics would balance the net carbon flux around a mean value of zero, but with important variability in fluxes from year to year Sierra et al., 2009). A deeper exploration of Assumptions 2 and 3 may help to explain the large difference between tree-and plot-level estimates of the time carbon spends in tropical ecosystems.
The fate of carbon through an ecosystem and the time it spends there, from photosynthesis until respiration, is well captured by the concept of transit time (Bolin & Rodhe, 1973;Rasmussen et al., 2016;Sierra et al., 2017;Thompson & Randerson, 1999). This concept quantifies the time it takes carbon atoms to travel through the entire ecosystem and links three main ecosystem processes: photosynthesis, storage and respiration. It can be expressed as a probability mass function that quantifies the time it takes to respire a proportion of carbon fixed at a given time. Under the assumption of equilibrium, the total carbon stock divided by the total input or output flux provides an estimate of the mean of the transit time distribution (Sierra et al., 2017). Therefore, estimates of the entire transit time distribution of carbon in tropical forests would help us to better understand not only the mean time carbon spends in the woody biomass, but also the time recent photosynthates spend in trees before being respired, and the time it takes for carbon that enters the soil to appear in the respiratory flux. This transit time distribution captures all these different processes over a wide range of time-scales.
In this manuscript, we provide an estimate of the transit time distribution of carbon in a tropical forest ecosystem using a data assimilation technique to parameterize a dynamic ecosystem model.
Our main hypothesis is that the shape of the transit time distribution reconciles estimates of the time carbon spends in ecosystems obtained from tree-and plot-level methods. Furthermore, we attempt to provide here the formal theory to not only obtain the transit time distribution, but also metrics to characterize the fate of carbon inputs through the entire ecosystem as well as the age of carbon in ecosystem pools. This theory is then used to present an alternative interpretation of the link between GPP, autotrophic respiration (Ra) and net primary production (NPP).

| THEO RY
The time that carbon spends in ecosystems can be obtained using the concept of transit time (Bolin & Rodhe, 1973;Rasmussen et al., 2016;Sierra et al., 2017;Thompson & Randerson, 1999). It characterizes the time carbon atoms spend in an ecosystem, from the time of carbon fixation through photosynthesis until release to the atmosphere through respiration in the absence of fire.
To compute transit times, we will consider a special case of the general mathematical representation of ecosystem carbon dynamics that follows the compartmental system representation proposed in Sierra et al. (2018). Since we are concerned in this manuscript with tropical old-growth forests at equilibrium, we will represent carbon dynamics with differential equations in multiple pools using a linear autonomous compartmental system of the form where the vector u represents total carbon inputs from the atmosphere to ecosystem pools, and the matrix B represents all cycling and transfer rates of carbon within the ecosystem. Linear first order models of differential equations are the most common representation of carbon dynamics in ecosystem and land surface models (Ceballos-Núñez et al., 2020;Huang et al., 2018;Luo & Weng, 2011;Luo et al., 2017). These linear autonomous compartmental systems at equilibrium have steady-state carbon stocks equivalent to At this equilibrium point, where inputs from photosynthesis are balanced by losses from ecosystem respiration, it is possible to compute (2) x * = −B −1 u. the fate of carbon inputs entering at an arbitrary time t 0 , defined as the trajectory of photosynthetically fixed carbon through the network of ecosystem compartments. This fate of carbon can be computed using the matrix exponential of the compartmental matrix . Explicitly, the mass of carbon remaining in the ecosystem after photosynthetic fixation can be obtained as where e (t − t 0 )B is the matrix exponential. In other words, photosynthetic inputs are lost from the ecosystem according to an exponential term that takes into account possible transfers of matter among compartments that are encapsulated in the matrix B.
Carbon that is lost from each pool and that is not transferred to other pools is lost from the system as respiration. Therefore, the rate of respiratory losses can be obtained as the sum of all column elements of the compartmental matrix as where ⊺ is the transpose operator and − 1 ⊺ is a row vector containing 1 (i.e. by this multiplication the column sum of B is obtained). Therefore, z ⊺ is a row vector of rates of carbon loss from each pool. Total respiratory losses are thus proportional to the amount of carbon stored at any time t. If we focus on the fate of inputs entering at t 0 , we can thus obtain the amount of respiratory losses as This function represents how carbon that enters at a particular time t 0 is lost from the system. This equation is virtually similar to the transit time distribution function derived by Metzler and Sierra (2018) and expressed as Assuming that = t − t 0 , we can see that Equations (5) and (6) are identical, with the only difference that f T ( ) is a density function that integrates to the value of one, while R(t) is a mass function that integrates to the total input mass ‖ u ‖. The symbol ‖ ‖ represents the sum of all elements inside the vector.
We can see now that the transit time distribution can be interpreted as the time it takes for carbon entering the ecosystem as GPP to appear in the respiratory flux. Rasmussen et al. (2016) have previously shown that the mean transit time is composed by the contribution to respiration of ecosystem carbon pools with specific mean ages. It is therefore of interest to compute the age distribution for each individual pool and for the entire ecosystem. According to Metzler and Sierra (2018), the vector of density distributions of age for individual pools can be obtained as where X * = diag x * 1 , x * 2 , …, x * n is the diagonal matrix with the steady-state vector of carbon stocks as components. The age distribution function for the entire system is given by These age distributions can help us to better understand how carbon of different ages contributes to the total respiratory flux in an ecosystem. We used data previously collected on above-and below-ground biomass, the biomass of fine and coarse roots, the mass of fine litter and coarse woody debris, and soil carbon stocks up to 30-cm depth (Table 1). We used data from 33 plots from secondary forests where we have a comprehensive inventory of all major carbon stocks, using locally derived biomass equations for trees, palms and coarse roots, and measurements of individual trees with diameter at breast height > 1 cm (del Valle et al., 2011;Sierra et al., 2007a;Yepes et al., 2010). We also used estimates of carbon stocks for the old-growth forests were similar measurements were conducted.

| MATERIAL S AND ME THODS
Together, these observations were used in a data assimilation procedure to fit a linear compartmental system of the form of Equation (1), using as carbon inputs satellite-derived estimates of GPP for the region as reported in Tramontana et al. (2016) and Ryu et al. (2011) (updated in Jiang & Ryu, 2016). In particular, we used the average ± (3) ‖u‖ . ( The model has seven pools, x 1 : foliage, x 2 : wood, x 3 : fine roots, x 4 : coarse roots, x 5 : fine litter, x 6 : coarse woody debris, and x 7 : soil carbon from 0 to 30 cm depth ( Figure 1). In the model, all carbon fixed as GPP enters through the foliage compartment; that is, u 1 = GPP, and from there carbon is transferred to the x 2 , x 3 , and x 4 pools according to transfer coefficients i,j that represent the proportional transfers of material from pool j to pool i. We make the implicit assumption that photosynthetically fixed carbon stored as non-structural carbohydrates in the foliage can be mobilized and allocated to wood, fine and coarse roots.
Transfers from the vegetation pools to the litter and soil pools were also represented using transfer coefficients i,j . In particular, the dynamic model has the form where the cycling rates for each pool i are denoted as k i , and the transfer coefficients from a pool j to a pool i are denoted as i,j .
Measurements of above-ground tree biomass and palm biomass were aggregated and transformed to foliage biomass using a fraction of foliage of 0.08 (Zapata & del Valle, 2001). This foliage fraction is based on site-level measurements used for the development of local biomass equations (Sierra et al., 2007a). Measurements of biomass of herbaceous vegetation were added to this foliage biomass pool.
To obtain values for the wood biomass pool, we used the aggregated values of tree and palm above-ground biomass multiplied by a fraction of wood biomass of 0.92.
The data assimilation procedure used random variates of GPP and carbon stocks in old-growth forests sampled from a normal distribution of mean values with their corresponding standard deviation. We used 1,000 random variates for GPP and 33 random variates (equivalent to the original sample size) for the old-growth carbon stocks, which were used to find 1,000 sets of parameter x 3 x 4 x 5 x 6 x 7 x 1 x 2 x 3 x 4 x 5 x 6 x 7 , values for the model using the Levenberg-Marquardt optimization algorithm (Soetaert & Petzoldt, 2010). The algorithm finds parameter values that minimize the difference between model predictions and the join set of observations of carbon stocks for all pools.
Using the average of the entire set of parameter values, we computed representative distributions of age and transit time using Equations (7), (8) and (6). We also obtained estimates of autotrophic (Ra) and heterotrophic respiration (Rh)  Atmosphere Wood respiration estimates, we then computed net primary production NPP as the difference GPP − Ra.

| Model data assimilation
We obtained 1,000 sets of parameter values of the dynamic model that provide the best fit between predictions and observations, taking into account the uncertainty and variability in GPP and steadystate carbon stocks. These parameter sets were used to compute uncertainty ranges for the predictions of the dynamic model, and to obtain one average parameter set considered as representative for the entire ensemble of parameters. Averages of the obtained parameter values, together with their uncertainty, are shown in Table 2.
Observations of carbon stocks along the successional sequence, together with possible values of GPP and carbon stocks in oldgrowth forests, provided relatively good fit to a linear autonomous compartmental system with seven pools (Figure 2). The variability in model predictions was much lower for the wood and the coarse root biomass pools than for other ecosystem pools. Except for soil carbon, the model predicts rapid accumulation of carbon in all compartment during succession consistent with previous analyses for this chronosequence (del Valle et al., 2011;Sierra et al., 2007aSierra et al., , 2012Yepes et al., 2010).
The model predicts a steady-state carbon stock of 263.9 ± 2.0 MgC/ha, which is within the upper range of the observations of total carbon stocks (with soil carbon up to 30-cm depth) of 252.4 ± 20.2 for the primary forests of the region (Sierra et al., 2007a).

| Fate of gross primary production
Using the set of average parameter values (Table 2), we obtained a representative function for the fate of carbon once it enters the ecosystem; that is, the amount of remaining carbon after photosynthetic fixation computed using Equation (3) (Figure 3). The model predicts that once carbon is fixed and incorporated in the foliage mass, it is lost within a third of a year (k 1 = 2.978/year), due to autotrophic respiration (55%) and to transfers to other pools (45%). In particular, about 25% of the losses from the foliage pool are transferred to the fine root pool ( 5,1 ), and about 16% to the wood pool ( 2,1 ) ( Table 2); however, carbon is lost quickly from the fine litter pool while it stays for longer in the wood pool (Figure 3).
Within a few years after fixation, carbon is transferred to the soil pool where it can remain for some decades. However, the model predicts that 100 years after photosynthetic fixation, most of the carbon is lost and very small proportions remain in situ.

| Age and transit time distributions
We obtained probability distributions for the age of carbon in individual pools and for the entire ecosystem using Equations (7) and Although the coarse woody debris pool has a very similar mean age (32.95 ± 1.24 years), the shape of the distribution is very different TA B L E 2 Mean and standard deviation (SD) of parameter values obtained from the 1,000 iterations of the optimization procedure than the distribution of other pools, with an age delay of a few years due to the time carbon spends in wood and coarse roots before entering this pool. The pool with the oldest mean age was the soil carbon pool, with a mean value of 61.85 ± 8.73 years, and a relatively long tail indicating that some carbon can stay for hundreds of years in the soil.
The mean age of carbon for the entire ecosystem was predicted by the model as 43.15 ± 3.33 years, and the median age was 28.6 ± 2.4 years, but clearly there is carbon that can be much older than these mean or median values. The model predicts that 95% of the carbon stored in the ecosystem is younger than 134.9 ± 10.0 years (95% quantile of the system age distribution).
We also obtained the transit time distribution of carbon for these forests at equilibrium (Figure 5a). The obtained distribution shows that 50% of the carbon that is fixed at any given year is lost in < 0.50 ± 0.14 year (median transit time), while 95% of the carbon is lost in < 68.60 ± 5.53 years. The mean transit time for the system, which can also be obtained dividing carbon stocks at equilibrium by

| D ISCUSS I ON
Our results indicate that carbon fixed during photosynthesis in a tropical forest returns back to the atmosphere at a wide range of time-scales, a property that is captured by the transit time distribution. We found that in old-growth tropical forests of the Porce region in Colombia, most of the fixed carbon is respired very quickly, with 50% of total GPP returning back to the atmosphere in half of a year after fixation. Smaller proportions of the annually fixed carbon are transferred to other ecosystem pools, and they are also gradually lost from the system. Quantiles of the transit time distribution show that 95% of the annual photosynthesis is lost in less than 69 years, and very small proportions may remain in wood, coarse roots or soil carbon for longer times.
The concept of transit time distribution as presented here, helps to reconcile different types of studies on the time-scales at which carbon is cycled in tropical forests. Previous studies with healthy tropical trees using radiocarbon techniques have shown that respired carbon is generally a few years old (Muhr et al., 2013(Muhr et al., , 2018, while mean residence time estimates based on the above-ground biomass of inventory plots are around 50 years or higher (Galbraith et al., 2013;Malhi et al., 2013Malhi et al., , 2015. However, these different estimates can be better explained in the context of an underlying distribution of transit (residence) times that can capture the fast dynamics of respiratory processes as well as the slow dynamics due to carbon transfers among compartments (e.g. from live biomass to Carbon stock (Mg C/ha) Carbon stock (Mg C/ha) coarse woody debris after tree mortality) and stabilization in slow cycling pools such as soil carbon. Previous radiocarbon studies in tropical soils have shown that soil carbon and heterotrophic respiration is mostly young, with small proportions that can persist in soils for hundreds of years (Trumbore, 1993;Trumbore & Barbosa De Camargo, 2013), in agreement with our results.
For the old-growth tropical forests of the Porce region, we estimated a mean transit time of carbon of 11 years, but the underlying transit time distribution showed, at one extreme, fast carbon losses within the first year after fixation, and at the other extreme, small amounts being respired only after several decades. Therefore, the transit time distribution has a shape with a strong initial decline, suggesting that most metabolic processes responsible for sustaining biomass stocks operate at short (intra-annual) time-scales ( Figure 5).
These processes are not well captured by mean transit (residence) time estimates such as those obtained from inventory plots alone, or dividing total carbon stocks by GPP.
The model data assimilation approach introduced here allowed us to estimate important ecosystem-level metrics that are very difficult to obtain from measurements alone such as Ra and Rh (Chambers et al., 2004). In particular, we obtained an estimate of NPP of 7.0 ± 1.5 MgC ha − 1 year − 1 by subtracting Ra from GPP. Commonly, NPP is quantified in tropical forests by measuring litter production, fine-root growth and changes in biomass from inventory plots, but this type of estimates can largely deviate from NPP as defined by the difference between GPP and Ra (Clark et al., 2001). Due to this deviation, plot-based estimates are often called NPP * to differentiate them from the flux-based definition of NPP (Clark et al., 2001).
Indeed, the inventory-based estimate of NPP * for old-growth forests of the Porce region was reported as 12.76 ± 1.36 MgC ha − 1 year − 1 in Sierra et al. (2007b). This large difference between NPP and NPP * can be due to overestimations of the inventory-based methods such as the accounting of ingrowth of new trees to inventory plots; or due to overestimations of GPP from the satellite-based products, which can lead to large estimates of autotrophic respiration in the data assimilation procedure. Independent of the reason for the disagreement, our results confirm the assertion by Clark et al. (2001) that these two type of approaches can give largely different estimates of net primary production.
We obtained an average value of 0.3 for the ratio NPP:GPP for the forests at equilibrium, a ratio that is often called carbon use efficiency (CUE) (Chambers et al., 2004;DeLucia et al., 2007;Gifford, 2003;Malhi et al., 2015). According to common interpretation, this ratio would suggests that 30% of the photosynthetically fixed carbon is used for biomass production. Similar values for CUE with similar interpretations are also given by Chambers et al. (2004) and Malhi et al. (2013), although larger variability in CUE is reported in Doughty et al. (2018). However, we believe that this common interpretation of CUE has problems since, as our transit time distribution showed, autotrophic respiration is composed of carbon that spends some time in biomass before being respired. The amount of time carbon stays in plant cells can vary from hours to decades, but photosynthates have to be metabolized from living cells (biomass) for CO 2 production to occur. Thus, autotrophic respiration originates from biomass already produced; however, most of this metabolism occurs very quickly as the transit time distribution suggests, giving the false impression that a large proportion of carbon was not used to produce biomass.
As other authors have shown (DeLucia et al., 2007;Gifford, 2003), estimates of CUE depend largely on whether estimates are made on short or long periods of time, and the transit time distribution provides good support for avoiding an interpretation of this ratio out of the context of the time-scales involved.
We prefer to interpret the ratio NPP:GPP as the proportion of total photosynthesized carbon metabolized and respired by heterotrophs, and not by autotrophs. This interpretation emerges by the simple relations assuming that at equilibrium, GPP and ecosystem respiration are equal, so GPP = Ra + Rh (Gifford, 2003;Raich & Nadelhoffer, 1989).
For the old-growth forests of the Porce region, we can thus infer that 30% of total photosynthate is respired by heterotrophic organisms, and 70% by autotrophic organisms. This interpretation has little to do with an efficiency concept for biomass production,   Age (year) Density Ecosystem flux, about 70% of the carbon follows a pathway across the network of ecosystem carbon compartments that leads to respiration by autotrophs, while 30% follows a pathway that leads to respiration by heterotrophs.

F I G U R E 3
In comparison with traditional methods that estimate mean residence times in biomass, we offer here a new perspective to integrate multiple ecosystem processes using the age of respired carbon, that is, the transit time distribution, as a unifying concept. This approach also provides a new perspective for interpreting the ratio NPP:GPP, not as an efficiency of biomass production, but as the proportion of photosynthetic products that are not respired by autotrophs.

DATA AVA I L A B I L I T Y S TAT E M E N T
All code and data needed to reproduce all results in this manuscript have been permanently archived in Zenodo with the digital object identifier https://doi.org/10.5281/zenodo.4893606 (Sierra, 2021).