Models representing exchange of carbon between the atmosphere and the terrestrial biosphere include a large variety of processes and mechanisms, and have increased in complexity in the last decades. These models are no exception of the simulation versus understanding conundrum previously articulated for models of the physical climate, which states that increasing detail in process representation in models, and the simulations they produce, hinders understanding of holistic system behavior. However, recent theoretical progress on the mathematical representation of the carbon cycle in ecosystems may help to provide a general framework for the qualitative understanding of models without compromising detail in process representation. Here we (1) briefly review recent ideas on the theory of transient dynamics of the terrestrial carbon cycle and its matrix representation, pointing out issues of interpretation, (2) show that these ideas can be further generalized in the mathematical concept of nonautonomous compartmental systems, and (3) provide thoughts on how this framework can be used to address a new set of questions in carbon cycle science.

Abstract. 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 modeled three systems of vegetation compartments and assessed their performance by calculating the age of the carbon in vegetation system and within each compartment, and 5 the overall transit time of C in the system. We used these diagnostics to assess the influence of three different carbon allocation schemes on the rates of C cycling in vegetation. First, we used published measurements of ecosystem C compartments from the Harvard Forest Environmental Measurement Site to find the best set of parameters for the different model structures. Second, we calculated C stocks, release fluxes, radiocarbon values based on the bomb spike, ages, and transit times. We found a good fit of the three model structures to the available data, but the time series of C in foliage and wood need to be complemented 10 with other ecosystem compartments in order to reduce the high parameter collinearity that we observed, and reduce model equifinality. Differences in model structures had a small impact on predicting C stocks in ecosystem compartments, but overall they 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 10-20 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, 15 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 on the temporal dynamics of carbon use, storage, and allocation to plant 20 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. 1Biogeosciences Discuss., https://doi

The representation of carbon allocation (CA) in ecosystem differs tremendously among models, resulting in diverse responses of carbon cycling and storage to global change. Several studies have highlighted discrepancies between empirical observations and model predictions, attributing these differences to problems of model structure. We analyzed the mathematical representation of CA in models using concepts from dynamical systems theory; we reviewed a representative sample of models of CA in vegetation and developed a model database within the Python package bgc-md. We asked whether these representations can be generalized as a linear system, or whether a more general framework is needed to accommodate nonlinearities. Some of the vegetation systems simulated with the reviewed models have a fixed partitioning of photosynthetic products, independent of environmental forcing. Vegetation is often represented as a linear system without storage compartments. Yet, other structures with nonlinearities have also been proposed, with important consequences on the temporal trajectories of ecosystem carbon compartments. The proposed mathematical framework unifies the representation of alternative CA schemes, facilitating their classification according to mathematical properties as well as their potential temporal behaviour. It can represent complex processes in a compact form, which can potentially facilitate dialog among empiricists, theoreticians, and modellers.

Abstract. Carbon allocation in vegetation is an important process in the terrestrial carbon cycle; it determines the fate of photoassimilates, and it has an impact on the time carbon spends in the terrestrial biosphere. Although previous studies have highlighted important conceptual issues in the definition and metrics used to assess carbon allocation, very little emphasis has been placed on the distinction between the allocation of carbon from gross primary production (GPP) and the allocation from net primary production (NPP). An important number of simulation models and conceptual frameworks are based on the concept that C is allocated from NPP, which implies that C is respired immediately after photosynthetic assimilation. However, empirical work that estimates the age of respired CO2 from vegetation tissue (foliage, stems, roots) shows that it may take from years to decades to respire previously produced photosynthates. The transit time distribution of carbon in vegetation and ecosystems, a metric that provides an estimate of the age of respired carbon, indicates that vegetation pools respire carbon of a wide range of ages, on timescales that are in conflict with the assumption that autotrophic respiration only consumes recently fixed carbon. In this contribution, we attempt to provide compelling evidence based on recent research on the age of respired carbon and the theory of timescales of carbon in ecosystems, with the aim to promote a change in the predominant paradigm implemented in ecosystem models where carbon allocation is based on NPP. In addition, we highlight some implications for understanding and modeling carbon dynamics in terrestrial ecosystems.

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.