Comparisons among ecosystem models or ecosystem dynamics along environmental gradients commonly rely on metrics that integrate different processes into a useful diagnostic. Terms such as age, turnover, residence, and transit times are often used for this purpose; however, these terms are variably defined in the literature and in many cases, calculations ignore assumptions implicit in their formulas. The aim of this opinion piece was i) to make evident these discrepancies and the incorrect use of formulas, ii) highlight recent results that simplify calculations and may help to avoid confusion, and iii) propose the adoption of simple and less ambiguous terms.

Linear compartmental models are commonly used in different areas of science, particularly in modeling the cycles of carbon and other biogeochemical elements. The representation of these models as linear autonomous compartmental systems allows different model structures and parameterizations to be compared. In particular, measures such as system age and transit time are useful model diagnostics. However, compact mathematical expressions describing their probability distributions remain to be derived. This paper transfers the theory of open linear autonomous compartmental systems to the theory of absorbing continuous-time Markov chains and concludes that the underlying structure of all open linear autonomous compartmental systems is the phase-type distribution. This probability distribution generalizes the exponential distribution from its application to one-compartment systems to multiple-compartment systems. Furthermore, this paper shows that important system diagnostics have natural probabilistic counterparts. For example, in steady state the system's transit time coincides with the absorption time of a related Markov chain, whereas the system age and compartment ages correspond with backward recurrence times of an appropriate renewal process. These relations yield simple explicit formulas for the system diagnostics that are applied to one linear and one nonlinear carbon-cycle model in steady state. Earlier results for transit-time and system-age densities of simple systems are found to be special cases of probability density functions of phase-type. The new explicit formulas make costly long-term simulations to obtain and analyze the age structure of open linear autonomous compartmental systems in steady state unnecessary.

Many processes in nature are modeled using compartmental systems (reservoir/pool/box systems). Usually, they are expressed as a set of first-order differential equations describing the transfer of matter across a network of compartments. The concepts of age of matter in compartments and the time required for particles to transit the system are important diagnostics of these models with applications to a wide range of scientific questions. Until now, explicit formulas for transit-time and age distributions of nonlinear time-dependent compartmental systems were not available. We compute densities for these types of systems under the assumption of well-mixed compartments. Assuming that a solution of the nonlinear system is available at least numerically, we show how to construct a linear time-dependent system with the same solution trajectory. We demonstrate how to exploit this solution to compute transit-time and age distributions in dependence on given start values and initial age distributions. Furthermore, we derive equations for the time evolution of quantiles and moments of the age distributions. Our results generalize available density formulas for the linear time-independent case and mean-age formulas for the linear time-dependent case. As an example, we apply our formulas to a nonlinear and a linear version of a simple global carbon cycle model driven by a time-dependent input signal which represents fossil fuel additions. We derive time-dependent age distributions for all compartments and calculate the time it takes to remove fossil carbon in a business-as-usual scenario.

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. Ecosystems play a fundamental role in climate change mitigation by photosynthetically fixing carbon from the atmosphere and storing it for a period of time in organic matter. Although climate impacts of carbon emissions by sources can be quantified by global warming potentials, the appropriate formal metrics to assess climate benefits of carbon removals by sinks are unclear. We introduce here the climate benefit of sequestration (CBS), a metric that quantifies the radiative effect of fixing carbon dioxide from the atmosphere and retaining it for a period of time in an ecosystem before releasing it back as the result of respiratory processes and disturbances. In order to quantify CBS, we present a formal definition of carbon sequestration (CS) as the integral of an amount of carbon removed from the atmosphere stored over the time horizon it remains within an ecosystem. Both metrics incorporate the separate effects of (i) inputs (amount of atmospheric carbon removal) and (ii) transit time (time of carbon retention) on carbon sinks, which can vary largely for different ecosystems or forms of management. These metrics can be useful for comparing the climate impacts of carbon removals by different sinks over specific time horizons, to assess the climate impacts of ecosystem management, and to obtain direct quantifications of climate impacts as the net effect of carbon emissions by sources versus removals by sinks.