Analytic derivation of a carbon turnover time stabilization model from a standard two-pool model

Multi-compartment first-order kinetics models are commonly employed to represent soil organic carbon (SOC) dynamics. In such framework, SOC is partitioned into distinct theoretical pools, each characterized by its own first-order constant that determines its intrinsic potential decomposition rate. Models with only two interacting dynamic soil carbon compartments—such as RothC or ICBM—are commonly utilized in national inventories and carbon farming initiatives due to their simplicity and ease of initialization and parameter identifiability. Alternatively, streamlined soil carbon models can treat the potential fractional turnover rate of the soil layer (ρ)—the reciprocal of its turnover time—as a state variable, further minimizing the number of parameters required. Considering the two-pool model matrices, the derivative of the ratio between the compartment stocks produces a quadratic Riccati-type differential equation which can then be further algebraically manipulated to yield a quadratic equation of the variation of fractional turnover rate, i.e. dρ/dt = aρ² + bρ + c. This continuous formulation is particularly relevant because it allows carbon decomposability to be represented as a state variable, rather than as a fixed property associated with discrete theoretical compartments. Consequently, SOC dynamics can be represented by combining the directly measurable total SOC stock with an evolving scalar descriptor of decomposability, yielding an analytical reduced-form solution that reproduces the same trajectories as the original multi-compartment model. This formulation embeds the full mechanistic behavior of pool-based kinetics into a lower-dimensional state space, providing substantial computational and statistical advantages by reducing parameter dimensionality, improving identifiability, and stabilizing estimation and data assimilation procedures.