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. Multicompartmental models can be represented via a matrix approach as a linear dynamical system, such as dC/dt = Bu + AKC, where C is the carbon stock vector, u represents external inputs, B the partition vector of input material, the matrix A defines the partitioning of carbon decomposed in each pool which is lost as CO₂ or transferred to other pools. K defines each pool’s potential fractional turnover rate. Such formulation explicitly encodes both carbon exchanges between compartments, and so SOC stabilization, as well as its losses to the atmosphere. Considering the two-pool 2×2 model matrices with parameters A = [-1, a₁₂, a₂₁, -1], K = [k₁, 0, 0, k₂], and their product AK = [-k₁, a₁₂k₂, a₂₁k₁, -k₂], the derivative of the ratio between the compartment stocks, r = C₁/C₂, produces a quadratic Riccati-type differential equation dr/dt = a₁₂k₂ - (k₁ - k₂)r - a₂₁k₁r² 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 captured using only the measurable SOC stock coupled to an evolving potential fractional turnover rate (decomposability), enhancing model identifiability and initialization since total soil carbon is directly measurable in the field.