Exact solutions to the continuous-quality equation for soil organic matter turnover

All living systems depend on transformations of elements between different states. In particular, the transformation of dead organic matter in the soil (SOM) by decomposers (microbes) releases elements incorporated in SOM and makes the elements available anew to plants. A major problem in analysing and describing this process is that SOM, as the result of the decomposer activity, is a mixture of a very large number of molecules with widely differing chemical and physical properties. The continuous-quality equation (CQE) is a general equation describing this complexity by assigning a continuous-quality variable to each carbon atom in SOM. The use of CQE has been impeded by its complicated mathematics. Here, we show by deriving exact solutions that, at least for some specific cases, there exist solutions to CQE. These exact solutions show that previous approximations have overestimated the rate by which litter decomposes and as a consequence underestimated steady state SOM amounts. The exact and approximate solutions also differ with respect to the parameter space in which they yield finite steady-state SOM amounts. The latter point is important because temperature is one of the parameters and climatic change may move the solution from a region of the parameter space with infinite steady-state SOM to a region of finite steady-state SOM, with potentially large changes in soil carbon stores. We also show that the solution satisfies the Chapman-Kolmogorov theorem. The importance of this is that it provides efficient algorithms for numerical solutions.