Fitting litter decomposition datasets to mathematical curves: Towards a generalised exponential approach

Abstract

The use of exponential functions to fit decomposition datasets is common in scientific literature. Olson's exponential equation ( X t = X 0 e -kt ) is widely used, but when strong curvatures are observed, the decomposing organic matter is commonly split into two compartments ( Labile and Recalcitrant), thus obtaining double-exponential equations that often provide a good fit. Nevertheless, to correlate the so-calculated pools with quantifiable organic fractions is often very difficult, if not impossible. This suggests that even though these equations fit the experimental data well, they do not necessarily reflect what really happens in the decomposition process. The alternative is to apply models in which the organic matter, instead of being split into labile and recalcitrant compartments, is taken as a single pool whose decomposition rate is not constant. Here we propose a general approach, which considers a single organic compartment. While the original exponential function that fits the basic equation is d X/d t = - kdt, here we substitute the constant k by a function, f ( t), i.e. the decomposition rate is assumed to vary with time. Whatever function we choose, the remaining organic matter at time t is: X t = X 0 ⋅ e − ∫ 0 t f ( t ) d t and thus the problem being addressed is how to integrate the function that describes the change in the decomposition rate. In this paper we study four possible dynamics for such a change: (1) an exponential decay, (2) a wave-form change, simulating seasonal rhythms, (3) a sigmoidal increase or decrease, and (4) a rational-type dynamics, involving an increase in the initial phase, followed by a gradual decrease. For each one, the integrated form is calculated, and some practical examples are given. Given its flexibility, our approach allows a good fit for a wide number of datasets, including those that well fit a single-exponential function, the classic Olson's function strictly being a specific case of the general equation we suggest.