General Overview

This report is the result of the use of the Python 3.4 package Sympy (for symbolic mathematics), as means to translate published models to a common language. It was created by Holger Metzler (Orcid ID: 0000-0002-8239-1601) on 22/03/2016, and was last modified on lm.

About the model

The model depicted in this document considers soil organic matter decomposition. It was originally described by Fontaine & Barot (2005).

Abstract

Soil organic matter (SOM) models are based on the equation \(dC/dt = −kC\) which states that the decomposition rate of a particular carbon (C) pool is proportional to the size of the pool and the decomposition constant \(k\). However, this equation does not adequately describe the decomposition of recalcitrant SOM compounds. We present an alternative theory of SOM dynamics in which SOM decay rate is controlled by the size and the diversity of microbe populations and by the supply of energy-rich litter compounds. We show that the SOM pool does not necessarily reach equilibrium and may increase continuously, which explains how SOM can accumulate over thousands of years. However, the simulated SOM accumulation involves the sequestration of available nutrients. How can plants persist? This question is explored with two models that couple the C cycle with a limiting nutrient. The first model considers a single type of microbe whereas the second includes two functional types in competition for energy and nutrient acquisition. The condition for plant persistence is the presence of these two competing microbial types.

Keywords

differential equations, linear, time invariant, analytic

Principles

substrate dependence of decomposition, heterogeneity of speed of decay, internal transformations of organic matter, substrate interactions

State Variables

The following table contains the available information regarding this section:

Decomposition Coefficients

The following table contains the available information regarding this section:

Additional Parameters

The following table contains the available information regarding this section:

Input Components

The following table contains the available information regarding this section:

Components

The following table contains the available information regarding this section:

Pool model representation

Flux description

Figure 1: Pool model representation

Input fluxes \(C_{f}: \Phi_{l}\) \(N: \Phi_{i} -\Phi_{o} -\Phi_{up}\) Output fluxes \(C_{f}: C_{f}\cdot y\cdot\left(\alpha -\beta\right)\) \(C_{ds}: C_{ds}\cdot r\cdot\left(-\alpha + 1\right)\) \(C_{df}: C_{df}\cdot r\) \(N: N\cdot i\) Internal fluxes \(C_{s} > C_{ds}: A\cdot C_{ds}\) \(C_{f} > C_{ds}: C_{f}\cdot y\) \(C_{f} > N: C_{f}\cdot y\cdot\left(-\alpha +\beta\right)\) \(C_{ds} > C_{s}: C_{ds}\cdot s\) \(C_{ds} > N: C_{ds}\cdot\alpha\cdot r\) \(C_{df} > C_{s}: C_{df}\cdot s\) \(C_{df} > C_{f}: -\frac{C_{df}\cdot\alpha\cdot r}{\alpha -\beta}\) \(N > C_{f}: -\frac{N\cdot i}{\alpha -\beta}\) \(N > C_{df}: \frac{N\cdot i}{\alpha -\beta}\)

The right hand side of the ODE

\(\left[\begin{matrix}- A\cdot C_{ds} + C_{df}\cdot s + C_{ds}\cdot s\\-\frac{C_{df}\cdot\alpha\cdot r}{\alpha -\beta} - C_{f}\cdot y -\frac{N\cdot i}{\alpha -\beta} +\Phi_{l}\\A\cdot C_{ds} + C_{ds}\cdot\left(- r - s\right) + C_{f}\cdot y\\C_{df}\cdot\left(\frac{\alpha\cdot r}{\alpha -\beta} - r - s\right) +\frac{N\cdot i}{\alpha -\beta}\\C_{ds}\cdot\alpha\cdot r + C_{f}\cdot y\cdot\left(-\alpha +\beta\right) - N\cdot i +\Phi_{i} -\Phi_{o} -\Phi_{up}\end{matrix}\right]\)

The Jacobian (derivative of the ODE w.r.t. state variables)

\(\left[\begin{matrix}0 & 0 & - A + s & s & 0\\0 & - y & 0 & -\frac{\alpha\cdot r}{\alpha -\beta} & -\frac{i}{\alpha -\beta}\\0 & y & A - r - s & 0 & 0\\0 & 0 & 0 &\frac{\alpha\cdot r}{\alpha -\beta} - r - s &\frac{i}{\alpha -\beta}\\0 & y\cdot\left(-\alpha +\beta\right) &\alpha\cdot r & 0 & - i\end{matrix}\right]\)

References