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 09/03/2016, and was last modified on lm.
The model depicted in this document considers soil organic matter decomposition. It was originally described by Andrén & Kätterer (1997).
A two-component model was devised, comprising young and old soil C, two decay constants, and parameters for litter input, ‘‘humification,’’ and external influences. Due to the model’s simplicity, the differential equations were solved analytically, and parameter optimizations can be made using generally available nonlinear regression programs. The calibration parameter values were derived from a 35-yr experiment with arable crops on a clay soil in central Sweden. We show how the model can be used for medium-term (30 yr) predictions of the effects of changed inputs, climate, initial pools, litter quality, etc., on soil carbon pools. Equations are provided for calculating steady-state pool sizes as well as model parameters from litter bag or \(^{14}\)C-labeled litter decomposition data. Strategies for model parameterization to different inputs, climatic regions, and soils, as well as the model’s relations to other model families, are briefly discussed.
differential equations, linear, time invariant, analytic
mass balance, substrate dependence of decomposition, heterogeneity of speed of decay, internal transformations of organic matter, environmental variability effects
| Abbreviation | Source |
|---|---|
| Bare fallow | Andrén & Kätterer (1997) |
| +N +straw | Andrén & Kätterer (1997) |
| -N +straw | Andrén & Kätterer (1997) |
| -N -straw | Andrén & Kätterer (1997) |
| +N -straw | Andrén & Kätterer (1997) |
| Farmyard manure | Andrén & Kätterer (1997) |
| Sewage sludge | Andrén & Kätterer (1997) |
| Abbreviation | Source |
|---|---|
| Bare fallow | Andrén & Kätterer (1997) |
| +N +straw | Andrén & Kätterer (1997) |
| -N +straw | Andrén & Kätterer (1997) |
| -N -straw | Andrén & Kätterer (1997) |
| +N -straw | Andrén & Kätterer (1997) |
| Farmyard manure | Andrén & Kätterer (1997) |
| Sewage sludge | Andrén & Kätterer (1997) |
The following table contains the available information regarding this section:
| Name | Description | Units | Values Bare fallow |
+N +straw | -N +straw | -N -straw | +N -straw | Farmyard manure | Sewage sludge |
|---|---|---|---|---|---|---|---|---|---|
| \(Y\) | young pool of soil carbon | \(kgCm^{-2}\) | \(0.3\) | \(0.3\) | \(0.3\) | \(0.3\) | \(0.3\) | \(0.3\) | \(0.3\) |
| \(O\) | old pool of soil carbon | \(kgCm^{-2}\) | \(3.96\) | \(4.11\) | \(4.05\) | \(3.99\) | \(4.02\) | \(3.99\) | \(4.14\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Units | Values Bare fallow |
+N +straw | -N +straw | -N -straw | +N -straw | Farmyard manure | Sewage sludge |
|---|---|---|---|---|---|---|---|---|---|---|
| \(k_{1}\) | decomposition rate of young pool | parameter | \(yr^{-1}\) | \(0.8\) | \(0.8\) | \(0.8\) | \(0.8\) | \(0.8\) | \(0.8\) | \(0.8\) |
| \(k_{2}\) | decomposition rate of old pool | parameter | \(yr^{1}\) | \(0.00605\) | \(0.00605\) | \(0.00605\) | \(0.00605\) | \(0.00605\) | \(0.00605\) | \(0.00605\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Units | Values Bare fallow |
+N +straw | -N +straw | -N -straw | +N -straw | Farmyard manure | Sewage sludge |
|---|---|---|---|---|---|---|---|---|---|---|
| \(i\) | mean annual carbon input | parameter | \(kgC m^{-2}yr^{-1}\) | \(0\) | \(0.285\) | \(0.248\) | \(0.057\) | \(0.091\) | \(0.272\) | \(0.296\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Values Bare fallow |
+N +straw | -N +straw | -N -straw | +N -straw | Farmyard manure | Sewage sludge |
|---|---|---|---|---|---|---|---|---|---|
| \(h\) | humification coefficient | parameter | \(0.13\) | \(0.125\) | \(0.125\) | \(0.125\) | \(0.125\) | \(0.25\) | \(0.34\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Values Bare fallow |
+N +straw | -N +straw | -N -straw | +N -straw | Farmyard manure | Sewage sludge |
|---|---|---|---|---|---|---|---|---|---|
| \(r\) | climatic and edaphic factors | parameter | \(1.32\) | \(1.0\) | \(1.22\) | \(1.17\) | \(1.07\) | \(1.1\) | \(0.97\) |
The following table contains the available information regarding this section:
| Name | Units | Values Bare fallow |
+N +straw | -N +straw | -N -straw | +N -straw | Farmyard manure | Sewage sludge |
|---|---|---|---|---|---|---|---|---|
| \(t\) | \(yr\) | - | - | - | - | - | - | - |
The following table contains the available information regarding this section:
| Name | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}Y\\O\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}i\\0\end{matrix}\right]\) |
| \(\xi\) | environmental effects multiplier | \(\xi=r\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0\\h & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}k_{1} & 0\\0 & k_{2}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+\xi\,T\,N\,C\) |
| Flux description | |
|---|---|
 Figure 1: Pool model representation |
Input fluxes\(Y: i\) Output fluxes\(Y: Y\cdot k_{1}\cdot r\cdot\left(- h + 1\right)\) Internal fluxes\(Y > O: Y\cdot h\cdot k_{1}\cdot r\) |
\(\left[\begin{matrix}- Y\cdot k_{1}\cdot r + i\\- O\cdot k_{2}\cdot r + Y\cdot h\cdot k_{1}\cdot r\end{matrix}\right]\)
\(\left[\begin{matrix}- k_{1}\cdot r & 0\\h\cdot k_{1}\cdot r & - k_{2}\cdot r\end{matrix}\right]\)
\(Y = \frac{i}{k_{1}\cdot r}\)
\(O = \frac{h\cdot i}{k_{2}\cdot r}\)
\(Y: 0.0\), \(O: 0.0\)
\(\lambda_{1}: -0.008\)
\(\lambda_{2}: -1.056\)
\(Y: 0.356\), \(O: 5.888\)
\(\lambda_{1}: -0.006\)
\(\lambda_{2}: -0.800\)
\(Y: 0.254\), \(O: 4.2\)
\(\lambda_{1}: -0.976\)
\(\lambda_{2}: -0.007\)
\(Y: 0.061\), \(O: 1.007\)
\(\lambda_{1}: -0.936\)
\(\lambda_{2}: -0.007\)
\(Y: 0.106\), \(O: 1.757\)
\(\lambda_{1}: -0.856\)
\(\lambda_{2}: -0.006\)
\(Y: 0.309\), \(O: 10.218\)
\(\lambda_{1}: -0.880\)
\(\lambda_{2}: -0.007\)
\(Y: 0.381\), \(O: 17.149\)
\(\lambda_{1}: -0.776\)
\(\lambda_{2}: -0.006\)
Andrén, O., & Kätterer, T. (1997). ICBM: The introductory carbon balance model for exploration of soil carbon balances. Ecological Applications, 7(4), 1226–1236.