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 10/03/2016, 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 D. S. Jenkinson & Rayner (1977).

Abstract

Data are assembled from the Rothamsted classical field experiments on the effects of long-continued cropping and manuring on the amount of organic matter in soil, on the age of this soil organic matter, on the amount of microbial biomass in the soil, and on the rate at which plant residues decompose in these soils. These data were then fitted to a model in which soil organic matter was separated into five compartments: decomposable plant material (DPM, half-life 0.165 years); resistant plant material (RPM, 2.31 years); soil biomass (BIO, 1.69 years); physically stabilized organic matter (POM, 49.5 years) and chemically stabilized organic matter (COM, 1980 years). For unitary input of plant material (1 t fresh plant \(C ha^{-1} year^{-1}\)) under steady-state conditions, after 10,000 years, the model predicts that the soil will contain 0.01 t C in DPM, 0.47 t in RPM, 0.28 t in BIO, 11.3 t in POM, and 12.2 t in COM. The predicted radiocarbon age is 1240 years (equivalent age). The fit between predicted and measured data is sufficiently good to suggest that the model is a useful representation of the turnover of organic matter in cropped soils.

Keywords

differential equations, linear, time variant

Principles

mass balance, substrate dependence of decomposition, heterogeneity of speed of decay, internal transformations of organic matter, environmental variability effects

Space Scale

plot, field, catchment, regional, national, global

Available parameter values

State Variables

The following table contains the available information regarding this section:

Decomposition Rates

The following table contains the available information regarding this section:

Additional Parameters

The following table contains the available information regarding this section:

Auxiliary Variables

The following table contains the available information regarding this section:

Input Components

The following table contains the available information regarding this section:

Transfer Coefficients

The following table contains the available information regarding this section:

Environmental Coefficients

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 \(C_{1}: \frac{DR\cdot J}{DR + 1}\) \(C_{2}: J\cdot\left(-\frac{DR}{DR + 1} + 1\right)\) Output fluxes \(C_{1}: \frac{C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\cdot\left(3.0895\cdot e^{0.0786\cdot pClay} + 2.672\right)\) \(C_{2}: \frac{C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\cdot\left(3.0895\cdot e^{0.0786\cdot pClay} + 2.672\right)\) \(C_{3}: \frac{C_{3}\cdot f_{T}\cdot f_{W}\cdot k_{3}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\cdot\left(3.0895\cdot e^{0.0786\cdot pClay} + 2.672\right)\) \(C_{4}: \frac{C_{4}\cdot f_{T}\cdot f_{W}\cdot k_{4}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\cdot\left(3.0895\cdot e^{0.0786\cdot pClay} + 2.672\right)\) Internal fluxes \(C_{1} > C_{3}: \frac{0.46\cdot C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}\cdot e^{0.0786\cdot pClay}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\) \(C_{1} > C_{4}: \frac{0.54\cdot C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}\cdot e^{0.0786\cdot pClay}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\) \(C_{2} > C_{3}: \frac{0.46\cdot C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}\cdot e^{0.0786\cdot pClay}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\) \(C_{2} > C_{4}: \frac{0.54\cdot C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}\cdot e^{0.0786\cdot pClay}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\) \(C_{3} > C_{4}: \frac{0.54\cdot C_{3}\cdot f_{T}\cdot f_{W}\cdot k_{3}\cdot e^{0.0786\cdot pClay}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\) \(C_{4} > C_{3}: \frac{0.46\cdot C_{4}\cdot f_{T}\cdot f_{W}\cdot k_{4}\cdot e^{0.0786\cdot pClay}}{4.0895\cdot e^{0.0786\cdot pClay} + 2.672}\)

The right hand side of the ODE

\(\left[\begin{matrix}- C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1} +\frac{DR\cdot J}{DR + 1}\\- C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2} + J\cdot\left(-\frac{DR}{DR + 1} + 1\right)\\\frac{0.46\cdot C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} +\frac{0.46\cdot C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} + C_{3}\cdot f_{T}\cdot f_{W}\cdot\left(- k_{3} +\frac{0.46\cdot k_{3}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\right) +\frac{0.46\cdot C_{4}\cdot f_{T}\cdot f_{W}\cdot k_{4}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\\\frac{0.54\cdot C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} +\frac{0.54\cdot C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} +\frac{0.54\cdot C_{3}\cdot f_{T}\cdot f_{W}\cdot k_{3}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} + C_{4}\cdot f_{T}\cdot f_{W}\cdot\left(- k_{4} +\frac{0.54\cdot k_{4}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\right)\\0\end{matrix}\right]\)

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

\(\left[\begin{matrix}- f_{T}\cdot f_{W}\cdot k_{1} & 0 & 0 & 0 & 0\\0 & - f_{T}\cdot f_{W}\cdot k_{2} & 0 & 0 & 0\\\frac{0.46\cdot f_{T}\cdot f_{W}\cdot k_{1}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} &\frac{0.46\cdot f_{T}\cdot f_{W}\cdot k_{2}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} & f_{T}\cdot f_{W}\cdot\left(- k_{3} +\frac{0.46\cdot k_{3}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\right) &\frac{0.46\cdot f_{T}\cdot f_{W}\cdot k_{4}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} & 0\\\frac{0.54\cdot f_{T}\cdot f_{W}\cdot k_{1}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} &\frac{0.54\cdot f_{T}\cdot f_{W}\cdot k_{2}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} &\frac{0.54\cdot f_{T}\cdot f_{W}\cdot k_{3}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} & f_{T}\cdot f_{W}\cdot\left(- k_{4} +\frac{0.54\cdot k_{4}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\right) & 0\\0 & 0 & 0 & 0 & 0\end{matrix}\right]\)

Steady state formulas

\(C_{1} = \frac{DR\cdot J}{f_{T}\cdot f_{W}\cdot k_{1}\cdot\left(DR + 1.0\right)}\)
\(C_{2} = \frac{J}{f_{T}\cdot f_{W}\cdot k_{2}\cdot\left(DR + 1.0\right)}\)
\(C_{3} = \frac{5.50898203592814\cdot J\cdot\left(4.35845179839506\cdot 10^{18}\cdot e^{0.0786\cdot pClay} + 3.26027527396786\cdot 10^{19}\cdot e^{0.1572\cdot pClay} + 9.75003544221277\cdot 10^{19}\cdot e^{0.2358\cdot pClay} + 1.45709229508337\cdot 10^{20}\cdot e^{0.3144\cdot pClay} + 1.08814080708274\cdot 10^{20}\cdot e^{0.393\cdot pClay} + 3.24846079083974\cdot 10^{19}\cdot e^{0.4716\cdot pClay}\right)}{f_{T}\cdot f_{W}\cdot k_{3}\cdot\left(1.20455080421033\cdot 10^{21}\cdot e^{0.0786\cdot pClay} + 4.32631319287619\cdot 10^{21}\cdot e^{0.1572\cdot pClay} + 8.2702084578855\cdot 10^{21}\cdot e^{0.2358\cdot pClay} + 8.87329207447322\cdot 10^{21}\cdot e^{0.3144\cdot pClay} + 5.06562843927484\cdot 10^{21}\cdot e^{0.393\cdot pClay} + 1.20193049261071\cdot 10^{21}\cdot e^{0.4716\cdot pClay} + 1.39470457548642\cdot 10^{20}\right)}\)
\(C_{4} = \frac{6.46706586826347\cdot J\cdot\left(152615747584.0\cdot e^{0.0786\cdot pClay} + 700735890432.0\cdot e^{0.1572\cdot pClay} + 1072477329312.0\cdot e^{0.2358\cdot pClay} + 547142719339.0\cdot e^{0.3144\cdot pClay}\right)}{f_{T}\cdot f_{W}\cdot k_{4}\cdot\left(28070331154432.0\cdot e^{0.0786\cdot pClay} + 60246502483968.0\cdot e^{0.1572\cdot pClay} + 57190228203392.0\cdot e^{0.2358\cdot pClay} + 20244280615543.0\cdot e^{0.3144\cdot pClay} + 4883703922688.0\right)}\)
\(C_{5} = C_{5}\)

Steady states (potentially incomplete), according jacobian eigenvalues, damping ratio

Parameter set: Set1

\(C_1: \frac{0.100327868852459}{f_{T}\cdot f_{W}}\), \(C_2: \frac{2.3224043715847}{f_{T}\cdot f_{W}}\), \(C_3: \frac{0.337161881627308}{f_{T}\cdot f_{W}}\), \(C_4: \frac{13.061358109997}{f_{T}\cdot f_{W}}\), \(C_5: C_{5}\)

\(\lambda_{1}: 0\)
\(\lambda_{2}: - 0.01733\cdot f_{T}\cdot f_{W}\)
\(\lambda_{3}: - 0.593\cdot f_{T}\cdot f_{W}\)
\(\lambda_{4}: - 10.0\cdot f_{T}\cdot f_{W}\)
\(\lambda_{5}: - 0.3\cdot f_{T}\cdot f_{W}\)

References