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 17/03/2016, 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 S. D. Allison, Wallenstein, & Bradford (2010).

Further references

Li, Wang, Allison, Mayes, & Luo (2014): The notation here is adapted to the one in this paper. Temperature sensitivities for \(V\), \(V_U\), \(K\), and \(K_U\) follow the Arrhenius relationship.

Abstract

Most ecosystem models predict that climate warming will stimulate microbial decomposition of soil carbon, producing a positive feedback to rising global temperatures. Although field experiments document an initial increase in the loss of CO\(_2\) from soils in response to warming, in line with these predictions, the carbon dioxide loss from soils tends to decline to control levels within a few years. This attenuation response could result from changes in microbial physiological properties with increasing temperature, such as a decline in the fraction of assimilated carbon that is allocated to growth, termed carbon-use efficiency. Here we explore these mechanisms using a microbial-enzyme model to simulate the responses of soil carbon to warming by 5°C. We find that declines in microbial biomass and degradative enzymes can explain the observed attenuation of soil-carbon emissions in response to warming. Specifically, reduced carbon-use efficiency limits the biomass of microbial decomposers and mitigates the loss of soil carbon. However, microbial adaptation or a change in microbial communities could lead to an upward adjustment of the efficiency of carbon use, counteracting the decline in microbial biomass and accelerating soil-carbon loss. We conclude that the soil-carbon response to climate warming depends on the efficiency of soil microbes in using carbon.

Keywords

differential equations, nonlinear, time variant, Michaelis-Menten

Principles

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

State Variables

The following table contains the available information regarding this section:

Constants

The following table contains the available information regarding this section:

Parameters

The following table contains the available information regarding this section:

Additional Variables

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 \(S: I_{S}\) \(D: I_{D}\) Output fluxes \(S: -\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\) \(D: -\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0} - 1\right)\) Internal fluxes \(S > D: \frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\) \(S > B: \frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\) \(D > B: \frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0}\right)\) \(B > S: B\cdot a_{BS}\cdot r_{B}\) \(B > D: B\cdot r_{B}\cdot\left(- a_{BS} + 1\right)\) \(B > E: B\cdot r_{E}\) \(E > D: E\cdot r_{L}\)

The right hand side of the ODE

\(\left[\begin{matrix}B\cdot a_{BS}\cdot r_{B} -\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} + I_{S}\\-\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T} + B\cdot r_{B}\cdot\left(- a_{BS} + 1\right) +\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} + E\cdot r_{L} + I_{D}\\\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0}\right) + B\cdot\left(- r_{B} - r_{E}\right) +\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\\B\cdot r_{E} - E\cdot r_{L}\end{matrix}\right]\)

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

\(\left[\begin{matrix}\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{\left(K_{0} + K_{s}\cdot T + S\right)^{2}} -\frac{E\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} & 0 & a_{BS}\cdot r_{B} & -\frac{S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\\-\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{\left(K_{0} + K_{s}\cdot T + S\right)^{2}} +\frac{E\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} &\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{\left(D + K_{U0} + K_{Us}\cdot T\right)^{2}} -\frac{B\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T} & -\frac{D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T} + r_{B}\cdot\left(- a_{BS} + 1\right) &\frac{S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} + r_{L}\\-\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{\left(K_{0} + K_{s}\cdot T + S\right)^{2}} +\frac{E\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} & -\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{\left(D + K_{U0} + K_{Us}\cdot T\right)^{2}}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0}\right) +\frac{B\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0}\right) &\frac{D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0}\right) - r_{B} - r_{E} &\frac{S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\\0 & 0 & r_{E} & - r_{L}\end{matrix}\right]\)

References