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 Verónika Ceballos-Núñez (Orcid ID: 0000-0002-0046-1160) on 16/3/2016, and was last modified on lm.

About the model

The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Murty & McMurtrie (2000).

Abstract

For closed canopy forests, both above-ground net primary productivity (ANPP) and wood yield decline as stands age. However the physiological mechanisms responsible for the decline are not well understood. Understanding of the causes of the decline and incorporation of aging mechanisms into models of forest production are essential both for sound forest management and for reliable prediction of changes in terrestrial carbon storage under altered climates. To investigate causes for declining net primary productivity (NPP) an ecosystem model G'DAY was modified to include aging mechanisms associated with three main current hypotheses for NPP decline. These hypotheses are: (1) sapwood maintenance respiration increases as stands age, reducing the availability of carbon to support growth; (2) stomatal conductance and hence photosynthetic rates decline; and (3) soil nitrogen availability declines due to nitrogen (N) accumulation in woody litter. A model-based method was developed for determining the relative importance of three mechanisms for NPP decline in forest stands. The method involves a decomposition of simulated model output into three components, each related to one aging mechanism. The method is illustrated by parameterizing G'DAY for young (40 year-old) and mature (245 year-old) stands of Pinus contorta in Colorado USA. Results from this method of analysis indicate that: 1. The G'DAY model without aging mechanisms cannot reproduce the observed changes in NPP as stands age. When the above three aging mechanisms are switched off, G'DAY shows only transient changes in NPP lasting less than 20 years. When the aging mechanisms are incorporated in G'DAY, the model simulates declining NPP on a scale similar to that observed in the field. 2. The gradual decline in NPP following canopy closure is sensitive to assumptions about aging mechanisms and is particularly sensitive to assumptions about soil N availability and declining photosynthetic rates. We identified key areas of model uncertainty requiring further experimental clarification. Here we highlight two inadequately understood processes: soil N immobilization associated with woody litter accumulation and changes in carbon allocation as stands develop. (C) 2000 Elsevier Science B.V.

Space Scale

global

Time

The following table contains the available information regarding this section:

State Variables

The following table contains the available information regarding this section:

Additional Variables

The following table contains the available information regarding this section:

Respiration Parameters

The following table contains the available information regarding this section:

Photosynthetic Parameters

The following table contains the available information regarding this section:

Partitioning coefficients

The following table contains the available information regarding this section:

Cycling Rates

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 \(C_{f}: a_{f}\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases} - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\) \(C_{r}: a_{r}\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases} - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\) \(C_{w}: \left(- a_{f} - a_{r} + 1\right)\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases} - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\) Output fluxes \(C_{f}: C_{f}\cdot\gamma_{f}\) \(C_{r}: C_{r}\cdot\gamma_{r}\) \(C_{w}: C_{w}\cdot\gamma_{w}\)

The right hand side of the ODE

\(\left[\begin{matrix}- C_{f}\cdot\gamma_{f} + a_{f}\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases} - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\\- C_{r}\cdot\gamma_{r} + a_{r}\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases} - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\\- C_{w}\cdot\gamma_{w} +\left(- a_{f} - a_{r} + 1\right)\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases} - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\end{matrix}\right]\)

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

\(\left[\begin{matrix}I_{0}\cdot a_{f}\cdot k\cdot\sigma\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\left(\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases}\right)\cdot e^{- C_{f}\cdot k\cdot\sigma} -\gamma_{f} & 0 & -\frac{0.007487172}{C_{w}^{0.23}}\cdot Q_{010}^{\frac{T_{a}}{10}}\cdot a_{f}\\I_{0}\cdot a_{r}\cdot k\cdot\sigma\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\left(\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases}\right)\cdot e^{- C_{f}\cdot k\cdot\sigma} & -\gamma_{r} & -\frac{0.007487172}{C_{w}^{0.23}}\cdot Q_{010}^{\frac{T_{a}}{10}}\cdot a_{r}\\I_{0}\cdot k\cdot\sigma\cdot\left(- a_{f} - a_{r} + 1\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t_{1} < t\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{for}\: t{\geq} t_{2}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\left(\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{f} < n_{crit}\\1 &\text{for}\: n_{f} > n_{crit}\end{cases}\right)\cdot e^{- C_{f}\cdot k\cdot\sigma} & 0 & -\frac{0.007487172}{C_{w}^{0.23}}\cdot Q_{010}^{\frac{T_{a}}{10}}\cdot\left(- a_{f} - a_{r} + 1\right) -\gamma_{w}\end{matrix}\right]\)

References