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 11/8/2015, and was last modified on lm.
The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Van Der Werf, Enserink, Smit, & Booij (1993).
In this paper we model allocation of carbon and nitrogen to roots and leaves as a function of the nitrogen status of a plant. Under steady-state conditions, allocation of carbon and nitrogen to leaves is exponentially (positively) correlated with plant nitrogen concentration, whereas allocation to roots is correlated negatively, also in an exponential manner. Allocation functions derived under steady-state conditions are used to simulate biomass partitioning under non-steady-state nutrient conditions. Upon nitrogen deprivation, measured and simulated values are rather similar with time, suggesting that allocation functions derived under steady-state conditions also hold under non-steady-state conditions.
growth chamber
The following table contains the available information regarding this section:
| Name | Description |
|---|---|
| \(W_{l}\) | Dry weight of leaf blades |
| \(W_{s}\) | Dry weight of leaf sheaths |
| \(W_{r}\) | Dry weight of roots |
The following table contains the available information regarding this section:
| Name | Description | Expressions | Type |
|---|---|---|---|
| \(C_{cl}\) | Carbon content of leaf blades | - | parameter |
| \(C_{cs}\) | Carbon content of leaf sheaths | - | parameter |
| \(C_{cr}\) | Carbon content of roots | - | parameter |
| \(r_{m}\) | Amount of carbon lost per day in root respiration for maintenance processes | - | parameter |
| \(c_{\nu}\) | Amount of carbon lost per unit of nitrogen taken up | - | parameter |
| \(c_{g}\) | Amount of carbon lost per unit of root biomass produced | - | parameter |
| \(\sigma\) | Rate of nitrogen uptake per unit of root weight | - | parameter |
| \(N_{p}\) | Rate of nitrogen uptake | \(N_{p}=\sigma\,W_{r}\) | variable |
The following table contains the available information regarding this section:
| Name | Description | Type |
|---|---|---|
| \(\phi_{g}\) | Rate of gross photosynthesis per unit of leaf weight | variable |
The following table contains the available information regarding this section:
| Name | Description | Type |
|---|---|---|
| \(\alpha_{cl}\) | Allocation of carbon to leaf blades | parameter |
| \(\alpha_{cs}\) | allocation of carbon to leaf sheaths | parameter |
| \(\alpha_{cr}\) | allocation of carbon to roots | parameter |
The following table contains the available information regarding this section:
| Name | Description | Entry Author Orcid | Expressions | Type |
|---|---|---|---|---|
| \(Q_{l}\) | rate of respiration per unit of leaf | 0000-0002-0046-1160 | - | parameter |
| \(Q_{s}\) | rate of respiration per unit of stem | 0000-0002-0046-1160 | - | parameter |
| \(Q_{r}\) | Respiration rates per unit of roots -variable formulated by us, for generalization purposes- | 0000-0002-0046-1160 | \(Q_{r}=\frac{r_{m}+\sigma\,c_{\nu}}{1+\frac{c_{g}}{C_{cr}}}\) | variable |
The following table contains the available information regarding this section:
| Name | Description | Expressions |
|---|---|---|
| \(x\) | vector of states for vegetation | \(x=\left[\begin{matrix}W_{l}\\W_{s}\\W_{r}\end{matrix}\right]\) |
| \(u\) | scalar function of photosynthetic inputs | \(u=\phi_{g}\) |
| \(c\) | carbon contents per compartment | \(c=\left[\begin{matrix}\frac{1}{C_{cl}} & 0 & 0\\0 &\frac{1}{C_{cs}} & 0\\0 & 0 &\frac{1}{C_{cr}}\end{matrix}\right]\) |
| \(A\) | vector of respiration rates | \(A=\left[\begin{matrix}- Q_{l} & 0 & 0\\0 & - Q_{s} & 0\\0 & 0 & - Q_{r}\end{matrix}\right]\) |
| \(b\) | vector of partitioning coefficients of photosynthetically fixed carbon | \(b=\left[\begin{matrix}\alpha_{cl}\\\alpha_{cs}\\\frac{\alpha_{cr}}{1 +\frac{c_{g}}{C_{cr}}}\end{matrix}\right]\) |
| \(f_{v}\) | the righthandside of the ode | \(f_{v}=u\cdot x_{0, 0} c b + A c x\) |
\(\left[\begin{matrix}-\frac{Q_{l}}{C_{cl}}\cdot W_{l} +\frac{W_{l}}{C_{cl}}\cdot\alpha_{cl}\cdot\phi_{g}\\-\frac{Q_{s}}{C_{cs}}\cdot W_{s} +\frac{W_{l}}{C_{cs}}\cdot\alpha_{cs}\cdot\phi_{g}\\\frac{W_{l}\cdot\alpha_{cr}\cdot\phi_{g}}{C_{cr}\cdot\left(1 +\frac{c_{g}}{C_{cr}}\right)} -\frac{W_{r}\cdot\left(c_{\nu}\cdot\sigma + r_{m}\right)}{C_{cr}\cdot\left(1 +\frac{c_{g}}{C_{cr}}\right)}\end{matrix}\right]\)
\(\left[\begin{matrix}-\frac{Q_{l}}{C_{cl}} +\frac{\alpha_{cl}}{C_{cl}}\cdot\phi_{g} & 0 & 0\\\frac{\alpha_{cs}}{C_{cs}}\cdot\phi_{g} & -\frac{Q_{s}}{C_{cs}} & 0\\\frac{\alpha_{cr}\cdot\phi_{g}}{C_{cr}\cdot\left(1 +\frac{c_{g}}{C_{cr}}\right)} & 0 & -\frac{c_{\nu}\cdot\sigma + r_{m}}{C_{cr}\cdot\left(1 +\frac{c_{g}}{C_{cr}}\right)}\end{matrix}\right]\)
\(W_{l} = 0\)
\(W_{s} = 0\)
\(W_{r} = 0\)
Van Der Werf, A., Enserink, T., Smit, B., & Booij, R. (1993). Allocation of carbon and nitrogen as a function of the internal nitrogen status of a plant: Modelling allocation under non-steady-state situations. Plant and Soil, 155-156(1), 183–186. http://doi.org/10.1007/BF00025014