This report is the result of the use of the python package bgc_md, as means to translate published models to a common language. The underlying yaml file was created by Verónika Ceballos-Núñez (Orcid ID: 0000-0002-0046-1160) on 29/7/2015.
The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Hilbert & Reynolds (1991).
individual
| Abbreviation | Description | Source |
|---|---|---|
| set1 | Original dataset of publication | @Hilbert1991Annals_of_Botany |
| Abbreviation | Source |
|---|---|
| Original plots | Hilbert & Reynolds (1991) |
| Name | Description |
|---|---|
| \(W_{p}\) | Mass of leaf proteins |
| \(W_{s}\) | Mass of leaf structural components |
| \(W_{r}\) | Mass of roots |
| \(W_{C}\) | Substrate carbon |
| \(W_{N}\) | Substrate nitrogen |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(W_{g}\) | Plant biomass | \(W_{g}=W_{p} + W_{r} + W_{s}\) | - |
| \(\kappa\) | growth rate coefficient | - | - |
| \(\sigma_{r}\) | specific root activity | - | \([g N (g root)^{-1} d^{-1}]\) |
| \(f_{C}\) | Proportion of carbon | - | - |
| \(f_{N}\) | Proportion of nitrogen | \(f_{N}=\frac{W_{r}\cdot f_{C}\cdot\sigma_{r}}{A\cdot\sigma_{c}}\) | - |
| \(B\) | Target whole plant nitrogen:carbon ratio | - | - |
| \(f_{cp}\) | Proportion of carbon in leaf proteins | - | - |
| \(f_{cs}\) | Proportion of carbon in leaf structural components | - | - |
| \(f_{cr}\) | Proportion of carbon in roots | - | - |
| \(f_{np}\) | Proportion of nitrogen in leaf proteins | - | - |
| \(f_{ns}\) | Proportion of nitrogen in leaf structural components | - | - |
| \(f_{nr}\) | Proportion of nitrogen in roots | - | - |
| \(C\) | Substrate carbon concentration | \(C=\frac{W_{C}}{W_{g}}\) | - |
| \(N\) | Substrate nitrogen concentration | \(N=\frac{W_{N}}{W_{g}}\) | - |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(t\) | time step | - | \(day\) |
| \(I_{dens}\) | (I) photon flux density. Photosynthetically active radiation. | - | \([\mu mol\, m^{-2}\,s^{-1}]\) |
| \(h_{max}\) | leaf max. thickness | - | \([m]\) |
| \(h_{half}\) | \(h_{0.5}\) leaf half thickness | - | - |
| \(\rho\) | leaf density | - | - |
| \(h\) | \(h=\frac{I_{dens}\cdot h_{max}}{I_{dens} + h_{half}}\) | - | |
| \(A\) | Area | \(A=\frac{W_{s}}{h\cdot\rho}\) | - |
| \(N_{a}\) | Leaf nitrogen concentration | \(N_{a}=\frac{W_{p}\cdot f_{np}}{A}\) | \(gN\cdot m^{-2}\) |
| \(C_{i}\) | Intercellular CO_2 concentration | - | \([\mu l\, l^{-1}]\) |
| \(V_{cmax}\) | Maximum carboxylation velocity | \(V_{cmax}=35.76\cdot N_{a} + 12.42\) | \([\mu mol CO_2\, m^{-2}\,s^{-1}]\) |
| \(J_{max}\) | Maximum rate of electron transport | \(J_{max}=92.55\cdot N_{a} + 13.85\) | \([\mu Eq\, m^{-2}\,s^{-1}]\) |
| \(J\) | Rate of electron transport | \(J=\frac{I_{dens}\cdot J_{max}}{I_{dens} + 2.1\cdot J_{max}}\) | - |
| \(R_{d}\) | Dark respiration rate | \(R_{d}=0.775\cdot N_{a} - 0.238\) | \([\mu mol CO_2\, m^{-2}\,s^{-1}]\) |
| \(A_{1}\) | RuBP saturated portion of the carbon dioxide response curve | \(A_{1}=- R_{d} +\frac{V_{cmax}\cdot\left(C_{i} - 31\right)}{C_{i} + 827}\) | \([\mu mol CO_2\, m^{-2}\,s^{-1}]\) |
| \(A_{2}\) | RuBP limited portion of the carbon dioxide response curve | \(A_{2}=\frac{J\cdot\left(C_{i} - 31\right)}{4.5\cdot C_{i} + 325.5} - R_{d}\) | \([\mu mol CO_2\, m^{-2}\,s^{-1}]\) |
| \(\sigma_{c}\) | Photosynthetic rate per unit leaf | \(\sigma_{c}=\min\left(A_{1}, A_{2}\right)\) | - |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(P\) | \(P=\frac{W_{r}\cdot f_{C}\cdot\sigma_{r}}{A\cdot f_{N}\cdot\sigma_{c}}\) | - | |
| \(Q\) | \(Q=\frac{f_{N}}{B\cdot f_{C}}\) | - | |
| \(\lambda_{p}\) | \(\lambda_{p}=\frac{P}{P + Q + 1}\) | - | |
| \(\lambda_{s}\) | \(\lambda_{s}=\frac{Q}{P + Q + 1}\) | - | |
| \(\lambda_{r}\) | \(\lambda_{r}=\frac{1}{P + Q + 1}\) | - |
| Name | Description | Expression |
|---|---|---|
| \(x\) | vector of states for vegetation | \(x=\left[\begin{matrix}W_{N}\\W_{C}\\W_{p}\\W_{s}\\W_{r}\end{matrix}\right]\) |
| \(u\) | scalar function of photosynthetic inputs | \(u=\left[\begin{matrix}A\cdot\sigma_{c}\\W_{r}\cdot\sigma_{r}\\0\\0\\0\end{matrix}\right]\) |
| \(b\) | partitioning coefficient of system inputs | \(b=1\) |
| \(A_{x}\) | matrix of turnover (cycling) rates | \(A_{x}=\left[\begin{matrix}\frac{W_{N}\cdot\kappa\cdot\left(- W_{p}\cdot f_{cp}\cdot\lambda_{p} - W_{r}\cdot f_{cr}\cdot\lambda_{r} - W_{s}\cdot f_{cs}\cdot\lambda_{s}\right)}{W_{g}^{2}} & 0 & 0 & 0 & 0\\0 &\frac{W_{C}\cdot\kappa\cdot\left(- W_{p}\cdot f_{np}\cdot\lambda_{p} - W_{r}\cdot f_{nr}\cdot\lambda_{r} - W_{s}\cdot f_{ns}\cdot\lambda_{s}\right)}{W_{g}^{2}} & 0 & 0 & 0\\0 &\frac{N\cdot W_{p}\cdot\kappa\cdot\lambda_{p}}{W_{g}} & 0 & 0 & 0\\0 &\frac{N\cdot W_{s}\cdot\kappa\cdot\lambda_{s}}{W_{g}} & 0 & 0 & 0\\0 &\frac{N\cdot W_{r}\cdot\kappa\cdot\lambda_{r}}{W_{g}} & 0 & 0 & 0\end{matrix}\right]\) |
| \(f_{v}\) | the righthandside of the ode | \(f_{v}=b u + A_{x} x\) |
\(W_{N}: \frac{W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(-\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{\left(C_{i} - 31\right)\cdot\left(\frac{35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 12.42\right)}{C_{i} + 827} + 0.238, -\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{I_{dens}\cdot\left(C_{i} - 31\right)\cdot\left(\frac{92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 13.85\right)}{\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(\frac{194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + I_{dens} + 29.085\right)} + 0.238\right)}{I_{dens}\cdot h_{max}\cdot\rho}\)
\(W_{C}: W_{r}\cdot\sigma_{r}\)
\(W_{N}: \frac{W_{N}^{2}\cdot W_{s}\cdot\kappa\cdot\left(B\cdot W_{p}\cdot f_{cp}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + B\cdot W_{r}\cdot f_{cr}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{r}\cdot f_{cs}\cdot h_{max}\cdot\rho\cdot\sigma_{r}\right)}{\left(2\cdot B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
\(W_{C}: -\frac{W_{C}\cdot W_{s}\cdot\kappa\cdot\left(B\cdot W_{N}\cdot W_{p}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + B\cdot W_{N}\cdot W_{r}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{N}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r} - W_{C}\cdot\left(B\cdot W_{p}\cdot f_{np}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + B\cdot W_{r}\cdot f_{nr}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{r}\cdot f_{ns}\cdot h_{max}\cdot\rho\cdot\sigma_{r}\right)\right)}{\left(2\cdot B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
\(W_{C} \rightarrow W_{p}: \frac{W_{C}\cdot W_{N}\cdot W_{p}\cdot\kappa}{\left(2 +\frac{I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}}{B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(-\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{\left(C_{i} - 31\right)\cdot\left(\frac{35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 12.42\right)}{C_{i} + 827} + 0.238, -\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{I_{dens}\cdot\left(C_{i} - 31\right)\cdot\left(\frac{92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 13.85\right)}{\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(\frac{194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + I_{dens} + 29.085\right)} + 0.238\right)}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
\(W_{C} \rightarrow W_{s}: \frac{I_{dens}\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot h_{max}\cdot\kappa\cdot\rho\cdot\sigma_{r}}{\left(2\cdot B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
\(W_{C} \rightarrow W_{r}: \frac{W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa}{\left(2 +\frac{I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}}{B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(-\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{\left(C_{i} - 31\right)\cdot\left(\frac{35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 12.42\right)}{C_{i} + 827} + 0.238, -\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{I_{dens}\cdot\left(C_{i} - 31\right)\cdot\left(\frac{92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 13.85\right)}{\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(\frac{194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + I_{dens} + 29.085\right)} + 0.238\right)}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
Hilbert, D. W., & Reynolds, J. F. (1991). A model allocating growth among leaf proteins, shoot structure, and root biomass to produce balanced activity. Annals of Botany, 68(5), 417–425. Retrieved from http://aob.oxfordjournals.org/content/68/5/417.abstract