General Overview


Logo

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.

About the model

The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Hilbert & Reynolds (1991).

Space Scale

individual

Available parameter values

Information on given parameter sets
Abbreviation Description Source
set1 Original dataset of publication @Hilbert1991Annals_of_Botany

Available initial values

Information on given sets of initial values
Abbreviation Source
Original plots Hilbert & Reynolds (1991)
state_variables
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
additional_variables
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 \(N=\frac{W_{N}}{W_{g}}\) -
photosynthetic_parameters
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 \(\sigma_{c}=\min\left(A_{1}, A_{2}\right)\) -
allocation_coefficients
Name Description Expression Unit
\(P\) \(P=\frac{W_{r}\cdot f_{C}\cdot\sigma_{r}}{A\cdot f_{N}\cdot\sigma_{c}}\) -
\(Q\) \(Q=\frac{f_{N}}{B\cdot f_{C}}\) -
\(\lambda_{p}\) \(\lambda_{p}=\frac{P}{P + Q + 1}\) -
\(\lambda_{s}\) \(\lambda_{s}=\frac{Q}{P + Q + 1}\) -
\(\lambda_{r}\) \(\lambda_{r}=\frac{1}{P + Q + 1}\) -
components
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 \(f_{v}=b u + A_{x} x\)

Pool model representation


Figure 1
Figure 1: Pool model representation

Input fluxes

\(W_{N}: \frac{W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(-\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{\left(C_{i} - 31\right)\cdot\left(\frac{35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 12.42\right)}{C_{i} + 827} + 0.238, -\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{I_{dens}\cdot\left(C_{i} - 31\right)\cdot\left(\frac{92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 13.85\right)}{\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(\frac{194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + I_{dens} + 29.085\right)} + 0.238\right)}{I_{dens}\cdot h_{max}\cdot\rho}\)
\(W_{C}: W_{r}\cdot\sigma_{r}\)

Output fluxes

\(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}}\)

Internal fluxes

\(W_{C} \rightarrow W_{p}: \frac{W_{C}\cdot W_{N}\cdot W_{p}\cdot\kappa}{\left(2 +\frac{I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}}{B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(-\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{\left(C_{i} - 31\right)\cdot\left(\frac{35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 12.42\right)}{C_{i} + 827} + 0.238, -\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{I_{dens}\cdot\left(C_{i} - 31\right)\cdot\left(\frac{92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 13.85\right)}{\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(\frac{194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + I_{dens} + 29.085\right)} + 0.238\right)}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
\(W_{C} \rightarrow W_{s}: \frac{I_{dens}\cdot W_{C}\cdot W_{N}\cdot W_{r}\cdot W_{s}\cdot h_{max}\cdot\kappa\cdot\rho\cdot\sigma_{r}}{\left(2\cdot B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(C_{i} + 827\right) + W_{s}\cdot\left(0.238\cdot C_{i} + 196.826\right)\cdot\left(I_{dens} + h_{half}\right) +\left(C_{i} - 31\right)\cdot\left(35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 12.42\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(C_{i} + 827\right)\cdot\left(I_{dens} + h_{half}\right)},\frac{- 0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right) + I_{dens}\cdot W_{s}\cdot\left(C_{i} - 31\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + 13.85\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\right) + 0.238\cdot W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}{W_{s}\cdot\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(I_{dens} + h_{half}\right)\cdot\left(194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho + W_{s}\cdot\left(I_{dens} + 29.085\right)\cdot\left(I_{dens} + h_{half}\right)\right)}\right) + I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)
\(W_{C} \rightarrow W_{r}: \frac{W_{C}\cdot W_{N}\cdot W_{r}\cdot\kappa}{\left(2 +\frac{I_{dens}\cdot W_{r}\cdot h_{max}\cdot\rho\cdot\sigma_{r}}{B\cdot W_{s}\cdot\left(I_{dens} + h_{half}\right)\cdot\min\left(-\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{\left(C_{i} - 31\right)\cdot\left(\frac{35.76\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 12.42\right)}{C_{i} + 827} + 0.238, -\frac{0.775\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} +\frac{I_{dens}\cdot\left(C_{i} - 31\right)\cdot\left(\frac{92.55\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + 13.85\right)}{\left(4.5\cdot C_{i} + 325.5\right)\cdot\left(\frac{194.355\cdot I_{dens}\cdot W_{p}\cdot f_{np}\cdot h_{max}\cdot\rho}{W_{s}\cdot\left(I_{dens} + h_{half}\right)} + I_{dens} + 29.085\right)} + 0.238\right)}\right)\cdot\left(W_{p} + W_{r} + W_{s}\right)^{2}}\)

References

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