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 26/1/2016.
The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Foley et al. (1996).
global
| Abbreviation | Source |
|---|---|
| Tropical evergreen trees | Foley et al. (1996) |
| Name | Description |
|---|---|
| \(C_{il}\) | Carbon in leaves of plant functional type (PFT) i |
| \(C_{is}\) | Carbon in transport tissue (mainly stems) of PFT\(_{i}\) |
| \(C_{ir}\) | Carbon in fine roots of PFT\(_{i}\) |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(t\) | time step | - | \(s\) |
| \(Q_{p}\) | Flux density of photosynthetically active radiation absorbed by the leaf | - | \(Einstein\cdot m^{-2}\cdot s^{-1}\) |
| \(\alpha_{3}\) | Intrinsic quantum efficiency for CO_2 uptake in C_3 plants | - | \(mol CO_2\cdot Einstein^{-1}\) |
| \(\alpha_{4}\) | - | - | |
| \(O_{2}\) | Atmospheric [O_2] (value: 0.209) | - | \(mol\cdot mol^{-1}\) |
| \(\tau\) | Ratio of kinetic parameters describing the partitioning of enzyme activity to carboxylase or oxygenase function | - | - |
| \(\Gamma\) | Gamma^* is the compensation point for gross photosynthesis | \(\Gamma=\frac{O_{2}}{2\cdot\tau}\) | \(mol\cdot mol^{-1}\) |
| \(C_{i}\) | [CO_2] in the intercellular air spaces of the leaf | - | \(mol\cdot mol^{-1}\) |
| \(J_{e}\) | Light-limited rate of photoynthesis | \(J_{e}=\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\Gamma\right)}{C_{i} + 2\cdot\Gamma}\) | - |
| \(J_{e4}\) | Rubisco-limited rate of photosynthesis (C4 plants) | \(J_{e4}=V_{m}\) | - |
| \(V_{m}\) | Maximum capacity of Rubisco to perform the carboxylase fuction | - | \(mol CO_2\cdot m^{-2}\cdot s^{-1}\) |
| \(K_{c}\) | Michaelis-Menten coefficient for CO\(_{2}\) | - | \(mol\cdot mol^{-1}\) |
| \(K_{o}\) | Michaelis-Menten coefficient for O\(_{2}\) | - | \(mol\cdot mol^{-1}\) |
| \(J_{c}\) | Rubisco-limited rate of photosynthesis | \(J_{c}=\frac{V_{m}\cdot\left(C_{i} -\Gamma\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\) | - |
| \(k\) | - | - | |
| \(J_{c4}\) | CO\(_{2}\)-limited rate of photosynthesis at low [CO\(_{2}\)] (C4 plants) | \(J_{c4}=C_{i}\cdot k\) | - |
| \(T\) | Rate of triose phosphate utilization | \(T=0.121951219512195\cdot V_{m}\) | - |
| \(J_{p}\) | see section 6a | - | - |
| \(J_{s}\) | Triose phosphate-limited rate of photosynthesis | \(J_{s}=3\cdot T\cdot\left(1 -\frac{\Gamma}{C_{i}}\right) +\frac{\Gamma\cdot J_{p}}{C_{i}}\) | - |
| \(J_{i}\) | Light-limited rate of photosynthesis (C4 plants) | \(J_{i}=Q_{p}\cdot\alpha_{4}\) | - |
| \(A_{g}\) | Gross photosynthesis rate per unit of area | \(A_{g}=\min\left(J_{c}, J_{e}, J_{s}\right)\) | \(mol CO_2\cdot m^{-2}\cdot s^{-2}\) |
| \(\gamma\) | Leaf respiration cost of Rubisco acivity | - | - |
| \(B_{stem}\) | Maintenance respiration coefficient defined at 15°C | - | - |
| \(B_{root}\) | Maintenance respiration coefficient defined at 15°C | - | - |
| \(\lambda_{sapwood}\) | Sapwood fraction of the total stem biomass (estimated from an assumed sap velocity and the maximum rate of transpiration experienced during the previous year) | - | - |
| \(E_{0}\) | Temperature sensitivity factor | - | - |
| \(T_{0}\) | Set to absolute zero (-273.16 °C) | - | - |
| \(T_{stem}\) | Stem temperature | - | \(°C\) |
| \(T_{soil}\) | Temperature of the soil in the rooting zone | - | \(°C\) |
| \(fT_{stem}\) | f(T) is the Arrenhius temperature function | \(fT_{stem}=e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)}\) | - |
| \(fT_{soil}\) | f(T) is the Arrenhius temperature function | \(fT_{soil}=e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)}\) | - |
| \(R_{leaf}\) | Leaf maintenance respiration | \(R_{leaf}=V_{m}\cdot\gamma\) | \(mol CO_2\cdot m^{-2}\cdot s^{-1}\) |
| \(R_{stem}\) | Stem maintenance respiration | \(R_{stem}=B_{stem}\cdot C_{is}\cdot fT_{stem}\cdot\lambda_{sapwood}\) | - |
| \(R_{root}\) | Root maintenance respiration | \(R_{root}=B_{root}\cdot C_{ir}\cdot fT_{soil}\) | - |
| \(A_{n}\) | Net leaf assimilation rate | \(A_{n}=A_{g} - R_{leaf}\) | \(mol CO_2\cdot m^{-2}\cdot s^{-1}\) |
| \(GPP - i\) | Gross primary productivity | \(GPP_{i}=A_{g}\cdot t\) | - |
| \(\eta\) | Fraction of carbon lost in the construction of net plant material because of growth respiration (value 0.33) | - | - |
| \(NPP_{i}\) | Net Primary Production for PFT\(_{i}\) | \(NPP_{i}=t\cdot\left(1 -\eta\right)\cdot\left(A_{g} - R_{leaf} - R_{root} - R_{stem}\right)\) | - |
| Name | Description |
|---|---|
| \(a_{il}\) | Fraction of annual NPP allocated to leaves for PFT\(_{i}\) |
| \(a_{is}\) | Fraction of annual NPP allocated to stem for PFT\(_{i}\) |
| \(a_{ir}\) | Fraction of annual NPP allocated to roots for PFT\(_{i}\) |
| Name | Description | Unit |
|---|---|---|
| \(\tau_{il}\) | Residence time of carbon in leaves for PFT\(_{i}\) | - |
| \(\tau_{is}\) | Residence time of carbon in stem for PFT\(_{i}\) | - |
| \(\tau_{ir}\) | Residence time of carbon in roots for PFT\(_{i}\) | - |
| Name | Description | Expression |
|---|---|---|
| \(x\) | vector of states for vegetation | \(x=\left[\begin{matrix}C_{il}\\C_{is}\\C_{ir}\end{matrix}\right]\) |
| \(u\) | scalar function of photosynthetic inputs | \(u=NPP_{i}\) |
| \(b\) | vector of partitioning coefficients of photosynthetically fixed carbon | \(b=\left[\begin{matrix}a_{il}\\a_{is}\\a_{ir}\end{matrix}\right]\) |
| \(A\) | matrix of turnover (cycling) rates | \(A=\left[\begin{matrix}-\frac{1}{\tau_{il}} & 0 & 0\\0 & -\frac{1}{\tau_{is}} & 0\\0 & 0 & -\frac{1}{\tau_{ir}}\end{matrix}\right]\) |
| \(f_{v}\) | the righthandside of the ode | \(f_{v}=u b + A x\) |
\(C_{il}: a_{il}\cdot\left(1 -\eta\right)\cdot\left(\begin{cases} t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} +\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}} - V_{m}\cdot\gamma\right) &\text{for}\:\left(\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\vee\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\\t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} - V_{m}\cdot\gamma +\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\right) &\text{for}\:\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\right)\vee\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\vee\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\\t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} - V_{m}\cdot\gamma + 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right) &\text{otherwise}\end{cases}\right)\)
\(C_{is}: a_{is}\cdot\left(1 -\eta\right)\cdot\left(\begin{cases} t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} +\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}} - V_{m}\cdot\gamma\right) &\text{for}\:\left(\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\vee\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\\t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} - V_{m}\cdot\gamma +\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\right) &\text{for}\:\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\right)\vee\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\vee\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\\t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} - V_{m}\cdot\gamma + 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right) &\text{otherwise}\end{cases}\right)\)
\(C_{ir}: a_{ir}\cdot\left(1 -\eta\right)\cdot\left(\begin{cases} t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} +\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}} - V_{m}\cdot\gamma\right) &\text{for}\:\left(\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\vee\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\\t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} - V_{m}\cdot\gamma +\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\right) &\text{for}\:\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq}\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}\right)\vee\left(\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\wedge\frac{Q_{p}\cdot\alpha_{3}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} +\frac{O_{2}}{\tau}}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right)\vee\frac{V_{m}\cdot\left(C_{i} -\frac{O_{2}}{2\cdot\tau}\right)}{C_{i} + K_{c}\cdot\left(1 +\frac{O_{2}}{K_{o}}\right)}{\leq} 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\\t\cdot\left(- B_{root}\cdot C_{ir}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{soil}} +\frac{1}{15 - T_{0}}\right)} - B_{stem}\cdot C_{is}\cdot\lambda_{sapwood}\cdot e^{E_{0}\cdot\left(-\frac{1}{- T_{0} + T_{stem}} +\frac{1}{15 - T_{0}}\right)} - V_{m}\cdot\gamma + 0.365853658536585\cdot V_{m}\cdot\left(1 -\frac{O_{2}}{2\cdot C_{i}\cdot\tau}\right) +\frac{J_{p}\cdot O_{2}}{2\cdot C_{i}\cdot\tau}\right) &\text{otherwise}\end{cases}\right)\)
\(C_{il}: \frac{C_{il}}{\tau_{il}}\)
\(C_{is}: \frac{C_{is}}{\tau_{is}}\)
\(C_{ir}: \frac{C_{ir}}{\tau_{ir}}\)
Foley, J. A., Prentice, I. C., Ramankutty, N., Levis, S., Pollard, D., Sitch, S., & Haxeltine, A. (1996). An integrated biosphere model of land surface processes, terrestrial carbon balance, and vegetation dynamics. Global Biogeochemical Cycles, 10(4), 603–628. http://doi.org/10.1029/96gb02692