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 14/3/2016, 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 Wang, Law, & Pak (2010).
Carbon storage by many terrestrial ecosystems can be limited by nutrients, predominantly nitrogen (N) and phosphorus (P), in addition to other environmental constraints, water, light and temperature. However the spatial distribution and the extent of both N and P limitation at the global scale have not been quantified. Here we have developed a global model of carbon (C), nitrogen (N) and phosphorus (P) cycles for the terrestrial biosphere. Model estimates of steady state C and N pool sizes and major fluxes between plant, litter and soil pools, under present climate conditions, agree well with various independent estimates. The total amount of C in the terrestrial biosphere is 2767 Gt C, and the C fractions in plant, litter and soil organic matter are 19%, 4% and 77%. The total amount of N is 135 Gt N, with about 94% stored in the soil, 5% in the plant live biomass, and 1% in litter. We found that the estimates of total soil P and its partitioning into different pools in soil are quite sensitive to biochemical P mineralization. The total amount of P (plant biomass, litter and soil) excluding occluded P in soil is 17 Gt P in the terrestrial biosphere, 33% of which is stored in the soil organic matter if biochemical P mineralization is modelled, or 31 Gt P with 67% in soil organic matter otherwise. This model was used to derive the global distribution and uncertainty of N or P limitation on the productivity of terrestrial ecosystems at steady state under present conditions. Our model estimates that the net primary productivity of most tropical evergreen broadleaf forests and tropical savannahs is reduced by about 20% on average by P limitation, and most of the remaining biomes are N limited; N limitation is strongest in high latitude deciduous needle leaf forests, and reduces its net primary productivity by up to 40% under present conditions.
global
| Abbreviation | Source |
|---|---|
| Evergreen needle leaf forest | Wang et al. (2010) |
| Evergreen broadleaf forest | Wang et al. (2010) |
| Deciduous needle leaf forest | Wang et al. (2010) |
| Deciduous broadleaf forest | Wang et al. (2010) |
| Mixed forest | Wang et al. (2010) |
| Shrub land (open and close shrubland) | Wang et al. (2010) |
| Woddy savannah | Wang et al. (2010) |
| Savannah | Wang et al. (2010) |
| Grassland | Wang et al. (2010) |
| Crop land (cropland mosaic was aggregated into this term) | Wang et al. (2010) |
| Barren or sparse vegetation | Wang et al. (2010) |
The following table contains the available information regarding this section:
| Name | Description |
|---|---|
| \(C_{leaf}\) | Plant (carbon) pool Leaf |
| \(C_{root}\) | Plant (carbon) pool Root |
| \(C_{wood}\) | Plant (carbon) pool Wood |
The following table contains the available information regarding this section:
| Name | Description | Expressions | Type | Units | Values Evergreen needle leaf forest |
Evergreen broadleaf forest | Deciduous needle leaf forest | Deciduous broadleaf forest | Mixed forest | Shrub land (open and close shrubland) | Woddy savannah | Savannah | Grassland | Crop land (cropland mosaic was aggregated into this term) | Barren or sparse vegetation |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\Delta_{t}\) | Time step of model integration | - | parameter | \(d\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
| \(N_{min}\) | Amount of mineral N in soil | - | variable | \(gN\cdot m^{-2}\) | - | - | - | - | - | - | - | - | - | - | - |
| \(P_{lab}\) | Amount of labile P in soil | - | variable | \(gP\cdot m^{-2}\) | - | - | - | - | - | - | - | - | - | - | - |
| \(F_{nupmin}\) | Minimum amount of N uptake required to sustain a given NPP | - | parameter | - | - | - | - | - | - | - | - | - | - | - | - |
| \(F_{pupmin}\) | Minimum amount of P uptake required to sustain a given NPP | - | parameter | - | - | - | - | - | - | - | - | - | - | - | - |
| \(x_{nup}\) | Nitrogen uptake limitation on NPP | \(x_{nup}=\operatorname{Min}\left(1, \frac{N_{min}}{F_{nupmin}\,\Delta_{t}}\right)\) | variable | - | - | - | - | - | - | - | - | - | - | - | - |
| \(x_{pup}\) | Phosphorus uptake limitation on NPP | \(x_{pup}=\operatorname{Min}\left(1, \frac{P_{lab}}{F_{pupmin}\,\Delta_{t}}\right)\) | variable | - | - | - | - | - | - | - | - | - | - | - | - |
| \(x_{npup}\) | Nutrient uptake limiting factor | \(x_{npup}=\operatorname{Min}\left(x_{nup}, x_{pup}\right)\) | variable | - | - | - | - | - | - | - | - | - | - | - | - |
| \(n_{leaf}\) | N:C ratio of leaf biomass | - | parameter | \(gN/gC\) | \(\frac{1}{42}\) | \(\frac{1}{21}\) | \(\frac{1}{50}\) | \(\frac{1}{21}\) | \(\frac{1}{28}\) | \(\frac{1}{33}\) | \(\frac{1}{21}\) | \(\frac{1}{21}\) | \(\frac{1}{42}\) | \(\frac{1}{21}\) | \(\frac{1}{17}\) |
| \(p_{leaf}\) | P:C ratio of leaf biomass | - | parameter | \(gP/gC\) | \(\frac{1}{408}\) | \(\frac{1}{400}\) | \(\frac{1}{405}\) | \(\frac{1}{333}\) | \(\frac{1}{278}\) | \(\frac{1}{293}\) | \(\frac{1}{354}\) | \(\frac{1}{492}\) | \(\frac{1}{833}\) | \(\frac{1}{333}\) | \(\frac{1}{167}\) |
| \(k_{n}\) | Empirical constant | - | parameter | \(gN\cdot (gC)^{-1}\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) | \(0.01\) |
| \(k_{p}\) | Empirical constant | - | parameter | \(gP\cdot (gC)^{-1}\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) | \(0.0006\) |
| \(x_{nleaf}\) | - | \(x_{nleaf}=\frac{n_{leaf}}{n_{leaf}+k_{n}}\) | parameter | - | - | - | - | - | - | - | - | - | - | - | - |
| \(x_{pleaf}\) | - | \(x_{pleaf}=\frac{p_{leaf}}{p_{leaf}+k_{p}}\) | parameter | - | - | - | - | - | - | - | - | - | - | - | - |
| \(x_{npleaf}\) | Nutrient concentration limiting factor | \(x_{npleaf}=\operatorname{Min}\left(x_{nleaf}, x_{pleaf}\right)\) | parameter | - | - | - | - | - | - | - | - | - | - | - | - |
| \(F_{cmax}\) | Nutrient unlimited NPP | - | variable | \(gC\cdot m^{-2}\cdot d^{-1}\) | - | - | - | - | - | - | - | - | - | - | - |
| \(F_{c}\) | Net Primary Productivity (flux) | \(F_{c}=x_{npleaf}\,x_{npup}\,F_{cmax}\) | variable | \(gC\cdot m^{-2}\cdot d^{-1}\) | - | - | - | - | - | - | - | - | - | - | - |
The following table contains the available information regarding this section:
| Name | Description | Type | Values Evergreen needle leaf forest |
Evergreen broadleaf forest | Deciduous needle leaf forest | Deciduous broadleaf forest | Mixed forest | Shrub land (open and close shrubland) | Woddy savannah | Savannah | Grassland | Crop land (cropland mosaic was aggregated into this term) | Barren or sparse vegetation |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(a_{leaf}\) | Fraction of NPP allocated to plant pool Leaf | parameter | \(0.42\) | \(0.25\) | \(0.4\) | \(0.3\) | \(0.35\) | \(0.4\) | \(0.3\) | \(0.2\) | \(0.3\) | \(0.3\) | \(0.2\) |
| \(a_{root}\) | Fraction of NPP allocated to plant pool Root | parameter | \(0.25\) | \(0.65\) | \(0.3\) | \(0.5\) | \(0.25\) | \(0.45\) | \(0.6\) | \(0.7\) | \(0.7\) | \(0.7\) | \(0.6\) |
| \(a_{wood}\) | Fraction of NPP allocated to plant pool Wood | parameter | \(0.33\) | \(0.1\) | \(0.3\) | \(0.2\) | \(0.4\) | \(0.15\) | \(0.1\) | \(0.1\) | \(0\) | \(0\) | \(0.2\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Units | Values Evergreen needle leaf forest |
Evergreen broadleaf forest | Deciduous needle leaf forest | Deciduous broadleaf forest | Mixed forest | Shrub land (open and close shrubland) | Woddy savannah | Savannah | Grassland | Crop land (cropland mosaic was aggregated into this term) | Barren or sparse vegetation |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\mu_{leaf}\) | Turnover rate of plant pool Leaf | parameter | \(year^{-1}\) | \(\frac{1}{2}\) | \(\frac{2}{3}\) | \(\frac{4503599627370496}{3602879701896397}\) | \(\frac{1}{8}\) | \(\frac{4503599627370496}{5404319552844595}\) | \(\frac{4503599627370496}{5404319552844595}\) | \(\frac{2}{3}\) | \(\frac{2}{3}\) | \(1\) | \(1\) | \(1\) |
| \(\mu_{root}\) | Turnover rate of plant pool Root | parameter | \(year^{-1}\) | \(\frac{1}{18}\) | \(\frac{1}{10}\) | \(\frac{1}{10}\) | \(\frac{1}{10}\) | \(\frac{1}{10}\) | \(\frac{1}{5}\) | \(\frac{1}{5}\) | \(\frac{1}{3}\) | \(\frac{1}{3}\) | \(\frac{9007199254740992}{8106479329266893}\) | \(\frac{1}{4}\) |
| \(\mu_{wood}\) | Turnover rate of plant pool Wood | parameter | \(year^{-1}\) | \(\frac{1}{70}\) | \(\frac{1}{60}\) | \(\frac{1}{80}\) | \(\frac{1}{40}\) | \(\frac{1}{50}\) | \(\frac{1}{40}\) | \(\frac{1}{40}\) | \(\frac{1}{40}\) | \(1\) | \(1\) | \(\frac{1}{5}\) |
The following table contains the available information regarding this section:
| Name | Description | Expressions | Units |
|---|---|---|---|
| \(x\) | vector of states for vegetation | \(x=\left[\begin{matrix}C_{leaf}\\C_{root}\\C_{wood}\end{matrix}\right]\) | - |
| \(u\) | scalar function of photosynthetic inputs | \(u=F_{c}\) | - |
| \(b\) | vector of partitioning coefficients of photosynthetically fixed carbon | \(b=\left[\begin{matrix}a_{leaf}\\a_{root}\\a_{wood}\end{matrix}\right]\) | - |
| \(A\) | matrix of turnover (cycling) rates | \(A=\left[\begin{matrix}-\mu_{leaf} & 0 & 0\\0 & -\mu_{root} & 0\\0 & 0 & -\mu_{wood}\end{matrix}\right]\) | - |
| \(f_{v}\) | the righthandside of the ode | \(f_{v}=u\,b+A\,x\) | \(gC\cdot m^{-2}\cdot d^{-1}\) |
| Flux description | |
|---|---|
 Figure 1: Pool model representation |
Input fluxes\(C_{leaf}: F_{cmax}\cdot a_{leaf}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\) Output fluxes\(C_{leaf}: C_{leaf}\cdot\mu_{leaf}\)\(C_{root}: C_{root}\cdot\mu_{root}\) \(C_{wood}: C_{wood}\cdot\mu_{wood}\) |
\(\left[\begin{matrix}- C_{leaf}\cdot\mu_{leaf} + F_{cmax}\cdot a_{leaf}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\\- C_{root}\cdot\mu_{root} + F_{cmax}\cdot a_{root}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\\- C_{wood}\cdot\mu_{wood} + F_{cmax}\cdot a_{wood}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\end{matrix}\right]\)
\(\left[\begin{matrix}-\mu_{leaf} & 0 & 0\\0 & -\mu_{root} & 0\\0 & 0 & -\mu_{wood}\end{matrix}\right]\)
\(C_{leaf} = \frac{F_{cmax}}{\mu_{leaf}}\cdot a_{leaf}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\)
\(C_{root} = \frac{F_{cmax}}{\mu_{root}}\cdot a_{root}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\)
\(C_{wood} = \frac{F_{cmax}}{\mu_{wood}}\cdot a_{wood}\cdot\min\left(\frac{n_{leaf}}{k_{n} + n_{leaf}},\frac{p_{leaf}}{k_{p} + p_{leaf}}\right)\cdot\min\left(1,\frac{N_{min}}{\Delta_{t}\cdot F_{nupmin}},\frac{P_{lab}}{\Delta_{t}\cdot F_{pupmin}}\right)\)
\(C_leaf: 0.591549295774648\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 3.16901408450704\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 16.2676056338028\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.014\)
\(\lambda_{2}: -0.056\)
\(\lambda_{3}: -0.500\)
\(C_leaf: 0.30241935483871\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 5.24193548387097\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 4.83870967741935\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.017\)
\(\lambda_{2}: -0.667\)
\(\lambda_{3}: -0.100\)
\(C_leaf: 0.213333333333333\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 2.0\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 16.0\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -1.250\)
\(\lambda_{2}: -0.100\)
\(\lambda_{3}: -0.013\)
\(C_leaf: 1.98347107438017\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 4.13223140495868\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 6.61157024793388\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.100\)
\(\lambda_{2}: -0.025\)
\(\lambda_{3}: -0.125\)
\(C_leaf: 0.328125\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 1.953125\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 15.625\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.020\)
\(\lambda_{2}: -0.100\)
\(\lambda_{3}: -0.833\)
\(C_leaf: 0.360902255639098\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 1.69172932330827\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 4.51127819548872\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.200\)
\(\lambda_{2}: -0.025\)
\(\lambda_{3}: -0.833\)
\(C_leaf: 0.371164632134608\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 2.47443088089739\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 3.29924117452986\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.200\)
\(\lambda_{2}: -0.667\)
\(\lambda_{3}: -0.025\)
\(C_leaf: 0.231624459542928\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 1.62137121680049\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 3.08832612723904\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.667\)
\(\lambda_{2}: -0.333\)
\(\lambda_{3}: -0.025\)
\(C_leaf: 0.200026670222696\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 1.40018669155888\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 0.0\)
\(\lambda_{1}: -0.333\)
\(\lambda_{2}: -1.000\)
\(C_leaf: 0.247933884297521\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 0.520661157024793\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 0.0\)
\(\lambda_{1}: -1.111\)
\(\lambda_{2}: -1.000\)
\(C_leaf: 0.170940170940171\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_root: 2.05128205128205\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\), \(C_wood: 0.854700854700855\cdot F_{cmax}\cdot\min\left(1.0,\frac{N_{min}}{F_{nupmin}},\frac{P_{lab}}{F_{pupmin}}\right)\)
\(\lambda_{1}: -0.200\)
\(\lambda_{2}: -0.250\)
\(\lambda_{3}: -1.000\)
Wang, Y. P., Law, R. M., & Pak, B. (2010). A global model of carbon, nitrogen and phosphorus cycles for the terrestrial biosphere. Biogeosciences, 7(7), 2261–2282. http://doi.org/10.5194/bg-7-2261-2010