Competition and environmental limitations
Light climate
A single tree on a patch receives full incoming radiation. An increasing number of individual trees of differing heights on a patch results in shading within the canopy. Higher trees intercept radiation, which is not available for smaller individuals. Thus, with decreasing height from the canopy down to the ground, radiation is decreasing. We call this vertical distribution of light availability within a patch ’light climate’.
To calculate the light availability in different heights within the canopy, the vertical discretization of the above-ground space is used (i.e. height layers of constant width $\Delta h$). For each patch and height layer, the leaf area accumulated by all trees on the patch is calculated. Each tree contributes parts of its crown leaf area to those height layers, which are occupied by its crown (i.e. height layers from $l_{min}$ to $l_{max}$). These limits are determined by the individual’s crown length $C_{L}$ and its height $H$: \[ l_{max}= \left\lfloor \frac{H}{\Delta h} \right\rfloor\\ l_{min}= \left\lfloor \frac{H-C_L}{\Delta h} \right\rfloor.\]
The number of height layers a tree is occupying by its crown $n_{layer}$ can then be calculated by: \[n_{layer} = l_{max}-l_{min}\]
For those height layers between $l_{min}$ and $l_{max}$, an individual’s leaf area contributes equally to each layer $i$: \[ \bar{L_{i}}=\frac{LAI \cdot C_A}{n_{layer}}, \]
whereby $ \overline{L_{i}} $ $[m²]$ represents the contribution of an tree’s leaf area to the layer $i$, $LAI~[-]$ is the leaf area index of the tree (see section B.6) and $C_{A}~[m²]$ is crown projection area of the tree’s crown. The multiplication of $LAI$ by $C_{A}$ results in the leaf area in $[m²]$ of a single tree .
Summing up all contributions of the trees’ leaf area per patch to their respective occupied height layers and relative to the patch area, results in the patch-based leaf area index $\hat{L_{i}}~[-]$ per layer $i$:
\[\hat{L_{i}} =\frac{1}{A_{patch}} \sum_{\ all\ individuals \\ with\ l_{min} \leq i \leq l_{max}} \hat{L_{i}},\] where $\hat{L_{i}}$ [m²] represents the leaf area contribution of an tree to the height layer $i$ and $A_{patch}~[m²]$ denotes the area of a patch.
Using this information, the radiation each tree is able to intercept can be determined. Light attenuation through the canopy is calculated using the approach of . The incoming radiation $I_{ind}$ on top of a tree (i.e. on top of the height layer $l_{max}$ the tree is reaching) is calculated by: \[ I_{ind}=I_{0}\cdot exp \left(-k \cdot \sum_{i>l_{max}}\hat{L_{i}}\right), \]
where the sum in the exponent accumulates the patch-based leaf area indices of all height layers above the individual’s height. The parameter $k$ denotes the light extinction coefficient $[-]$ of a tree, $I_{0}$ $[\mu mol (photons)/m² s]$ is the daily radiation above canopy averaged from sunrise to sunset during the vegetation period and $\hat{L_{i}}$ [-] represents the patch-based leaf area index of height layer $i$.
Figure 6: Illustration of the light climate on the example of two single trees . The limits of each crown are shown by $l_{min} (Tree1)$, $l_{max} (Tree1)$, $l_{min} (Tree2)$, $l_{max} (Tree2)$. The vertically discretized aboveground space into height layers of width $\Delta h$ $[m]$ is coloured differently according to the available radiation. The lighter the colour is, the more attenuated the radiation is, which results from the absorption by higher individuals' leaves. On the right hand side the decrease of available light from the canopy to the floor is illustrated by the grey polygon. Thereby, attenuation is greatest in the height layer both trees occupy by their crowns (i.e. layer $l_{min} (Tree1)$ and $l_{max} (Tree2)$.
By determining the available radiation for each single tree (at the top of the crown), competition for light between trees is considered.
Water cycle and soil water limitation
Individual trees take up soil water resources to fulfill the requirements for their gross productivity. We determine an individual's uptake of soil water based on its demand and on the total available soil water.
Firstly, the soil water content $\Theta_{soil}$ is computed preliminary on an hourly basis using a differential equation, which quantifies preliminary hourly changes in the soil water content per patch depending on precipitation $PR$, interception $IN$ and run-off $RO$ (see figure below, cf. Kumagai et al. 2004 ): \begin{equation} \tag{55} \frac{d \Theta_{soil}}{dt} = PR(t) - IN(t) - RO(t) .\ \label{SWC} \end{equation} The resulting soil water content represents the total available soil water before soil water uptake by individuals. Uptake of soil water resources by trees is modelled equal to their transpiration and subtracted from $\Theta_{soil}$ later within the timestep (see eqn. \ref{soilwater}).
The interception $IN~[mm/h]$ is calculated dependent on the total leaf area index per patch (i.e. $ \sum_{i} \hat{L_{i}}$ in $[-]$, cf. Liang et al. 1994): \begin{equation} \tag{56} IN(t) = \text{min}\left(K_{L} \cdot \left(\sum_{i} ~ \hat{L_{i}}\right), PR(t)\right), \label{in} \end{equation} where $K_{L}~[mm/h]$ is the interception constant and $PR~[mm/h]$ denotes the precipitation.
On the ground surface of a patch, we consider two different run-offs: surface run-off and subsurface run-off:
\begin{equation}
RO(t) = RO_{\rightarrow}(t) + RO_{\downarrow}(t),\
\end{equation}
where surface run-off $RO_{\rightarrow}~[mm/h]$ is defined in the following way:
\begin{equation}
RO_{\rightarrow} = \text{max} \left(0, \Theta _{soil}(t) + PR(t) - IN(t) - POR \right)
\label{rosur}
\end{equation}
with $POR~[mm/h]$ denoting the soil porosity (i.e. defined as the maximum water intake of the soil per patch).All additional incoming water is assumed to be surface run-off.
Figure 7: Illustration of the water cycle on the example of a single tree.
For the calculation of the subsurface run-off $RO_{\downarrow}$ due to gravitation, we use the Brooks-Corey relation (cf. Liang et al. 1994): \begin{equation} RO_{\downarrow} = K_{s} \cdot \left( \frac{\Theta_{\text{soil}} (t)-\Theta_{\text{res}}}{POR-\Theta_{\text{res}}} \right)^{\frac{2}{\lambda}+3}, \label{rosub} \end{equation} where $K_{s}[mm/h]$ is the fully saturated conductivity, $\Theta_{res}[mm/h]$ the residual water content, and $\lambda~[-]$ the pore size distribution index.
The preliminary soil water content $\Theta_{soil}$ represents the soil water content, which is available for the individuals' uptake or transpiration. To calculate the transpiration $TR~[mm/h]$ of all trees per patch, we use the water-use-efficiency concept (cf. Lambers et al. 2008): \begin{equation} TR=\frac{1}{A_{patch}} ~\sum_{\text{all trees}} \frac{GPP}{WUE}, \end{equation} whereby $GPP$ in $[g_{\text{ODM}}/h]$ denotes the hourly gross primary production of an individual on the patch (see section F). Please note, that we simulate $GPP$ per time step $t_y$. To calculate $GPP$ on an hourly basis, we divide $GPP$ $[g_{\text{ODM}}/\Delta t]$ by the number of hours within the time step $\Delta t$. The constant type-specific value $WUE$ in $[g_{\text{ODM}}/kg_{H_2O}]$ represents the water-use-efficiency parameter and $A_{\text{patch}} \text{m}^2$ the area of a patch.
The resulting transpiration $TR$ may be limited in three ways calculated in a serial way:
PET limitation
Transpiration can be limited by the potential evapotranspiration $PET \tfrac{\text{mm}}{\text{h}}$ and the interception $IN \tfrac{\text{mm}}{\text{h}}$ (calculated by eqn. \ref{in}):
\begin{equation}
TR_{new} =
\begin{cases}
TR(t) & \text{,} TR(t) \leq PET(t) - IN(t)\\
PET(t) - IN(t) & \text{,} TR(t) > PET(t) - IN(t)
\end{cases}
\end{equation}
Soil water limitation Transpiration can be limited by the preliminary soil water content $\Theta_{soil}~[mm/h]$ (calculated by eqn. \ref{SWC}) and the permanent wilting point $\Theta_{pwp}~[mm/h]$: \begin{equation} TR_{new}(\Theta_{soil})= \begin{cases} TR(t) & \text{,} \Theta_{soil}(t) - TR(t) \geq \Theta_{pwp} \\ \Theta_{soil}(t) - \Theta_{pwp} & \text{,} \Theta_{soil}(t) - TR(t) < \Theta_{pwp} \\ 0 & \text{,} \Theta_{soil}(t) \leq \Theta_{pwp} \end{cases}. \end{equation}
Competition for water Competition between trees can limit the transpiration in the following way: \begin{equation} TR = \varphi_{W}(\Theta_{soil}) \cdot TR(t), \label{actTR} \end{equation} where $\varphi_{W}$ [-] represents a reduction factor ranging between 0 and 1, depending on the actual soil water content.
The reduction factor $\varphi_{W}$ is calculated using the approach of Granier et al. 1999, which is based on the preliminary soil water content (calculated by eqn. \ref{SWC}): \begin{equation} \varphi_{W}(\Theta_{soil})= \begin{cases} 0 & \text{,} \Theta_{soil}(t) \leq \Theta_{pwp} \\ \frac{\Theta_{soil}(t)- \Theta_{pwp}}{ \Theta_{msw}-\Theta_{pwp}} & \text{,} \Theta_{pwp} < \Theta_{soil}(t) < \Theta_{msw} \\ 1 & \text{,} \Theta_{soil}(t) \leq \Theta_{msw} \end{cases}, \tag{64} \label{phiW} \end{equation} where $\Theta_{pwp}$ is the permanent wilting point in $[V\%]$ and $\Theta_{msw}$ is the minimum soil water content in $[V\%]$. For the purpose of the calculation of eqn. \ref{phiW} only, $\Theta_{soil}$ needs to be converted from $[mm/h]$ to $[V\%]$. Thereby, the soil is modelled down to a constant depth $[m]$ defined prior to the start of the simulation.
The minimum soil water content ($\Theta_{msw}$) is determined according to Granier et al. 1999 by:
\begin{equation}
\Theta_{msw} = \Theta_{pwp} + 0.4 (\Theta_{fc}-\Theta_{pwp})
\label{msw}
\end{equation}
whereby $\Theta_{fc}$ denotes the field capacity in $[V\%]$.
The soil water content in the next day step is then calculated by the difference between the preliminary soil water content (calculated by eqn. \ref{SWC}) and the (eventually limited) transpiration $TR$: \begin{equation} \tag{66} \frac{d\Theta_{soil}}{dt} = \Theta_{soil}(t) – TR(t). \label{soilwater} \end{equation}
Figure 8: Water limitation function. a) limitation of TR due to PET. b) limitation of TR due to Soil water. c) $\varphi_{water}$ as function of Soil water.
Temperature
The gross primary production $GPP~[t_{ODM}/t_y]$ of a tree (see section F) may be influenced by phenology (esp. in the temperate zone) and air temperature. Respiration for maintenance purposes of an individual (see section F) may also be affected by air temperature. The influence on both - gross productivity and respiration, is modelled using limitation factor, by which they are simply multiplied (see section F). In the following, we describe the calculations of these limitation factors:
Phenology
Individual trees make photosynthesis only during their photosynthetic active period. In the temperate zone, we distinguish between broad-leaf and needle-leaf trees. Only deciduous broad-leaf trees have two phenology phases: (i) a dormant phase during winter and (ii) a photosynthetic active period of $\varphi_{act}~[days]$ after bud-burst until fall (i.e. the vegetation period).
The date of bud-burst is reached, if the temperature sum (daily mean air temperatures $> 5°C$) since 1 January is higher than a critical temperature $T_{crit}$ (Sato et al. 2007): \begin{equation} T_{crit}= -68+638 ~e^{-0.01\cdot n}, \label{budburst} \end{equation} where $n$ is the number of days per time step $\Delta t$ with an air temperature below $5 °$ since 1 November of the previous year. This algorithm is based on the global distribution of leaf onset dates estimated from remote sensing data (Botta et al. 2000). The photosynthetic active period stops if the 10-day moving average of daily mean air temperatures falls below $9 °C$ (Sato et al. 2007).
In contrast to the broad-leaf trees, the photosynthetic active period $\varphi_{act}$ of needle-leaf trees amounts a complete year of 365 days (without any dormant phase).
In the tropical zone, we assume for all individuals irrespective of their type a complete photosynthetic active period with $\varphi_{act}=365$ days.
Temperature limitation of gross productivity
The gross primary production of a tree can be reduced due to air temperatures. A corresponding limitation factor $\varphi_{T}$ is calculated by averaging the reduction factors over the whole time step $\Delta t$: \begin{equation} \varphi_{T}= \frac{1}{n} ~ \sum_{1}^{n} \varphi_{T,l} \cdot \varphi_{T,h}, \label{lim_temp} \end{equation} where $n$ is the number of days per time step $\Delta t$ and the values $\varphi_{T,l}$ and $\varphi_{T,h}$ are the daily inhibition factors for low and high air temperatures (Gutiérrez & Huth 2012, Haxeltine & Prentice 1996).\
The reduction factor for low air temperatures $\varphi_{T,l}~[°C]$ is calculated by: \begin{equation} \varphi_{T,l}= \left(1+ e^{k_{0} \cdot k_{1} - T }\right)^{-1}, \label{temp_low} \end{equation} where $T~[°C]$is the daily mean air temperature and $k_{0}$ and $k_{1}$ are type-specific parameters.
These parameters $k_{0}$ and $k_{1}$ are calculated by: \begin{equation} k_{0} = \frac{2 ~ln(0.01/0.99)}{T_{CO_{2},l}-T_{cold}} \label{k0} \end{equation}
\begin{equation} k_{1} = 0.5 ~(T_{CO_{2},l}+ T_{cold}) \label{k1} \end{equation} where $T_{CO_{2},l}$ [$°C$] and $T_{cold}$ [$°C$] are type-specific parameters representing the lowest temperature limit for CO$_{2}$ assimilation and the monthly mean air temperature of the coldest month an individual can cope with, respectively.\
Similarly, the \textbf{inhibition factor for high air temperatures} $\varphi_{T,h}$ in $°C$ is calculated by: \begin{equation} \varphi_{T,h} = 1 - 0.01 \cdot e^{k_{2} ~ (T-T_{hot}) } \label{temp_high} \end{equation} where $k_{2}$ is a type-specific parameter, $T$ [$°C$] is the daily mean temperature and $T_{hot}$ [$°C$] is the type-specific mean temperature of the hottest month an individual can occur.\
The parameter $k_{2}$ is calculated as: \begin{equation} k_{2} = \frac{ln(0.99/0.01)}{T_{CO_{2},h}-T_{hot}}, \label{k2} \end{equation} whereby $T_{CO_{2},h}$ [$°C$] and $T_{hot}$ [$°C$] are type-specific parameters representing the higher temperature limit for CO$_{2}$ assimilation and the monthly mean air temperature of the warmest month an individual can cope with, respectively.
Temperature limitation of maintenance respiration
Maintenance respiration is assumed to change exponentially with air temperature represented by the limitation factor $\kappa_{T}$ (Prentice et al. 1993):
\begin{equation} \kappa_{T} = \frac{1}{n} ~ \sum_{1}^{n} Q_{10}^{\left(\frac{T-T_{ref}}{10}\right)}, \label{r_main} \end{equation} where $n$ is the number of days per time step $t_y$, $T$ $[°C]$ is the daily mean air temperature, $Q_{10}$ $[-]$ and $T_{ref}$ $[°C]$ are constant parameters, irrespective of type. $T_{ref}$ represents the reference temperature, at which maintenance respiration is not influenced. Air temperatures below $T_{ref}$ result in a decrease of maintenance respiration ($\kappa_{T} < 1$) and those above $T_{ref}$ in an increase of maintenance respiration ($\kappa_{T} > 1$).
Figure 9: Temperature limitation function. a) limitation factor of Photosynthesis. b) limitation factor of maintenance respiration.