Mortality

• Modelling tree mortality
• General mortality
• Crowding mortality
• Tree fall mortality
• Change of mortality due to fragmentation
• Overall change in number of trees per cohort

Modelling tree mortality

In FORMIND trees can die due to various reasons. The following different types of mortality occur in a serial way:

• background mortality $M_B$
• mortality dependent on an individual's stem diameter $M_D$
• mortality dependent on an individual's diameter increment $M_I$
• crowding mortality due to limited space
• mortality due to damage by a falling tree
• mortality due to fragmentation

Individual trees of the same type and size that are located in the same patch are summarized in this section by a so-called cohort. Each cohort is uniquely described by its type, the number of identical trees ($N$), their age and the size of one single tree (i.e. aboveground biomass). In this section, the number of identical trees in a cohort changes due to mortality processes. Below, we describe the different types of mortality in greater detail.

General mortality

In contrast to the event-driven forms of mortality we will describe later, there is a general mortality rate per tree which is active in each time step $t_y$. This mortality rate $M$ is calculated as the sum of the background mortality rate $M_B$ and two further mortality rates dependent on the stem diameter $M_D$ as well as its increment $M_I$:

\[ M = M_B + M_D + M_I .\]

The background mortality $M_B$ $[\frac{1}{\text{yr}}]$ is a type-specific constant input parameter.

The mortality rate $M_D$ depends on the stem diameter $D \, [\text{m}]$ and provides the possibility to give older trees (with a bigger stem diameter) a higher mortality rate than younger trees or vice versa. The rate is calculated via

\[M_D(D) = m_{d0} \cdot D^{m_{d1}},\]

where $m_{d0}$ and $m_{d1}$ are type-specific parameters.

The mortality rate $M_I$ depends on the increment of the stem diameter $D$ $[\text{mm}]$ per time step $t_y$ and provides the possibility to include a higher mortality for older trees or those under stress. It is modelled by the functional relationship

\[M_I(\Delta D) = m_{i0} + m_{i1} \Delta D + m_{i2} {\Delta D}^2 , \]

where $ m_{i0}$, $ m_{i1}$ and $ m_{i2}$ are type-specific parameters. The increment of the stem diameter from time $t$ to time $t + ty$ is denoted as $\Delta D$.

The trees per patch die according to their mortality rate $M$ - either stochastically or deterministically.

Deterministic dying is active if the number of individuals per cohort is greater than a predefined number $N_M$ and if the stem diameter of each individual is smaller than a predefined threshold $D_M$. In this case, the number of dying trees per cohort is determined by

\[N_Y = N \cdot M, \]

where $N$ is the number of trees in this cohort, $N_Y$ is the number of dying trees per cohort and $M$ is the calculated mortality rate per time step $t_y$. The number of dying trees $N_Y$ is rounded to $\lfloor N_Y + 0.5 \rfloor$.

In the contrary case (i.e. $N < N_M$ or $D > D_M$), deaths occur stochastically. That means, for each tree the mortality rate $M$ represents its probability of dying (i.e. by comparing a random number from a uniform distribution in the range of $[0;1]$ with the mortality rate $M$):

\[N_Y = \sum_{j=1}^N \delta_{rM},\]

where $N$ is the number of trees per cohort, $N_Y$ is the number of dying trees per cohort, $M$ is the calculated mortality rate per time step $t_y$ and $r$ is a random number from a uniform distribution in the range of $[0;1]$. The symbol $\delta_{rM}$ is defined as

\[\delta_{rM} = \begin{cases} 1 & \text{, } r \leq M \\ 0 & \text{, } r > M \end{cases}.\]

Crowding mortality

Crowding occurs if at any height layer the cumulative crown area of all trees on a patch exceeds $A_\text{patch}$. At first, the cumulative crown area $CCA$ $[\frac{m^2}{m^2}]$ of all trees on a patch is calculated for each height layer $i$ relative to the patch area $A_\text{patch}$:

\[CCA_i = \frac{1}{A_\text{patch}} \cdot \sum_{\text{all individuals} \ \text{with } l_{min} \leq i \leq l_{max}} C_A,\]

where $C_A$ is the crown projection area of a tree (see section B). Each tree occupies only a limited amount of height layers (i.e. between layer $l_\text{min}$ and $l_\text{max}$) defined by the individual's crown length $C_L [\text{m}]$ and its height $H [\text{m}]$:

\[l_\text{max}= \left\lfloor \frac{H}{\Delta h} \right\rfloor \ l_\text{min}= \left\lfloor \frac{H-C_L}{\Delta h} \right\rfloor \]

Mortality due to crowding is calculated per tree and represented by a reduction factor $Rc$ $[-]$. This individual reduction factor is calculated based on those height layers that the individual's crown is occupying (see Figure below).

Figure: Illustration of crowding on the example of two single trees. The limits of each crown are shown by $l_\text{min}(\text{Tree 1})$, $l_\text{max}(\text{Tree 1})$, $l_\text{min}(\text{Tree 2})$ and $l_\text{max}(\text{Tree 2})$. The vertically discretized aboveground space into height layers of width $\Delta h$ $[\text{m}]$ is coloured differently according to the sum of the crown projection areas of both individuals occupying the layers. The darker the colour is, the more crowns occupy the respective height layer. This is calculated by the cumulative crown area $CCA$ $[-]$ relative to the patch area, which is illustrated on the right side. The maximum of $CCA$ is used to calculate the reduction factor $R_c$ for each individual. In this example, the reduction factor for each of both trees is calculated based on the 5th height layer from the bottom (equal to layer $l_\text{min}(\text{Tree 1})$ and $l_\text{max}(\text{Tree 2})$).

The reduction factor $R_c$ is determined by the reciprocal of the maximum cumulative crown area $CCA$ corresponding to the height layers between the individual limits $l_\text{min}$ and $l_\text{max}$:

\begin{align} R_c = \frac{1}{\max\limits_{i \in [l_\text{min}, ..., l_\text{max}]} (CCA_i)}. \end{align}

If the maximum cumulative crown area of any height layer that the individual's crown is occupying exceeds $A_\text{patch}$ (i.e. $CCA_i > 1$), the individual reduction factor $R_c$ falls below the threshold of $0.99$. In this case, the number of dying identical trees per cohort $N_C$ is calculated by

\[N_C = N~(1-R_c).\]

Mortality due to crowding (or self-thinning) can be interpreted as competition for space. Besides crowding, the vertical discretization of the aboveground space is also important for the light climate calculations. To save computation time, the calculation of $R_c$ is coupled to that of the light climate, which is explained in Chapter E.

Tree fall mortality

If a tree falls, neighboring trees can be destroyed. A dying tree falls down with probability $f_{fall}$. The falling target patch depends on the falling direction and on the tree height $H$. The falling direction $DIR$ (drawn from a uniform distribution in the range of $[0, 360]$) is chosen randomly. The target coordinates of the falling tree $(x_{fall}; y_{fall})$ are determined in the following way:

\begin{align}
x_\text{fall} &= x_\text{tree}+H\sin\left(2~\pi~\frac{DIR}{360}\right) \\ y_\text{fall} &= y_\text{tree}+H\cos\left(2~\pi~\frac{DIR}{360}\right) \end{align}

where $(x_\text{tree}; y_\text{tree})$ is the standing position of the falling tree. These target coordinates are used to determine the affected patch. All smaller trees ($\text{tree height} < H$) in this target patch are dying with a damage probability $M_\text{dam}$:

\begin{align} M_\text{dam} = \frac{C_A}{A_\text{patch}} , \end{align}

where $C_A$ is the crown area of the falling tree and $A_\text{patch}$ the area of the target patch.

The trees in the target patch die according to the damage rate $M_\text{dam}$ - either stochastically or deterministically. Deterministic dying is active if the number of trees per cohort is greater than $100$. In this case, the number of dying trees per cohort $N_F$ is determined by multiplying the number of trees $N$ per cohort with the damage rate $M_\text{dam}$.

\begin{align} N_F = N \cdot M_\text{dam}, \label{NF} \end{align}

The number of dying trees $N_F$ is rounded to $\left\lfloor N_F + 0.5 \right\rfloor$.

In the contrary case (less than $100$ trees per cohort), stochastic dying is performed. That means, for each tree the damage rate $M_\text{dam}$ represents its probability of dying (i.e. by comparing a random number from a uniform distribution in the range of $[0;1]$ with the damage rate).

\begin{align} N_F = \sum_{j=1}^N \delta_{rM_\text{dam}} \text{,} \label{NF2} \end{align}

where $N$ is the number of trees per cohort, $N_F$ is the number of dying trees per cohort, $M_\text{dam}$ is the damage rate per time step $t_y$ and $r$ is a random number from a uniform distribution in the range of $[0;1]$. The symbol $\delta_{rM_\text{dam}}$ is defined as:

\begin{align} \delta_{rM_\text{dam}} = \begin{cases} 1 & \text{, } r \leq M_\text{dam} \\ 0 & \text{, } r > M_\text{dam} \end{cases} \label{deltarMdam} \end{align}

Change of mortality due to fragmentation

It has been observed that mortality is increased and tree species richness is reduced at forest edges, and that large trees are often missing in small fragments. The extent of forest edges varies between forest regions. For example, increased edge mortality could be measured up to $100 \text{m}$ into the forest interior in the Amazon.

In FORMIND we model increased mortality at the edge of forest fragments by multiplying the general mortality $M$ with a fragmentation variable $m_\text{frag}$. Thus, the additional mortality due to fragmentation can be calculated as

\begin{align} M_\text{frag}=M \cdot (m_\text{frag} - 1). \end{align}

We assume that the fragmentation-induced mortality $M_\text{frag}$ is higher at forest edges ($< 100 \,m$) than in the interior. Thus, the value of $m_\text{frag}$ is modelled dependent on the distance to the fragment edges (see Table below). In addition, large trees ($D > 60 \,\text{cm}$) can suffer an increased mortality.

Table: Mortality increase due to fragmentation, dependent on the distance to a fragment edge and on the stem diameter $D \, [\text{cm}]$ of a tree.

If this type of mortality is activated, we recommend to choose a patch size of $20 \,\text{m}$ x $20 \,\text{m}$ (i.e. $A_\text{patch} = 400 \,\text{m}^2$) according to the distance classes in the Table above.

The number of additional trees that die due to fragmentation effects can be calculated as:

\begin{align} N_\text{frag}=N \cdot M_\text{frag}. \end{align}

Overall change in number of trees per cohort

Overall, per time step $\Delta t$ and for each cohort the change in the number of trees per cohort $N$ is determined by:

\begin{align} \Delta N = -(N_Y + N_C +N_F + N_\text{frag}), \end{align}

where $N_Y$ is the number of trees dying due to regular mortality, $N_C$ is the number of trees dying due to crowding, $N_F$ is the number of trees dying due to damages caused by a falling tree and $N_\text{frag}$ is the number of trees dying due to increased mortality near fragment edges.

The amount of aboveground carbon $S_\text{mort}$ $[\frac{t_{C}}{\text{ha}}]$ resulting from the death of trees within the current time step is calculated by

\begin{align} S_\text{mort} = 0.44 \cdot \sum_\text{all cohorts} (N_Y + N_C + N_F + N_\text{frag})\cdot B, \end{align}

where $B$ is the aboveground biomass of the tree (see chapter Geometry). We assume that $1 \text{g}$ organic dry matter contains $44 \%$ carbon Larcher 2001.