Based on the allocation of biomass increment to the stem compartment tree dimensions are updated dynamically, accounting for the influence of light competition on the stem growth strategy of trees.

height to diameter ratio of stem increment

Altering the partitioning regime between height and diameter increment is a major tree response to competition for light. Ecological theory (Grime 1979) suggests that height growth is favored over diameter increment with decreasing light levels for the individual, as a strategy to improve its competitive status in the canopy. This is also supported by empirical findings, documenting a non-constant allometric relationship between height and diameter growth of individual trees as a result of their competitive status (e.g., Henry and Aarssen 1999). Niklas (1995), for instance, found the rate of height growth to diameter increment greater for trees growing in stands compared to open-grown trees.

To simulate this behavior as emerging property of the model (cf. also the analysis of Seidl et al. 2010) we extend the approach of Bossel (1996), who defined two height-diameter bounds (with and without competition) for allocation of annual increments (see also Peng et al. 2002). In iLand, diameter-dependent h/d-thresholds are defined per species. Upper limits (h/d_max)are derived from densly stocked, even-aged stands (e.g., yield table stands), while lower limits (h/d_min) are parameterized from growth records for open-grown trees (e.g., Hasenauer et al. 1997). The specific h/d-ratio for the increment of a tree at a given diameter (r_h/d) is subsequently derived using LRI as measure of light competition (Eq. 1a)

\[\begin{aligned} r_{h/d}=h/d_{min}\cdot LRI+h/d_{max}\cdot (1-LRI) \end{aligned} \]

Eq. 1a

For an open-grown tree LRI→1, i.e. r_h/d will be close to its h/d_min, while for conditions of intense light competition LRI→0 and r_h/d→h/d_max.

Note for the parameterization of species The h/d_min and h/d_max are typically expressed as an allometric relationship:

\[\begin{aligned} \frac{H}{D} = a \cdot dbh^b \end{aligned} \]

Eq. 1b

The equation in this form describes the state of the H/D-ratio of a tree with a given diameter (this expression is used e.g., for the generation of light influence patterns). However, the equation needs to be reformulated, when used dynamically for the H/D increment. Rearranging for height and differentiating with respect to dbh yields:

\[\begin{aligned} H=a \cdot dbh^{b+1} \end{aligned} \]

Eq. 1c

\[\begin{aligned} \frac{d H}{d dbh}={a}{(b+1)}dbh^b \end{aligned} \]

Using the equation in the form a (b+1) dbh^b for the H/D increment makes sure, that the realized H/D ratios (e.g., for an open grown tree) are consistent with the specified H/D equation (a dbh^b):

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stem update

The biomass of an individual stem (W_s) is defined as Eq. 2

\[\begin{aligned} W_{s}=dbh^2\cdot H \cdot \phi \end{aligned} \]

Eq. 2

with \[\begin{aligned} \phi=\frac{\rho\varphi \pi }{4} \end{aligned} \]

, H the tree height, ρ the species-specific wood density, and φ; the form factor (cf. Bossel 1996). Substituting Eq. 1 into Eq. 2 and differentiating to derive stem increment yields

\[\begin{aligned} \frac{\delta W_s}{\delta dbh} = \phi dbh^2(2\frac{H}{dbh}+r_{h/d}) \end{aligned} \]

Eq. 3

This can be solved for dbh increment (Eq. 4):

\[\begin{aligned} \delta dbh = \frac{\delta W_s}{\phi dbh^2(2\frac{H}{dbh}+r_{h/d})} \end{aligned} \]

Eq. 4

and linearized

\[\begin{aligned} \Delta dbh=\frac{1}{\phi dbh^2(2\frac{H}{dbh}+r_{h/d})}w_{s} \end{aligned} \]

Eq. 5

with

\[\begin{aligned} \Delta H=\Delta dbh\cdot r_{h/d} \end{aligned} \]

Eq. 6

A linearization as given in Eq. 5 is erroneous due to the high non-linear nature of the stem growth derivate (but see Peng et al. 2002). We thus use the previous years dbh increment as well as a redistribution of residuals to ultimately derive Δdbh and ΔH, granting consistency with overall w_f but avoiding a computationally expensive iterative soluation. Eq. 7 shows the inclusion of the dbh increment of last year:

\[\begin{aligned} \Delta dbh = \frac{1}{\phi (dbh + \Delta dbh_{t+1})^2(2\frac{H}{dbh}+r_{h/d})}w_s \end{aligned} \]

Eq. 7

The correction of the estimate after the update is provided by Eq. 8 (note that W_s in Eq. 8 is the stem wood biomass including w_s):

\[\begin{aligned} \Delta dbh = \Delta dbh - \frac{\Delta dbh}{w_s}(\phi (dbh + \Delta dbh)^2)\cdot(h+\Delta dbh \cdot r_{h/d})-W_s) \end{aligned} \]

Eq. 8

Only if after these steps of approximating a solution the estimated stemwood increment resulting from the estimated diameter and height increment differs substantially from the stemwood increment expected from allocation of biomass an iterative solution is sought.