# ForGEM - principle equations

 FORGEM

Principal equations in ForGEM

An overview is presented of the principal state and empirical equations used in ForGEM. See (Kramer, 2001, Kramer et al., 2008) for details and references.

## number of trees

${\displaystyle {\frac {dN_{x}}{dt}}=NNew_{x}-NMrt_{x}}$

with:

N: number (# trees plot-1)

x: seeds, seedlings, trees

New: new individual seed, seedling or tree in the population

Mrt: mortality of individual tree, or tree-cohort in case of seeds and seedlings

## weight

${\displaystyle {\frac {dW_{y}}{dt}}=f_{y}\cdot NPP-T_{y}}$

with:

W: weight (kg tree-1)

y: foliage, branches, heartwood, sapwood, coarse roots, fine roots, reserves

f: fraction

NPP: net primary production (kg tree-1 d-1)

T: turnover (d-1)

### allocation

Empirical equations are used for both allocation $f_{y$

and the rate of change of structural features. Empirical coefficients of these equations are indicated as ${\displaystyle C_{n}}$, with ${\displaystyle {n}}$ a numeric identifier. See  allocation parameters for the values of ${\displaystyle C_{n}}$ for different tree species.


${\displaystyle \ln \left({\frac {Wfl}{Wst}}\right)=C1+C2\cdot \ln \left(Wsh\right)+C3\cdot \ln \left(Wsh\right)^{2}+C4\cdot \ln \left(Wsh\right)^{3}}$

${\displaystyle \ln \left({\frac {Wbr}{Wst}}\right)=C5+C6\cdot \ln \left(Wsh\right)+C7\cdot \ln \left(Wsh\right)^{2}+C8\cdot \ln \left(Wsh\right)^{3}}$

${\displaystyle \Rightarrow {\frac {Wst}{Wsh}}={\frac {1}{1+\ln \left({\frac {Wfl}{Wst}}\right)+\ln \left({\frac {Wbr}{Wst}}\right)}}}$

with:

fl: foliage; br: branches; st: stem (= heartwood + sapwood); sh: shoot (foliage+branches+stem)

The fraction of NPP allocated to the plant components $f_{y$

are derived such that the tree strives for partitioning ratios between plant organs, ${\displaystyle y}$. A fixed fraction of NPP is allocated to the roots.


## structural features

Structural features simulated by the ForGEM model are stem volume, tree height and stem diameter

### stem volume

${\displaystyle {\frac {dV}{dt}}={\frac {f_{st}}{\rho _{st}}}\cdot NPP}$

With:

$f{}_{st$

allocation of NPP to stem

${\displaystyle \rho _{st}}$: wood density

### height

${\displaystyle H=H_{\max }\cdot \left(1-e^{C7\cdot t}\right)^{C8}}$

${\displaystyle \Rightarrow {\frac {dH}{dt}}=C7\cdot C8\cdot H\cdot \left({\frac {e^{C7\cdot t}}{1-e^{C7\cdot t}}}\right)}$

with:

H: tree height (m) t: tree age (yr)

See height parameters for the values of ${\displaystyle C_{n}}$ for different tree species.

### stem diameter

${\displaystyle V=D^{C1}\cdot H^{C2}\cdot e^{C3}}$

${\displaystyle \Rightarrow {\frac {dD}{dt}}={\frac {D}{C_{1}}}x\,\left\{{\frac {1}{V}}\cdot {\frac {dV}{dt}}-{\frac {C_{1}}{H}}\cdot {\frac {dH}{dt}}\right\}}$

with:

V: volume (dm^3 stem^-1)

D: stem diameter at breast height (cm)

e: exponent of the natural logarithm.