On the Ecology of Mountainous Forests in a Changing Climate: A ...
On the Ecology of Mountainous Forests in a Changing Climate: A ...
On the Ecology of Mountainous Forests in a Changing Climate: A ...
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Analysis <strong>of</strong> exist<strong>in</strong>g forest gap models 15<br />
x(t+∆t) = ƒ( x(t), u(t) ) (2.1)<br />
Eq. 2.1 implies that <strong>in</strong> <strong>the</strong> simulation model <strong>the</strong> follow<strong>in</strong>g must be avoided: Imag<strong>in</strong>e that<br />
a variable x 1 currently has <strong>the</strong> value x 1 (t) and is updated to x 1 (t+∆t). Later dur<strong>in</strong>g <strong>the</strong><br />
same time step, <strong>the</strong> variable x 2 is updated from x 2 (t) to x 2 (t+∆t). Now, if x 2 is a function<br />
<strong>of</strong> x 1 , Eq. 2.1 is violated because<br />
x 2 (t+∆t) = ƒ( x(t), x 1 (t+∆t), u(t) )<br />
(2.1')<br />
Many gap models work on variables which are constantly be<strong>in</strong>g updated (e.g. Botk<strong>in</strong> et<br />
al. 1972, Shugart & West 1977, Pastor & Post 1985, Kienast 1987, Leemans & Prentice<br />
1989). For <strong>in</strong>stance, <strong>the</strong> FORECE model (Kienast 1987) features <strong>the</strong> procedure sequence<br />
BIRTH, GROW, and KILL, which removes some <strong>of</strong> <strong>the</strong> sapl<strong>in</strong>gs added dur<strong>in</strong>g <strong>the</strong> same<br />
time step, although <strong>the</strong>y would formally enter <strong>the</strong> system only <strong>in</strong> <strong>the</strong> next time step<br />
(Fig. 2.1 left). Moreover, some gap models repeatedly calculate auxiliary variables with<strong>in</strong><br />
one time step, such as <strong>the</strong> leaf area <strong>in</strong>dex, although <strong>the</strong>y would formally depend only<br />
on x(t) and u(t) (Kienast 1987).<br />
Given states and <strong>in</strong>puts at time t, <strong>the</strong> follow<strong>in</strong>g computational sequence results <strong>in</strong> a correct<br />
updat<strong>in</strong>g <strong>of</strong> <strong>the</strong> new states at time t+∆t: (1) determ<strong>in</strong><strong>in</strong>g which trees will die, (2)<br />
calculat<strong>in</strong>g <strong>the</strong> growth <strong>in</strong>crement <strong>of</strong> <strong>the</strong> trees which will survive, and (3) establishment <strong>of</strong><br />
sapl<strong>in</strong>gs with<strong>in</strong> ∆t (Fig. 2.1 right). However, most forest gap models do not conform to<br />
this scheme (Tab. 2.1). S<strong>in</strong>ce a correct update mechanism avoids repeated calculation <strong>of</strong><br />
some variables with<strong>in</strong> <strong>the</strong> same ∆t, e.g. leaf area <strong>in</strong>dex, simulations become more efficient:<br />
In <strong>the</strong> case <strong>of</strong> <strong>the</strong> FORECE model, <strong>the</strong> version with a correct updat<strong>in</strong>g is approximately<br />
25% faster.<br />
Establishment<br />
Mortality<br />
Growth<br />
Growth<br />
Mortality<br />
Establishment<br />
Fig. 2.1: Sequence <strong>of</strong> procedure calculations as <strong>in</strong>corporated <strong>in</strong> <strong>the</strong> simulation model<br />
FORECE (left) lead<strong>in</strong>g to <strong>in</strong>consistencies, and a corrected sequence (right). Arrows to <strong>the</strong><br />
left and <strong>the</strong> right symbolize <strong>the</strong> transition from one time step to <strong>the</strong> next; <strong>the</strong> o<strong>the</strong>r arrows<br />
<strong>in</strong>dicate <strong>the</strong> sequence <strong>of</strong> calculation with<strong>in</strong> a time step.