On the Ecology of Mountainous Forests in a Changing Climate: A ...

Analysis of existing forest gap models 15
x(t+∆t) = ƒ( x(t), u(t) ) (2.1)
Eq. 2.1 implies that in the simulation model the following must be avoided: Imagine that
a variable x 1 currently has the value x 1 (t) and is updated to x 1 (t+∆t). Later during the
same time step, the variable x 2 is updated from x 2 (t) to x 2 (t+∆t). Now, if x 2 is a function
of x 1 , Eq. 2.1 is violated because
x 2 (t+∆t) = ƒ( x(t), x 1 (t+∆t), u(t) )
(2.1')
Many gap models work on variables which are constantly being updated (e.g. Botkin et
al. 1972, Shugart & West 1977, Pastor & Post 1985, Kienast 1987, Leemans & Prentice
1989). For instance, the FORECE model (Kienast 1987) features the procedure sequence
BIRTH, GROW, and KILL, which removes some of the saplings added during the same
time step, although they would formally enter the system only in the next time step
(Fig. 2.1 left). Moreover, some gap models repeatedly calculate auxiliary variables within
one time step, such as the leaf area index, although they would formally depend only
on x(t) and u(t) (Kienast 1987).
Given states and inputs at time t, the following computational sequence results in a correct
updating of the new states at time t+∆t: (1) determining which trees will die, (2)
calculating the growth increment of the trees which will survive, and (3) establishment of
saplings within ∆t (Fig. 2.1 right). However, most forest gap models do not conform to
this scheme (Tab. 2.1). Since a correct update mechanism avoids repeated calculation of
some variables within the same ∆t, e.g. leaf area index, simulations become more efficient:
In the case of the FORECE model, the version with a correct updating is approximately
25% faster.
Establishment
Mortality
Growth
Growth
Mortality
Establishment
Fig. 2.1: Sequence of procedure calculations as incorporated in the simulation model
FORECE (left) leading to inconsistencies, and a corrected sequence (right). Arrows to the
left and the right symbolize the transition from one time step to the next; the other arrows
indicate the sequence of calculation within a time step.