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cients of the model described above are functions of the
environmental parameters. An account of the effect of the environment on the
dynamics of the tree growth is realized, as a rule, within the principle of the Libikh
limiting factors. PAR, carbon dioxide, temperature, humidity, as well as mineral
salts in the soil are among the factors of the environment. By the principle of
limiting factors, a plant grows until the
Of course, the coef
of some factor reach the limit.
An introduction of this principle is connected with many uncertainties, and there-
fore in concrete situations it must be tested using experimental estimates.
More developed models of the tree growth instead of a set of empirical
parameters whose uncertainties are often a great barrier to practical application of
the models, include the functional dependences of the elements of the tree growth
on the environmental factors. Among the most widespread and often applied is the
Libikh criterion according which the growth of a plant is con
“
resources
”
ned to an element,
whose concentration is at a minimum. By this criterion a plant or its organs grow
until the resources of some element, for instance, photosynthetically active radiation
(PAR), moisture, temperature, carbon dioxide, nutritious salts are exhausted. There
are other criteria testifying to a weighed in
uence of all environmental factors, these
weights are characteristic indicators of the plant type.
The process of tree growing can be presented as a result of interaction of three
processes of growth: change of the leaf (needle) mass, growth of the trunk, and
development of the root system. This process is schematically shown in Figs.
8.10
and
8.11
.
Denote the biomass of the canopy, trunk, and roots of the tree as x
1
, x
2
, and x
3
,
respectively. Then, the balance equation of a tree model is written as
fl
x
i
ð
t
þ D
t
Þ
¼x
i
ðÞþe
i
ðÞ
R
x
T
x
f
gD
t
;
ð
i ¼ 1
;
2
;
3
Þ;
where
t is the characteristic time step, R
x
is the gross productivity of a tree due to
photosynthesis, T
x
is the biomass expenditure on respiration,
ʔ
ʵ
i
is the share of new
biomass moving into the biomass of the ith organ of the tree.
The
ʵ
i
parameters are functions of time as well as other parameters of the
environment. Following the principle of maximum survival, a supposition can be
made that the principle of maximum primary productivity is valid:
Y
ð
t
þ D
t
Þ
¼max
e
i
ð
t
fg
Yx
i
ð
t
Þþe
i
ð
t
Þ
Yx
i
ð
t
Þ;
f
½
E
ð
t
; u; kÞ;
C
A
;
T
A
D
t
;
E
;
C
A
;
T
A
g
One of the important limiting factors of the tree growth is water, the movement of
which in the soil and the tree body determines the dynamics of the tree biomass
change. Following Kirilenko (1990), let us denote the soil as a homogeneous porous
layer [0, z
g
], where z
g
is the depth of ground water. Moreover, the following nota-
tions are introduced: W
relative soil moisture (kg H
2
O/m
3
/[kg soil/m
3
],
—
ρ—
water
density,
volume density of soil. A system of coordinates (x, y, z) is also intro-
duced, where the z-axis is directed down from the soil surface (z = 0). Then, the
movement of water through the soil can be described by Darsi
ρ
s
—
'
s 3-D equation:
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