Annals of Botany 84 : 195–205, 1999
Article No. anbo.1999.0908, available online at http :\\www.idealibrary.com on
Modelling Stem Height and Diameter Growth in Plants
J O H N H. M. T H O R N L E Y*
Institute of Terrestrial Ecology (Edinburgh), Bush Estate, Penicuik, Midlothian EH26 0QB, UK
Received : 18 September 1998
Returned for revision : 11 March 1999
Accepted : 27 April 1999
A model of stem height and diameter growth in plants is developed. This is formulated and implemented within the
framework of an existing tree plantation growth model : the ITE Edinburgh Forest Model. It is proposed that the
height : diameter growth rate ratio is a function of a within-plant allocation ratio determined by the transportresistance model of partitioning, multiplied by a foliage turgor pressure modifier. First it is demonstrated that the
method leads to a stable long-term growth trajectory. Diurnal and seasonal dynamics are also examined. Predicted
time courses over 20 years of stem mass, stem height, height : diameter ratio, and height : diameter growth rate ratio
are presented for six treatments : control, high nitrogen, increased atmospheric carbon dioxide concentration,
increased planting density, increased temperature and decreased rainfall. High nitrogen and increased temperature
give initially higher stem height : diameter ratios, whereas high CO gives an initially lower stem height : diameter ratio.
#
However, the responses are complex, reflecting interactions between factors which often have opposing influences on
height : diameter ratios, for example : stem density, competition for light and for nitrogen ; carbon dioxide and
decreased water stress ; rainfall, leaching and nitrogen nutrition. The approach relates stem height and diameter
growth variables via internal plant variables to environmental and management variables. Potentially, a coherent view
of many observations which are sometimes in apparent conflict is provided. These aspects of plant growth can be
considered more mechanistically than has hitherto been the case, providing an alternative to the empirical or
teleonomic methods which have usually been employed.
# 1999 Annals of Botany Company
Key words : Plant, stem, height, diameter, growth, model, forest, plantation, trees.
INTRODUCTION
The height and diameter growth of plant stems are of wide
concern to agronomists, foresters and agro-foresters who
are pursuing practical and economic problems, to ecologists
and ecophysiologists interested in climate and environment,
and to evolutionary theorists and ecosystem modellers who
are constructing mathematical representations for general
and specific applications. Many studies of stem height and
diameter growth have focused strongly on trees and forests,
where these relationships are economically important, rather
than on plants of agricultural importance, although in the
latter area breeding plants which retain short stems under
well-fertilized conditions has been a very significant and
highly successful activity.
My principal objective here is to construct a mechanistic
submodel of stem height and diameter growth in plants
based on internal plant variables, rather than depending
directly on environment and management variables. This
should give a unified view of what sometimes appears
dauntingly complex—the responses of height and diameter
growth to nitrogen, CO , temperature, rainfall and planting
#
density. Moreover, the submodel must be suitable for use in
what is perhaps the most common type of crop and plant
* Reprint requests to Prof. Melvin Cannell at ITE. Fax : j44 0131
445 3943.
0305-7364\99\080195j11 $30.00\0
growth simulator : these simulators use aggregated biochemical categories where plant dry mass is divided into
structure and substrates\storage, and where organs are
similarly grouped into ‘ foliage ’, ‘ stem ’, ‘ coarse roots ’ etc.
The plant stem height : diameter growth submodel can only
be constructed and evaluated within the context of a plant
simulator : for this purpose I use the ITE Edinburgh Forest
Model (henceforth, EFM). The EFM takes account of all
pools and fluxes of C, N and water in a fully coupled soilplant-atmosphere system ; it uses a within-day time-step,
and various management options, such as fertilizer application and thinning, can be applied. The EFM provides
a suitable framework which has been documented, applied
and validated over some years (Thornley, 1991 ; Thornley
and Cannell, 1992, 1996). This account of the problem is
therefore centred on trees and forest plantations, although
the method is equally applicable to grassland or other
plants.
Three principal modelling approaches to quantifying and
predicting height and diameter growth have been employed,
which may be described as empirical, teleonomic, and
mechanistic. Empirical allometric equations for tree heightdiameter-mass relationships have been widely used (Ogawa
and Kira, 1977 ; Cannell, 1984 ; West et al., 1991 ; Sumida,
Ito and Isagi, 1997 ; see Causton, 1985, for a review).
Typical examples are
# )q#
, MsX l c (hstemd stem
(1)
MsX l c d qstem
" "
#
# 1999 Annals of Botany Company
196
Thornley—Modelling Stem Height and Diameter Growth
A.
Compartments of tree submodel
Light interception,
photosynthesis
Foliage
l
Litter
Branches
b
Coarse
roots
c
Stem
s
Fine roots,
mycorrhizas
f
Structure, X
Structure, X
Structure, X
Structure, X
Structure, X
Meristem, M
C substrate, Cl
Meristem, M
C substrate, Cb
Meristem, M
C substrate, Cs
Meristem, M
C substrate, Cc
Meristem, M
C substrate, Cf
N substrate, Nl
N substrate, Nb
N substrate, Ns
N substrate, Nc
N substrate, Nf
Photosynthetic, N, Nph
Soil mineral
nitrogen
NH4+, NO3–
CO2
Loading respiration
B.
N uptake
CO2
Respiration
linked to
N uptake
Processes for one compartment (not foliage)
Trannsport processes
Local maintenance respiration, CO2
Structure
X
Growth
Meristem
M
Carbon
substrate, C
Nitrogen
substrate, N
Growth
Litter
Growth
CO2, local growth respiration
F. 1. The tree growth and allocation submodel (Thornley, 1991) used as a basis for the present investigation. Photosynthetic acclimation to light,
nitrogen, temperature and CO is provided by the photosynthetic N pool in the foliage compartment (Thornley, 1998 c). The linked soil and water
#
submodels are described in Thornley (1998 b, chapters 5 and 6).
where MsX (kg structural dry matter per stem), hstem (m) and
dstem (m) are stem dry mass, height and diameter ; c , c , q
" # "
and q are constants that are specific to each species and
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situation. These models often give a good description of a
limited range of experimental data, but they usually fail to
account for the complexity of the observed responses to
environment and management when a wider range of
treatments and conditions is applied.
The second category of investigations of tree height,
diameter and dry mass relationships is based on concepts of
teleonomy or purpose. In the teleonomic approach the
capacity to perform some function is optimized. An early
paper by Paltridge (1973) considered the photosynthesis per
unit ground area averaged over the lifetime of the trees.
McMahon (1973), Mattheck, Bethge and Erb (1993) and
many others have studied mechanical criteria. The pipemodel theory of tree form proposed by Shinozaki et al.
(1964) has been applied by e.g. Valentine (1985) and Ma$ kela$
(1986). Ma$ kela$ and Sieva$ nen (1992) have examined a
strategy which balances self shading with structural costs in
determining height growth. Ludlow, Randle and Grace
(1990) give a useful discussion of the determinants and
benefits of height growth.
A teleonomic model requires three layers of contingent
assumptions : first, an optimum function (goal) whose choice
may appear plausible but which is often rather subjective ;
second, evolutionary pressures exist to fulfil the goal ; and
third, there are mechanisms which can fulfil the teleonomic
goal and are accessible by adaptation. Doubt has been
expressed as to whether teleonomic thinking is sound
scientific methodology (Monod, 1974). When a teleonomic
submodel fails, as all models do eventually, there seems no
logical way forward other than by ad hoc adjustment. In
some areas mechanistic (non-goal-seeking) models appear
to give optimized responses without being constrained to do
so [e.g. allocation (Ma$ kela$ and Sieva$ nen, 1987) ; phyllotaxis
(Thornley and Johnson, 1990)]. This may also be true for
height and diameter growth in trees. Finally, because goals
can only be achieved by means of mechanisms, and because
a mechanistic model permits step-by-step elaboration and
refinement as theory and experiment improve, it seems
preferable to proceed mechanistically. A mechanistic model
Thornley—Modelling Stem Height and Diameter Growth
can sometimes be simplified and approximately interpreted
in goal-seeking terms.
I am not aware of any mechanistic models of height
growth in trees. Ludlow (pers. comm.), in an unpublished
review of C and N allocation in Sitka spruce, outlined an
approach comparable to that used here when he suggested
that : ‘ The rate of shoot extension and xylem formation …
be made to depend on carbohydrate availability … ’. More
recently, and closely related to our approach, Deleuze and
Houllier (1997) have proposed a very interesting model of
tree ring width growth which is substrate-driven with
transport resistances along the stem. Their model predicts a
strong relationship between morphology and foliage : root
allocation, which is similar to the basis of the present model.
However, in their model, stem height is assumed to be
constant, and also, the treatment of the stem profile, using
partial differential equations, is too detailed for our purpose,
which focuses on the relation between stem height and mean
stem diameter. Nevertheless, Deleuze and Houllier’s (1997)
approach, using internal tree variables, with ‘ free ’ or
objective equations, offers a promising route for further
development. We explore here the extent to which some
simple assumptions, along the lines of Ludlow’s suggestions
(see above) and Deleuze and Houllier’s work (1997), can
‘ explain ’ observed responses. Since the submodel is to be
part of a plant ecosystem simulator, simplicity is important ;
but equally, the submodel must have one of the prime
virtues of a mechanistic submodel—that of being open to
rational modification and extension when it is found to fail.
The responses of interest concern the dynamics of stem
height, diameter and mass, and how these are affected by
environmental and management variables such as nitrogen
fertilization, atmospheric CO , planting density, tempera#
ture, and water. The version of the EFM used here is not
thinned during the 20-year growth period examined. Only
the relations and equations pertinent to the current
investigation are described. These equations, suitably
adapted, could be incorporated into other mechanistic plant
growth models of the same genre.
The traditional view of plant morphogenesis is that plant
form results from cellular activities, such as cell shape and
planes of cell division, and these may be controlled by plant
morphogens, possibly mediated by factors such as phytochrome. However, there is evidence to support the idea that
plant form is a result of the differential growth of parts of
the plant, and that this constrains cellular activities (Adler,
1974 ; Kaplan and Hagemann, 1991 ; Cooke and Lu, 1992).
It is easier to represent the latter view in a plant growth
model, and arguably the differential-growth approach may
account for many interesting morphological responses. We
therefore make no attempt to represent plant morphogens
or their possible effects on cellular-level activities. Such
considerations could be added to the currently proposed
framework if and when it is found to be inadequate.
Shoot extension has been widely researched, and it is
known that relevant factors can be nutrient status (Moorby
and Besford, 1983), turgor pressure (Van Volkenburgh,
Hunt and Davies, 1983), hormones (Thomas et al., 1981),
light quality and quantity (e.g. Cosgrove and Green, 1981 ;
Jones, 1992) and temperature. These factors can interact in
197
a complex manner which is not fully understood—for
instance, temperature affects nutrient status, turgor pressure
and usually all rate constants in the system ; light may
influence nutrient status but may also change hormone
levels. Smith, Casal and Jackson (1990) suggested that
phytochrome can enable plants to detect the proximity of
neighbouring vegetation, a mechanism which could be
important in forest plantations but which is ignored here.
We use internal plant variables taken from within the tree
submodel of Fig. 1, supplemented by a water submodel for
the soil-plant-atmosphere system (see for example Fig 6.1 of
Thornley, 1998 b) which calculates water potential and its
components in the tissue boxes of Fig. 1. These internal
variables are assumed to determine the stem height : diameter
growth rate ratio. The effects of the environment only
operate through the internal variables, and not directly.
While this restricts the scope of the height : diameter
submodel which can be constructed, it provides a valuable
focus and constraint, ensuring that the submodel can be
incorporated into, and evaluated within, a common type of
plant growth simulator.
THE MODEL
The tree submodel of the EFM is shown in Fig. 1. The core
plant submodel is described in Thornley (1991). One extra
foliage pool (‘ photosynthetic N ’) has been added to the
plant submodel to give photosynthetic acclimation to light,
nitrogen, CO and temperature (Thornley, 1998 c). More
#
important to the present application is the addition of soil
and water submodels (Thornley, 1998 b). The soil submodel
describes C and N cycles in the plant-litter-soil system,
including mineral N pools, a soluble C pool, biomass and
soil organic matter (SOM) pools. The processes represented
are competing N uptake by roots and biomass, C and N
exudation, C and N leaching, volatilization, nitrification,
denitrification, respiration, mineralization, immobilization,
and SOM transformations. The water submodel considers
water pools and fluxes in the soil-plant-atmosphere system ;
it is driven by the Penman-Monteith equation with water
flow down gradients of water potential. It is similar to the
model of Williams et al. (1996), but uses a variable
capacitance with elastic cell walls. Within the plant, the
osmotic and pressure components of water potential, with
relative water content, are calculated for the root and the
shoot. The two submodels for soil and water are important
in modulating the responses of the plant submodel in Fig. 1.
The representation of Fig. 1 gives good predictions of
allocation between the different tree compartments, based
on the variables shown for aggregated biochemical categories. Deleuze and Houllier (1997) found ‘ a strong
relationship between morphology and foliage : root allocation ’, and we use this same level of description to
construct a model of stem height and diameter growth. That
is, those factors which determine differential mass growth
(i.e. allocation) also determine the relationship between
stem height and diameter growth, with a modifier for turgor
pressure. Here just the equations relevant to the growth of
stem dry mass (DM), height and diameter are given.
198
Thornley—Modelling Stem Height and Diameter Growth
Stem growth
The differential equation determining growth of stem
structural DM [MsX (kg structural DM per stem)] is given
approximately by :
dMsX
l fTairkGM,s, cMpot,sAstem(CsNs)#
#!
dt
kGM,s, l 200 ([C] [N] d)−"
#!
cMpot,s l 1000 kg structural DM
(m# stem surface [C]
(2)
[N])−"
Here t is the time variable (d). fTair is a dimensionless
function of air temperature with value unity at 20 mC (figure
3n6, cubic, Thornley, 1998 b). Cs and Ns are the C and N
substrate concentrations in the stem meristem, with units of
kg C, N substrate (kg stem meristem structural dry matter)−"
[see eqn (9a) of Thornley, 1991]. [C] and [N] denote units of
substrate concentration, e.g. kg substrate C, N (kg structural
dry matter)−". kGM,s, is a stem meristem growth rate
#!
constant at 20 mC. cMpot,s is a constant describing the
dependence of meristem dry mass on stem surface area
(Astem) and stem nutrient status (CsNs).
Equation (2) has been simplified for presentation here, in
order that an explicit equation for the rate of growth of stem
structural DM can be written down. The details of this
simplication are unimportant ; the important relationships
define how the stem structural DM growth is partitioned
between height growth and diameter growth.
Obligate relationships between stem growth and height and
diameter growth
Some equations must be satisfied to meet the requirements
of geometry and conservation. Variable stem shape factors
are ignored (but see Dewar, 1993 ; Deleuze and Houllier,
1997). Assuming that the stem is cylindrical with height hstem
(m) and a constant diameter dstem (m), stem volume Vstem
(m$) is given by
M
# l sX
Vstem l "hstemπd stem
%
ρ
stem
(3)
ρstem l 350 kg m−$
ρstem is the stem structural dry mass density. Taking
logarithms of each side of MsX l "ρstemπhstemdstem# and
%
differentiating with respect to time t gives :
1 dMsX
1 dhstem
l
MsX dt
hstem dt
j
(4)
2 ddstem
dstem dt
The ratio of the instantaneous height and diameter
growth rates is denoted by :
λh/d l
dhstem
ddstem
(5)
If it is assumed that this ratio is determined by internal tree
variables, then the height and diameter growth rates are
calculable from the stem dry matter growth rate [combining
eqn (5) with eqn (4)] :
0
1
dhstem
λh/d
l
dt
λh/d\hstemj2\dstem
1 dMsX
i
MsX dt
0
(6)
1
ddstem
1
l
dt
λh/d\hstemj2\dstem
1 dMsX
i
MsX dt
The forest model calculates a value for the stem dry matter
growth rate, dMsX\dt [eqn (2)]. The assumptions described
below define the ratio λh/d which then allows stem height
and diameter growth rates to be obtained using eqn (6). For
example, if the height : diameter ratio is not changing, i.e.
λh/d l dhstem\ddstem l hstem\dstem, so that the proportional
rate of height growth equals the proportional rate of
diameter growth (dhstem\hstem l ddstem\dstem), therefore
1 dhstem
1 ddstem
l
hstem dt
dstem dt
l
(7)
1 dMsX
3MsX dt
Stem height and diameter growth
I assume that there are two factors determining the
height : diameter increment ratio of eqn (5) : (1) relative
‘ nutrient ’ status, for which meristem specific growth rates
are a surrogate ; and (2) water status at the top of the stem,
for which foliage pressure (turgor) potential is used.
The gross specific growth rates of the meristems of foliage
(l ) and stem (s) are µM,l, µM,s (d−"), with [Thornley 1991,
eqns (3a), (9b)]
µM,l l kGM,l, ClNl
#!
µM,s l kGM,s, CsNs
(8)
#!
kGM,l, l kGM,s, l 200 ([C] [N] d)−"
#!
#!
k is a rate parameter at 20 mC for growth of meristem in
foliage and stem. Temperature and water-status modifiers
are omitted from eqn (8) for simplicity (Thornley, 1998 b).
The gross specific growth rate of a meristem depends on its
nutrient status (l the product of C and N substrate
concentrations), and this determines plant form.
The ratio of these two gross specific growth rate constants
is :
µ
(9)
rµM,l/s l M,l
µM,s
This ratio depends [eqn (8)] on the relative nutrient status in
the foliage (ClNl) and the stem (CsNs). The local nutrient
status is an important factor in the transport-resistance
model of allocation (Thornley, 1998 a), and here the concept
199
Thornley—Modelling Stem Height and Diameter Growth
Height increment/diameter increment, λh/d,G = dh/dd
100
qh/d,G = 4
80
2
60
1
40
20
0
0·5
0·0
1·0
1·5
Meristem activity ratio: foliage/stem, rµM, l/s
2·0
F. 2. Equation (10) is drawn for three values of the parameter qh/d. Other parameters are given in eqn (10).
is extended by assuming that a nutrient status ratio also
determines the ratio between height growth and diameter
growth.
The main determinant of the ratio of height to diameter
growth rates [eqn (5)] is assumed to depend on the meristem
activity ratio of eqn (9) according to
λh/d,G l
rh/d,minjrh/d,max(rµM,l/s)qh/d,G
1j(rµM,l/s)qh/d,G
(10)
rh/d,min l 0, rh/d,max l 120, qh/d,G l 1
There are three parameters in eqn (10) : the minimum and
maximum values of λh/d, obtained at rµM,l/s l 0 and _, and
a steepness parameter qh/d,G, which alters the steepness of the
response in the middle range, and could be used to amplify
the response. This equation is drawn in Fig. 2 for three
values of the parameter qh/d. The meristem activity ratio
[eqn (9)] predicted by the EFM is found to vary over the
range 0n5 to 1n5, under the conditions applied below.
Following Lockhart’s suggestion (1965) that a minimum
turgor pressure is required for cell wall extension to occur,
with a strong dependence of extension on the excess turgor
pressure, I assume that foliage pressure potential, ψP,l [J
(kg water)−"], modifies the height : diameter growth ratio
multiplicatively by the factor :
fh/d,P,l l
ψqP,lP,l
q
P,l
Kψ,P,ljψqP,lP,l
qP,l l 4
(11)
Kψ,P,l l 500 J (kg water)−"
This equation is similar to eqn (10), illustrated in Fig. 2.
With adequate water availability and a high turgor pressure,
this modifier equals unity, falling to 0n5 when ψP,l l Kψ,P,l.
Most tree physiologists would agree that height growth
limitation in trees is the result of water stress at the top of
the tree. Here it is assumed that there is a differential effect
of water stress on height and diameter growth, affecting the
former more than the latter. The plausibility of this
assumption may be addressed at the cell level (see
Discussion), as well as observing that height growth and
diameter growth seem to be at least partially independent
processes, spatially and perhaps also temporally. However,
in young trees and other plants which are well supplied with
water, it seems unlikely that water stress can play a
significant role in determining height and diameter growth.
The final expression used for the instantaneous ratio of
the height and diameter growth rates [eqn (5)] is obtained by
multiplying the nutrient status expression of eqn (10) with
the turgor modifier of eqn (11) :
λh/d l λh/d,G fh/d,P,l
(12)
Height and diameter growth are then calculated with eqn
(6). The nutrient status factor is operative under all
conditions, reflecting the great plasticity of root : shoot
ratios in plants, whereas the turgor modifier is important
only under conditions of foliage water stress, which includes
trees approaching their limiting height.
SIMULATIONS
The model is programmed in the continuous system
simulation language, ACSL (Mitchell and Gauthier, 1993).
The program for the EFM is available by using a file
200
Thornley—Modelling Stem Height and Diameter Growth
A. Stem mass
B. Stem height
Stem mass, MsX
(kg XDM stem–1)
20
30
15
10
5
0
5
10
15
20
C. Stem height:diameter ratio
120
Height:diameter ratio
4
3
2
0
10
4
3
2
Initial
height:
diameter
ratio:
120
60
30
–1
0
Initial height:diameter
ratio:
100
80
120
60
60
40
30
20
Stem height, hstem (m)
Initial height:diameter ratio: 120 60
0
5
10
Time (year)
15
20
10
Height:diameter growth rate ratio,
∆hstem/∆dstem
25
0
5
10
15
20
D. Height:diameter growth rate ratio
55
Initial height:diameter
ratio:
53
51
120
49
60
30
47
0
5
10
Time (year)
15
20
F. 3. The tree growth model (Fig. 1) is run from the same initial values except for the stem height : diameter ratio which is assigned initial values
of 30, 60 and 120. Equations (10), (11) and (12) with parameters as given are used to model growth in height and diameter. XDM denotes
structural dry mass. In D, the growth rate ratio is calculated from the annual increments in height and diameter.
transfer protocol utility : ftp :\\budbase.nbu.ac.uk\pub\
tree\Forest (the most recent version of the EFM is
FOREST.CSL). Euler’s method is used for integration with
an interval of approx. 10 min. The ‘ standard ’ environment
is for Eskdalemuir in northern Britain, as described in
Thornley and Cannell (1997, Table 1). Initial values for all
simulations are for an equilibrium managed plantation in
the standard environment, with a 60-year rotation and a
defined thinning regime (Thornley and Cannell, 1996). The
60-year standard rotation has a yield class of 13n5 m$ ha−"
year−" and a final leaf area index (LAI) of 8n1. Thinning is
not applied in the 20-year simulations described below, so
that stem density remains constant at its initial value.
Only the 20-year control simulations are in a long-term
equilibrium situation.
Long-term dynamic stability of algorithm
If the algorithm is to work satisfactorily, then it must give
results for a given environment which are independent of
the initial values of stem height and stem diameter, after a
suitable time period has elapsed. Therefore the time course
in the standard environment, beginning with three different
initial values of stem height : diameter (h : d) ratio, is
investigated. The results are shown in Fig. 3. Note that stem
mass is hardly affected by the initial h : d value (Fig. 3A).
The height : diameter growth rate ratio [Fig. 3D, eqns (5)
and (12)], calculated by integrating height and diameter
growth separately over successive years, is immediately
stable within 2 %—i.e. like stem mass, it is little affected by
the initial height : diameter value. During the first 5 years of
growth, stem height (Fig. 3B) is affected by the initial h : d
value, as is the height : diameter ratio (Fig. 3C). However,
after about five years, the height and h : d trajectories (Fig.
3B and C) are not much changed by the initial values.
Diurnal and seasonal changes in height and diameter
growth
The model is driven by diurnally changing light, air
temperature and humidity, and seasonally changing light,
daylength, air and soil temperatures, humidity and rainfall.
This gives rise to the diurnal and seasonal changes in height
growth rate and height : diameter growth rate ratio shown in
Fig. 4.
Figure 4A illustrates the diurnal response for 3 d in April.
The height growth rate is responding mostly to the changing
air temperature, with a small (approx. 3 %) modulation of
the height : diameter growth rate ratio arising from diurnal
changes in substrate concentrations. In the model, it is
assumed that foliage and stem [the two components of eqns
(8) and (9)] are both at air temperature.
201
Thornley—Modelling Stem Height and Diameter Growth
63
0·20
62
0·15
61
0·10
60
Height
growth
rate
0·05
0·0
59
Ratio
465
466
11 Apr
467
Time, t (day)
Height:diameter growth rate ratio, dhstem/ddstem
Height growth rate, dhstem/dt (10–3 m d–1)
A. Diurnal responses
0·25
57
468
B. Seasonal responses
Height growth rate, ∆hstem/∆t (10–3 m d–1)
Ratio
1·2
70
1·0
Height
growth
rate
60
0·8
0·6
50
0·4
40
0·2
0·0
0·0
0·5
1·0
1·5
Time (year)
2·0
2·5
3·0
Height:diameter growth rate ratio, ∆hstem/∆dstem
80
1·4
30
F. 4. Diurnal and seasonal responses of stem height growth rate (——) and height : diameter growth rate ratio (– – –) are simulated using eqns
(10), (11) and (12) with parameters as given. A, Instantaneous values for 3 d in April. B, Daily increments (∆t l 1 d) over 3 years. The year
starts on 1 January.
In Fig. 4B, as in Fig. 4A for the diurnal changes, the
height growth rate tracks seasonal air temperature quite
closely. In this case the height : diameter growth rate ratio
varies more strongly seasonally than diurnally (Fig. 4A).
Also, the seasonal changes in growth rate ratio are out of
phase with seasonal changes in growth rate, where temperature is the most important modulator. This reflects the
seasonal importance of other factors such as mineralization
and soil mineral N levels. For example, there is a dip in the
stem height : diameter growth rate ratio in spring to
midsummer when N is normally limiting, directing allocation towards the root, and decreasing the meristem
activity ratio of eqn (9) and hence the height : diameter
growth rate ratio via eqn (10).
Effect of fiŠe treatments on height growth and
height : diameter ratios
Relative to the standard ‘ control ’ environment, five
treatments have been simulated : (1) nitrogen deposition is
increased from 20 to 200 kg N ha−" year−") (jnitrogen) ; (2)
ambient CO concentration is increased from 350 to 700 vpm
#
(jCO ) ; (3) planting density is increased from 0n25 to 1
#
202
Thornley—Modelling Stem Height and Diameter Growth
60
A. Stem mass
B. Stem height
8
+ nitrogen
+ nitrogen
40
Stem height: hstem (m)
Stem mass, MsX
(kg XDM stem–1)
50
+T
30
+ CO2
20
control
– water
10
+ nstems
0
10
15
20
+ nstems
2
25
+ nitrogen
52
+ CO2
48
control
+ CO2
+ nitrogen
+ nstems
– water
44
+T
0
5
10
15
Time (year)
20
25
0
Height:diameter growth rate ratio,
∆hstem/∆dstem
5
C. Stem height:diameter ratio
Height:diameter ratio
4
0
0
40
+T
+ CO2
control
– water
6
55
5
10
15
20
25
D. Height:diameter growth rate ratio
50
45
40
+ CO2
control
+ nstems
35
+ nitrogen
+T
– water
30
0
5
10
15
Time (year)
20
25
F. 5. Stem height and diameter growth are simulated using eqn (10) with qh/d l 1. Five treatments are imposed : (1) jnitrogen : nitrogen
deposition is increased from 20 to 200 kg N ha−" year−" ; (2) jCO : ambient CO concentration is increased from 350 to 700 vpm ; (3) jnstems :
#
#
planting density is increased from 0n25 to 1 stems m−# (stem density is constant throughout each 20-year simulation) ; (4) jT : air and soil
temperatures are increased by 3 mC ; (5) kwater : rainfall is decreased by 75 %. XDM denotes structural dry mass. In D, the growth rate ratio is
calculated for annual increments in height and diameter.
stems m−# (jnstems) ; (4) air and soil temperatures are
increased by 3 mC (jT) ; and (5) rainfall is decreased by
75 % (kwater). The 20-year trajectories occurring in these
treatments are illustrated in Fig. 5. To help interpret Fig. 5,
some pertinent variables are provided in Fig. 6.
It can be seen in Fig. 5A and B that applying jnitrogen,
jtemperature and jCO all lead to increases in stem mass
#
and stem height whereas jnstems and kwater decrease both
quantities. However, in the first 13 years, kwater increases
stem mass and height very slightly ; this is due to decreased
leaching and improved N nutrition (Fig. 6B and C) opposing
the effects of increased water stress. The effects of the
treatments on the two ratios of Fig. 5C and D are much
more complex, suggesting that there are no simple answers
to questions like ‘ What is the effect of … ’. The answer
depends on the particular conditions of the experiment and
the time scale.
Increased N increases the height : diameter ratio (Fig. 5C
and D), although the higher LAI leads to early (relative to
the control) water stress, stomatal closure (Fig. 6A) and
decreased foliage turgor (Fig. 6D), which decreases the
height : diameter growth rate ratio [eqn (11)].
Increased temperature produces increased growth (Fig.
5A and B) and increased height : diameter ratio (Fig. 5C and
D), although these increases are partly due to increased
mineralization and higher mineral N concentrations (Fig.
6C). After about five years, the increased water stress (Fig.
6A and D), the increased system N losses (Fig. 6B) and the
resulting decreased soil mineral N (Fig. 6C), all combine to
depress the height : diameter ratio (Fig. 5C).
Increased CO increases growth (Fig. 5A and B), but
#
decreases the height : diameter ratio (Fig. 5C). However,
after some years, the decreased water stress (Fig. 6A and D),
the decreased N losses and relative increases in soil mineral
N (Fig. 6B and C), lead to steadily increasing
height : diameter ratios (Fig. 5C and D).
Increased planting density in this case gives rise to
competition for N before competition for light. Stem height
(jnstems) keeps pace with the control for the first 5 years
(Fig. 5B), but by then soil mineral N is appreciably depleted
(Fig. 6C), and the height : diameter growth rate ratio is
decreasing accordingly (Fig. 5D). Increased water stress
(Fig. 6A and D) is also a factor in decreasing height growth.
Decreasing rainfall produces three effects, as noted earlier.
Decreased leaching (Fig. 6B) gives improved soil mineral N
(Fig. 6C) which enhances height : diameter ratio. Decreased
foliage turgor (Fig. 6D) gives decreased height : diameter
ratio. This can be partially offset by decreased stomatal
conductance (Fig. 6A) causing decreased photosynthesis
which increases height : diameter growth ratio.
203
Thornley—Modelling Stem Height and Diameter Growth
B. Annual N leaching
Stomatal conductance,
15 h,1 Jul (m s–1)
+ nitrogen
0·0045
+ nstems
0·0040
control
0·0035
0·0030
+T
+ CO2
0·0025
– water
0·0020
5
0
10
15
20
Leaching rate (kg N m–2 yr–1)
A. Stomatal conductance (15 h, 1 Jul)
0·0050
+ nitrogen
0·0015
control
+ CO2
0·0005
– water
0·0000
0
25
0·0015
+ nitrogen
0·0010 + n
stems
control
+ CO2
0·0005
0·0000
0
5
10
15
Time (year)
20
25
Foliage pressure potential
[J (kg water)–1)]
Soil mineral N (kg N m–2)
1300
– water
+ nstems
0·0010
C. Soil mineral nitrogen
+T
+T
5
10
15
20
25
D. Foliage pressure potential
+ nitrogen
control and –water
1100
+ CO2
900
– water
700
+ nstems
500
0
5
10
15
Time (year)
+T
20
25
F. 6. For the five treatments described in Fig. 5, the responses of some variables helpful to understanding stem height and diameter growth are
illustrated. In C and D, annual averages are given. 1 J (kg water)−" 1000 Pa.
where γ (m−") is a taper factor and dstem, (m) is the base
!
diameter. Stem volume then becomes [cf. eqn (3)] :
DISCUSSION
In the model described above, a minimum hypothesis has
been proposed in order to determine the scope of the effects
predicted. The relative responses in Fig. 5 seem qualitatively
reasonable. Their magnitude could be increased by assuming
higher values of the parameter qh/d,G of eqn (10), or changing
the parameters of eqn (11). The approach proposed can be
divided into two parts for consideration. Equations (10) and
(11) are clearly quite subjective and could be modified in
many ways : with different equation forms, different components of metabolic activity or water potential, as well as
different parameter values. A different forest model would
require its own surrogates for these quantities. However, the
analysis from eqn (3) to eqn (6) is relatively model-free, that
is, applicable to all models. This suggests that these
equations can be used in general for attacking the
height : diameter problem, even if the particular implementation here [eqns (10) and (11)] with the Edinburgh Forest
Model (EFM) does not seem appropriate. Indeed, in so far
as I have tried different formulations of eqns (10) and (11),
the method appears pleasingly robust.
It is assumed that the stem is cylindrical : see eqn (3) and
above. This assumption could be relaxed by taking (say) a
negative exponential dependence of stem diameter (dstem) on
stem height (hstem) :
dstem l dstem, e−γhstem
!
(13)
Vstem l "πdstem,#
%
!
(1ke−#γhstem)
2γ
(14)
The analysis from eqns (3) to (6) is no longer possible unless
γ is assumed to be small. It seems doubtful that the method
proposed can be applied to situations where stem shape is a
significant factor, at least, in its present formulation.
The height : diameter submodel operates using the values
of current internal tree variables. Although these are
influenced to an extent by the past environment, including
that of the previous year, in the EFM as currently structured,
this connection is weak. In some species, height growth
depends mainly on the growing conditions of the previous
year (e.g. during August to October for some pines) whereas
the radial increment is mostly sensitive to the current
environment. The model may therefore not always perform
well in relation to yearly predictions where there is
substantial year-to-year variation in the environment, and
where seasonal memory effects are pronounced.
Our assumption that height : diameter growth ratio depends on nutrient status can be argued for in several ways.
The effects of say nitrogen fertilization or CO enrichment,
#
which affect nutrient status and plant morphology in
different ways, are well-documented (e.g. Moorby and
Besford, 1983 ; Newton, 1991) : usually (loc. cit.), nitrogen
204
Thornley—Modelling Stem Height and Diameter Growth
fertilization promotes height growth, whereas increased
CO promotes diameter growth. At the level of the single
#
cell, it has been found that in E. coli and Pseudomonas
putida both cell size and shape depend on growth rate and
nutrient supply (Domach et al., 1984 ; Givskov et al., 1994).
In plant cells it has been observed that the planes of cell
division respond to growth conditions (Lyndon, 1976 ;
Yeoman, 1976). Another possible rationale is that height
growth results more from metabolism in the region of the
top of the stem (see also Ludlow et al., 1990 ; Ma$ kela$ , 1990),
for which foliage meristem activity is a surrogate, in relation
to diameter growth which occurs over the whole length of
the stem, including the base of the stem, for which stem
meristem activity is a surrogate. However, whatever the
strength or otherwise of plausibility arguments, our assumption is only justified if it gives a helpful description of
tree growth. There may be other internal tree variables
which can fulfil this purpose as well as or even better than
those chosen in eqns (8) to (12). Also, it is assumed here that
the stem is cylindrical, with a constant height : diameter
ratio. This assumption precludes easy comparison of model
predictions with field data.
Lockhart (1965) proposed the concept that a minimum
turgor pressure is needed for cell wall extension to occur,
that is, to be ‘ switched-on ’. Boyer (1968, Fig. 4) examined
the relationship between an externally applied pressure and
the specific area growth rate of a sunflower leaf. Boyer’s
results suggest that a turgor pressure potential of about 150
J (kg water)−" ( 0n15 MPa l 1n5 bar) is needed to initiate
cell wall extension. Bunce (1977) however found a linear
relationship between elongation rate and turgor pressure
above a threshold value for soybean leaves. Thus turgor
pressure may alter the balance between cell extension
(possibly height growth) and cell wall thickening (possibly
diameter growth). Equation (11) describes the dependence
of height : diameter growth rate ratio on turgor pressure
(ψP,l) assumed here. This is a ‘ phenomenological ’ equation
with parameters whose values have been chosen to give
satisfactory operation in the context of the water submodel
used in the EFM (Thornley, 1998 b). It may be noted that
increasing CO or increasing N can, under different
#
conditions, either increase or decrease turgor (Fig. 6D).
Phytochrome and light quality have been shown to be
related to morphogenesis and stem extension (e.g. Smith,
1981). Another possible modification of eqn (12) for
height : diameter growth rate ratio is to introduce a
morphogen-dependent factor, whose level depends on light
quality. This could be needed if the responses of the
currently suggested model to planting density are found to
be unsatisfactory. However, an additional submodel describing the synthesis, degradation and possibly the within-plant
transport of a morphogen, with its dependence on environmental and plant variables, and its morphogenetic
effect, would be a considerable addition to the EFM.
‘ Validation ’ is still a contentious issue with many
biologists, although at last, arguably, its appeal is beginning
to diminish, following difficulties with model validation
across the whole spectrum of science (e.g. Hopkins and
Leipold, 1996). Statistical models, which make contact with
reality at few points, and where parameters are uncon-
strained by scientific principles, can often be shown to work
satisfactorily within a given arena. Without doubt this can
be a useful matter to demonstrate. Mechanistic simulators,
with their multiple and variously weighted objectives, are
another proposition entirely. ‘ Evaluation ’ is then a more
helpful term than ‘ validation ’, allowing a more flexible view
of the ingredients of the model to be taken, as well as a more
sympathetic tasting of the resulting dish. At the qualitative
level, the height : diameter model behaves reasonably. At the
quantitative level, an experiment which would allow a
meaningful comparison with observational data has not yet
been performed. Indeed, such an experiment and comparison would be a massive undertaking, and of doubtful
value (Hopkins and Leipold, 1996).
There are, inevitably, some quite important features
missing from the height : diameter growth model as currently
proposed, as have already been mentioned. No models
would be constructed at all if it were not acceptable to build
models which only tell part of the story. Any explanation
must stop somewhere, leaving an unexplained residue. The
scope of a model is always limited. The predictions of a
model are always approximate. The failure of a model will
usually indicate how that model needs to be changed and
perhaps extended. A simple model can be valuable for
indicating, or contra-indicating, the utility of a general
approach. This has been an important aim in this
investigation.
A C K N O W L E D G E M E N TS
I thank Melvin Cannell, Roddy Dewar and François
Houllier for comments and suggestions. The work has been
supported by NERC through its TIGER (Terrestrial
Initiative in Global Environment Research) programme,
the Department of the Environment Contract No PECD
7\12\79 on carbon sequestration, by the EC MEGARICH
programme, and is a contribution to COST action 619.
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