Directory UMM :Data Elmu:jurnal:T:Tree Physiology:Vol16.1996:

Tree Physiology 16, 161--171
© 1996 Heron Publishing----Victoria, Canada

Analytical model of stemwood growth in relation to nitrogen supply
RODERICK C. DEWAR and ROSS E. MCMURTRIE
School of Biological Science, University of New South Wales, Sydney NSW 2052, Australia

Received March 2, 1995

Summary We derived a simplified version of a previously
published process-based model of forest productivity and used
it to gain information about the dependence of stemwood
growth on nitrogen supply. The simplifications we made led to
the following general expression for stemwood carbon (cw) as
a function of stand age (t), which shows explicitly the main
factors involved:
c w (t) =

λe−µ wt − µwe−λ t 
η w G∗ 
1 −

,
λ − µw
µw 


where ηw is the fraction of total carbon production (G) allocated to stemwood, G* is the equilibrium value of G at canopy
closure, λ describes the rate at which G approaches G*, and µw
is the combined specific rate of stemwood maintenance respiration and senescence. According to this equation, which describes a sigmoidal growth curve, cw is zero initially and
asymptotically approaches ηwG*/µw with the rate of approach
dependent on λ and µw. We used this result to derive corresponding expressions for the maximum mean annual stemwood volume increment (Y) and optimal rotation length (T).
By calculating the quantities G* and λ (which characterize the
variation of carbon production with stand age) as functions of
the supply rate of plant-available nitrogen (Uo), we estimated
the responses of Y and T to changes in Uo. For a plausible set
of parameter values, as Uo increased from 50 to 150 kg N ha −1
year −1, Y increased approximately linearly from 8 to 25 m3
ha −1 year −1 (mainly as a result of increasing G*), whereas T
decreased from 21 to 18 years (due to increasing λ). The
sensitivity of Y and T to other model parameters was also
investigated.

The analytical model provides a useful basis for examining
the effects of changes in climate and nutrient supply on sustainable forest productivity, and may also help in interpreting
the behavior of more complex process-based models of forest
growth.
Keywords: canopy closure, mean annual increment, optimal
rotation length, plantation, sustained yield.

Introduction
Quantifying the relationship between forest nutrition and forest productivity is an important concern of plantation manag-

ers. Over the last decade, attempts to improve on yield predictions based on empirical site quality indices (e.g., Lewis et al.
1976) have led to the development of models based on the
underlying biological processes that govern the relationship
between productivity and nutrition (e.g., Dixon et al. 1990).
However, interpreting the behavior of biologically realistic
models is often difficult because of the complexity of the
processes involved. To facilitate such interpretation, and therefore the translation of information from complex models to
guidelines for forest practice, it is useful to examine the behavior of relatively simple process-based models that are amenable to analytical solution, even though quantitative accuracy is
sacrificed to some extent for qualitative insight.
McMurtrie and Wolf (1983) and McMurtrie (1985, 1991)

developed a process-based model of stand growth that simulates the effect of nitrogen supply on forest productivity. Recently, the model has been combined with the soil
carbon--nitrogen model of Parton et al. (1987) and Schimel et
al. (1990). The combined model (called G’DAY) has been used
to examine the long-term response of unmanaged forest ecosystems to increasing CO2 concentration (Comins and
McMurtrie 1993, Kirschbaum et al. 1994).
The aim of this study was to derive a simplified, analytically
tractable version of the plant production part of G’DAY, and
use it to gain insights into the general relationship between
stemwood growth and nitrogen supply in managed forests. In
particular, the model was used to predict how the maximum
mean annual stemwood volume increment and optimal rotation length can be expected to vary in response to changes in
the supply of nitrogen from net mineralization, fertilizer additions, fixation and atmospheric deposition.

Simple stand growth model
Figure 1 shows the general structure of the model. Four dynamic variables are represented: foliage carbon (cf, kg C m −2),
foliage nitrogen (nf), stemwood carbon (cw, which includes
stems, branches and coarse roots), and fine root carbon (cr).
Table 1 lists the symbol definitions and units. A list of parameter values is given in Table 2. Some of these parameter values
have been taken from Comins and McMurtrie (1993), others
are biologically reasonable estimates. As such, the values have

been chosen to illustrate the general behavior of the model, and
do not reflect a particular species or data set.

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DEWAR AND MCMURTRIE

allocated to foliage, stemwood and fine roots according to
fixed allocation fractions ηi (i = f, w, r), with Σi ηi = 1. Losses
of carbon due to foliage, stemwood and fine root senescence
occur at rates γi (i = f, w, r). Maintenance respiration in
stemwood and fine roots (specific rates rw and rr, respectively)
is also explicitly represented, whereas foliage maintenance
respiration and all components of plant growth respiration are
included implicitly in the definition of G. In Equation 1b, the
rate of increase of foliage nitrogen is the balance between plant
nitrogen uptake (U, kg N m−2 year −1), the allocation of nitrogen
to stemwood and fine roots (derived from the respective carbon
allocation fractions ηw and ηr and N/C ratios νw and νr, assumed constant), and the rate of return of nitrogen to the soil
in leaf fall (for simplicity we ignored nitrogen retranslocation).

Carbon production is assumed proportional to absorbed
photosynthetically active radiation (φabs, GJ m−2 year −1):
Figure 1. General structure of the stand growth model, showing dynamic variables (boxes), carbon fluxes (solid lines) and nitrogen
fluxes (broken lines).

The model equations are based on those of McMurtrie and
Wolf (1983) and McMurtrie (1985, 1991):

G = εφ abs .

In the G’DAY model (Comins and McMurtrie 1993), the
light utilization coefficient ε (kg C GJ−1 PAR) is assumed to be
proportional to the foliage N/C ratio (νf = nf / cf) when nitrogen
is limiting:

dcf
= η f G − γf c f
dt

(1a)

ε(νf) = εo

dnf
= U − (ηw νw + ηrνr)G − γfnf
dt

(1b)

= εo,

dcw
= ηw G − (γw + rw )cw
dt

(1c)

dcr
= ηrG − (γr + r r)cr .
dt

(1d)

In Equation 1, carbon production (G, kg C m −2 year −1) is

(2)

νf
,
νo

if νf < ν o

(3a)

if ν f ≥ νo

(3b

where νo is the foliage N/C ratio below which carbon production is nitrogen-limited, and εo is the maximum value of ε

obtained when nitrogen is non-limiting. Some other forms for
ε(νf) are considered by McMurtrie (1991); Equation 3 is
adopted here for simplicity.
McMurtrie and Wolf (1983) and McMurtrie (1985, 1991)

Table 1. Definition of symbols for main variables used (other symbols used in Appendices A−C are defined there). Abbreviations: units involving
m − 2 are per unit ground area; C = carbon; N = nitrogen; PAR = photosynthetically active radiation; X* = equilibrium value of X. Relevant equation
numbers are given in square brackets. (See Table 2 for parameter definitions and values.)
Symbol

Definition

Units

cf, cf*
cr, cr*
cw, cw*
G, G*
nf
t

T
U, U*
Vw*
Y
Yg
ε
φabs
λ
µi, i = w and r
νf, νf*

Foliage C [1a, B6]
Fine root C [1d, B4]
Stemwood C [1c, 8, 9]
Total plant C production (net of canopy maintenance and plant growth respiration) [2, 7, B9]
Foliage N [1b]
Stand age
Optimal rotation length [11, 12, 14a]
Rate of N uptake by trees [5, B10]
Equilibrium stemwood volume [Table 3]

Maximum mean annual stemwood volume increment [11, 13a, 14b]
Gross stemwood volume productivity at canopy closure [13b]
Canopy light utilization coefficient [2, 3]
PAR absorbed by the canopy [2, 4]
Rate of approach of G toward equilibrium [7, C5]
Sum of maintenance respiration and senescence rates [10, B5]
Foliage N/C ratio [3, B3]

kg C m − 2
kg C m − 2
kg C m − 2
kg C m − 2 year −1
kg N m − 2
year
year
kg N m − 2 year − 1
m3 ha −1
m3 ha −1 year − 1
m3 ha −1 year − 1
kg C GJ−1 PAR

GJ m − 2 year −1
year − 1
year − 1
kg N kg−1 C

STEMWOOD GROWTH AND NITROGEN SUPPLY

163

Table 2. Standard parameter values and initial values of state variables. Parameter values are either based on Comins and McMurtrie (1993) or
otherwise represent reasonable estimates. Abbreviations as in Table 1; also f = foliage, w = stemwood, r = fine roots, dm = dry matter. Relevant
equation numbers are given in square brackets.
Parameter

Definition

Standard value

fcw
k
Kf
Kr
ri, i = w, r
Uo
γi, i = f, w, r
εo
φo
ηi, i = f, w, r
νi, i = w, r
νo
ρ
σ

Fractional C content of stemwood dm [11b]
Beer’s law canopy extinction coefficient [4b]
Hyperbolic light absorption coefficient [4]
N Uptake coefficient [5]
Maintenance respiration rates [1]
Supply rate of available mineral N [5]
Senescence rates [1]
Maximum canopy light utilization coefficient [3]
Incident PAR [4]
Carbon production allocation fractions [1]
N/C Ratio [1]
Foliage N/C ratio below which carbon production is N-limited [3]
Stemwood density [11b]
Specific leaf area [4b]

0.45 kg C kg − 1 dry matter
0.5
0.139 kg C m − 2
0.05 kg C m −2
0.005, 0.000 year − 1
0.010 kg N m − 2 year −1
0.333, 0.005, 1.000 year −1
1.0 kg C GJ − 1 PAR
3.0 GJ PAR m − 2 year −1
0.2, 0.6, 0.2
0.002, 0.02 kg N kg −1 C
0.04 kg N kg − 1 C
600 kg dm m −3
10 m2 leaf kg − 1 foliage C

State variable
ci, i = f, w, r
nf

Definition
C In plant compartment i
Foliage N

Initial value
0.01, 0.00, 0.01 kg C m − 2
0.0 kg N m − 2

assumed a Beer’s law function for the relationship between φabs
and foliage carbon. To make the model analytically tractable,
we approximate Beer’s law using a hyperbolic function:
φ abs (cf) = φ o

cf
,
c f + Kf

(4a)

where φo is the amount of incident radiation at the top of the
canopy, and Kf is the value of foliage carbon at which 50% of
the incident light is absorbed. If k is the Beer’s law extinction
coefficient and σ (m2 leaf kg−1 C) is the specific leaf area (i.e.,
φabs /φo = 1 − exp(−kσcf)), then matching Equation 4a to Beer’s
law at the point where the canopy absorbs 50% of the incident
light gives the parameter correspondence:
Kf =

ln2
,
kσ

Additional simplifying assumptions
(4b)

Equations 4a and 4b give a good numerical approximation to
Beer’s law up to 50% absorption, but underestimate the
amount of light absorbed by denser canopies by up to 15%
(Figure 2a).
Following McMurtrie (1985), we assume that the rate of
nitrogen uptake by trees depends on the rate at which soil
mineral nitrogen is made available (Uo) and on root carbon:
U = Uo

cr
.
cr + K r

which 50% of the available nitrogen is taken up. Appendix A
shows how Kr may be interpreted in terms of the nitrogen
absorption capacity of tree roots, the intensity of competition
for nitrogen from other vegetation, and the rate of nitrogen loss
from the system through leaching and gaseous emissions.
Equations 1--5 describe the basic stand growth model. With
the parameter values given in Table 2, numerical simulation of
the model (data not shown) generates a realistic pattern of
stand growth in which foliage carbon, foliage nitrogen and root
carbon attain equilibrium values approximately 7--8 years after planting, corresponding to canopy closure. After canopy
closure, stemwood carbon continues to increase, taking much
longer to reach equilibrium. Some additional stand growth
characteristics are given in Table 3.

(5)

Here, Uo represents the flux of nitrogen made available to trees
from net mineralization, fertilizer application, fixation and
atmospheric deposition, and Kr is the value of root carbon at

Two additional approximations are now introduced into Equations 1--5 to allow the model to be solved analytically in terms
of the model parameters. A general expression can then be
obtained for the variation of stemwood carbon with stand age.
Approximation 1: foliage N/C ratio is a fast variable relative
to foliage C
From Equations 1a and 1b it may be shown that the effective
relaxation time for the foliage N/C ratio (νf) is much shorter
than that for cf (by a factor of 10, with the parameter values of
Table 2). By ‘‘relaxation time’’ we mean the time scale within
which a dynamic variable would return to equilibrium following a small perturbation away from equilibrium. The relaxation
time for cf is governed by the leaf senescence term in Equation
1a, and is proportional to 1/γf. The effective relaxation time for
νf is much smaller than this because of two additional loss

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DEWAR AND MCMURTRIE
Table 3. Stand growth characteristics predicted by the analytical model
using parameter values in Table 2. See Table 1 for symbol definitions.
Stand characteristic

Equation used

cf*
cr*
cw*
G*
Yg
nf*
T
U*
Vw*
Y
λ
2.3/λ(1)
µw, µr
νf*

B6
B4
9
B9
13b
nf* = νf*. cf*
12
B10
Vw* = cw*.104/ρfcw
13a
C5
C5
10, B5
B3

1

Standard value
0.57 kg C m − 2
0.19 kg C m − 2
56.9 kg C m − 2
0.95 kg C m − 2 year −1
21.1 m3 ha −1 year − 1
9 × 10−3 kg N m −2
18.8 year
7.9 × 10−3 kg N m − 2 year −1
2107 m3 ha − 1
17.7 m3 ha −1 year − 1
0.64 year − 1
3.6 year
0.01, 1.00 year −1
0.016 kg N kg − 1 C

Time at which G = 0.9G* (see Equation 7).

will lie close to its ‘‘fast-equilibrium’’ value, given by the
solution to
dν f
=0
dt

(6a)

at each value of cf. As Figure 2b shows, this is already the case
within the first year after planting. Note that the fast-equilibrium value of νf changes with cf, and is not the same as the
long-term equilibrium νf* attained at canopy closure (see
Equation B3, Appendix B and Table 3). Equation 6a can be
restated as
Figure 2. Illustration of model approximations. (a) The fraction of
incident light absorbed by the canopy (φabs /φo) as a function of foliage
carbon (cf), calculated according to Beer’s law (φabs /φo = 1 − exp(−
kσcf), solid line) and a rectangular hyperbolic approximation (Equation 4a, broken line); the two responses have been matched at 50%
absorption according to Equation 4b. (b) Changes in foliage N/C ratio
(νf = nf /cf) with time since planting, predicted from numerical simulations of Equations 1--5 (basic model, solid line) and Equations 1--6
(assuming that νf is a ‘‘fast’’ variable, broken line). (c) Carbon production (G) as a function of time since planting (t), predicted from
numerical simulations of Equations 1--5 (solid line) and from the
analytical approximation given by Equation 7 (broken line). The
values for G* and λ in Equation 7 were calculated using the analytical
expressions given in Appendices B and C. Parameter values in (a) to
(c) are given in Table 2.

terms in the dynamic equation for νf (derived from Equations
1a and 1b), associated with (i) export of nitrogen to stemwood
and fine roots, and (ii) the negative effect on νf of dilution by
foliage carbon growth. In other words, νf is a ‘‘fast’’ variable
relative to cf.
This means that, as cf increases during the canopy-building
phase, the value of νf (when measured on the time scale 1/γf)

dcf
dnf
= νf
.
dt
dt

(6b)

The advantage of this approximation is that Equation 6b enables nf to be eliminated from Equation 1, which therefore
reduces to a set of three dynamic equations for cf, cw and cr.
Approximation 2: G approaches equilibrium on a single
characteristic time scale
In numerical simulations of Equations 1--5, foliage carbon and
nitrogen attain their equilibrium values (corresponding to canopy closure) approximately 7--8 years after planting. This time
scale is determined principally by the inverse of the leaf senescence rate (e.g., the time to reach 90% canopy closure is
approximately 2.3/γf = 6.9 years). Simulations show that the
carbon production rate (G) equilibrates on a faster time scale
(about 4 years after planting), because G (Equations 2--4) is a
saturating function of cf (Figure 2a), and νf is already close to
its long-term equilibrium value νf* after about 4 years (Figure
2b, Table 3). The approach of G toward its equilibrium value,
G*, can be approximated by the following expression:
G = G∗(1 − e−λ t),

(7)

STEMWOOD GROWTH AND NITROGEN SUPPLY

165

where λ determines the rate of approach. Equation 7 is increasingly accurate the closer G is to equilibrium. The approximation consists of extending this expression back to the initial
time of planting (t = 0) when G = 0, implying in effect that the
dynamics of G can be characterized by a single time scale
(1/λ). As shown in the next section, the advantage of Equation 7 is that it allows the dynamic equation for stemwood
carbon (cw, Equation 1c) to be integrated explicitly, giving an
analytical expression for cw as a function of stand age (t),
which involves the carbon production constants G* and λ.
In Appendix B, G* is derived analytically as a function of
the model parameters (see Equation B9). With the aid of
Approximation 1 (Equation 6), λ can also be calculated explicitly in terms of the model parameters (see Equation C5, Appendix C). With standard parameter values (Table 2), we find G*
= 0.95 kg C m−2 year −1 and λ = 0.64 year −1 (so that G reaches
90% of its equilibrium value at time t = 2.3/λ = 3.6 years after
planting). As shown in Figure 2c, Equation 7 gives a reasonable approximation for the variation of carbon production with
stand age, although it slightly underestimates G, especially
during years 2--4 after planting.
General factors affecting stemwood growth
The constants G* and λ, which characterize the variation of
carbon production with stand age according to Equation 7, are
calculated as functions of the model parameters in Appendices
B and C. Here, however, we examine some general consequences of Equation 7 for the relationship of stemwood growth
to G* and λ that will be of use in interpreting the response of
stemwood growth to changes in nitrogen supply.
Substituting Equation 7 into Equation 1c allows Equation
1c to be integrated explicitly, giving the following expression
for stemwood carbon as a function of stand age (t):

λe−µw t − µwe−λt 
c w (t) = c w ∗ 1 −
,
λ − µw



(8)

in which cw* is the equilibrium value of stemwood carbon,
given by
cw∗ =

ηwG ∗
µw

(9)

and where µw is given by
µ w = γ w + rw .

(10)

Equation 8 describes a sigmoidal growth curve (Figure 3a),
with cw increasing from zero toward an asymptotic value,
given by Equation 9, that is proportional to carbon production
at canopy closure, G*. The shape of the sigmoidal curve is
determined by λ and µw; in general λ >> µw because the time
scale for carbon production to equilibrate is normally much
less than the time scales associated with stemwood maintenance respiration and senescence; in the example from Table 3,
λ = 0.64 year −1 and µw = 0.01 year −1.

Figure 3. (a) Stemwood carbon (cw) as a function of time since
planting, calculated from the analytical stemwood model (Equation 8,
solid line) and from numerical simulation of the full model (Equations
1--5, broken line). (b) Mean annual stemwood volume increment
(solid line) and current annual stemwood volume increment (broken
line) as functions of time since planting, predicted from the analytical
stemwood model (Equation 8). Standard parameter values (Tables 2
and 3) apply in both figures.

We now use Equation 8 to calculate the relationship of
maximum mean annual stemwood volume increment (Y, m3
ha −1 year −1) and optimal rotation length (T) to the carbon
production constants, G* and λ. In carbon terms, the mean
annual increment (MAI, kg C m−2 year −1) at time t is defined
by cw(t)/t. The optimal rotation length is the time when MAI
has a maximum, which coincides with the time when MAI
equals the current annual increment (dcw/dt, Equation 1c), as
illustrated in Figure 3b (in terms of equivalent volume increments). Therefore, Y (m3 ha −1 year −1) and T satisfy the conditions:
ρfcw
cw (T ) dcw
Y=
=
(t = T ),
T
dt
10 4

(11)

where ρ is the stemwood density (kg dry matter m−3), fcw is the
stemwood fractional carbon content (kg C kg−1 dry matter),
and the factor of 10−4 converts from ha −1 to m −2. Substituting
Equation 1c for dcw/dt, and using Equations 7 and 8 to express
G and cw as functions of time, the following implicit equation
for T can be derived from the second equality in Equation 11:

166

DEWAR AND MCMURTRIE

1−

λe−µw T − µwe−λ T
µwT
=
(1 − e−λT).
1 + µw T
λ − µw

(12)

This equation may be solved numerically for T for given
values of λ and µw. From Equations 8, 9, 11 and 12, Y is given
by:
Y = Yg

1 − e−λT
,
1 + µw T

(13a)

where
Y g = ηw G ∗

10 4
ρfcw

(13b)

is the rate at which carbon production is allocated to stemwood
at canopy closure, expressed on a stemwood volume basis (m3
ha −1 year −1), i.e., the gross stemwood volume production at
canopy closure. The solution for T from Equation 12 may then
be substituted into Equation 13a. With standard parameter
values (Table 2), Equations 12 and 13 predict T = 18.8 years
and Y = 17.7 m3 ha −1 year −1 (Table 3), which are typical values
for fast-growing Eucalyptus species in Australia (West and
Mattay 1993).
Although Equations 12 and 13 must be solved numerically,
useful analytical approximations to the numerical solutions
can be derived by exploiting the fact that λ is typically much
greater than µw. As shown in Appendix D, these approximations are given by the following simple expressions:
T≈√
2
λµw

(14a)

and
1

Y ≈ Yg
1+

√2µλ

,

(14b)

w

which give T ≈ 17.7 year and Y ≈ 17.9 m3 ha −1 year −1 with
standard parameter values (cf. Table 3). These results show
explicitly how the optimal rotation length (T) and maximum
MAI (Y) can be expected to vary as functions of (i) the time for
carbon production to reach equilibrium (∝ 1/λ), (ii) the combined rates of stemwood maintenance respiration and senescence (µw, Equation 10), and (iii) the gross stemwood volume
production rate at canopy closure (Yg, Equation 13b). The
value of T depends only on the first two of these factors. As
Figure 4 shows, the analytical expressions given by Equation
14 are reasonable approximations to the numerical solutions of
Equations 12 and 13.
According to Equation 14a, T increases if carbon production
equilibrates more slowly (i.e., if λ decreases), or if the specific
rates of stemwood maintenance and senescence decrease (i.e.,
if µw decreases), as shown in Figures 4a and 4b. Equation 14b
implies that Y increases if carbon production equilibrates more

Figure 4. Variation of optimal rotation length (T ) and maximum mean
annual stemwood volume increment (Y ) with (a) λ (∝ 1/equilibration
time for G), (b) µw (Equation 10), and (c) Yg (gross stemwood volume
production at canopy closure) (for Y only), predicted by numerical
solution of Equations 12 and 13a (solid lines) and by the analytical
approximation given by Equations 14a and 14b (adjacent broken
lines). In each plot, one parameter varies while the other two parameters are held fixed at their standard values (Table 3).

rapidly (i.e., if λ increases), or if stemwood maintenance/senescence rates (µw) decrease (Figures 4a and 4b); Y is also
directly proportional to Yg (Figure 4c), and hence (see Equation 13b) to carbon production at canopy closure (G*). In
quantitative terms, Y and T are relatively insensitive to changes
in λ above λ = 1.0 year −1, corresponding to equilibration times
for G shorter than about 2 years (Figure 4a). The optimal
rotation length is particularly sensitive to changes in µw below
about µw = 0.01 year −1, corresponding to specific maintenance
and senescence rates of less than 1% per year (Figure 4b).
It is worth noting that Equation 7 also enables analytical

STEMWOOD GROWTH AND NITROGEN SUPPLY

167

solutions to be found for foliage and fine root carbon as
functions of stand age. These solutions have the same form as
Equations 8 and 9 for stemwood carbon, with (i) ηw replaced
by ηf and ηr, and (ii) µw replaced by γf and µr, respectively.

Dependence of maximum MAI and optimal rotation
length on model parameters
According to the above analysis, the sensitivities of Y and T to
a given parameter of the model can be interpreted in terms of
its influence on each of the three factors λ, µw and Yg. In
Appendices B and C, analytical expressions are derived for Yg
and λ as functions of the model parameters, and the results of
a sensitivity analysis are shown in Table 4.
Maximum MAI (Y)
The parameters to which Y is most sensitive are the stemwood
density (ρ), stemwood carbon fraction (fcw) and nitrogen supply rate (Uo). The effect on Y of increases in ρ and fcw is
straightforwardly explained in terms of their effect on Yg
(Equation 13b).
Of major interest to plantation managers is the dependence
of Y on Uo. The sensitivity of stemwood growth to changes in
Uo is therefore shown more generally in Figure 5. Increasing
Uo by 10% leads to a 9% increase in Y, mainly due to its effect
on Yg through G* (Table 4, Figure 5a). Over the range of Uo
from 50 to 150 kg N ha −1 year −1 there is an approximately
linear increase in Y with Uo from 8 to 25 m3 ha −1 year −1,
representing an average of 0.17 m3 stemwood volume gained
Table 4. Predicted percentage changes (∆) in Yg (Equation 13b), λ
(Equation C5), Y (Equation 13a) and T (Equation 12) due to a 10%
increase in individual model parameters from the reference case described by Tables 2 and 3. See Tables 1 and 2 for symbol definitions.
Sensitivity to the three allocation parameters was assessed by varying
ηf and ηr with ηw = 1 − ηf − ηr as indicated below. Note that ∆T ≈
−∆λ/2 in accordance with Equation 14a.
Parameter

∆Yg

∆λ

∆Y

∆T

fcw
Kf
Kr
Uo
γf
εo
φo
ηf(1)
ηr(2)
µw
µr
νw
νr
νo
ρ

−9.0
−0.7
−1.9
+8.9
−0.7
+3.3
+3.3
−5.5
−5.4
0
−1.9
−1.3
−4.3
−3.3
−9.0

0
−0.8
−2.9
+1.4
+3.0
+2.0
+2.0
−1.3
+3.1
0
+2.4
+0.1
+0.3
−2.0
0

−9.0
−0.7
−2.1
+9.1
−0.4
+3.4
+3.4
−5.6
−5.1
−0.8
−1.7
−1.3
−4.2
−3.5
−9.0

0
+0.4
+1.6
−0.8
−1.6
−1.0
−1.0
+0.7
-1.6
−4.3
−1.3
0
−0.2
+1.1
0

1
2

Allocation fractions {ηf, ηw, ηr}= {0.22, 0.58, 0.20}
Allocation fractions {ηf, ηw, ηr}= {0.20, 0.58, 0.22}

Figure 5. The predicted dependence of (a) Yg (gross stemwood volume
productivity at canopy closure, solid line), and λ (∝ 1/equilibration
time for G, broken line), and (b) Y (solid line) and T (broken line) on
nitrogen supply rate (Uo), with all other parameters fixed at their
standard values (Table 2).

for each extra kg of nitrogen supplied (Figure 5b).
Maximum MAI is moderately sensitive to the carbon production parameters (εo, φo, νo), allocation fractions (ηf, ηr) and
root N/C ratio (νr).
Optimal rotation length (T)
The optimal rotation length is predicted to be generally much
less sensitive than maximum MAI to changes in parameter
values (Table 4), including Uo. Over the range of Uo from 50
to 150 kg N ha −1 year −1, T decreases by 14% from 21 to 18
years (Figure 5b), as a result of a 26% increase in λ from 0.53
to 0.67 year −1 (Figures 4a and 5a). The value of T is moderately
sensitive to a change in µw (Table 4), the sum of stemwood
maintenance respiration and senescence rates, especially at
small values of µw (Figure 4b).

Discussion
The relationship between stemwood growth and nitrogen
supply
Equations 8 and 9 describe a simple, process-based model of
stemwood growth, showing explicitly the main factors involved. The variation of stemwood carbon with stand age
[cw(t)] is determined by: (i) ηw, the stemwood allocation fraction, (ii) G*, the equilibrium value of carbon production (G),

168

DEWAR AND MCMURTRIE

(iii) λ, the rate at which G approaches G*, and (iv) µw, the
combined specific rate of stemwood maintenance respiration
and senescence. The simple expressions for the optimal rotation length (T) and maximum MAI (Y), given by Equation 14,
are of particular relevance for forest management.
The effect of nitrogen supply (Uo) on maximum MAI and
optimal rotation length occurs through its effect on the carbon
production constants, G* and λ, which we were able to calculate as analytical functions of the model parameters. The effect
of Uo on maximum MAI (Y) occurs mainly through its effect
on G*, the equilibrium carbon production at canopy closure.
The results in Figures 4 and 5 show that G* (and therefore Y)
responds linearly to Uo. This response is related to our assumption that the light utilization coefficient (ε) is proportional to
the foliage N/C ratio when nitrogen is limiting (Equation 3a),
and the fact that nitrogen remained limiting (i.e., νf < νo) over
the range of Uo considered. The effect of Uo on the optimal
rotation length (T) occurs only through its effect on λ, the rate
at which carbon production equilibrates. It is worth noting
from Appendices B and C that G* and λ are not independent,
and that an increase in equilibrium carbon production is associated with a decrease in the time to reach that equilibrium
(Figure 5a).
The value of T is much less sensitive than Y to a change in
nitrogen supply, Uo (Figure 5b). As noted earlier, this result
follows from the prediction that the optimal rotation length
responds to nitrogen supply only via changes in the equilibration rate for carbon production (λ, see Equation 14a), and that
λ is relatively insensitive to Uo (Figure 5a). With reference to
Equation 14a, T may be expected to decrease more rapidly
with increasing Uo if stemwood maintenance respiration (µw)
increases with Uo due to changes in tissue nitrogen concentration (cf. Ryan 1991, 1995). As the model stands, however, the
predicted behavior is broadly consistent with general yield
tables for Pinus radiata D. Don in South Australia (see Lewis
et al. 1976, their Figure V.2) and with some examples of
Eucalyptus yield tables from several countries given in FAO
(1979, Chapter 11), which show that the effect of changes in
site quality on the age of maximum MAI is much smaller than
the effect on the value of the maximum MAI itself.
The analytical expressions for Y and T (Equation 14) become invalid at very low nitrogen supply rates, when λ becomes so small that the condition λ >> µw no longer holds. For
the parameter values in Table 2, λ = µw when Uo ≈ 15 kg N ha −1
year −1, at which point the stemwood growth curve is no longer
sigmoidal, and the concepts of maximum MAI and optimal
rotation length break down.
Critical assumptions of the model
There are four key simplifying assumptions in our analysis that
should be borne in mind when using the model as the basis for
specific applications (e.g., Dewar and McMurtrie 1996, following article). First, the validity of Equations 8 and 14 depends on the assumption that the approach of G toward its
equilibrium value can be characterized by a single time scale
(1/λ), as expressed by Equation 7. For the model of McMurtrie
and Wolf (1983) and McMurtrie (1991) that we started with

(Equations 1--5), this assumption appears reasonable (Figure
2c). If the behavior of G predicted by more complex processbased models can also be characterized this way, then Equations 8 and 14 provide a useful general framework with which
to interpret the output of these models.
Second, the present model assumes that the decrease in
stemwood volume increment in old stands (Figure 3b) is
caused by increasing stemwood maintenance respiration and
senescence (described by the parameter µw), with foliage
biomass (cf) and carbon production (G, Figure 2c) remaining
constant after canopy closure. This interpretation of the observed decline in the aboveground productivity of older stands,
though widely accepted for many years, has not been rigorously tested and has been questioned (Ryan and Waring 1992).
Other hypotheses have been proposed (e.g., Murty et al. 1996),
including reduced stomatal conductance (thereby reducing ε)
and decreased nutrient availability (Uo). It is clearly important
to examine the implications of these alternative hypotheses for
predictions of stemwood growth, maximum MAI and optimal
rotation lengths.
Third, it is assumed that the allocation fractions are fixed
constants during stand development. This is clearly not the
case in real stands, particularly during the years before canopy
closure. Numerous studies of coniferous species have shown
that partitioning of aboveground dry matter production to
wood tends to increase at the expense of foliage as stands
approach canopy closure, but is approximately constant after
canopy closure (see reviews by Cannell 1985, Gower et al.
1994). There are few data on changes in allocation between
roots and foliage for different-aged stands of the same species.
However, fertilization studies and comparisons between trees
grown on fertile and nutrient-poor sites (Gower et al. 1994)
suggest that allocation to root growth may increase if nutrient
availability declines with stand age, for example due to immobilization of nutrients by woody litter in old stands (Pearson et
al. 1987). There are several approaches to modeling allocation
dynamically (Cannell and Dewar 1994). However, it is unlikely that dynamic allocation could be incorporated into the
present analytical framework, and its implications for stemwood growth would probably require numerical simulation.
The assumption of constant allocation fractions provides a
useful first approximation, but its limitations are acknowledged and will be critically examined elsewhere.
Fourth, the model ignores the effects of litter and soil organic matter decomposition on the supply of mineral nitrogen,
assuming instead that Uo is a constant, externally imposed
parameter. Nevertheless, this may be a reasonable assumption
to make for managed plantation systems. Over successive
rotations, however, Uo may change as a result of nitrogen
losses from the system associated with harvesting and fire
management regimes. The model therefore provides a useful
basis for examining the long-term sustainable forest productivity, and its relationship to nutrient management and climate
change. Such an application is described in a companion paper
(Dewar and McMurtrie 1996).

STEMWOOD GROWTH AND NITROGEN SUPPLY
Acknowledgments
This work was supported by the UK NERC through its TIGER
(Terrestrial Initiative in Global Environmental Research) programme
(grant GST/91/15), the Australia-New Zealand-UK Tripartite Agreement on Climate Change Research, the NGAC Dedicated Greenhouse
Research Grants Scheme and the Australian Research Council.]

169

Schimel, D.S., W.J. Parton, T.G.F. Kittel, D.S. Ojima and C.V. Cole.
1990. Grassland biogeochemistry: links to atmospheric processes.
Clim. Change 17:13--25.
West, P.W. and J.P. Mattay. 1993. Yield prediction models and comparative growth rates for six Eucalyptus species. Aust. For. 56:211-225.

References

Appendix A. Nitrogen uptake by trees (U, Equation 5)

Cannell, M.G.R. 1985. Dry matter partitioning in tree crops. In Attributes of Trees as Crop Plants. Eds. M.G.R. Cannell and J.E. Jackson.
Inst. Terrestrial Ecology, Huntingdon, UK, pp 160--193.
Cannell, M.G.R. and R.C. Dewar. 1994. Carbon allocation in trees: a
review of concepts for modelling. Adv. Ecol. Res. 25:59--104.
Comins, H.N. and R.E. McMurtrie. 1993. Long-term biotic response
of nutrient-limited forest ecosystems to CO2-enrichment: equilibrium behaviour of integrated plant-soil models. Ecol. Appl. 3:666-681.
Dewar, R.C. and R.E. McMurtrie. 1996. Sustainable stemwood yield
in relation to the nitrogen balance of forest plantations: a model
analysis. Tree Physiol. 16:173--182.
Dixon, R.K., R.S. Meldahl, G.A. Ruark and W.G. Warren. 1990.
Process modeling of forest growth responses to environmental
stress. Timber Press, Portland, Oregon, USA. 441 p.
FAO. 1979. Eucalypts for Planting. FAO Forestry Series No. 11. Food
and Agriculture Organization of the United Nations, Rome, 677 p.
Gower, S.T., H.L. Gholz, K. Nakane and V.C. Baldwin. 1994. Production and carbon allocation patterns of pine forests. Ecol. Bull.
43:115--135.
Kirschbaum, M.U.F., D.A. King, H.N. Comins, R.E. McMurtrie, B.E.
Medlyn, S. Pongracic, D. Murty, H. Keith, R.J. Raison, P.K. Khanna
and D.W. Sheriff. 1994. Modelling forest response to increasing
CO2 concentration under nutrient-limited conditions. Plant Cell
Environ. 17:1081--1099.
Lewis, N.B., A. Keeves and J.W. Leech. 1976. Yield regulation in
South Australian Pinus radiata plantations. Bull. No. 23. Woods
and Forests Dept., SA, Australia.
McMurtrie, R.E. 1985. Forest productivity in relation to carbon partitioning and nutrient cycling: a mathematical model. In Attributes of
Trees as Crop Plants. Eds. M.G.R. Cannell and J.E. Jackson. Inst.
Terrestrial Ecology, Huntingdon, UK, pp 194--207.
McMurtrie, R.E. 1991. Relationship of forest productivity to nutrient
and carbon supply----a modeling analysis. Tree Physiol. 9:87--99.
McMurtrie, R.E. and L. Wolf. 1983. Above- and below-ground growth
of forest stands: a carbon budget model. Ann. Bot. 52:437--448.
Murty, D., R.E. McMurtrie and M.G. Ryan. 1995. Declining forest
productivity in ageing forest stands----a modeling analysis of alternative hypotheses. Tree Physiol. 16:187--200.
Parton, W.J., D.S. Schimel, C.V. Cole and D.S. Ojima. 1987. Analysis
of factors controlling soil organic matter levels in Great Plains
grasslands. Soil Sc. Soc. Am. J. 51:1173--1179.
Pearson, J.A., D.H. Knight and T.J. Fahey. 1987. Biomass and nutrient
accumulation during stand development in Wyoming lodgepole
pine forests. Ecology 68:1966--1973.
Ryan, M.G. 1991. The effects of climate change on plant respiration.
Ecol. Appl. 1:157--167.
Ryan, M.G. 1995. Foliar maintenance respiration of subalpine and
boreal trees and shrubs in relation to nitrogen content. Plant Cell
Environ. 18:765--772.
Ryan, M.G. and R.H. Waring. 1992. Maintenance respiration and
stand development in a subalpine lodgepole pine forest. Ecology
73:2100--2108.

Let Uo be the rate of nitrogen input to the soil mineral nitrogen
pool (Nmin ) due to net mineralization, fertilizer application,
atmospheric deposition and fixation. The rate of change of the
soil mineral nitrogen pool is therefore:
dN min
= Uo − U − U v − L,
dt

(A1)

where U is the rate of nitrogen uptake by trees, Uv is the rate
of nitrogen uptake by other vegetation and micro-organisms,
and L is the external loss of nitrogen from the ecosystem due
to leaching and gaseous emission losses. The rate U is assumed
to be proportional to the amount of root present (cr) and to Nmin:
U = σcrNmin ,

(A2)

where σ describes the nitrogen uptake capacity of tree roots.
Uv and L are also assumed to be proportional to Nmin :
U v = σv Nmin

(A3)

L = ln Nmin ,

(A4)

where σv and ln are rate constants, so that Equation A1 becomes:
dN min
= Uo − (σcr + σv + ln )Nmin .
dt

(A5)

Therefore, for given values of Uo and cr, Nmin will tend toward
the equilibrium value
N min ∗ =

Uo
.
σc r + σ v + l n

(A6)

If the effective relaxation time 1/(σ cr + σv + ln) for Nmin in
Equation A5 is sufficiently small compared to the time scales
on which changes in Uo and cr occur, then Nmin can be taken as
a ‘‘fast’’ variable relative to Uo and cr. The uptake rate U can
then be approximated by replacing Nmin in Equation A2 with
its fast-equilibrium value Nmin*. Substituting Equation A6 into
Equation A2 and dividing by σ gives the rate of nitrogen
uptake by roots as:
U=

U oc r
.
cr + (σv + ln)/σ

(A7)

This is equivalent to Equation 5, in which the constant Kr is

170

DEWAR AND MCMURTRIE

where the three dimensionless parameter combinations α, β
and δ are given by:

given by
Kr =

σv + l n
σ

(A8)

α = (ηwν w + ηrν r)

and so reflects the nitrogen absorption capacity of tree roots,
the intensity of competition for nitrogen from other vegetation,
and the intrinsic loss rate of nitrogen out of the system.

β=
δ=

Appendix B. Derivation of G* (equilibrium carbon production) as a function of the model parameters

γ f cf ∗
.
ηf

(B1)

Substituting the expressions for ε(νf) and φabs (cf), given by
Equations 3a and 4a, into Equation 2 for G, this relationship
can be re-written as:
ε oφ ο
νo

ν f∗

γf
1
=
.
∗
c f + K f ηf

(B2)

The value of νf∗, the foliage N/C ratio at canopy closure, can
be calculated by combining Equations 1a and 1b to give:
ν f∗ =

(ηw νw + ηrνr)
U∗
−
,
∗
γ f cf
ηf

(B3)

ν o (K f γf )2

U∗ =

(B8b)
(B8c)

.

K f γf
x.
ηf

(B9)

U ox
.
x+β

(B10)

Appendix C. Derivation of λ (rate of equilibration of
carbon production) as a function of the model parameters
Equations 1a, 1b and 1d governing the dynamics of cf, nf and
cr may be replaced by an equivalent system of equations for G,
nf and U using the relationships given by Equations 2, 3a, 4a
and 5. Using approximation 1 (foliage N/C is a ‘‘fast’’ variable)
to eliminate nf, this system can be reduced to the following two
coupled dynamic equations for G and U alone:

ηf G 2
dG 
= σ c −

dt
K
[U
−
(η
ν
+
η
ν
)
G]
f
w w
r r



where
(B5)

η f ε o φο U o

(B8a)

Equation B9 may be inserted into Equation 15b to obtain Yg.
The corresponding expression for U* can be obtained by
combining Equations 5, B4 and B6, to give:

(B4)

µ r = γ r + rr .

η fKrµ r
η rKfγ f

G∗ =

where U* is the nitrogen uptake rate at canopy closure. From
Equation 5, U* depends on the equilibrium root carbon, cr*,
which, from Equations 1a and 1d, is simply related to equilibrium foliage carbon by:
γf η r
c f ∗,
c r∗ =
µr η f

νo K f γ f

Finally, Equation B6 may be substituted into Equation B1 to
give

From Equation 1a, G* is related to the amount of foliage at
canopy closure, cf*, by
G∗ =

ε o φο


ηf 2 
G
U − τG −
K
fσ c



(C1a)

dU U o − U
=
[ηrG(U o − U ) − KrµrU],
K rUo
dt

(C1b)

where the parameter combinations σc and τ are given by

Combining Equation 5 with Equations B2--B4 then leads to a
quadratic equation for cf*, the solution to which can be written
as:

σc =

ε o φο

(C2a)

ν oK f

and
cf∗ = K fx.

(B6)
τ = ηw ν w + η r ν r +

The dimensionless quantity x is given by:
x=

1+α+β
2





4[δ − β( 1 + α)]

√
1+
(1 + α + β)
2



− 1 ,


(B7)

γf
.
σc

(C2b)

Near canopy closure, Equations C1 can be linearized about the
equilbrium values G = G* and U = U* (calculated in Appendix

STEMWOOD GROWTH AND NITROGEN SUPPLY

B). Mathematically, the parameter λ in Equation 7 is then equal
to −λ1, where λ1 is the least negative eigenvalue of the dynamics of the linearized G--U system.
Standard eigenvalue analysis gives:
λ1 =

−µr − A + √

(µr − A)2 + 4B
2

(C3a)

by exploiting the fact that λ is typically much greater than µw.
Substituting the values of λ, µw and T from Table 3 into the
left-hand side of Equation 12, the term µwexp(−λT) ≈ 6 × 10 −8
year −1 is negligible compared to the term λexp(−µwT) ≈ 0.5
year −1. In addition, on the right-hand side of Equation 12, the
term exp(−λT) ≈ 6 × 10 −6 is negligible compared to 1. Therefore, Equation 12 may be approximated by:

and
−µr − A − √

(µr − A) + 4B
2

1−

2

λ2 =

γ f(2x + α + 1)(x + β)
δ − α(x + β)

(C4a)

and
γfµrβδ
,
(x + β)(δ − α(x + β))

(C4b)

where x, α, β and δ are given in Appendix B (Equations
B7--B8). The least negative eigenvalue is λ1 (Equation C3a),
and so, setting λ = −λ1, we have:
λ=

µr + A − √

(µr − A)2 + 4B
.
2

(D1)

Noting that µwT ≈ 0.19, this equation can be simplified further
by expanding each side as a power series up to second order in
µwT, to give:

1−

B=

µwT
λe−µw T
=
.
λ − µw 1 + µw T

(C3b)

in which
A=

171

(C5)

With the standard parameter values in Table 2, we find λ1 =
−0.640 and λ2 = − 1.512. Although λ2 only differs from λ1 by a
factor of 2.3, the use of the single exponential term exp(−λt) in
Equation 7 appears to be justified by the close agreement
between the calculated stemwood growth curve based on
Equation 7 (i.e., Equation 8) and that based on the full numerical simulation of Equations 1--5, as shown in Figure 3.
Appendix D. Derivation of analytical expressions for T
and Y (Equation 14)
The implicit equation for T (Equation 12) may be simplified

λ 
(µw T)2 
2
1 − µwT + 2  = µw T − (µw T) .
λ − µw 


(D2)

This is a quadratic equation for µwT, which may be solved to
give the solution for T as:

T=


1

λ − 2µw 


2λ
√


− 3 − 1 .
µ


(D3)

w

Because λ >> µw, we have λ − 2µw ≈ λ and the expression in
square brackets may be approximated by √

2λ/µw , giving the
approximation:
T≈√
2 .
λµw

(D4)

Substituting this result into the equation for Y (Equation 13a)
(in which the term exp(−λT) can similarly be neglected) then
gives:
Yg

Y≈
1+

√2µλ


.
w

(D5)