Directory UMM :Data Elmu:jurnal:T:Tree Physiology:Vol16.1996:
Tree Physiology 16, 173--182
© 1996 Heron Publishing----Victoria, Canada
Sustainable stemwood yield in relation to the nitrogen balance of forest
plantations: a model analysis
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 used an existing analytical model of stemwood growth in relation to nitrogen supply, which we describe
in an accompanying paper, to examine the long-term effects of
harvesting and fire on tree growth. Our analysis takes into
account the balance between nitrogen additions from deposition, fixation, and fertilizer applications, and nitrogen losses
from stemwood harvesting, regeneration burning, leaching and
gaseous emissions. Using a plausible set of parameter values
for Eucalyptus, we conclude that nitrogen loss through fire is
the main factor limiting sustainable yield, defined as the maximum mean annual stemwood volume increment obtained in
the steady state, if management practices are continued indefinitely. The sustainable yield is 30 m3 ha −1 year −1 with harvesting only, 15 m3 ha −1 year −1 with harvesting and regeneration
burning, and 13 m3 ha −1 year −1 with harvesting, fire, leaching
and gaseous emissions combined.
Our approach uses a simple graphical analysis that provides
a useful framework for examining the factors affecting sustainable yield. The graphical analysis is also useful for extending
the application of the present model to the effects of climate
change on sustainable yield, or for interpreting the behavior of
other models of sustainable forest growth.
Keywords: fire, growth, harvesting, mean annual increment,
sustainability.
Introduction
In managed forests, the balance between inputs and outputs of
nutrients over successive rotations is a major ecological constraint on the sustainability of site productivity. As a major
nutrient for plant growth, nitrogen has been extensively studied in this context. Inputs of nitrogen may occur through wet
and dry atmospheric deposition, symbiotic and non-symbiotic
fixation, and fertilizer applications; nitrogen may be lost
through leaching and gaseous emissions, removal of harvested
biomass (Turner 1981, Raison et al. 1982, Stewart et al. 1985)
and fire (O’Connell et al. 1981, Grove et al. 1986, Walker et al.
1986, Raison et al. 1990, 1993). In Australian forests, fire is
widely used to reduce accumulated fuel (and thus the risk of
uncontrolled fires), to clear slash debris and to regenerate
seedlings.
Understandably, most studies of the effects of fire on nitro-
gen budgets and cycling have been limited to time scales of
months to a few years (e.g., Grove et al. 1986, Raison et al.
1990). The immediate effects, such as release of mineral nitrogen through oxidation of organic matter, are usually beneficial
to plant nutrition, but they occur in association with direct
nitrogen losses to the atmosphere from volatilization and ash
transport, in addition to indirect losses such as leaching of
released nitrogen or erosion of surface soil and ash. Little is
known of the cumulative effect of fire and harvesting on forest
nutrition and growth over time scales spanning several rotations. As a result, models of the interactions between forest
growth, fire and harvesting regimes, and nutrient balance provide valuable tools for extrapolating over these longer time
scales.
In the preceding article (Dewar and McMurtrie 1996), we
described a simple process-based model of stemwood growth
in relation to plant-available nitrogen supply (Uo, kg N m −2
year −1); the model predicts the variation of stemwood volume
(or carbon equivalent) with stand age for an undisturbed stand.
The supply rate, Uo, refers to external inputs as described
above, in addition to nitrogen supplied through mineralization
of organic matter. In this paper, we combine that model with
simple assumptions about external nitrogen inputs and losses,
focusing on removals in harvesting, regeneration burning,
leaching and gaseous emissions. We use the combined model
to define the term ‘‘sustainable yield’’ in an unambiguous way,
and to quantify its dependence on the individual contributions
to nitrogen removal. Our approach exploits a simple graphical
analysis, which provides a useful general framework for examining the factors affecting sustainable yield.
The model
Stemwood growth in relation to nitrogen supply
Dewar and McMurtrie (1996, this issue) (hereafter referred to
as DM) presented a model of stemwood growth in relation to
nitrogen supply, and applied the model to predict the variation
of stemwood volume (or carbon equivalent) with stand age for
an unharvested stand. A complete mathematical description of
the model is given in DM, and only a brief summary is given
here. Symbols used, their definitions and units are listed in
Table 1. In the model, which is based on the work of McMur-
174
DEWAR AND MCMURTRIE
Table 1. Symbol definitions. Abbreviations: dm = dry matter;m −2 = per unit ground area (N fluxes are re-expressed on a per hectare basis in Figures
3b--5); X* = equilibrium value of X at canopy closure. Subscripts: f = foliage; w = stemwood; r = fine roots; l = aboveground litter; fl = foliage
and aboveground litter.
Symbol
Definition
Units
A
cf, cf*
cl, cl*
cw, cr
cw(n)
fh
fcw
fiburn ; i = f, l, w
Ffl, Fw
G, G*
H
Kr
L
M
MAI
nf
t
T
Tn
U
Uo
Y
Ys
γf, γl
ηi, i = f, w, r
λ
µi, i = w, r
νf, νf*, νl, νw
Ω
ρ
ξ
ψ
Addition rate of N from fertilizer inputs, atmospheric deposition and fixation
Foliage C
Aboveground litter C
Stemwood and fine root C
Value of cw in the nth rotation (n = 1, 2, ...)
Fraction of harvested stemwood removed from site
Fractional C content of stemwood dm
Fraction of N lost in regeneration burn
Amount of N lost in regeneration burn
Total C production by trees (net of canopy maintenance and plant growth respiration)
Amount of N removed by stemwood harvesting
Fine root N uptake coefficient
Rate of N loss from leaching and gaseous emissions
Net mineralization rate
Mean annual stemwood volume increment
Foliage N
Stand age
Optimal rotation length
Value of T in the nth rotation (n = 1, 2, ...)
Rate of N uptake by trees
Supply rate of plant-available N
Maximum MAI in a given rotation
Sustainable yield (steady-state value of Y)
Specific senescence rate
Carbon production allocation fraction
Rate of approach of G towards equilibrium
Sum of specific maintenance respiration and senescence rates
N/C Ratio
Rotation-averaged rate of N loss from burning foliage and litter, leaching and gaseous emissions
Stemwood density
Fraction of available N not taken up by trees that is lost by leaching and gaseous emissions
Ratio of aboveground litter N/C to live foliage N/C
trie and Wolf (1983) and McMurtrie (1985, 1991), four dynamic variables are represented: stemwood carbon (cw, kg C
m−2, which includes stems, branches and coarse roots), foliage
carbon (cf) and nitrogen (nf), and fine root carbon (cr). Stemwood and fine roots are assumed to have fixed N/C ratios.
Total carbon production by trees (G, kg C m −2 year−1) is
given by a nitrogen-dependent light utilization coefficient multiplied by absorbed photosynthetic radiation, and is allocated
to each of the three tree components in fixed proportions.
Maintenance respiration in stemwood and fine roots is explicitly represented, as is senescence of all tree components; foliage maintenance respiration and all components of plant
growth respiration are included implicitly in the definition of
G. Nitrogen uptake by trees (U, kg N m−2 year −1) depends on
root mass and on the rate at which plant-available nitrogen is
supplied (Uo):
U = Uo
cr
,
cr + K r
(1)
kg N m −2 year −1
kg C m −2
kg C m −2
kg C m −2
kg C m −2
kg C kg−1 dm
kg N m −2
kg C m −2 year −1
kg N m −2
kg C m −2
kg N m −2 year −1
kg N m −2 year −1
m3 ha−1 year −1
kg N m −2
year
year
year
kg N m −2 year −1
kg N m −2 year −1
m3 ha−1 year −1
m3 ha−1 year −1
year −1
year −1
year −1
kg N kg−1 C
kg N m −2 year −1
kg dm m −3
where Kr is the value of root carbon at which 50% of the
available nitrogen is taken up. As discussed by DM (their
Appendix A), the constant 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. The rate Uo represents nitrogen supplied by net
mineralization (M) within the system, plus additions of nitrogen from outside the system (A) due to fertilizer inputs, atmospheric deposition, and fixation:
U O = M + A.
(2)
In DM, Uo is taken to be a constant parameter throughout the
life of the stand; for constant external inputs (A), this implies
that M is also constant so that the model ignores variations in
M with stand age associated with the dynamics of litter and soil
organic matter decomposition. In this study, in order to exploit
the results of DM, we will also assume that Uo is constant
within a given rotation. However, in the sustainability analysis
MODEL OF SUSTAINABLE YIELD
presented below, we will examine the long-term consequence
of changes in M (and therefore Uo) over successive rotations.
By introducing some simplifying approximations into the
basic growth model outlined above, DM derived various analytical results describing the relationship of stemwood growth
to nitrogen supply (Uo). For example, the mean annual stemwood volume increment (MAI, m3 ha −1 year −1) at stand age t
(years) is defined by:
MAI(t) =
10 4 cw(t)
,
ρfcw t
(3a)
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 104 converts from m −2 to ha −1. Figure 1 shows
the variation of stemwood carbon [cw(t)] and mean annual
increment [MAI(t)] with stand age, predicted analytically by
the model when Uo = 100 kg N ha −1 year −1, with other parameter values given in DM. Stemwood carbon follows a
realistic sigmoidal growth pattern, so that MAI first increases
to a maximum rate (here, 18 m3 ha −1 year −1 at t ≈ 19 years) and
then gradually declines as the stand ages. In the model, this
decline is caused by increasing stemwood maintenance respiration and senescence, with foliage biomass and carbon production remaining fixed after canopy closure. Alternative
hypotheses for the mechanisms underlying the decline in productivity of old stands are examined by Murty et al. (1996, this
issue).
To optimize the average volume output of the stand, the
forest should be harvested at the age of maximum MAI. Although economic factors may be more important to a forest
manager deciding when to harvest, the concept of maximum
MAI continues to be widely used as a characteristic of stemwood growth in forestry yield tables. In DM it was shown that
the dependence of maximum MAI (Y) on nitrogen supply
(U o) is given approximately by:
175
Y(U o) = maximum MAI
≈
10 4
ηw G∗(Uo )
ρfcw
1
1+
√
2 µw
λ(Uo )
,
(3b)
in which ηw is the fraction of total carbon production (G)
allocated to stemwood, G* is the equilibrium value of G at
canopy closure, λ (year −1) describes the rate at which G approaches G*, and µw (year −1) is the combined specific rate of
stemwood maintenance respiration and senescence.
The corresponding optimal rotation length (T = stand age at
maximum MAI) is given approximately by:
2
T(U o) ≈ √
.
λ(U o)µw
(4)
According to these equations, Y and T are functions of Uo
through the quantities G* and λ (which also depend on all the
other model parameters). Dewar and McMurtrie (1996) obtained analytical expressions for the functions G*(Uo) and
λ(Uo) (see their Appendices B and C); their results show that λ
(and therefore T) is relatively insensitive to Uo, but that G*
(and therefore Y) varies approximately linearly with Uo.
Nitrogen losses due to harvesting, regeneration burn and
‘‘leakage’’
We now consider the nitrogen losses that occur over one
rotation of a managed plantation that is harvested at maximum
mean annual increment (i.e., when stand age = T), then subjected to a regeneration burn. The amount of nitrogen removed
in harvesting (H, kg N ha −1) is given by:
H = cw (T)νw fh ,
(5)
where cw(T) is the amount of stemwood carbon at the end of
the rotation, νw is the (constant) stemwood N/C ratio, and fh is
the fraction of stemwood removed from the site. Dewar and
McMurtrie (1996) calculated cw(T) for a given value of nitrogen supply (Uo) during the rotation (their Equation 8).
During the regeneration burn, nitrogen is lost to the atmosphere by the combustion of standing foliage, aboveground
litter, and stemwood slash that has not been removed from the
site. In each case, we assume that the amount of nitrogen lost
is proportional to the total fuel N content. Therefore, the
nitrogen lost from burning foliage and aboveground litter (Ffl,
kg N m −2) is:
Ffl = cf(T )ν f(T ) ffburn + cl(T )νl(T ) flburn ,
Figure 1. Stemwood carbon [cw(t), solid line] and mean annual stemwood volume increment [MAI(t), broken line] as a function of stand
age (t), as predicted by the analytical stemwood growth model of
Dewar and McMurtrie (1996), for a nitrogen supply rate Uo = 100 kg
N ha −1 year −1 (with other parameter values as given in their paper).
(6a)
where cf(T) and cl(T) are the amounts of foliage carbon and
aboveground litter carbon at the end of the rotation, νf (T) and
νl(T) are the corresponding N/C ratios, and ffburn and flburn are
the fractions of nitrogen lost when foliage and aboveground
litter are burnt. Similarly, the amount of nitrogen lost from
burning stemwood slash (Fw, kg N m −2) is:
176
DEWAR AND MCMURTRIE
Fw = cw (T )νw (1 − fh )fwburn ,
(6b)
where fwburn is the corresponding fraction of nitrogen lost. Like
H (Equation 5), Fw can be calculated from the results of DM.
In order to calculate Ffl, we make the following three simplifying assumptions. First, as the predictions of the growth model
show (DM), it is reasonable to assume that foliage carbon has
reached its equilibrium value, cf*, corresponding to canopy
closure, before the end of the rotation, so that:
cf(T ) = cf∗ =
η fG∗
,
γf
(7a)
where ηf is the fraction of carbon production allocated to
foliage, and γf is the foliage senescence rate. Second, we make
a similar assumption for aboveground leaf litter:
cl(T ) = cl∗ =
cf∗γ f
,
γl
(7b)
where cl* is the equilibrium value of litter carbon, and γl is the
leaf litter turnover rate. (In the present study, we ignore nitrogen losses from burning woody litter and understorey vegetation.) Third, we assume that the aboveground litter N/C ratio
is proportional to the foliage N/C ratio, and that the foliage N/C
ratio is also at its equilibrium value, νf*, at the end of the
rotation (as suggested by the results of DM), so that:
ν l(T ) = ψν f(T ) = ψνf∗,
(7c)
where ψ is a constant. Substituting Equations 7 into Equation
6a then gives:
ffburn flburn ψ
Ffl =
+
ηf ν f ∗ G ∗ ,
γl
γf
(8)
in which νf* and G* may be calculated for a given value of Uo
using the analytical results in DM (their Appendix B).
Throughout the rotation, a constant fraction ξ of the nitrogen
supply not taken up by trees is lost from the system through
‘‘leakage,’’ i.e., leaching and gaseous emissions, the remaining
Figure 2. Ecosystem inputs and outputs of nitrogen in a forest plantation, included in the present study.
fraction being taken up by other vegetation. Therefore, the
leakage rate (L, kg N m −2 year −1) is:
L = ξ (Uo − U) = ξ Uo
Kr
,
cr + K r
(9)
where we have used Equation 1. The value of L is greatest
during the early growth years when root biomass is low. In the
notation of DM (their Appendix A), ξ is given by ln /(ln + σv ),
where ln and σv are parameters describing the intrinsic leakage
rate and the intensity of competition for nitrogen from other
vegetation, respectively. The value of ξ is generally small in
the dry conditions that exist in many Australian forests.
Equation 3b gives the stemwood yield at harvest (of which
a fraction fh is removed from site), and Equations 5--9 describe
the nitrogen losses from the system, during a single rotation for
a given constant nitrogen supply rate (Uo). In the following
section we will consider the long-term consequences of these
nitrogen losses for stemwood yield, due to changes in Uo over
successive rotations.
Sustainable yield
Definition
We adopt a definition of sustainable yield (Ys) that is based
solely on the ecophysiological constraints on the long-term
productivity of a given forest stand. Our definition does not
address issues of conservation of biological and genetic diversity (e.g., Franklin et al. 1989), or wider socioeconomic aspects of sustainability, such as energy costs associated with
fertilizer inputs, harvest operations, transport and processing
of wood products.
Figure 2 depicts the overall nitrogen balance of the system.
External inputs of nitrogen from fertilizer additions, fixation
and atmospheric deposition are assumed to occur at a constant
rate (A), and the system is subjected to a repeated management
cycle consisting of harvesting at maximum MAI, followed by
regeneration burn and re-planting. Under these conditions, the
supply of available nitrogen (Uo, Equation 2) will in general
vary from one rotation to the next, due to the effect of nitrogen
removal by harvesting and fire on the input of litter to the soil
(and, subsequently, on the mineralization rate M). As a result,
the stemwood growth curve and the optimal rotation length in
the nth rotation (cw(n)(t) and Tn , respectively), will also vary
with rotation number (n = 1, 2, ...).
Our aim here is not to estimate these dependences on the
rotation number (n) explicitly, but only to examine the situation in the long-term limit as n → ∞. In this limit, we suggest
that the system asymptotically approaches a periodic steady
state in which total nitrogen losses due to harvesting, fire and
leakage equal total nitrogen inputs integrated over one rotation. Thus:
T
T A = H + Ffl + Fw + ∫Ldt (for n → ∞).
0
(10)
MODEL OF SUSTAINABLE YIELD
In the steady state, the values of M and Uo, and therefore the
stemwood growth curve [cw(t)] and the value of the optimal
rotation length (T), are repeated from one rotation to the next
(barring wild fire events). The long-term sustainable yield
(Ys, m3 ha −1 year −1) is defined as the resulting steady-state
value of maximum MAI. Thus, from Equation 3a:
177
balance condition (Equation 10) re-expressed as a relationship
between sustainable stemwood yield and nitrogen supply. We
call Y(Uo) the ‘‘growth response constraint on yield.’’ This
constraint is satisfied in each rotation. Sustainable yield must
satisfy both of these ecological constraints.
Graphical analysis
(n)
10 cw (Tn)
.
Ys = lim
Tn
n → ∞ ρfcw
4
(11)
By inserting the expressions for H and Fw given by Equations
5 and 6b into the steady-state nitrogen balance condition
(Equation 10) and re-arranging, we obtain an expression for
cw(T)/T (valid for n → ∞) which may be substituted into
Equation 11, giving the sustainable yield as:
1
Ffl + ∫Ldt
T
T
A−
Ys =
0
10 4
,
ρfcw [fh + (1 − fh)fwburn ]ν w
(12)
where the values of T, Ffl and L are those in the steady state
(n = ∞).
In order to calculate Ys using Equation 12, we need to know
the steady-state values of the optimal rotation length (T), the
amount of nitrogen lost when foliage and aboveground litter
are burnt (Ffl), and the integral of the leakage rate (L) over a
rotation length. Each of these quantities may be expressed in
terms of the steady-state nitrogen supply rate (Uo), using Equations 1, 4, 8 and 9 in combination with the analytical results of
DM. In other words, Equation 12 may be expressed as a
relationship between Ys and Uo, which holds in the steady state:
Ys (Uo ) =
A − Ω(Uo)
10 4
.
ρfcw [fh + (1 − fh )fwburn]ν w
Equation 14 may be solved graphically by finding the intersection point of the two curves defined by the steady-state N
balance constraint and the growth response constraint. Figure 3a illustrates the general form of the analysis.
The growth response constraint (Y(Uo), Equation 3b) describes a curve with positive slope; in any given rotation, the
system must lie somewhere on this curve. In general, the
steady-state N balance constraint (Ys(Uo), Equation 13a) describes a curve with negative slope. There is a negative relationship between Ys and the steady-state value of Uo because,
when external inputs (A) are constant, there must be a trade-off
between the mean rate of nitrogen loss from foliage and litter
burning and from leakage [Ω(Uo), Equation 13b] on the one
hand, and stemwood harvesting and burning [(H + Fw)/T ∝ Ys]
on the other (Figure 2). An increase in Uo is associated with an
increase in Ω(Uo) and hence (in the steady state) a decrease
in Ys.
Above the steady-state N balance constraint curve (e.g., at
the point marked A1 in Figure 3a), total nitrogen losses exceed
inputs; if, as a result, the nitrogen supply (Uo) in the next
rotation is reduced, then the system will move down the growth
response curve (e.g., to point A2). Below the N balance constraint curve, the opposite occurs (e.g., giving the sequence B1,
B2, ...). In either case, the system eventually approaches a
steady state given by the point of intersection (point SS) of the
two constraint curves; the sustainable yield is given by the
height of SS above the horizontal axis.
(13a)
Numerical examples for three rates of nitrogen removal
Here, Ω(Uo) is the rotation-averaged rate of nitrogen loss from
burning foliage and aboveground litter, and from leakage:
T(U o)
1
Ω(U o) =
Ffl(U o) + ∫L(Uo)dt ,
T(Uo )
0
(13b)
which depends positively on nitrogen supply (Uo). Appendix
A discusses the evaluation of Ω(Uo) in more detail. It remains
to find the steady-state value of Uo. This may be obtained by
simultaneously solving Equation 13a for Ys(Uo) (valid only in
the steady state) and Equation 3b for Y(Uo), describing the
growth response of stemwood yield to nitrogen supply in any
given rotation. Therefore, the steady-state value of Uo is given
by the solution to:
Ys (Uo ) = Y(Uo ).
(14)
We call Ys(Uo) ‘‘the steady-state N balance constraint on
yield.’’ This constraint is simply the steady state nitrogen
We now examine the yield sustained by a given rate of nitrogen
additions (A) for the following set of scenarios representing
increasing amounts of nitrogen removal: (Case 1) harvesting
only, (Case 2) harvesting + regeneration burn, and (Case 3)
harvesting + regeneration burn + leakage. The graphical solution to Equation 14 for each case is shown in Figure 3b, where
the parameter values in Table 2 have been used. These parameter values have been taken mostly from studies of Eucalyptus
diversicolor F.J. Muell. and Eucalyptus marginata J. Donn.
growing in Western Australia. However, the values used have
been chosen to illustrate the general behavior of the model, and
the model outputs do not closely represent any particular
species.
Case 1: harvesting only
In this case, Ω(Uo) = 0 and fwburn = 0, so that Equation 13a
reduces to:
Ys =
10 4 A
,
ρfcw fhνw
(15)
178
DEWAR AND MCMURTRIE
Table 2. Standard parameter values (for notation, see Table 1). These
parameter values have been taken mostly from studies of Eucalyptus
diversicolor and Eucalyptus marginata growing in Western Australia.
All other parameter values of the model are given by Dewar and
McMurtrie (1996). Case 1 = harvesting only; Case 2 = harvesting and
regeneration burn, no leakage.
Parameter Value
A
fh
ffburn
flburn
fwburn
γl
ξ
ψ
Source
0.008 kg N m −2 year −1
O’Connell and Grove (1991)
0.5
Hingston et al. (1979)
0.5 (zero in Case 1)
O’Connell and Grove (1991)
0.5 (zero in Case 1)
O’Connell and Grove (1991)
0.3 (zero in Case 1)
Illustrative example
0.333 year −1
O’Connell (1987, 1988)
0.05 (zero in Cases 1 and 2) Illustrative example
0.5
Baker (1983)
gen addition rate (A), as indicated in Figure 4a. With standard
parameter values (Table 2), Equation 15 predicts a sustainable
yield of Ys = 29.6 m3 ha−1 year −1; this corresponds to the point
marked A in Figure 3b, where Uo = 181 kg N ha−1 year −1 and
so (from Equation 2 and Table 2) the mineralization rate is
M = 173 kg N ha−1 year −1.
Case 2: harvesting and regeneration burn
In this case, Ω(Uo) is given by:
Ω(Uo ) =
Figure 3. (a) General scheme of the graphical analysis of long-term
sustainable yield. The growth response constraint describes the positive relationship between maximum MAI (Y) and plant-available nitrogen supply (Uo) in any given rotation (Equation 3b), where Uo
(Equation 2) includes external inputs (assumed constant) and net mineralization (which varies over successive rotations, but is assumed
constant within each rotation). The steady-state N balance constraint is
the relationship between the steady-state values of maximum MAI (Ys)
and Uo, defined by the condition that N inputs to the system equal N
losses summed over a rotation (Equation 13a); above this curve, N
outputs exceed N inputs, so that N supply and maximum MAI will
decrease over successive rotations (e.g., sequence A1, A2, ...); below
this curve, the opposite occurs (e.g., sequence B1, B2, ...). In either
case, the system approaches the steady state (point SS) where both
constraints are satisfied; the sustainable yield is given by the height of
SS above the horizontal axis. (b) Numerical examples for three scenarios of N removal: harvesting only, harvesting + regeneration burn, and
harvesting + regeneration burn + leaching and gaseous emissions;
points A, B and C indicate the sustainable steady states, respectively.
Parameter values are given in Table 2.
independent of Uo, giving a horizontal line for the steady-state
N balance constraint curve (Figure 3b). This reflects that, in the
absence of Uo-dependent nitrogen losses from fire and leakage,
Equation 10 implies that the steady-state, rotation-averaged
rate of nitrogen loss from harvesting (i.e., H/T, which, from
Equations 5 and 11, is proportional to Ys) must equal the nitro-
Ffl(U o)
,
T(U o)
(16)
which describes an increasing function of Uo (mainly due to
increasing Ffl rather than decreasing T, according to the results
of DM). Therefore, the steady-state N balance constraint curve
(Equation 13a) has a negative slope (Figure 3b), as discussed
above.
At the intersection of the two constraint curves (point B,
Figure 3b), Ys = 15.0 m3 ha−1 year −1. Therefore, compared to
Case 1 (harvesting only), the extra losses of nitrogen associated with regeneration burning reduce the sustainable yield by
a factor of 2. The steady-state nitrogen budget of the system is
shown in Figure 4b. In this example, the rotation-averaged
nitrogen loss rates from harvesting (4.1 kg N ha−1 year −1) and
from regeneration burning (3.9 kg N ha−1 year −1) are approximately equal.
Table 3 shows the sensitivity of sustainable yield to changes
in the values of individual model parameters, with Case 2 as
the reference case. The effect of an increase in the rate of
nitrogen addition (A) or litter turnover (γl) is to raise the height
of the N balance constraint curve, and thus the intersection
point with the growth response constraint curve. Similarly,
increasing the foliage senescence rate (γf) raises the height of
the N balance constraint curve (by reducing the amount of fuel
available to burn), but also lowers the growth response constraint curve (by reducing foliage biomass and carbon production at canopy closure, see DM); however, the net effect is an
increase in sustainable yield. Increases in the other parameters
MODEL OF SUSTAINABLE YIELD
179
Table 3. Predicted percentage change in sustainable yield (∆Ys, Equations 13a, 14 and 3b) due to a 10% increase in individual model
parameters from their values in reference Case 2 (harvesting and
regeneration burn, no leakage), as given in Table 2 (with ξ = 0). The
sensitivities to stemwood N/C ratio (νw) and foliage turnover rate (γf)
are also shown; the reference values are νw = 0.002 kg N kg −1 C and
γf = 0.333 year −1, as given by Dewar and McMurtrie (1996).
Parameter increased by 10%
∆Ys (%)
A
fh
ffburn
flburn
fwburn
γl
ψ
νw
γf
+7.6
−2.7
−1.7
−0.9
−1.2
+0.8
−0.9
−4.9
+0.8
duction. Table 3 indicates that, for Case 2 as the reference case,
Ys is most sensitive to changes in A and νw, moderately sensitive to the harvest fraction (fh), and least sensitive to changes
in the parameters defining nitrogen losses from burning litter
(flburn, γl, ψ), live foliage (ffburn , γf) and woody slash (fwburn).
Case 3: harvesting, regeneration burn and leakage
The inclusion of leaching and gaseous emission losses (which
are positively related to nitrogen supply, see Equation 9) lowers the steady-state N balance constraint curve (which also
becomes slightly steeper). As a result, the sustainable yield is
reduced to 13.2 m3 ha−1 year −1 (point C, Figure 3b). In this
example, leakage accounts for 15% of the nitrogen loss in the
steady state (Figure 4c).
Nitrogen additions required to sustain a given yield
Figure 4. Ecosystem nitrogen budgets for the sustainable steady states
of Figure 3b corresponding to (a) point A (harvesting only), (b) point
B (harvesting + regeneration burn), and (c) point C (harvesting +
regeneration burn + leakage). Numbers next to the arrows are the
steady-state, rotation-averaged fluxes of nitrogen (kg N ha −1 year −1)
associated with external inputs (A = fertilizer + fixation + deposition),
removal of harvested stemwood (= H/T ), burning of foliage and
aboveground litter (= Ffl /T ) and stemwood slash (= Fw /T ), and
leakage [(1/T )∫ L dt]. Also shown are the steady-state values of
sustainable yield (Ys, m3 ha −1 year −1), available nitrogen supply
(Uo, kg N ha−1 year −1), net mineralization (M, kg N ha −1 year −1), and
optimal rotation length (T, year).
in Table 3 lower the height of the N balance curve; in addition,
an increase in the stemwood N/C ratio (νw) lowers the height
of the growth response curve also (e.g., see DM, their Table 4)
by reducing the amount of nitrogen available for foliage pro-
The above analysis may be inverted to estimate the nitrogen
addition rate (A) that is required to sustain a given maximum
MAI in the long-term (Ys), including the case of maintaining
current yield. In the case of harvesting only (Case 1), from
Equation 15 we obtain:
A(Ys ) =
ρfcw
fh νw Ys,
10 4
(17)
which predicts a linear relationship between A and Ys (Figure 5). More generally, Equation 13a gives:
A(Ys ) =
ρfcw
[ fh + (1 − fh )fwburn ]ν wYs + Ω(Uo).
10 4
(18)
To calculate Ω(Uo) for a given value of Ys, the required
nitrogen supply rate is just the value of Uo at which Y(Uo) = Ys,
which may be calculated numerically by inverting Equation 3b. This value of Uo is then substituted into Equation 13b
to calculate Ω(Uo) (see Appendix A). Equation 18 is illustrated
180
DEWAR AND MCMURTRIE
Figure 5. The predicted nitrogen addition rate (A) required to sustain
a given long-term yield (Ys), as a function of Ys for two scenarios of N
removal: harvesting only (Equation 17), and harvesting + regeneration
burn (Equation 18). Parameter values are given in Table 2.
in Figure 5 for Case 2 (harvesting + regeneration burn). In this
case, in order to sustain a given yield, the required nitrogen
addition rate is approximately double the value required when
the only nitrogen loss is from removal of harvested biomass
(Case 1).
Discussion
Few observational studies exist of the long-term effects of
harvesting and fire on tree growth. What little has been reported includes positive, negative and null responses, possibly
because measurements have been made on different time
scales (e.g., Balneaves 1990, Raison et al. 1990). Short-term
responses to fire are usually favorable to plant nutrition (which
may be reflected in growth). Removal of woody litter and slash
at harvest (e.g., through regeneration burn) may also be beneficial to plant nutrition in the short term; this is so because
woody litter has a low N/C ratio and can immobilize large
amounts of soil N (e.g., Pearson et al. 1987). Reported longerterm growth responses tend to be more negative (e.g., Balneaves 1990). However, Abbott and Loneragan (1986) found
no detectable effect of regular prescribed burning on the
growth of jarrah forest. O’Connell and Grove (1991) suggested
that this may reflect the fact that many ecosystems are currently buffered against disturbance by nutrient reserves in the
soil.
In the long term, however, it seems inescapable that the
cumulative effect of a negative net balance of nitrogen must be
a reduction in the supply of plant-available nitrogen, and
consequently a decrease in tree growth, until a steady state is
reached in which nitrogen inputs balance outputs (e.g., Figure 3a, sequence A1, A2, ..., SS). An analogous conclusion
applies if nitrogen additions were to exceed current losses,
resulting in an increase in tree growth toward the steady state
(e.g., Figure 3a, sequence B1, B2, ..., SS).
We have examined sustainable yield purely as a steady-state
problem. As a result, we can say nothing about how long the
system takes to approach the steady state, or about the likely
magnitude of changes in nitrogen supply and productivity
from one rotation to the next (the sequences A1, A2, ... and B1,
B2, ... in Figure 3a are purely schematic examples). To address
these questions would require using a model that links plant
and soil nutrient cycling in a dynamic fashion (e.g., King
1995). The advantage of the steady-state analysis, however, is
that we do not need to know any details about the dynamics of
nitrogen cycling in the soil and its response to harvesting and
fire. Instead, the condition for steady-state N balance (Equation 10), when combined with the growth response predicted
by DM (Equation 3b), provides a sufficient constraint on the
steady-state mineralization rate to determine the sustainable
yield.
Our analysis suggests that, for Eucalyptus, the most important factor that limits long-term sustainable yield is likely to be
fire (here, regeneration burning). In the examples illustrated in
Figure 3b, harvesting alone results in a relatively large sustainable yield (≈ 30 m3 ha −1 year −1), which is decreased by a factor
of 2 when N removals from regeneration burning are included.
Leaching and gaseous emissions (which represent relatively
small fluxes in many Australian forests) play a minor role. In
addition, the sensitivity analysis (Table 3) shows that reliable
estimates of the nitrogen addition rate (A) and stemwood N/C
ratio (νw) are required to quantify sustainable yield.
Figure 3 is analogous to Comins and McMurtrie’s (1993)
graphical prediction of the equilibrium net primary productivity of unmanaged forests (see their Figure 6). Their ‘‘N-cycling
constraint’’ is equivalent to our steady-state N balance constraint, both being based on the conservation of nitrogen at
equilibrium (or rotation-averaged steady state, as here). Their
‘‘photosynthetic constraint’’ is analogous to our growth response constraint, although the former is based on the conservation of carbon at equilibrium, whereas the latter is based on
the relationship between stemwood growth and nitrogen supply that exists in any given rotation (not just in the steady state).
In Comins and McMurtrie (1993), the constraint curves are
related to leaf N/C ratio (νf) rather than to nitrogen supply.
As with any model, a number of assumptions underlie the
predictions illustrated in Figures 3--5 (see Discussion in DM).
One important assumption is that Uo (Equation 2) is constant
within a rotation, so that the mineralization rate (M) represents
an effective rotation-averaged value. As a result, the predicted
response of tree growth to nitrogen supply does not take into
account within-rotation variations in M (associated with the
dynamics of litter and soil decomposition), which may be
important for nitrogen uptake by trees, especially around canopy closure. The numerical predictions do not include the
effects of mechanical disturbance and compaction of soil,
prescribed fuel-reduction burning, or feedbacks between fire
and the activity of nitrogen-fixing legumes (here, A is a constant). Also, we have assumed that standard silvicultural practice applies whereby the stand is harvested at maximum MAI,
and have not considered the implications of treating the rotation length as a free parameter.
Nevertheless, the graphical analysis presented here is sufficiently general to provide a useful framework for incorporating
MODEL OF SUSTAINABLE YIELD
some of these factors and examining their effects on sustainable yield. This analysis may also be useful for extending the
application of the present model to the effects of climate
change and increases in atmospheric CO2 concentrations on
sustainable yield, or for interpreting the behavior of more
complex models of sustainable forest growth (e.g., King
1995).
Acknowledgments
This work was supported by the NGAC Dedicated Greenhouse Research Grants Scheme and the Australian Research Council. We are
grateful to Tony O’Connell and John Raison for providing references
on their field studies of forest fire ecology.
References
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marginata) in the northern jarrah forest of Western Australia.
CALM, Perth, WA, Bull. No. 1, 137 p.
Baker, T.G. 1983. Dry matter, nitrogen, and phosphorus content of
litterfall and branch fall in Pinus radiata and Eucalyptus forests.
N.Z. J. For. Sci. 13:205--221.
Balneaves, J.M. 1990. Maintaining site productivity in second rotation
crops, Canterbury Plains, New Zealand. In Impact of Intensive
Harvesting on Forest Site Productivity. Eds. W.J. Dyck and C.A.
Mees. Proc. IEA/BE A3 Workshop, South Island, New Zealand,
March 1989. IEA/BE T6/A6 Report No. 2., For. Res. Inst., Rotorua,
N.Z., FRI Bull. No. 159, pp 73--83.
Comins, H.N. and R.E. McMurtrie. 1993. Long-term biotic response
of nutrient-limited forest ecosystems to CO2-enrichment: equilibrium behavior of integrated plant--soil models. Ecol. Appl. 3:666-681.
Dewar, R.C. and R.E. McMurtrie. 1996. Analytical model of stemwood growth in relation to nitrogen supply. Tree Physiol. 16:161-171.
Franklin, J.F., D.A. Perry, T.D. Schowalter, M.E. Harmon, A. McKee
and T.A. Spies. 1989. Importance of ecological diversity in maintaining long-term site productivity. In Maintaining the Long-Term
Productivity of Pacific Northwest Forest Ecosystems. Eds. D.A.
Perry, R. Meurisse, B. Thomas, R. Miller, J. Boyle, J. Means, C.R.
Perry and R.F. Powers. Timber Press, Portland, Oregon, pp 82--97.
Grove, T.S., A.M. O’Connell and G.M. Dimmock. 1986. Nutrient
changes in surface soils after an intense fire in jarrah (Eucalyptus
marginata Donn ex Sm.) forest. Aust. J. Ecol. 11:303--317.
Hingston, F.J., A.G. Turton and G.M. Dimmock. 1979. Nutrient distribution in karri (Eucalyptus diversicolor F. Muell.) ecosystems in
southwest Western Australia. For. Ecol. Manage. 2:133--158.
King, D.A. 1995. A forest process model for evaluating the potential
productivity of Eucalyptus plantations in Southeastern Australia.
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Trees as Crop Plants. Eds. M.G.R. Cannell and J.E. Jackson. Inst.
Terrestrial Ecol., 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. 1996. Declining forest
productivity in ageing forest stands----a modeling analysis of alternative hypotheses. Tree Physiol. 16:187--200.
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nutritional resilience or vulnerability of jarrah and karri forests in
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Pearson, J.A., D.H. Knight and T.J. Fahey. 1987. Biomass and nutrient
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Raison, R.J., H. Keith and P.K. Khanna. 1990. Effects of fire on the
nutrient-supplying capacity of forest soils. In Impact of Intensive
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Turner, J. 1981. Nutrient supply in relation to immobilization in
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Appendix A. Evaluation of Ω(Uo)
The rate of nitrogen loss (Ω(Uo)) from burning foliage and
aboveground litter, and from leakage, averaged over one rotation, is given by Equation 13b:
T(U o)
F
(U
)
+
L
(t,U
)dt
Ω(Uo ) =
fl
o
o
.
∫
T(U(o )
0
1
(A1)
Here, Uo (kg N m −2 year −1) is the nitrogen supply rate (Equation 2), T (year) is the optimal rotation length, Ffl (kg N m −2)
is the amount of nitrogen lost in burning foliage and litter,
L (kg N m −2 year −1) is the rate of nitrogen loss from leaching
and gaseous emissions, and the integral is with respect to stand
age (T) over one rotation length. Each of the quantities T, Ffl
and L can be evaluated analytically as functions of Uo using the
results of the stemwood growth model of DM.
First, T(Uo) is given by Equation 4 in which λ(Uo), the rate
182
DEWAR AND MCMURTRIE
of approach of carbon production towards equilibrium at canopy closure, has been calculated by DM (their Appendix C,
Equation C5). Second, Ffl(Uo) is given by Equation 8, where
νf* and G* (the foliage N/C ratio and carbon production at
canopy closure, respectively) have been calculated as functions of Uo by DM (their Appendix B, Equations B3 and B9,
respectively). Finally, from Equations 1 and 9, L(t,Uo) is given
by:
ξUo Kr
,
L(t,Uo ) =
Kr + cr(t,U0)
(A2)
where cr(t, Uo) is the amount of fine root carbon at stand age t
and nitrogen supply rate Uo, and is given from the model of
DM by:
cr(t,Uo ) =
ηrG∗(Uo )
λ(U o) e−µ rt − µr e−λ(U o)t
,
1 −
µr
λ(U o) − µr
(A3)
where ηr is the fraction of carbon production allocated to fine
roots, and µr is the combined specific rate of fine root maintenance respiration and senescence. Combining Equations A2
and A3, the integration of L in Equation A1 can be performed
numerically for a given value of Uo.
© 1996 Heron Publishing----Victoria, Canada
Sustainable stemwood yield in relation to the nitrogen balance of forest
plantations: a model analysis
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 used an existing analytical model of stemwood growth in relation to nitrogen supply, which we describe
in an accompanying paper, to examine the long-term effects of
harvesting and fire on tree growth. Our analysis takes into
account the balance between nitrogen additions from deposition, fixation, and fertilizer applications, and nitrogen losses
from stemwood harvesting, regeneration burning, leaching and
gaseous emissions. Using a plausible set of parameter values
for Eucalyptus, we conclude that nitrogen loss through fire is
the main factor limiting sustainable yield, defined as the maximum mean annual stemwood volume increment obtained in
the steady state, if management practices are continued indefinitely. The sustainable yield is 30 m3 ha −1 year −1 with harvesting only, 15 m3 ha −1 year −1 with harvesting and regeneration
burning, and 13 m3 ha −1 year −1 with harvesting, fire, leaching
and gaseous emissions combined.
Our approach uses a simple graphical analysis that provides
a useful framework for examining the factors affecting sustainable yield. The graphical analysis is also useful for extending
the application of the present model to the effects of climate
change on sustainable yield, or for interpreting the behavior of
other models of sustainable forest growth.
Keywords: fire, growth, harvesting, mean annual increment,
sustainability.
Introduction
In managed forests, the balance between inputs and outputs of
nutrients over successive rotations is a major ecological constraint on the sustainability of site productivity. As a major
nutrient for plant growth, nitrogen has been extensively studied in this context. Inputs of nitrogen may occur through wet
and dry atmospheric deposition, symbiotic and non-symbiotic
fixation, and fertilizer applications; nitrogen may be lost
through leaching and gaseous emissions, removal of harvested
biomass (Turner 1981, Raison et al. 1982, Stewart et al. 1985)
and fire (O’Connell et al. 1981, Grove et al. 1986, Walker et al.
1986, Raison et al. 1990, 1993). In Australian forests, fire is
widely used to reduce accumulated fuel (and thus the risk of
uncontrolled fires), to clear slash debris and to regenerate
seedlings.
Understandably, most studies of the effects of fire on nitro-
gen budgets and cycling have been limited to time scales of
months to a few years (e.g., Grove et al. 1986, Raison et al.
1990). The immediate effects, such as release of mineral nitrogen through oxidation of organic matter, are usually beneficial
to plant nutrition, but they occur in association with direct
nitrogen losses to the atmosphere from volatilization and ash
transport, in addition to indirect losses such as leaching of
released nitrogen or erosion of surface soil and ash. Little is
known of the cumulative effect of fire and harvesting on forest
nutrition and growth over time scales spanning several rotations. As a result, models of the interactions between forest
growth, fire and harvesting regimes, and nutrient balance provide valuable tools for extrapolating over these longer time
scales.
In the preceding article (Dewar and McMurtrie 1996), we
described a simple process-based model of stemwood growth
in relation to plant-available nitrogen supply (Uo, kg N m −2
year −1); the model predicts the variation of stemwood volume
(or carbon equivalent) with stand age for an undisturbed stand.
The supply rate, Uo, refers to external inputs as described
above, in addition to nitrogen supplied through mineralization
of organic matter. In this paper, we combine that model with
simple assumptions about external nitrogen inputs and losses,
focusing on removals in harvesting, regeneration burning,
leaching and gaseous emissions. We use the combined model
to define the term ‘‘sustainable yield’’ in an unambiguous way,
and to quantify its dependence on the individual contributions
to nitrogen removal. Our approach exploits a simple graphical
analysis, which provides a useful general framework for examining the factors affecting sustainable yield.
The model
Stemwood growth in relation to nitrogen supply
Dewar and McMurtrie (1996, this issue) (hereafter referred to
as DM) presented a model of stemwood growth in relation to
nitrogen supply, and applied the model to predict the variation
of stemwood volume (or carbon equivalent) with stand age for
an unharvested stand. A complete mathematical description of
the model is given in DM, and only a brief summary is given
here. Symbols used, their definitions and units are listed in
Table 1. In the model, which is based on the work of McMur-
174
DEWAR AND MCMURTRIE
Table 1. Symbol definitions. Abbreviations: dm = dry matter;m −2 = per unit ground area (N fluxes are re-expressed on a per hectare basis in Figures
3b--5); X* = equilibrium value of X at canopy closure. Subscripts: f = foliage; w = stemwood; r = fine roots; l = aboveground litter; fl = foliage
and aboveground litter.
Symbol
Definition
Units
A
cf, cf*
cl, cl*
cw, cr
cw(n)
fh
fcw
fiburn ; i = f, l, w
Ffl, Fw
G, G*
H
Kr
L
M
MAI
nf
t
T
Tn
U
Uo
Y
Ys
γf, γl
ηi, i = f, w, r
λ
µi, i = w, r
νf, νf*, νl, νw
Ω
ρ
ξ
ψ
Addition rate of N from fertilizer inputs, atmospheric deposition and fixation
Foliage C
Aboveground litter C
Stemwood and fine root C
Value of cw in the nth rotation (n = 1, 2, ...)
Fraction of harvested stemwood removed from site
Fractional C content of stemwood dm
Fraction of N lost in regeneration burn
Amount of N lost in regeneration burn
Total C production by trees (net of canopy maintenance and plant growth respiration)
Amount of N removed by stemwood harvesting
Fine root N uptake coefficient
Rate of N loss from leaching and gaseous emissions
Net mineralization rate
Mean annual stemwood volume increment
Foliage N
Stand age
Optimal rotation length
Value of T in the nth rotation (n = 1, 2, ...)
Rate of N uptake by trees
Supply rate of plant-available N
Maximum MAI in a given rotation
Sustainable yield (steady-state value of Y)
Specific senescence rate
Carbon production allocation fraction
Rate of approach of G towards equilibrium
Sum of specific maintenance respiration and senescence rates
N/C Ratio
Rotation-averaged rate of N loss from burning foliage and litter, leaching and gaseous emissions
Stemwood density
Fraction of available N not taken up by trees that is lost by leaching and gaseous emissions
Ratio of aboveground litter N/C to live foliage N/C
trie and Wolf (1983) and McMurtrie (1985, 1991), four dynamic variables are represented: stemwood carbon (cw, kg C
m−2, which includes stems, branches and coarse roots), foliage
carbon (cf) and nitrogen (nf), and fine root carbon (cr). Stemwood and fine roots are assumed to have fixed N/C ratios.
Total carbon production by trees (G, kg C m −2 year−1) is
given by a nitrogen-dependent light utilization coefficient multiplied by absorbed photosynthetic radiation, and is allocated
to each of the three tree components in fixed proportions.
Maintenance respiration in stemwood and fine roots is explicitly represented, as is senescence of all tree components; foliage maintenance respiration and all components of plant
growth respiration are included implicitly in the definition of
G. Nitrogen uptake by trees (U, kg N m−2 year −1) depends on
root mass and on the rate at which plant-available nitrogen is
supplied (Uo):
U = Uo
cr
,
cr + K r
(1)
kg N m −2 year −1
kg C m −2
kg C m −2
kg C m −2
kg C m −2
kg C kg−1 dm
kg N m −2
kg C m −2 year −1
kg N m −2
kg C m −2
kg N m −2 year −1
kg N m −2 year −1
m3 ha−1 year −1
kg N m −2
year
year
year
kg N m −2 year −1
kg N m −2 year −1
m3 ha−1 year −1
m3 ha−1 year −1
year −1
year −1
year −1
kg N kg−1 C
kg N m −2 year −1
kg dm m −3
where Kr is the value of root carbon at which 50% of the
available nitrogen is taken up. As discussed by DM (their
Appendix A), the constant 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. The rate Uo represents nitrogen supplied by net
mineralization (M) within the system, plus additions of nitrogen from outside the system (A) due to fertilizer inputs, atmospheric deposition, and fixation:
U O = M + A.
(2)
In DM, Uo is taken to be a constant parameter throughout the
life of the stand; for constant external inputs (A), this implies
that M is also constant so that the model ignores variations in
M with stand age associated with the dynamics of litter and soil
organic matter decomposition. In this study, in order to exploit
the results of DM, we will also assume that Uo is constant
within a given rotation. However, in the sustainability analysis
MODEL OF SUSTAINABLE YIELD
presented below, we will examine the long-term consequence
of changes in M (and therefore Uo) over successive rotations.
By introducing some simplifying approximations into the
basic growth model outlined above, DM derived various analytical results describing the relationship of stemwood growth
to nitrogen supply (Uo). For example, the mean annual stemwood volume increment (MAI, m3 ha −1 year −1) at stand age t
(years) is defined by:
MAI(t) =
10 4 cw(t)
,
ρfcw t
(3a)
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 104 converts from m −2 to ha −1. Figure 1 shows
the variation of stemwood carbon [cw(t)] and mean annual
increment [MAI(t)] with stand age, predicted analytically by
the model when Uo = 100 kg N ha −1 year −1, with other parameter values given in DM. Stemwood carbon follows a
realistic sigmoidal growth pattern, so that MAI first increases
to a maximum rate (here, 18 m3 ha −1 year −1 at t ≈ 19 years) and
then gradually declines as the stand ages. In the model, this
decline is caused by increasing stemwood maintenance respiration and senescence, with foliage biomass and carbon production remaining fixed after canopy closure. Alternative
hypotheses for the mechanisms underlying the decline in productivity of old stands are examined by Murty et al. (1996, this
issue).
To optimize the average volume output of the stand, the
forest should be harvested at the age of maximum MAI. Although economic factors may be more important to a forest
manager deciding when to harvest, the concept of maximum
MAI continues to be widely used as a characteristic of stemwood growth in forestry yield tables. In DM it was shown that
the dependence of maximum MAI (Y) on nitrogen supply
(U o) is given approximately by:
175
Y(U o) = maximum MAI
≈
10 4
ηw G∗(Uo )
ρfcw
1
1+
√
2 µw
λ(Uo )
,
(3b)
in which ηw is the fraction of total carbon production (G)
allocated to stemwood, G* is the equilibrium value of G at
canopy closure, λ (year −1) describes the rate at which G approaches G*, and µw (year −1) is the combined specific rate of
stemwood maintenance respiration and senescence.
The corresponding optimal rotation length (T = stand age at
maximum MAI) is given approximately by:
2
T(U o) ≈ √
.
λ(U o)µw
(4)
According to these equations, Y and T are functions of Uo
through the quantities G* and λ (which also depend on all the
other model parameters). Dewar and McMurtrie (1996) obtained analytical expressions for the functions G*(Uo) and
λ(Uo) (see their Appendices B and C); their results show that λ
(and therefore T) is relatively insensitive to Uo, but that G*
(and therefore Y) varies approximately linearly with Uo.
Nitrogen losses due to harvesting, regeneration burn and
‘‘leakage’’
We now consider the nitrogen losses that occur over one
rotation of a managed plantation that is harvested at maximum
mean annual increment (i.e., when stand age = T), then subjected to a regeneration burn. The amount of nitrogen removed
in harvesting (H, kg N ha −1) is given by:
H = cw (T)νw fh ,
(5)
where cw(T) is the amount of stemwood carbon at the end of
the rotation, νw is the (constant) stemwood N/C ratio, and fh is
the fraction of stemwood removed from the site. Dewar and
McMurtrie (1996) calculated cw(T) for a given value of nitrogen supply (Uo) during the rotation (their Equation 8).
During the regeneration burn, nitrogen is lost to the atmosphere by the combustion of standing foliage, aboveground
litter, and stemwood slash that has not been removed from the
site. In each case, we assume that the amount of nitrogen lost
is proportional to the total fuel N content. Therefore, the
nitrogen lost from burning foliage and aboveground litter (Ffl,
kg N m −2) is:
Ffl = cf(T )ν f(T ) ffburn + cl(T )νl(T ) flburn ,
Figure 1. Stemwood carbon [cw(t), solid line] and mean annual stemwood volume increment [MAI(t), broken line] as a function of stand
age (t), as predicted by the analytical stemwood growth model of
Dewar and McMurtrie (1996), for a nitrogen supply rate Uo = 100 kg
N ha −1 year −1 (with other parameter values as given in their paper).
(6a)
where cf(T) and cl(T) are the amounts of foliage carbon and
aboveground litter carbon at the end of the rotation, νf (T) and
νl(T) are the corresponding N/C ratios, and ffburn and flburn are
the fractions of nitrogen lost when foliage and aboveground
litter are burnt. Similarly, the amount of nitrogen lost from
burning stemwood slash (Fw, kg N m −2) is:
176
DEWAR AND MCMURTRIE
Fw = cw (T )νw (1 − fh )fwburn ,
(6b)
where fwburn is the corresponding fraction of nitrogen lost. Like
H (Equation 5), Fw can be calculated from the results of DM.
In order to calculate Ffl, we make the following three simplifying assumptions. First, as the predictions of the growth model
show (DM), it is reasonable to assume that foliage carbon has
reached its equilibrium value, cf*, corresponding to canopy
closure, before the end of the rotation, so that:
cf(T ) = cf∗ =
η fG∗
,
γf
(7a)
where ηf is the fraction of carbon production allocated to
foliage, and γf is the foliage senescence rate. Second, we make
a similar assumption for aboveground leaf litter:
cl(T ) = cl∗ =
cf∗γ f
,
γl
(7b)
where cl* is the equilibrium value of litter carbon, and γl is the
leaf litter turnover rate. (In the present study, we ignore nitrogen losses from burning woody litter and understorey vegetation.) Third, we assume that the aboveground litter N/C ratio
is proportional to the foliage N/C ratio, and that the foliage N/C
ratio is also at its equilibrium value, νf*, at the end of the
rotation (as suggested by the results of DM), so that:
ν l(T ) = ψν f(T ) = ψνf∗,
(7c)
where ψ is a constant. Substituting Equations 7 into Equation
6a then gives:
ffburn flburn ψ
Ffl =
+
ηf ν f ∗ G ∗ ,
γl
γf
(8)
in which νf* and G* may be calculated for a given value of Uo
using the analytical results in DM (their Appendix B).
Throughout the rotation, a constant fraction ξ of the nitrogen
supply not taken up by trees is lost from the system through
‘‘leakage,’’ i.e., leaching and gaseous emissions, the remaining
Figure 2. Ecosystem inputs and outputs of nitrogen in a forest plantation, included in the present study.
fraction being taken up by other vegetation. Therefore, the
leakage rate (L, kg N m −2 year −1) is:
L = ξ (Uo − U) = ξ Uo
Kr
,
cr + K r
(9)
where we have used Equation 1. The value of L is greatest
during the early growth years when root biomass is low. In the
notation of DM (their Appendix A), ξ is given by ln /(ln + σv ),
where ln and σv are parameters describing the intrinsic leakage
rate and the intensity of competition for nitrogen from other
vegetation, respectively. The value of ξ is generally small in
the dry conditions that exist in many Australian forests.
Equation 3b gives the stemwood yield at harvest (of which
a fraction fh is removed from site), and Equations 5--9 describe
the nitrogen losses from the system, during a single rotation for
a given constant nitrogen supply rate (Uo). In the following
section we will consider the long-term consequences of these
nitrogen losses for stemwood yield, due to changes in Uo over
successive rotations.
Sustainable yield
Definition
We adopt a definition of sustainable yield (Ys) that is based
solely on the ecophysiological constraints on the long-term
productivity of a given forest stand. Our definition does not
address issues of conservation of biological and genetic diversity (e.g., Franklin et al. 1989), or wider socioeconomic aspects of sustainability, such as energy costs associated with
fertilizer inputs, harvest operations, transport and processing
of wood products.
Figure 2 depicts the overall nitrogen balance of the system.
External inputs of nitrogen from fertilizer additions, fixation
and atmospheric deposition are assumed to occur at a constant
rate (A), and the system is subjected to a repeated management
cycle consisting of harvesting at maximum MAI, followed by
regeneration burn and re-planting. Under these conditions, the
supply of available nitrogen (Uo, Equation 2) will in general
vary from one rotation to the next, due to the effect of nitrogen
removal by harvesting and fire on the input of litter to the soil
(and, subsequently, on the mineralization rate M). As a result,
the stemwood growth curve and the optimal rotation length in
the nth rotation (cw(n)(t) and Tn , respectively), will also vary
with rotation number (n = 1, 2, ...).
Our aim here is not to estimate these dependences on the
rotation number (n) explicitly, but only to examine the situation in the long-term limit as n → ∞. In this limit, we suggest
that the system asymptotically approaches a periodic steady
state in which total nitrogen losses due to harvesting, fire and
leakage equal total nitrogen inputs integrated over one rotation. Thus:
T
T A = H + Ffl + Fw + ∫Ldt (for n → ∞).
0
(10)
MODEL OF SUSTAINABLE YIELD
In the steady state, the values of M and Uo, and therefore the
stemwood growth curve [cw(t)] and the value of the optimal
rotation length (T), are repeated from one rotation to the next
(barring wild fire events). The long-term sustainable yield
(Ys, m3 ha −1 year −1) is defined as the resulting steady-state
value of maximum MAI. Thus, from Equation 3a:
177
balance condition (Equation 10) re-expressed as a relationship
between sustainable stemwood yield and nitrogen supply. We
call Y(Uo) the ‘‘growth response constraint on yield.’’ This
constraint is satisfied in each rotation. Sustainable yield must
satisfy both of these ecological constraints.
Graphical analysis
(n)
10 cw (Tn)
.
Ys = lim
Tn
n → ∞ ρfcw
4
(11)
By inserting the expressions for H and Fw given by Equations
5 and 6b into the steady-state nitrogen balance condition
(Equation 10) and re-arranging, we obtain an expression for
cw(T)/T (valid for n → ∞) which may be substituted into
Equation 11, giving the sustainable yield as:
1
Ffl + ∫Ldt
T
T
A−
Ys =
0
10 4
,
ρfcw [fh + (1 − fh)fwburn ]ν w
(12)
where the values of T, Ffl and L are those in the steady state
(n = ∞).
In order to calculate Ys using Equation 12, we need to know
the steady-state values of the optimal rotation length (T), the
amount of nitrogen lost when foliage and aboveground litter
are burnt (Ffl), and the integral of the leakage rate (L) over a
rotation length. Each of these quantities may be expressed in
terms of the steady-state nitrogen supply rate (Uo), using Equations 1, 4, 8 and 9 in combination with the analytical results of
DM. In other words, Equation 12 may be expressed as a
relationship between Ys and Uo, which holds in the steady state:
Ys (Uo ) =
A − Ω(Uo)
10 4
.
ρfcw [fh + (1 − fh )fwburn]ν w
Equation 14 may be solved graphically by finding the intersection point of the two curves defined by the steady-state N
balance constraint and the growth response constraint. Figure 3a illustrates the general form of the analysis.
The growth response constraint (Y(Uo), Equation 3b) describes a curve with positive slope; in any given rotation, the
system must lie somewhere on this curve. In general, the
steady-state N balance constraint (Ys(Uo), Equation 13a) describes a curve with negative slope. There is a negative relationship between Ys and the steady-state value of Uo because,
when external inputs (A) are constant, there must be a trade-off
between the mean rate of nitrogen loss from foliage and litter
burning and from leakage [Ω(Uo), Equation 13b] on the one
hand, and stemwood harvesting and burning [(H + Fw)/T ∝ Ys]
on the other (Figure 2). An increase in Uo is associated with an
increase in Ω(Uo) and hence (in the steady state) a decrease
in Ys.
Above the steady-state N balance constraint curve (e.g., at
the point marked A1 in Figure 3a), total nitrogen losses exceed
inputs; if, as a result, the nitrogen supply (Uo) in the next
rotation is reduced, then the system will move down the growth
response curve (e.g., to point A2). Below the N balance constraint curve, the opposite occurs (e.g., giving the sequence B1,
B2, ...). In either case, the system eventually approaches a
steady state given by the point of intersection (point SS) of the
two constraint curves; the sustainable yield is given by the
height of SS above the horizontal axis.
(13a)
Numerical examples for three rates of nitrogen removal
Here, Ω(Uo) is the rotation-averaged rate of nitrogen loss from
burning foliage and aboveground litter, and from leakage:
T(U o)
1
Ω(U o) =
Ffl(U o) + ∫L(Uo)dt ,
T(Uo )
0
(13b)
which depends positively on nitrogen supply (Uo). Appendix
A discusses the evaluation of Ω(Uo) in more detail. It remains
to find the steady-state value of Uo. This may be obtained by
simultaneously solving Equation 13a for Ys(Uo) (valid only in
the steady state) and Equation 3b for Y(Uo), describing the
growth response of stemwood yield to nitrogen supply in any
given rotation. Therefore, the steady-state value of Uo is given
by the solution to:
Ys (Uo ) = Y(Uo ).
(14)
We call Ys(Uo) ‘‘the steady-state N balance constraint on
yield.’’ This constraint is simply the steady state nitrogen
We now examine the yield sustained by a given rate of nitrogen
additions (A) for the following set of scenarios representing
increasing amounts of nitrogen removal: (Case 1) harvesting
only, (Case 2) harvesting + regeneration burn, and (Case 3)
harvesting + regeneration burn + leakage. The graphical solution to Equation 14 for each case is shown in Figure 3b, where
the parameter values in Table 2 have been used. These parameter values have been taken mostly from studies of Eucalyptus
diversicolor F.J. Muell. and Eucalyptus marginata J. Donn.
growing in Western Australia. However, the values used have
been chosen to illustrate the general behavior of the model, and
the model outputs do not closely represent any particular
species.
Case 1: harvesting only
In this case, Ω(Uo) = 0 and fwburn = 0, so that Equation 13a
reduces to:
Ys =
10 4 A
,
ρfcw fhνw
(15)
178
DEWAR AND MCMURTRIE
Table 2. Standard parameter values (for notation, see Table 1). These
parameter values have been taken mostly from studies of Eucalyptus
diversicolor and Eucalyptus marginata growing in Western Australia.
All other parameter values of the model are given by Dewar and
McMurtrie (1996). Case 1 = harvesting only; Case 2 = harvesting and
regeneration burn, no leakage.
Parameter Value
A
fh
ffburn
flburn
fwburn
γl
ξ
ψ
Source
0.008 kg N m −2 year −1
O’Connell and Grove (1991)
0.5
Hingston et al. (1979)
0.5 (zero in Case 1)
O’Connell and Grove (1991)
0.5 (zero in Case 1)
O’Connell and Grove (1991)
0.3 (zero in Case 1)
Illustrative example
0.333 year −1
O’Connell (1987, 1988)
0.05 (zero in Cases 1 and 2) Illustrative example
0.5
Baker (1983)
gen addition rate (A), as indicated in Figure 4a. With standard
parameter values (Table 2), Equation 15 predicts a sustainable
yield of Ys = 29.6 m3 ha−1 year −1; this corresponds to the point
marked A in Figure 3b, where Uo = 181 kg N ha−1 year −1 and
so (from Equation 2 and Table 2) the mineralization rate is
M = 173 kg N ha−1 year −1.
Case 2: harvesting and regeneration burn
In this case, Ω(Uo) is given by:
Ω(Uo ) =
Figure 3. (a) General scheme of the graphical analysis of long-term
sustainable yield. The growth response constraint describes the positive relationship between maximum MAI (Y) and plant-available nitrogen supply (Uo) in any given rotation (Equation 3b), where Uo
(Equation 2) includes external inputs (assumed constant) and net mineralization (which varies over successive rotations, but is assumed
constant within each rotation). The steady-state N balance constraint is
the relationship between the steady-state values of maximum MAI (Ys)
and Uo, defined by the condition that N inputs to the system equal N
losses summed over a rotation (Equation 13a); above this curve, N
outputs exceed N inputs, so that N supply and maximum MAI will
decrease over successive rotations (e.g., sequence A1, A2, ...); below
this curve, the opposite occurs (e.g., sequence B1, B2, ...). In either
case, the system approaches the steady state (point SS) where both
constraints are satisfied; the sustainable yield is given by the height of
SS above the horizontal axis. (b) Numerical examples for three scenarios of N removal: harvesting only, harvesting + regeneration burn, and
harvesting + regeneration burn + leaching and gaseous emissions;
points A, B and C indicate the sustainable steady states, respectively.
Parameter values are given in Table 2.
independent of Uo, giving a horizontal line for the steady-state
N balance constraint curve (Figure 3b). This reflects that, in the
absence of Uo-dependent nitrogen losses from fire and leakage,
Equation 10 implies that the steady-state, rotation-averaged
rate of nitrogen loss from harvesting (i.e., H/T, which, from
Equations 5 and 11, is proportional to Ys) must equal the nitro-
Ffl(U o)
,
T(U o)
(16)
which describes an increasing function of Uo (mainly due to
increasing Ffl rather than decreasing T, according to the results
of DM). Therefore, the steady-state N balance constraint curve
(Equation 13a) has a negative slope (Figure 3b), as discussed
above.
At the intersection of the two constraint curves (point B,
Figure 3b), Ys = 15.0 m3 ha−1 year −1. Therefore, compared to
Case 1 (harvesting only), the extra losses of nitrogen associated with regeneration burning reduce the sustainable yield by
a factor of 2. The steady-state nitrogen budget of the system is
shown in Figure 4b. In this example, the rotation-averaged
nitrogen loss rates from harvesting (4.1 kg N ha−1 year −1) and
from regeneration burning (3.9 kg N ha−1 year −1) are approximately equal.
Table 3 shows the sensitivity of sustainable yield to changes
in the values of individual model parameters, with Case 2 as
the reference case. The effect of an increase in the rate of
nitrogen addition (A) or litter turnover (γl) is to raise the height
of the N balance constraint curve, and thus the intersection
point with the growth response constraint curve. Similarly,
increasing the foliage senescence rate (γf) raises the height of
the N balance constraint curve (by reducing the amount of fuel
available to burn), but also lowers the growth response constraint curve (by reducing foliage biomass and carbon production at canopy closure, see DM); however, the net effect is an
increase in sustainable yield. Increases in the other parameters
MODEL OF SUSTAINABLE YIELD
179
Table 3. Predicted percentage change in sustainable yield (∆Ys, Equations 13a, 14 and 3b) due to a 10% increase in individual model
parameters from their values in reference Case 2 (harvesting and
regeneration burn, no leakage), as given in Table 2 (with ξ = 0). The
sensitivities to stemwood N/C ratio (νw) and foliage turnover rate (γf)
are also shown; the reference values are νw = 0.002 kg N kg −1 C and
γf = 0.333 year −1, as given by Dewar and McMurtrie (1996).
Parameter increased by 10%
∆Ys (%)
A
fh
ffburn
flburn
fwburn
γl
ψ
νw
γf
+7.6
−2.7
−1.7
−0.9
−1.2
+0.8
−0.9
−4.9
+0.8
duction. Table 3 indicates that, for Case 2 as the reference case,
Ys is most sensitive to changes in A and νw, moderately sensitive to the harvest fraction (fh), and least sensitive to changes
in the parameters defining nitrogen losses from burning litter
(flburn, γl, ψ), live foliage (ffburn , γf) and woody slash (fwburn).
Case 3: harvesting, regeneration burn and leakage
The inclusion of leaching and gaseous emission losses (which
are positively related to nitrogen supply, see Equation 9) lowers the steady-state N balance constraint curve (which also
becomes slightly steeper). As a result, the sustainable yield is
reduced to 13.2 m3 ha−1 year −1 (point C, Figure 3b). In this
example, leakage accounts for 15% of the nitrogen loss in the
steady state (Figure 4c).
Nitrogen additions required to sustain a given yield
Figure 4. Ecosystem nitrogen budgets for the sustainable steady states
of Figure 3b corresponding to (a) point A (harvesting only), (b) point
B (harvesting + regeneration burn), and (c) point C (harvesting +
regeneration burn + leakage). Numbers next to the arrows are the
steady-state, rotation-averaged fluxes of nitrogen (kg N ha −1 year −1)
associated with external inputs (A = fertilizer + fixation + deposition),
removal of harvested stemwood (= H/T ), burning of foliage and
aboveground litter (= Ffl /T ) and stemwood slash (= Fw /T ), and
leakage [(1/T )∫ L dt]. Also shown are the steady-state values of
sustainable yield (Ys, m3 ha −1 year −1), available nitrogen supply
(Uo, kg N ha−1 year −1), net mineralization (M, kg N ha −1 year −1), and
optimal rotation length (T, year).
in Table 3 lower the height of the N balance curve; in addition,
an increase in the stemwood N/C ratio (νw) lowers the height
of the growth response curve also (e.g., see DM, their Table 4)
by reducing the amount of nitrogen available for foliage pro-
The above analysis may be inverted to estimate the nitrogen
addition rate (A) that is required to sustain a given maximum
MAI in the long-term (Ys), including the case of maintaining
current yield. In the case of harvesting only (Case 1), from
Equation 15 we obtain:
A(Ys ) =
ρfcw
fh νw Ys,
10 4
(17)
which predicts a linear relationship between A and Ys (Figure 5). More generally, Equation 13a gives:
A(Ys ) =
ρfcw
[ fh + (1 − fh )fwburn ]ν wYs + Ω(Uo).
10 4
(18)
To calculate Ω(Uo) for a given value of Ys, the required
nitrogen supply rate is just the value of Uo at which Y(Uo) = Ys,
which may be calculated numerically by inverting Equation 3b. This value of Uo is then substituted into Equation 13b
to calculate Ω(Uo) (see Appendix A). Equation 18 is illustrated
180
DEWAR AND MCMURTRIE
Figure 5. The predicted nitrogen addition rate (A) required to sustain
a given long-term yield (Ys), as a function of Ys for two scenarios of N
removal: harvesting only (Equation 17), and harvesting + regeneration
burn (Equation 18). Parameter values are given in Table 2.
in Figure 5 for Case 2 (harvesting + regeneration burn). In this
case, in order to sustain a given yield, the required nitrogen
addition rate is approximately double the value required when
the only nitrogen loss is from removal of harvested biomass
(Case 1).
Discussion
Few observational studies exist of the long-term effects of
harvesting and fire on tree growth. What little has been reported includes positive, negative and null responses, possibly
because measurements have been made on different time
scales (e.g., Balneaves 1990, Raison et al. 1990). Short-term
responses to fire are usually favorable to plant nutrition (which
may be reflected in growth). Removal of woody litter and slash
at harvest (e.g., through regeneration burn) may also be beneficial to plant nutrition in the short term; this is so because
woody litter has a low N/C ratio and can immobilize large
amounts of soil N (e.g., Pearson et al. 1987). Reported longerterm growth responses tend to be more negative (e.g., Balneaves 1990). However, Abbott and Loneragan (1986) found
no detectable effect of regular prescribed burning on the
growth of jarrah forest. O’Connell and Grove (1991) suggested
that this may reflect the fact that many ecosystems are currently buffered against disturbance by nutrient reserves in the
soil.
In the long term, however, it seems inescapable that the
cumulative effect of a negative net balance of nitrogen must be
a reduction in the supply of plant-available nitrogen, and
consequently a decrease in tree growth, until a steady state is
reached in which nitrogen inputs balance outputs (e.g., Figure 3a, sequence A1, A2, ..., SS). An analogous conclusion
applies if nitrogen additions were to exceed current losses,
resulting in an increase in tree growth toward the steady state
(e.g., Figure 3a, sequence B1, B2, ..., SS).
We have examined sustainable yield purely as a steady-state
problem. As a result, we can say nothing about how long the
system takes to approach the steady state, or about the likely
magnitude of changes in nitrogen supply and productivity
from one rotation to the next (the sequences A1, A2, ... and B1,
B2, ... in Figure 3a are purely schematic examples). To address
these questions would require using a model that links plant
and soil nutrient cycling in a dynamic fashion (e.g., King
1995). The advantage of the steady-state analysis, however, is
that we do not need to know any details about the dynamics of
nitrogen cycling in the soil and its response to harvesting and
fire. Instead, the condition for steady-state N balance (Equation 10), when combined with the growth response predicted
by DM (Equation 3b), provides a sufficient constraint on the
steady-state mineralization rate to determine the sustainable
yield.
Our analysis suggests that, for Eucalyptus, the most important factor that limits long-term sustainable yield is likely to be
fire (here, regeneration burning). In the examples illustrated in
Figure 3b, harvesting alone results in a relatively large sustainable yield (≈ 30 m3 ha −1 year −1), which is decreased by a factor
of 2 when N removals from regeneration burning are included.
Leaching and gaseous emissions (which represent relatively
small fluxes in many Australian forests) play a minor role. In
addition, the sensitivity analysis (Table 3) shows that reliable
estimates of the nitrogen addition rate (A) and stemwood N/C
ratio (νw) are required to quantify sustainable yield.
Figure 3 is analogous to Comins and McMurtrie’s (1993)
graphical prediction of the equilibrium net primary productivity of unmanaged forests (see their Figure 6). Their ‘‘N-cycling
constraint’’ is equivalent to our steady-state N balance constraint, both being based on the conservation of nitrogen at
equilibrium (or rotation-averaged steady state, as here). Their
‘‘photosynthetic constraint’’ is analogous to our growth response constraint, although the former is based on the conservation of carbon at equilibrium, whereas the latter is based on
the relationship between stemwood growth and nitrogen supply that exists in any given rotation (not just in the steady state).
In Comins and McMurtrie (1993), the constraint curves are
related to leaf N/C ratio (νf) rather than to nitrogen supply.
As with any model, a number of assumptions underlie the
predictions illustrated in Figures 3--5 (see Discussion in DM).
One important assumption is that Uo (Equation 2) is constant
within a rotation, so that the mineralization rate (M) represents
an effective rotation-averaged value. As a result, the predicted
response of tree growth to nitrogen supply does not take into
account within-rotation variations in M (associated with the
dynamics of litter and soil decomposition), which may be
important for nitrogen uptake by trees, especially around canopy closure. The numerical predictions do not include the
effects of mechanical disturbance and compaction of soil,
prescribed fuel-reduction burning, or feedbacks between fire
and the activity of nitrogen-fixing legumes (here, A is a constant). Also, we have assumed that standard silvicultural practice applies whereby the stand is harvested at maximum MAI,
and have not considered the implications of treating the rotation length as a free parameter.
Nevertheless, the graphical analysis presented here is sufficiently general to provide a useful framework for incorporating
MODEL OF SUSTAINABLE YIELD
some of these factors and examining their effects on sustainable yield. This analysis may also be useful for extending the
application of the present model to the effects of climate
change and increases in atmospheric CO2 concentrations on
sustainable yield, or for interpreting the behavior of more
complex models of sustainable forest growth (e.g., King
1995).
Acknowledgments
This work was supported by the NGAC Dedicated Greenhouse Research Grants Scheme and the Australian Research Council. We are
grateful to Tony O’Connell and John Raison for providing references
on their field studies of forest fire ecology.
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Appendix A. Evaluation of Ω(Uo)
The rate of nitrogen loss (Ω(Uo)) from burning foliage and
aboveground litter, and from leakage, averaged over one rotation, is given by Equation 13b:
T(U o)
F
(U
)
+
L
(t,U
)dt
Ω(Uo ) =
fl
o
o
.
∫
T(U(o )
0
1
(A1)
Here, Uo (kg N m −2 year −1) is the nitrogen supply rate (Equation 2), T (year) is the optimal rotation length, Ffl (kg N m −2)
is the amount of nitrogen lost in burning foliage and litter,
L (kg N m −2 year −1) is the rate of nitrogen loss from leaching
and gaseous emissions, and the integral is with respect to stand
age (T) over one rotation length. Each of the quantities T, Ffl
and L can be evaluated analytically as functions of Uo using the
results of the stemwood growth model of DM.
First, T(Uo) is given by Equation 4 in which λ(Uo), the rate
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DEWAR AND MCMURTRIE
of approach of carbon production towards equilibrium at canopy closure, has been calculated by DM (their Appendix C,
Equation C5). Second, Ffl(Uo) is given by Equation 8, where
νf* and G* (the foliage N/C ratio and carbon production at
canopy closure, respectively) have been calculated as functions of Uo by DM (their Appendix B, Equations B3 and B9,
respectively). Finally, from Equations 1 and 9, L(t,Uo) is given
by:
ξUo Kr
,
L(t,Uo ) =
Kr + cr(t,U0)
(A2)
where cr(t, Uo) is the amount of fine root carbon at stand age t
and nitrogen supply rate Uo, and is given from the model of
DM by:
cr(t,Uo ) =
ηrG∗(Uo )
λ(U o) e−µ rt − µr e−λ(U o)t
,
1 −
µr
λ(U o) − µr
(A3)
where ηr is the fraction of carbon production allocated to fine
roots, and µr is the combined specific rate of fine root maintenance respiration and senescence. Combining Equations A2
and A3, the integration of L in Equation A1 can be performed
numerically for a given value of Uo.