A Carbon-Balance Model of Stand
Growth: A Derivation Employing the
Pipe-Model Theory and the Self
Thinning Rule
Valentine, H.T.
IIASA Working Paper
WP-87-056
June 1987
Valentine, H.T. (1987) A Carbon-Balance Model of Stand Growth: A Derivation Employing the Pipe-Model Theory and the
Self Thinning Rule. IIASA Working Paper. IIASA, Laxenburg, Austria, WP-87-056 Copyright © 1987 by the author(s).
http://pure.iiasa.ac.at/2996/
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WORKING PAPER
A CARBON-BALANCE MODEL OF
STAND GROWTH: A DERNATION EMPLOYING
THE PIPEMODEL THEORY AND THE
SELF-THINNING RULE
Harry T. Valentine
June 1987
WP-87-056
-
International Institute
for A p p l ~ e d Systems Analysis
A Carbon-Balance Model of Stand Growth:
a Derivation Employing the Pipe-Model Theory
and the Sell-Thinniry Rule
Harry T. Valentine
J u n e 1986
WP-87-56
Working Papers are interim r e p o r t s on work of t h e International Institute
f o r Applied Systems Analysis and have received only limited review. Views
o r opinions expressed herein do not necessarily r e p r e s e n t those of t h e
Institute o r of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
A-2361 Laxenburg, Austria
Preface
This p a p e r r e p r e s e n t s a n advance in t h e e f f o r t of IIASA's Acid Rain P r o j e c t t o
develop a model of t h e possible damage caused by acid r a i n t o forests. The p a p e r reports a derivation of a stand-level model t h a t describes how changes in t h e rate of
carbon fixation and p a t t e r n of allocation, as affected by pollutants, may affect t h e
growth and yield of a f o r e s t stand and i t s susceptibility t o decline syndrome. The pap e r also r e p r e s e n t s a contribution t o t h e field of f o r e s t ecology. In t h e course of t h e
derivation, connections among different theories and observations of stand s t r u c t u r e ,
self-thinning, and metabolism a r e demonstrated which previously had gone unnoticed.
Leen Hordi jk
Leader
Acid Rain P r o j e c t
Abstract
The pipe-model theory is used as a framework f o r t h e derivation of models
describing t h e growth of a v e r a g e s t e m length, total basal a r e a , and t o t a l volume of an
even-aged, self-thinning, mono-species stand. Variations of t h e models a r e derived f o r
two situations: (1) where t h e annual rates of s u b s t r a t e production and feeder-root
turnover can b e assumed constant o v e r time, and (2) where these rates a r e expected t o
change o v e r time, such as in polluted environments. The model describing t h e growth
of stand volume f o r t h e f i r s t situation has been studied previously and shows good
agreement with yield tables. Growth r a t e models applicable t o individual t r e e s are
described and p r e f e r r e d o v e r similar models derived previously by t h e author.
Table of Contents
1.
Introduction
2.
Model Framework
3.
Carbon Balance
4.
Stand-Level Model
5.
Discussion
References
A C a r b o n - B a l a n c e Model o f Stand Growth:
a D e r i v a t i o n Ehploying the Pipe-Model Theory
and the Sell-Thinning Rule
Harry T. Valentine
USDA Forest Service
5 1 Mill Pond Road,
Hamden, CT 06514, U.S.A.
1. MTRODUCTION
Carbon-balance modeling (.e.g., McMurtrie and Wolf, 1983; Makela and Hari, 1986), t h e
pipe-model theory (Shinozaki et al., 1 9 6 4 a , b ; Oohata and Shinozaki, 1979; Waring et
al., 1982), and t h e self-thinning of even-aged stands (e.g., Yoda et aL., 1963; White,
1981; Westoby. 1984) have been t h r e e important, though r a t h e r distinct, topics of
r e s e a r c h in f o r e s t ecology. Recently, however, t h e pipe-model theory w a s used in connection with derivations of carbon-balance models of t h e growth of individual trees
(Valentine, 1985; Makela, 1986). In t h e p r e s e n t a r t i c l e , I use t h e pipe-model theory as
a framework f o r t h e derivation of a model of t h e growth of a n even-aged, self-thinning,
mono-species stand. Two variations of t h e model are described: 1) f o r stands where
t h e annual rates of s u b s t r a t e production and feeder-root turnover can be considered
constant o v e r time, and 2 ) f o r stands where t h e s e rates are expected t o change o v e r
time due t o changing environments caused by a i r pollution/acid r a i n o r o t h e r factors.
The pipe-model theory of Shinozaki et aL. (1964a ,b ) provides a simple interpretation of t h e s t r u c t u r e s of plants and stands. In t h e p r e s e n t model, a stand of trees is assumed t o b e comprised of leaves, f e e d e r roots, active pipes, and disused pipes (Fig. 1 ) .
The woody components of t h e stand
-
branches, stems, and t r a n s p o r t r o o t s
-
are
modeled in aggregate a s a n assemblage of active and disused pipes. Active pipes extend from leaves t o f e e d e r r o o t s , and have both supporting and vascular functions. In
accordance with t h e pipe-model t h e o r y , a constant r a t i o i s maintained between total
foliar d r y matter and t h e total cross-sectional a r e a of t h e active pipes in t h e stand. In
addition, I assume t h a t feeder-root d r y matter is proportional t o foliar d r y matter.
Disused pipes have a supporting function but do not s e r v e a s vascular connections
between leaves and f e e d e r roots. An active pipe becomes disused when t h e foliage and
f e e d e r r o o t s attached t o i t die ( o r are not renewed). The distal portion of a disused
pipe i s lost from t h e stand when t h e branch of which it i s a p a r t , i s shed or pruned off.
However, t h e basal portion of a disused pipe remains a p a r t of a specific t r e e and a
p a r t of t h e stand until t h a t tree dies. Consequently, t h e basal a r e a of a stand equals
t h e aggregate basal area of i t s active pipes (active-pipe a r e a ) plus t h e aggregate
basal area of i t s disused pipes (disused-pipe a r e a ) at 1 . 3 7 m .
f i g u r e 1. In this pipe model, active pipes connect leaves t o f e e d e r r o o t s and have
both vascular and supporting functions. Disused pipes vestiges of old active pipes
no longer connect leaves to f e e d e r r o o t s and have only a supporting function. The active and disused pipes, in aggregate, r e p r e s e n t all of t h e woody components of t h e
stand: branches, boles, t r a n s p o r t roots.
-
-
The development of a n even-aged stand can b e divided into two s t a g e s according t o
foliar dynamics. In t h e f i r s t stage, t h e total foliar d r y matter of t h e stand increases
o v e r time. The growth of t h e basal area of t h e stand in t h i s s t a g e i s due, in l a r g e p a r t ,
t o t h e growth of active-pipe area which is proportional t o t h e growth of foliar d r y
matter. The second s t a g e of a stand's development begins when foliar d r y matter
r e a c h e s i t s maximum. In some stands t h e maximum may be r e a c h e d within four y e a r s
a f t e r establishment (Mohler et aL., 1978). Cross-sectional samples of stands have furnished evidence t h a t once t h e maximum of foliar d r y matter i s r e a c h e d , it tends t o be
sustained until senescence (Marks, 1974; Sprugel, 1974; Mohler et aL., 1978). Other
cross-sectional d a t a have been i n t e r p r e t e d a s indicating t h a t foliar d r y matter peaks
and then declines slightly, but rapidly. t o a constant level t h a t i s sustained until senescence (Forrest and Ovington, 1970; Kira and Shidei, 1967). Regardless of whether t h e
maximum of foliar d r y m a t t e r is a peak o r a plateau, t h e growth of t h e basal area of a
stand in t h e second s t a g e of development i s due t o t h e growth of t h e aggregate basal
a r e a of disused pipes. I n c r e a s e s in t h e active-pipe areas of some trees are compensated by reductions in t h e active-pipe areas of o t h e r trees. A tree dies when i t s activepipe a r e a d e c r e a s e s to zero.
2. MODEL FRAMEWORK
The p r e s e n t model applies to stands in t h e second stage of development, where foliar
d r y matter and active-pipe a r e a are maximal ( o r nearly s o ) and more o r less constant.
In deriving t h e model, I assume t h a t ingrowth into a n even-aged stand i s nil; all changes
in t h e tree count are negative and due t o tree death.
VariabLes
N = t h e number of trees in t h e stand.
A
= t h e total active-pipe area of t h e stand (m2).
5 = t h e a v e r a g e basal area p e r tree (m2).
B = t h e total basal area of t h e stand (m2).
L = t h e a v e r a g e length of a n active pipe (m).
V = t h e total volume of t h e stand (m3).
V' = t h e total aboveground volume of t h e stand (m3).
X = t h e total disused-pipe area of t h e stand (m2).
Growth r a t e of b a s a l a r e a
The growth r a t e of t h e basal a r e a of a stand equals t h e sum of t h e net growth r a t e s of
active-pipe and disused-pipe a r e a s :
= dA/dt+dX/dt
dB/dt
(1)
The net growth r a t e of active-pipe a r e a , d A / d t , can be split into a positive fraction
(dA + / d t ) and a negative fraction ( - d A - / d t ) , i.e., d A / d t
= dA + / d t - d A - / d t .
The
positive fraction is t h e r a t e of production of new active-pipe a r e a by the living t r e e s
in the stand, and in t h e absence of t r e e mortality, this rate equals t h e growth rate of
the total basal a r e a of t h e stand. The negative fraction, 4 . A - / d t ,
is t h e r a t e of
conversion of active-pipe a r e a t o disused-pipe a r e a . This conversion neither adds n o r
subtracts from the basal area of t h e stand. Similarly, t h e net growth r a t e of disusedpipe a r e a , dX/ d t , can be split into a positive fraction (dX+/ d t ) and a negative fraction (*-/
.
d t ), i.e., dX/ d t = d ~ + d/t a-/
d t The positive fraction equals dA -/ d t .
The negative fraction is the r a t e at which disused-pipe a r e a is lost from t h e stand t o
mortality. Because -dA -/ d t +dXf / d t
= 0, t h e growth rate of stand basal a r e a can be
rewritten as:
dB/dt = d~+/dt*-/dt
Growth r a t e of active-pipe Length
The growth r a t e of average, active-pipe length, dL / d t , also can be split into two f r a c tions: a metabolic fraction (dLM/ d t ) and a numerical o r non-metabolic fraction
(dLN/dt), i.e.,
dL/dt
= dLM/dt fdLN/dt
(3)
The metabolic fraction, d l g / d t , is t h e rate at which average, active-pipe length inc r e a s e s due t o the apical growth of shoots and roots. The non-metabolic fraction,
dLN/ d t , is t h e r a t e at which average, active-pipe length increases due t o t h e suppression and disuse of pipes t h a t a r e , on average, s h o r t e r than L .
A model of t h e metabolic r a t e , d L M / d t , is derived below, along with a model of
d A + / d t , with a carbon-balance approach. First, however, I show t h a t both t h e nonmetabolic r a t e , d L N / d t , and t h e rate of loss of disused-pipe a r e a t o mortality,
dX-/d t , can be expressed a s functions of
dA +/
dt
.
Disuse of s h o r t pipes
By definition, t h e growth rate of t h e active-pipe volume of a stand is
d(AL)/dt
= A(dLM/dt -MLN/dt)+L(dA + / d t - d A - / d t )
(4)
and t h e rate of production of new active-pipe volume, through metabolic processes, is
d t +AdLM/ d t . Disused pipes do not increase in length, s o if t h e a v e r a g e length
LdA + /
of a deactivating pipe is 19L (0
< 19 < I ) ,
then t h e rate of production of new disused-
pipe volume is -I9L dA -/ d t and conservation of active-pipe volume r e q u i r e s that:
-(l-.IP)L dA -/ d t +AdLN/ d t = 0
(5)
Substituting d A + / d t -dA / d t f o r dA -/ d t and solving f o r t h e non-metabolic fraction of
t h e growth rate of average, active-pipe length, t h e r e f o r e
dLN/dt
= [ ( 1 4 ) L / ~ ] ( d A + / d t- d A / d t )
Restricting consideration t o stands f o r which dA / d t
= 0 and A
(6)
A,,,
eqn (6) reduces
to
To derive a model f o r dX-/dt,
I use Reineke's (1933) p r e c u r s o r of t h e self-thinning
rule. The self-thinning of even-aged stands has been a n active topic of ecological
r e s e a r c h f o r t h e last two decades (e.g., Yoda et al.,1963; White, 1981; Westoby, 1984)
and a topic of f o r e s t r y r e s e a r c h f o r considerably longer. Numerous r e p o r t s in t h e
f o r e s t r y l i t e r a t u r e since t h e seminal p a p e r by Reineke (1933) indicate t h a t t h e number
of trees comprising an even-aged, mono-species stand, given an a v e r a g e basal a r e a p e r
tree,
g, is N 5 NR, where
and c (
0.8 f o r most species) and k a r e constants. Multiplying eqn (9), t h e time
derivative of eqn (8). by a n a r b i t r a r y s c a l a r , -$, and r e a r r a n g i n g gives:
-$SdlYR/dt
-$cNRG/dt
=O
By definition, B = i N and t h e r e f o r e
d ~ / d t= B m / d t + ~ d B / d t
(11)
Adding z e r o in t h e form of t h e left-hand side of eqn (10) t o t h e right-hand side of eqn
(11) gives t h e dynamics of B , B, and NR of a stand growing according t o eqns (8) and (9)
(i.e., a stand f o r which N = NR):
d~ / d t = ( 1+C
) N ~ ~ dBt +(I
/ +)B~w/
~dt
If $ is scaled such t h a t t h e a v e r a g e basal area of a dying tree i s (l-$)B,
(12)
then
( ~ + ) B c ~ ~ / di st t h e rate t h a t basal area is lost from t h e stand t o mortality which
equals 4 X - / d t
of eqn (2). Setting t h e positive and negative fractions of eqn (2) equal
to those of eqn (12), t h e r e f o r e :
d~ +/
d t = (l+c
4 - / d t
) ~ ~ t G / d t
=(~-$)Zc~~/dt
Solving eqns (13) and (14), respectively, f o r @ / d t
(13)
(14)
and cWR/dt, and inserting t h e
resultant expressions into (9),yields
dx-/ d t = cpdA / d t
+
where 9 = c (I-$)/
(15)
(1-c $) i s assumed constant. Substituting eqn (15) into (2). t h e r e -
fore
3. CARBON BALANCE
I use a carbon-balance a p p r o a c h (Thornley, 1976; Chapt. 6) to d e r i v e models of
dA / d t and dLM/ d t . Dry m a t t e r of foliage, f e e d e r roots, and pipes i s measured in kg
+
of equivalent COz as i s dry-matter s u b s t r a t e produced by t h e foliage and consumed in
respiration f o r construction o r maintenance of foliage, f e e d e r r o o t s , and active pipes.
FoLiar d r y m a t t e r
The following quantities p e r t a i n t o t h e growth, maintenance, and renewal of foliar d r y
matter:
z
= units of foliar d r y matter in mid-summer p e r unit active-pipe a r e a (kg
[co2l/mZ).
r, = units of d r y matter s u b s t r a t e consumed in t h e construction of a new unit of
foliar d r y matter.
6,
= units of dry-matter s u b s t r a t e consumed p e r y e a r f o r maintenance of a unit of
foliar d r y matter. In addition, (r, + l ) / T, units of dry-matter s u b s t r a t e are
consumed p e r unit of foliar d r y matter p e r y e a r f o r renewal, where
T, = t h e a v e r a g e ultimate leaf a g e (growing seasons) f o r t h e species.
I t follows from these definitions t h a t t h e total foliar d r y matter in t h e stand i s zA. The
rate of production of new foliar d r y matter i s zdA / d t and t h e rate of constructive
+
respiration f o r new foliar d r y matter i s zr,dA
+
/dt
.
The rates of production and con-
structive respiration a r e defined with d A / d t because i t is t h e positive fraction of
+
d A / d t . The negative fraction of dA / d t i s t h e rate of production of new disused-pipe
a r e a , f o r which t h e r e is no corresponding consumption of dry-matter substrate. The
rate of respiration f o r maintenance and renewal of extant foliar d r y matter on a n annual basis i s z [b, +(r, + I ) / T,
M.
Feeder-root d r y m a t t e r
Constants analogous t o z , r,,
b,,
and T,
a r e assumed to apply t o feeder-root d r y
matter and a r e denoted, respectively, by j', r f , bf, and Tf. Feeder-root d r y matter in
t h e stand is P A , t h e rate of production of new feeder-root d r y matter i s j'dA
+/
d t , the
rate of constructive respiration i s j'rf dA +/d t , and t h e rate of respiration f o r maintenance and renewal is j'[bf +(rf +1) / Tf
d r y matter a r e defined as follows
M.
Combined terms f o r foliar and feeder-root
z'=z+f
f
' = Z f z +fff
b ' = ~ [ b +, ( r z +I)/ Tz]+f[bf +(rf +I)/ Tf]
Thus, t h e r a t e s of production and constructive respiration of new foliar plus feederroot d r y matter a r e denoted, respectively, by z 'dA
+/
d t and r 'dA + /d t . The r a t e of
respiration f o r maintenance plus renewal of existing foliar plus feeder-root d r y matter
is b ' A .
Woody d r y m a t t e r
Quantities pertaining t o t h e growth and maintenance of active-pipe (woody) d r y matter
a r e defined as follows:
u = units of woody d r y matter p e r unit woody volume (kg [co2]/m3).
rp = units of s u b s t r a t e consumed in t h e construction of a new unit of woody d r y
matter.
bp
= units of dry-matter s u b s t r a t e consumed p e r y e a r f o r maintenance of a unit of
active-pipe d r y matter. Disused-pipe d r y matter i s assumed t o be devoid of
respiring cells.
The total active-pipe d r y matter in t h e stand i s uAL and t h e rate of respiration f o r
maintenance of t h e living woody d r y matter is ubpAL. The rate of production of new
active-pipe d r y matter i s u (L dA +/d t +AdLM / d t ) and, t h e rate of constructive
respiration f o r new active-pipe d r y matter i s urp (L dA + / d t
+ AdLM / d t ).
Dty-matter s u b s t r a t e
Production of dry-matter s u b s t r a t e by a stand is a function of t h e foliar d r y matter.
For t h e p r e s e n t model:
a = t h e maximum units of dry-matter s u b s t r a t e produced p e r y e a r p e r unit foliar
dry matter.
To account f o r a suboptimal environment f o r s u b s t r a t e production, t h e actual rate of
s u b s t r a t e production i s defined in terms of a scaling variable (I)s o t h a t
a l = t h e actual units of dry-matter s u b s t r a t e produced p e r y e a r p e r unit foliar d r y
matter.
Thus, I v a r i e s between 0 and 1 , and azAl is t h e r a t e of dry-matter s u b s t r a t e production by a stand.
Production rates
The sum of t h e rates of production and constructive respiration of new foliar, feederr o o t , and active-pipe d r y matter in t h e stand equals t h e rate of s u b s t r a t e production
minus t h e sum of t h e rates of maintenance respiration f o r existing foliar, feeder-root,
and active-pipe d r y matter
u(l+tp)AdLM/dt
+ [z'+r'+u(1+rp)L]dA+/dt = A [ a z l - b ' u b p L ]
(17)
A s was noted, branches, stem, and t r a n s p o r t r o o t s a r e modeled in a g g r e g a t e a s an assemblage of active and disused pipes.
The lengthening of existing active pipes
corresponds roughly t o apical growth, and t h e production of new active pipes
corresponds roughly t o cambial growth. Because apical growth tends t o o c c u r o v e r a
more o r less fixed portion of e a c h growing season f o r many species, i t s e e m s natural t o
assume t h a t a constant proportion of t h e available dry-matter s u b s t r a t e i s used f o r
t h i s growth and associated constructive respiration. Thus, eqn (17) i s split into 2
p a r t s according to t h e type of growth with constant partitioning coefficients, (A) and
(1-A), (0 < A
< 1):
u (l+rp)AdLM/ d t = (1-A)A [ a z l -b ' u b p L ]
[z ' + r ' + u ( l + r p )L ]dA / d t =AA [ a z l -b ' u b p L ]
+
Solving eqns (18) and (19), respectively, f o r dLM/ d t and dA / d t , t h e r e f o r e
+
where
4.
STAND-LEXEL MODEL
Active-pipe l e n g t h
The growth rate of a v e r a g e , active-pipe length i s obtained by substituting eqn ( 2 1 ) into
(6) and adding t h e r e s u l t t o eqn ( 2 0 ) , i.e.,
d L / d t = [aI-bs-bL][(l-A)+A(l-6)L/(z8
+L)]-[(I-6)L/A]dA/dt
When dA/ dt equals 0 , eqn ( 2 2 ) i s i n t e g r a b l e from
(22)
t t o t +At if I is assumed constant
o v e r t h a t span. However, t h e value of L / ( z ' +L ) should b e f a i r l y c o n s t a n t and c l o s e t o
1f o r l a r g e L , s o eqn ( 2 2 ) r e d u c e s t o
d L / d t = n o [ a l - b *- b ~ ]
Integration of t h i s simpler function yields
L ( t + A t ) = nl+n.$(t)+n3L ( t )
where
no
"119A
nl = -b ' ( 1 - n 3 ) / b
nz = a(1-n3)/ b
n3
= e x p ( -cob At )
It i s noted, o n t h e o n e hand, t h a t if I were constant f o r all
t , t h e n eqn ( 2 4 ) would
r e d u c e t o a Mitscherlich growth function in L . On t h e o t h e r hand, if t h e specific rate
of f e e d e r - r o o t t u r n o v e r (T;)
from
i s v a r i a b l e , b u t c a n b e assumed constant o v e r t h e s p a n
t t o t + A t , t h e n t h e constant nl in eqn ( 2 4 ) should b e r e p l a c e d by
t o give
L (t + A t ) = n4+n5/ Tf ( t ) + n z I ( t ) + n & .( t )
where
Basal a r e a
The growth rate of basal a r e a of a n actively self-thinning stand is obtained by substituting eqn (21) into (16), i.e.,
d B / d t = (1-cp)AArnaX[al-b8-+L]/(Z*+L)
(26)
Substituting eqn (23) into (26) furnishes t h e r a t e of change of B with r e s p e c t t o L
dB/dL = (1-cp)Urnax/[no(z*+L)l
(27)
Integration of eqn (27) from t t o t +At gives a function describing t h e annual growth of
t h e basal a r e a of t h e stand
B ( t +At)
where n6 = (l-cp)~,,,/
= ~ ( t ) + n ~ f l( nt +[ ~ t ) + ~ ' ] - l n [ ~ ( t ) + ~ * ] j
(28)
no. Substitution of e x p f [ ~ ()--cl]/
t
n6j z I, where c l i s a
constant, f o r L ( t ) into eqn (25) furnishes a model of t h e growth of stand basal a r e a a s
a function of t h e rates of feeder-root turnover and s u b s t r a t e production, i.e.,
B(t +At>
= n61nfn7+n8/ Tf(t ) + n d ( t )+n3exp[B(t
)/ n6] j
(29)
where
Total volume
The growth rate of t h e woody volume of a stand (dV/ d t ) equals t h e r a t e of production
of new active-pipe volume minus t h e rate t h a t disused-pipe volume i s lost t o mortality
and t h e shedding of branches. Let vlL (dA+/ d t -dA / d t ) denote t h e r a t e t h a t volume i s
lost t o shedding of branches in connection with t h e disuse of pipes and let v.&
(a-/
dt)
denote t h e r a t e t h a t disused-pipe volume is lost t o mortality s o t h a t
dV/dt
= LdA + / d t + ~ d L ~ / -L
d t [vl(dA+/dt
4 A /dt)+v2d.X-/dt]
(30)
where vl and vz a r e assumed constant. Eqn (15) applies t o a self-thinning stand where
dA / d t
= 0, in which c a s e eqn (30) r e d u c e s t o
The r a t e of change of V with r e s p e c t to L is obtained by substituting eqns (20), (21),
and (23) into (31), i.e.,
dV/ dL = (A / n o ) [ ( l-A) +(1--vl - v v Z ) U / (2 ' +L )I
Because A w A,,,
when dA / d t
= 0 , and L / ( z ' +L )
(32)
1 f o r l a r g e L , eqn (31) can be in-
tegrated t o yield
V ( t + A t ) = V(t)+nlo[L(t + A t ) - L ( t ) ]
(33)
where nlo = A,ax[1-(v1+~v2)A]/no. Substitution of [V(t)-c2]/ nlO, where c2 is a constant, f o r L ( t )into eqn (25) furnishes a model of growth of woody volume as a function
of t h e rates of feeder-root turnover and substrate production, i.e.,
~ (+ t ~) =
t nll+n12/ T~( t ) + q 3 l ( t)+n3v(t)
(34)
where nll = G ~ ~ ~ ~ + C n12
~ (=
~ n5n10,
- K ; and
~ ) G13
, = n2nlo.
The growth of volume, like t h e growth of L , is described by a Mitscherlich function if I ( t ) and Tf ( t ) a r e constant f o r all t . and if t h e assumptions leading t o eqn (34)
are not overly Procrustean.
Aboveground voLume
Eqn (34) applies t o aboveground plus belowground woody volume, but if aboveground
woody volume, V', i s assumed t o be a constant fraction ( v ) of total woody volume, then
t h e growth fraction of V' is
~ ' (+ t ~) =
t n14+n15/ T~(t )+n161(t)+n17V'(t)
where
K ~ ~ = v ~I cC ~ ~~ =, V K
and
~ ~ n16=Vn13.
,
(35)
In a n invariant environment, where I and Tf
are constant, t h e growth function of V' is
V'(t +At ) = nl8+nl7V'(t )
where nlB=n14+n15/Tf +n161.
5. DISCUSSION
Previously, Khil'mi (1957) used a model analogous t o eqn (36) t o describe t h e
aboveground growth of closed, self-thinning stands and obtained good agreement
between t h e model and yield tables. The parameter n17 w a s shown to b e more o r less
constant among stands of a given species in t h e same geographical region, r e g a r d l e s s
of s i t e quality, whereas t h e value of n18 increased f r o m poor to good sites. The assumptions of t h e p r e s e n t model are in accordance with t h e findings of Khil'mi inasmuch
as n17 is a constant, i.e.,
nI7
= vaexp[(l +A)
bp At / ( 1 + r p )]
and n18 i s a function of t h e rate of feeder-root turnover and t h e r a t e of s u b s t r a t e production which i s known t o v a r y among stands, depending on s i t e quality. These p r o p e r ties also may hold f o r some even-aged, mixed-species stands (Fig. 2).
Stand volume in year t (m3)
f i g u r e 2. Total predicted aboveground volume in y e a r t +10 vs. volume in y e a r t ( t =
10, 20, ......., 90) for mixed oak (@arcus spp.) stands of s i t e index 40, 60, or 80 in t h e
n o r t h e a s t e r n United States. D a t a are f r o m Schnur (1937: Table 12). The common slope
of t h e t h r e e regression lines i s 0.951. I n t e r c e p t s f o r s i t e indices 40, 60, and 80,
respectively a r e 21.1, 35.4, and 50.4 ( r 2 = 0.99; s e = 1.35).
Periodic increment (AV') from t t o t +At in a single self-thinning stand with cons t a n t Tf and I is obtained by subtracting V ' ( t ) from eqn ( 3 6 ) , i.e.,
AV'
= n18 + ( n I 7 -1)V'(t) = n14+n15/ Tf +n161 +(n17-l)V'(t
The asymptotic yield (V',,,)
(37)
of t h e stand equals V ' ( t ) when AV' equals z e r o
Vamax = n18/ (1-n17)
Corresponding to V',,,
)
= (n14+n15/ Tf + n l 6 1 ) / ( 1 - n I 7 )
is t h e asymptotic, average, active-pipe length (L,,,)
(38)
which is
obtained from eqn (25)
L ,,, = (rcl+n5/ Tf + n 2 1 ) / ( 1 - n g )
(39)
Unlike VBmaX,which depends on the values of t h e self-thinning and pruning parameters
(9,
'19,vl,v2), L
depends only on t h e fundamental carbon-balance parameters, i.e.,
Lmax
= IazI -z [ b , + ( r , + I ) / T , 1 - f [ b f + ( r f + I ) / Tf I{/ u b p
(40)
By the present theory, LmaX is t h e active-pipe length (i.e., average stem length from z
units of foliar d r y matter t o j' units of feeder-root d r y matter) at which t h e rate of
s u b s t r a t e production balances the rate of maintenance respiration s o t h a t t h e r e is no
f u r t h e r growth by t h e stand. With constant Tf and I, L,,,
is truly an asymptote, a
value that is approached, but never reached. However, in a changing environment,
where Tf and I o r both can b e expected t o vary in closed, self-thinning stands, a
b e t t e r definition f o r Lmax is potential active-pipe length.
It has been hypothesized t h a t a i r pollution/acid rain is causing o r contributing t o
certain f o r e s t declines in Europe, Japan, and p a r t s of North and South America, and
Southeast Asia. Bossel ( 1 9 8 6 ) noted t h a t such hypotheses fit into t w o categories: 1 )
those concerning d i r e c t damage t o foliage by pollutants with consequent deceleration
of the rate of s u b s t r a t e production, and 2) those concerning damage t o f e e d e r roots o r
mycorrhizae in acidifying soils with consequent acceleration of the r a t e of feeder-root
turnover. The present model suggests that e i t h e r kind of damage should cause dec r e a s e s in stand growth r a t e , potential active-pipe length and, correspondingly, potential yield. VamaX. If changes in T ~ o-r I~ were of such magnitude t h a t
[IC~+IC
Tf(t)+,cZ1(t
~/
)I/ (1-3)
< L (t
t h e n a c t u a l active-pipe length would exceed t h e potential and widespread dieback o r
t r e e death should o c c u r within t h e stand because t h e demand f o r s u b s t r a t e f o r t h e
maintenance of existing living tissue would not b e m e t . This situation should b e more
likely where L ( t ) i s l a r g e , which is consistent with observations t h a t dieback i s more
s e v e r e in o l d e r t h a n in younger stands.
The onset of dieback c r e a t e s a situation f o r which t h e p r e s e n t model does not apply. Self-thinning and t h e constancy of foliar d r y matter a r e d i s r u p t e d by t h e d e a t h o r
dieback of s u p p r e s s o r s , in addition t o s u p p r e s s e d individuals, a n d t h e r e s u l t a n t c r e a tion of g a p s in t h e canopy of t h e stand.
Moreover, t h e metabolic a s p e c t s of s t a n d
growth f o r instances where t h e rate of r e s p i r a t i o n e x c e e d s t h e rate of s u b s t r a t e production i s not formulated in t h e p r e s e n t carbon-balance model.
A situation f o r which t h e application of t h e p r e s e n t model could b e extended is t h e
period b e f o r e t h e canopy closes following t h e establishment of a stand (or following a
silvicultural thinning) where f o l i a r d r y m a t t e r i s increasing and self-thinning is nil o r
just beginning ( o r resuming). Such a n extension should r e q u i r e functions describing
where A
I ( t ) , dA / d t , and dX-/dt
< A,,.
Regarding t h e l a t t e r two r a t e s , it is ex-
pected t h a t in g e n e r a l
0 5 dA/dt 5 dA+/dt
0 5dX-/dt
5 cp(dAf/dt-dA/dt)
where cp is t h e constant of eqn (15) and (dA / d t -dA / d t ) equals t h e rate of production
+
of new disused-pipe a r e a , d X f / d t
.
The d e g r e e of c l o s u r e achieved by a stand b e f o r e
t h e onset of t h e production of disused-pipe a r e a (dA '/ d t
self-thinning (dX-/dt
> 0)
> dA / d t
) o r t h e onset of
depends largely on t h e variation in t h e sizes of t h e trees
and t h e i r spacial arrangement. In n a t u r a l stands of i r r e g u l a r l y s p a c e d t r e e s , t h e ons e t of self-thinning may o c c u r n e a r l y coincident with t h e onset of production of
disused-pipe a r e a , but in plantations of r e g u l a r l y spaced t r e e s , self-thinning may not
o c c u r until a f t e r disused-pipe area comprises a sizable f r a c t i o n of t h e stand's basal
area.
I n d i v i d u a l trees
Previously, I used a pipe-model framework t o derive a model of t h e growth r a t e of a n
individual t r e e (Valentine, 1985). Eqns ( I ) , (3) through (6), and (17) through (23) of
t h e present a r t i c l e also apply at t h e t r e e level, and are p r e f e r r e d o v e r t h e e a r l i e r
model. Because a n individual tree retains t h e basal portions of a l l of i t s disused pipes
within its bole, t h e rate of production of new active-pipe a r e a , d A + / d t , equals t h e
growth rate of t h e basal a r e a of t h e t r e e , d B / d t . Therefore, a t t h e t r e e level, eqn
(21) gives the basal a r e a growth rate, viz.
d B / d t = XA[aI-6' - 6 ~ ] / [ z ' + L ]
(41)
The growth rate of total volume a t t h e tree level is obtained by dropping t h e term involving dX-/ d t from eqn (30) and substituting dB / d t f o r dA +/d t , whence
dV/dt =L[vIdA/dt +(l--,l)dB/dt]-AdLM/dt
(42)
The growth rate of the fraction, w, of t h e total volume t h a t i s aboveground is simply
dV'/ d t = wdV/ d t
(43)
The rates dLg / d t and dL / d t , respectively, a r e described at t h e t r e e level by eqns
(21) and (23) without change. The foliar d r y matter of an individual tree and its rate of
substrate production p e r unit of foliar d r y matter a r e unlikely t o remain constant o v e r
time. Consequently, functions describing dA / d t and I ( t ) are needed t o complete t h e
individual tree model and remain t o be formulated.
REFERENCES
Bossel. H. (1986). Dynamics of f o r e s t dieback: systems analysis and simulation. Ecologi c a l Modelling, 34, 259-288.
Forrest, W.G. and Ovington, J.D. (1970). Organic matter changes in a n age s e r i e s of
Pinus radiata plantations. J. Applied Ecology, 7, 177-186.
Khil'mi, G.F. (1957). Teoreticheskaya biogeofizika Lesa. Izdatel'stvo Akademii Nauk
SSSR, Moskva (in Russian). (Theoretical Forest Biogeophysics. Translated from
Russian by I s r a e l Program f o r Scientific Translation, Jerusalem, 1962). 155 pp.
Kira, T. and Shidei, T. (1967). Primary production and t u r n o v e r of organic matter in
different f o r e s t ecosystems of t h e Western Pacific. J a p a n e s e J. Ecology, 1 7 ,
70-87.
~ a k e l a , ' A (1986).
.
Implications of t h e pipe model theory on d r y matter partitioning and
height growth in t r e e s . J. Theor. Biology, 1 2 3 , 103-120.
Makela, A. and Hari, P . (1986). Stand growth model based on c a r b o n uptake and allocation in individual trees. Ecological Modelling, 33, 205-229.
Marks, P.L. (1974). The role of pin c h e r r y ( P r u n u s p e n s y l v a n i c a L.) in t h e maintenance of stability in n o r t h e r n hardwood ecosystems. Ecological Monographs, 44,
73-88.
McMurtrie, R. and Wolf, L. (1983). Above- and below-ground growth of forest stands: a
c a r b o n budget model. A n n a l s ofBotany, 52, 437-448.
Mohler, C.L., Marks, P.L., and Sprugel, D.G. (1978). Stand s t r u c t u r e and allometry of
trees during self-thinning of p u r e stands. J. Ecology, 66, 599-614.
Oohata, S. and Shinozaki, K. (1979). A statistical model of plant f o r m - f u r t h e r
analysis of t h e pipe model theory. J a p a n e s e J. Ecology, 29, 323-335.
Reineke, L.H. (1933). Perfecting a stand density index f o r even-aged forests. J. Agricul. Res., 46, 627-638.
Schnur, G.L. (1937). Yield, stand, and volume tables for even-aged upland oak forests.
LBDepartment of A g r i c u l t u r e , Technical B u l l e t i n 560. Washington, D.C. 87 pp.
Shinozaki, K., Yoda, K., Hozumi, K., and Kira, T. (1964a). A quantitative analysis of
plant form
t h e pipe model theory. I. Basic analysis. J a p a n e s e J. Ecology, 1 4 ,
97-105.
Shinozaki, K., Yoda, K., Hozumi, K., and Kira, T. (1964b). A quantitative analysis of
plant f o r m
t h e pipe model theory. 11. F u r t h e r evidence of t h e t h e o r y and its
application in f o r e s t ecology. J a p a n e s e J. Ecology, 1 4 , 133-139.
Sprugel, D.G. (1974). Vegetation D y n a m i c s of Wave-Regenerated High-Altitude f i r
Forest. Ph.D. Thesis, Yale University, N e w Haven, CT, USA.
Thornley, J.H.M. (1976). Mathematical Models i n P l a n t Physiology. Academic P r e s s ,
London, 318 pp.
Valentine, H.T. (1985). Tree-growth models: derivations employing t h e pipe-model
theory. J. Theor. Biology, 1 1 7 , 579-585.
Waring, R.H., S c h r o e d e r , P.E., and Oren, R. (1982). Application of t h e pipe m o d e l
t h e o r y to p r e d i c t canopy leaf area. C a n a d i a n J. Forest Res., 1 2 , 556-560.
Westoby, M. (1984). The self-thinning rule. Advances i n Ecological Res., 1 4 , 167-225.
White, J. (1981). The allometric interpretation of t h e self-thinning rule. J. Theor.
Biology, 89, 475-500.
Yoda, K., Kim, T., Ogawa, H., and Hozumi, K. (1963). Self-thinning in overcrowded p u r e
stands under cultivated and natural conditions (inter-specific competition among
higher plants XI). J. Biology, O s a k a City U n i v e r s i t y , 1 4 , 107-129.
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