A Carbon-Balance Model of Stand Growth: A Derivation Employing the Pipe-Model Theory and the Self Thinning Rule

W O R K I N G P A P E R

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

-

I n t e r n a t i o n a l I n s t i t u t e 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 i t s 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 f o r e s t s . The p a p e r re- p o r t s a derivation of a stand-level model t h a t d e s c r i b e s 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 a f f e c t 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 pa- p e r a l s o 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 c o u r s e of t h e derivation, connections among different t h e o r i e s 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 t h e o r y 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, t o t a l basal a r e a , and t o t a l volume of a n 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 t u r n o v e r c a n b e assumed constant o v e r time, and (2) where t h e s e 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 a u t h o r .

Table of Contents

1. Introduction 2. Model Framework 3. Carbon Balance 4. Stand-Level Model 5 . Discussion

R e f e r e n c e s

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 S e r v i c e

5 1 Mill Pond Road, Hamden, CT 06514, U.S.A.

1. MTRODUCTION

Carbon-balance modeling (.e.g., McMurtrie a n d Wolf, 1983; Makela and Hari, 1986), t h e pipe-model t h e o r y (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 s t a n d s (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 t h e o r y w a s used in con- nection 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 t h e o r y 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 s t a n d s where t h e annual rates of s u b s t r a t e production and f e e d e r - r o o t t u r n o v e r c a n be considered constant o v e r time, and 2 ) f o r s t a n d s 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 f a c t o r s .

The pipe-model t h e o r y of Shinozaki et aL. (1964a ,b ) provides a simple i n t e r p r e t a - tion 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 s t a n d of trees is as- sumed t o b e comprised of leaves, f e e d e r roots, a c t i v e pipes, and disused pipes (Fig. 1 ) . The woody components of t h e s t a n d

-

b r a n c h e s , stems, 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 a n assemblage of a c t i v e and disused pipes. Active pipes ex- tend from leaves t o f e e d e r r o o t s , and have both supporting and vascular functions. In a c c o r d a n c e with t h e pipe-model t h e o r y , a constant r a t i o i s maintained between t o t a l

foliar d r y m a t t e r and t h e t o t a l cross-sectional a r e a of t h e a c t i v e pipes in t h e stand. In addition, I assume t h a t feeder-root d r y matter is proportional t o f o l i a r d r y matter.

Disused pipes have a supporting function but d o not s e r v e a s v a s c u l a r connections between leaves and f e e d e r r o o t s . An a c t i v e pipe becomes disused when t h e foliage and f e e d e r r o o t s a t t a c h e d t o i t d i e ( o r are not renewed). The distal portion of a disused pipe i s lost from t h e stand when t h e b r a n c h of which it i s a p a r t , i s s h e d 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 a g g r e g a t e basal area of i t s a c t i v e pipes (active-pipe a r e a ) plus t h e a g g r e g a t e 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 t h i s pipe model, a c t i v e pipes connect l e a v e s t o f e e d e r r o o t s and h a v e both vascular and supporting functions. Disused pipes

-

vestiges of old a c t i v e pipes

-

no longer connect leaves to f e e d e r r o o t s and have only a supporting function. The ac- tive and disused pipes, in a g g r e g a t e , 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 c a n 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 s t a g e , t h e total f o l i a r d r y m a t t e r of t h e stand i n c r e a s e s 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 f o l i a r d r y

matter. The second s t a g e of a stand's development begins when f o l i a r d r y matter r e a c h e s i t s maximum. In some stands t h e maximum may b e r e a c h e d within f o u r y e a r s a f t e r establishment (Mohler e t aL., 1978). Cross-sectional samples of s t a n d s h a v e fur- nished evidence t h a t once t h e maximum of f o l i a r d r y m a t t e r i s r e a c h e d , i t tends t o b e sustained until senescence (Marks, 1974; Sprugel, 1974; Mohler et aL., 1978). Other cross-sectional d a t a h a v e been i n t e r p r e t e d a s indicating t h a t f o l i a r d r y matter p e a k s and then declines slightly, but rapidly. t o a constant level t h a t i s sustained until senes- c e n c e ( F o r r e s t and Ovington, 1970; Kira and Shidei, 1967). Regardless of whether t h e maximum of f o l i a r 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 a g g r e g a t e 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 compensat- ed by reductions in t h e active-pipe areas of o t h e r trees. A tree d i e s when i t s active- pipe 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 s t a n d s in t h e second s t a g e of development, where f o l i a r d r y matter and active-pipe a r e a are maximal ( o r nearly s o ) and more o r l e s s 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 a c t i v e pipe (m).

V

=

t h e t o t a l volume of t h e stand (m3).

V'

=

t h e total aboveground volume of t h e stand (m3).

X

=

t h e t o t a l 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 n e t growth r a t e s of active-pipe and disused-pipe a r e a s :

d B / d t

=

d A / d t + d X / d t (1)

The n e t growth r a t e of active-pipe a r e a , d A / d t , can b e split into a positive fraction (dA + / d t ) and a negative f r a c t i o n ( - 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 t h e living t r e e s in the stand, and in t h e absence of t r e e mortality, t h i s rate equals t h e growth rate of t h e total basal a r e a of t h e stand. The negative fraction, 4 . A - / d t , i s 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 n e i t h e r adds n o r s u b t r a c t s from t h e basal area of t h e stand. Similarly, t h e n e t growth r a t e of disused- pipe a r e a ,

dX/

d t , can b e split into a positive fraction (dX+/ d t ) and a negative f r a c - tion (*-/ d t ), i.e., dX/ d t

=

d ~ + / d t

a-/

d t

.

The positive f r a c t i o n equals dA -/ d t

.

The negative f r a c t i o n i s 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:

d B / d t

=

d ~ + / d t * - / d t Growth r a t e of active-pipe Length

The growth r a t e of a v e r a g e , active-pipe length, dL / d t , also can b e split into two f r a c - tions: a metabolic f r a c t i o n (dLM/ d t ) and a numerical o r non-metabolic f r a c t i o n ( d L N / d t ) , i.e.,

d L / d t

=

d L M / d t f d L N / d t (3)

The metabolic fraction, d l g / d t , is t h e rate at which a v e r a g e , active-pipe length in- c r e a s e s due t o t h e apical growth of shoots and roots. The non-metabolic fraction, dLN/ d t , is t h e r a t e at which a v e r a g e , active-pipe length increases due t o t h e suppres- sion and disuse of pipes t h a t a r e , on a v e r a g e , 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 non- metabolic 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 +/ d t

.

D i s u s e 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 LdA + / d t +AdLM / d t

.

Disused pipes do not i n c r e a s e in length, s o if t h e a v e r a g e length of a deactivating pipe i s 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 a v e r a g e , active-pipe length, t h e r e f o r e

d L N / d t

=

[ ( 1 4 ) L / ~ ] ( d A + / d t - d A / d t ) (6) Restricting consideration t o stands f o r which dA / d t

=

0 and A A,,,, eqn (6) r e d u c e s t o

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 h a s been a n active topic of ecological r e s e a r c h f o r t h e l a s t two decades (e.g., Yoda e t 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 a n even-aged, mono-species stand, given a n a v e r a g e basal a r e a p e r t r e e ,

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 ~ ~ B / d t +(I + ) B ~ w ~ / d t (12) 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, t h e n ( ~ + ) B c ~ ~ / d t i s 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 f r a c t i o n s of eqn (2) equal to t h o s e of eqn (12), t h e r e f o r e :

d~ + / d t

=

( l + c ) ~ ~ t G / d t (13)

4 - / d t

=

( ~ - $ ) Z c ~ ~ / d t (14)

Solving eqns (13) and (14), r e s p e c t i v e l y , f o r @ / d t and cWR/dt, and inserting t h e r e s u l t a n t expressions into (9), yields

dx-/ _{d t}

=

cpdA ⁺ / d t (15) where 9

=

c (I-$)/ (1-c $) i s assumed constant. Substituting eqn (15) into (2). t h e r e - f o r e

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 r o o t s , 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 r e s p i r a t i o n f o r construction o r maintenance of foliage, f e e d e r r o o t s , and a c t i v e 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 f o l i a r d r y matter:

z

=

units of f o l i a r 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 f o l i a r 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 f o l i a r 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 f o l i a r 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 t h e s e definitions t h a t t h e t o t a l f o l i a r d r y matter in t h e stand i s zA. The rate of production of new f o l i a r d r y matter i s zdA ⁺ / d t and t h e rate of constructive r e s p i r a t i o n f o r new f o l i a r d r y matter i s zr,dA ⁺ / d t

.

The rates of production a n d con- s t r u c t i v e r e s p i r a t i o n a r e defined with d A ⁺ / d t because i t is t h e positive f r a c t i o n of d A / d t

.

The negative f r a c t i o n 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 s u b s t r a t e . The rate of r e s p i r a t i o n f o r maintenance and renewal of e x t a n t f o l i a r d r y matter on a n an- nual 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 , t h e rate of constructive r e s p i r a t i o n i s j'rf dA +/ d t

,

and t h e rate of r e s p i r a t i o n f o r mainte- nance and renewal is j'[bf +(rf +1) / Tf

M .

Combined t e r m s f o r f o l i a r and f e e d e r - r o o t d r y m a t t e r a r e defined as follows

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 r e s p i r a t i o n of new foliar plus f e e d e r - r o o t 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 f o l i a r 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 t o t a l active-pipe d r y matter in t h e stand i s uAL and t h e rate of r e s p i r a t i o n 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 m a t t e r i s u (L dA +/ d t +AdLM / d t ) and, t h e rate of constructive r e s p i r a t i o n f o r new active-pipe d r y matter i s urp (L dA + / d t

+

AdLM / d t ).

D t y - m a t t e r 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 f o l i a r 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 d r y m a t t e r .

To account f o r a suboptimal environment f o r s u b s t r a t e production, t h e a c t u a l rate of s u b s t r a t e production i s defined in t e r m s of a scaling variable (I) s o t h a t

a l

=

t h e a c t u a l 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 f o l i a r d r y matter.

Thus, I v a r i e s between 0 and 1 , and a z A l is t h e r a t e of dry-matter s u b s t r a t e produc- tion by a stand.

P r o d u c t i o n r a t e s

The sum of t h e rates of production and constructive r e s p i r a t i o n of new f o l i a r , f e e d e r - r o o t , and active-pipe d r y m a t t e r 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 r e s p i r a t i o n f o r existing f o l i a r , feeder-root, and active-pipe d r y matter

u ( l + t p ) A d L M / d t

+

[ z ' + r ' + u ( 1 + r p ) L ] d A + / d t

=

A [ a z l - b ' u b p L ] (17) A s was noted, b r a n c h e s , 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 as- semblage of a c t i v e a n d disused pipes. The lengthening of existing a c t i v e pipes c o r r e s p o n d s roughly t o a p i c a l growth, and t h e production of new a c t i v e pipes c o r r e s p o n d s roughly t o cambial growth. Because apical growth tends t o o c c u r o v e r a more o r l e s s fixed portion of e a c h growing season f o r many species, i t s e e m s n a t u r a l t o assume t h a t a constant p r o p o r t i o n 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 c o n s t r u c t i v e r e s p i r a t i o n . Thus, eqn (17) i s split into 2 p a r t s according to t h e t y p e 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), r e s p e c t i v e l y , 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 o b t a i n e d by substituting e q n ( 2 1 ) in t o (6) a n d 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 ] d A / d t ( 2 2 ) When dA/ d t e q u a l s 0 , e q n ( 2 2 ) i s i n t e g r a b l e from

t

t o

t

+At if I is assumed c o n s t a n t o v e r t h a t s p a n . 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 1 f o r l a r g e L , s o e q n ( 2 2 ) r e d u c e s t o

d L / d t

=

n o [ a l - b * - b ~ ]

I n t e g r a t i o n of t h i s s i m p l e r function yields

L

( t

+ A t )

=

nl+n.$(t)+n3L

( t )

w h e r e

no

"

^{1 1 9 A}

nl

=

-b ' ( 1 - n 3 ) / b

nz

=

a(1-n3)/ b n3

=

e x p ( -cob At )

I t i s noted, o n t h e o n e hand, t h a t if I were c o n s t a n t 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 s p e c i f i c rate of f e e d e r - r o o t t u r n o v e r (T;) i s v a r i a b l e , b u t c a n b e assumed c o n s t a n t o v e r t h e s p a n from

t

t o

t

+ A t , t h e n t h e c o n s t a n t nl in e q n ( 2 4 ) should b e r e p l a c e d by

t o give

w h e r e

L

(t

+ A t )

=

n4+n5/ Tf ( t ) + n z I ( t ) + n & .

( t )

B a s a l a r e a

The growth rate of basal a r e a of a n actively self-thinning stand is obtained by substi- tuting 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

d B / d L

=

(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)

=

~ ( t ) + n ~ f l n [ ~ ( t + ~ t ) + ~ ' ] - l n [ ~ ( t ) + ~ * ] j (28) where n6

=

(l-cp)~,,,/ no. Substitution of e x p f [ ~ ( t )--cl]/ 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 s t a n d basal a r e a a s a function of t h e rates of f e e d e r - r o o t t u r n o v e r and s u b s t r a t e production, i.e.,

B ( t + A t >

=

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 b r a n c h e s in connection with t h e disuse of pipes and l e t v.&

(a-/

^{d t )}

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

d V / d t

=

LdA + / d t + ~ d L ~ / d t -L [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 s t a n d 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 t o L i s 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

(32)

Because A w A,,, when dA / d t

=

0 , and L / ( z

'

+L ) 1 f o r l a r g e L , eqn (31) can b e in- t e g r a t e d 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 )- c 2 ] / nlO, where c 2 is a con- s t a n t , 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 t u r n o v e r and s u b s t r a t e production, i.e.,

~ ( t + ~ t ⁾

=

nll+n12/ T~ ( t ) + q 3 l ( t )+n3v(t ) (34)

where nll

=

G ~ ~ ~ ~ + C ~ ( ~ - K ; ~ ) , n12

=

n5n10, and G13

=

n2nlo.

The growth of volume, like t h e growth of L , i s described by a Mitscherlich func- tion 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 b e a constant fraction ( v ) of total woody volume, then t h e growth fraction of V' i s

~ ' ( t

+ ~ t )

=

n14 +n15/ T~

(t

)+n161(t )+n17V'(t ) (35) where K ~ ~ = v ~ c ~ ~ , I C ~ ~ = V K ~ ~ , and n16=Vn13. In a n invariant environment, where I and Tf are constant, t h e growth function of

V'

i s

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 d e s c r i b e t h e aboveground growth of closed, self-thinning stands and obtained good agreement

between t h e model and yield tables. The p a r a m e t e r n17 w a s shown to b e more o r l e s s constant among s t a n d s of a given s p e c i e s 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 i n c r e a s e d f r o m p o o r to good sites. The as- sumptions of t h e p r e s e n t model are in a c c o r d a n c e 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 f e e d e r - r o o t t u r n o v e r and t h e r a t e of s u b s t r a t e pro- duction 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 - t i e s a l s o 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 p r e d i c t e d aboveground volume in y e a r t +10 vs. volume in y e a r t ( t

=

10, 20,

...,

90) for mixed oak ( @ a r c u s 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 S t a t e s . D a t a are f r o m S c h n u r (1937: Table 12). The common slope of t h e t h r e e r e g r e s s i o n 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 con- s t a n t T f and I is obtained by subtracting V ' ( t ) from eqn ( 3 6 ) , i.e.,

AV'

=

n18 + ( n I 7 - 1 ) V ' ( t )

=

n 1 4 + n 1 5 / Tf +n161 + ( n 1 7 - l ) V ' ( t ) ( 3 7 ) The asymptotic yield (V',,,) of t h e stand equals V ' ( t ) when AV' equals z e r o

Vamax

=

n 1 8 / ( 1 - n 1 7 )

=

( n 1 4 + n 1 5 / T f + n l 6 1 ) / ( 1 - n I 7 ) (38) Corresponding to V',,, is t h e asymptotic, a v e r a g e , active-pipe length (L,,,) which is obtained from eqn (25)

L

,,, =

(rcl+n5/ Tf + n 2 1 ) / ( 1 - n g ) ( 3 9 ) Unlike V B m a X , which depends on t h e values of t h e self-thinning and pruning p a r a m e t e r s (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 t h e p r e s e n t theory, LmaX is t h e active-pipe length (i.e., a v e r a g e 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 t h e rate of maintenance r e s p i r a t i o n s o t h a t t h e r e i s no f u r t h e r growth by t h e stand. With constant Tf and I, L,,, i s truly a n asymptote, a value t h a t i s approached, but n e v e r reached. However, in a changing environment, where Tf and I o r both c a n b e expected t o v a r y in closed, self-thinning stands, a b e t t e r definition f o r Lmax i s potential active-pipe length.

I t has been hypothesized t h a t a i r pollution/acid r a i n is causing o r contributing t o c e r t a i n 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 f i t into t w o categories: 1 ) those concerning d i r e c t damage t o foliage by pollutants with consequent deceleration of t h e 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 r o o t s o r mycorrhizae in acidifying soils with consequent acceleration of t h e r a t e of feeder-root turnover. The p r e s e n t model suggests t h a t e i t h e r kind of damage should cause de- c r e a s e s in stand growth r a t e , potential active-pipe length and, correspondingly, poten- tial yield. VamaX. If changes in T ~ - ~ o r I were of such magnitude t h a t

[ I C ~ + I C ~ / Tf(t)+,cZ1(t

) I /

(1-3)

<

^{L ( t}

t h e n a c t u a l active-pipe length would e x c e e d t h e potential a n d widespread dieback o r t r e e d e a t h should o c c u r within t h e s t a n d b e c a u s e 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 t i s s u e would not b e m e t . This situation should b e more likely w h e r e L ( t ) i s l a r g e , which is consistent with o b s e r v a t i o n s 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 s t a n d s .

The o n s e t of d i e b a c k c r e a t e s a situation f o r which t h e p r e s e n t model d o e s not ap- ply. Self-thinning a n d t h e constancy of f o l i a r d r y m a t t e r 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 i n s t a n c e s 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 p r o - duction 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 e x t e n d e d is t h e p e r i o d b e f o r e t h e canopy c l o s e s following t h e establishment of a s t a n d ( o r following a silvicultural thinning) w h e r e f o l i a r d r y m a t t e r i s i n c r e a s i n g a n d self-thinning is nil o r just beginning ( o r resuming). Such a n extension should r e q u i r e functions d e s c r i b i n g I ( t ) , dA / d t , a n d dX-/dt w h e r e A

<

^A,,,. Regarding t h e l a t t e r two r a t e s , i t is ex- p e c t e d t h a t in g e n e r a l

0 5 d A / d t 5 d A + / d t 0 5 d X - / d t 5 c p ( d A f / d t - d A / d t )

w h e r e cp is t h e c o n s t a n t of eqn (15) a n d (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 a c h i e v e d by a s t a n d b e f o r e t h e o n s e t of t h e production of disused-pipe a r e a (dA '/ d t

>

dA / d t ) o r t h e o n s e t of self-thinning (dX-/dt

>

0 ) d e p e n d s l a r g e l y on t h e v a r i a t i o n in t h e s i z e s of t h e trees a n d t h e i r s p a c i a l a r r a n g e m e n t . In n a t u r a l s t a n d s 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 on- s e t of self-thinning may o c c u r n e a r l y coincident with t h e o n s e t of production of disused-pipe a r e a , b u t in plantations of r e g u l a r l y s p a c e d t r e e s , self-thinning may n o t 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 s t a n d ' s b a s a l a r e a .

I n d i v i d u a l t r e e s

Previously, I used a pipe-model framework t o d e r i v e 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 p r e s e n t 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 r e t a i n s 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 t h e basal a r e a growth rate, ^viz.

d B / d t

=

XA[aI-6' - 6 ~ ] / [ z ' + L ] (41) The growth rate of t o t a l volume a t t h e tree level i s obtained by dropping t h e term in- volving dX-/ d t from eqn (30) and substituting d B / d t f o r dA +/ d t , whence

dV/dt = L [ v I d A / d t +(l--,l)dB/dt]-AdLM/dt (42) The growth rate of t h e f r a c t i o n , w, of t h e t o t a l 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 a n individual tree and i t s rate of s u b s t r a t e 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 b e formulated.

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