Implications of the Pipe Model
Theory on Dry Matter Partitioning
and Height Growth in Trees
Maekelae, A.
IIASA Working Paper
WP-85-089
December 1985
Maekelae, A. (1985) Implications of the Pipe Model Theory on Dry Matter Partitioning and Height Growth in Trees. IIASA
Working Paper. IIASA, Laxenburg, Austria, WP-85-089 Copyright © 1985 by the author(s). http://pure.iiasa.ac.at/2614/
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IMPLICATIONS OF THE PIPE MODEL THEORY
ON DRY HATTER PARTITIONING
AND HEIGHT GROWTH IN TREES
Annikki Makela
December 1985
WP-85-89
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 or of i t s National Member
Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
2361 Laxenburg, Austria
PREFACE
IIASA's Acid Rain project is currently extending its model system,
RAINS (Regional Acidification INformation and Simulation) t o t h e possible
damage caused by acid rain t o forests. Like all submodels of RAINS, the
forest impact submodel will be based upon a simple description of basic
dynamics, while t h e complexity of RAINS originates in spatial and temporal
extent and integration of disciplines. Building a simple model, however,
often requires a thorough, understanding of t h e complexities of t h e system,
s o a s t o maintain t h e essential relationships of the r e a l object in t h e model.
Our understanding of t h e response of forests t o a i r pollution is still
limited not only a s r e g a r d s t h e pathways of pollutant impact themselves, but
also in relation t o some basic phenomena of t r e e growth and development.
One of t h e missing links in t h e chain from pollutants t o damage is how t h e
physiological processes, immediately affected by a i r pollution, interact a t
the whole-tree level, and how this interaction evolves a s t h e t r e e grows
bigger.
This p a p e r describes a development of a dynamic individual-tree model
which is based on requirements of metabolic balance and t h e i r interaction
with t r e e structure. Because of its generality, the model can be applied to'
various situations, also giving insight into t h e impacts of a changing
environment. An interesting contribution of t h e paper is t h a t it provides a
quantitative relationship between sensitivity t o environmental s t r e s s , ecophysiological parameters and tree size.
Leen Hordijk
Project Leader
ACKNOWLEDGEMENTS
The author wishes to acknowledge the contributions of Pertti Hari and
Leo Kaipiainen to the ideas presented in this paper. Special thanks are due
to Pekka Kauppi, Ram Oren and Risto Sievanen who have reviewed the
manuscript and given their fruitful comments.
ABSTRACT
A dynamic growth model i s developed f o r f o r e s t trees where t h e partitioning of growth between foliage and wood i s performed so as to fulfill t h e
assumptions of t h e p i p e model t h e o t y (Shinozaki et al. 1964a), i.e to maintain sapwood:foliage r a t i o constant. Partitioning of growth to f e e d e r roots
i s t r e a t e d using t h e principle of f u n c t i o n a l balance (White 1935, Brouwer
1964, Davidson 1969 and Reynolds and Thornley 1982). The model uses t h e
time resolution of one y e a r and i t applies to t h e life-time of t h e tree. The
consequences of t h e pipe model t h e o r y are examined by studying t h e lifetime growth dynamics of trees in different environments as functions of
length growth p a t t e r n s of t h e woody organs. I t i s shown t h a t t h e r e i s a
species-specific, environment-dependent u p p e r limit f o r a sustainable
length of t h e woody organs, and t h a t t h e diameter of a tree at a c e r t a i n
height depends upon t h e rate at which t h a t height h a s been achieved. These
r e s u l t s are applied to t h e analysis of deceleration of growth, response t o
environmental stress and height growth p a t t e r n s in a tree population.
F u r t h e r , t h e possible f a c t o r s t h a t allow c e r t a i n coniferous species in t h e
Pacific Northwest region to maintain a n almost unlimited height growth patt e r n are discussed in relation to t h e pipe model theory.
TABLE OF CONTENTS
1.
INTRODUCTION
2.
TREE MODEL WITH PARTITIONING OF GROWTH
2.1
Mass Balance Model of Growth
2.2
Partitioning of Growth
2.2.1 Description of T r e e S t r u c t u r e
2.2.2Foliage: Wood Ratio
2.2.3 Foliage: F e e d e r Root Ratio
2.2.4Partitioning Coefficients
3.
2.3
Summary of t h e Model
2.4
Extension of t h e Model t o Changing Environment
PROPERTIES OF THE MODEL
3.1
Root-Foliage Relationships
3.2
Length Variables and Growth
4.
5.
SOME IMPLICATIONS OF THE MODEL
4.1
Deceleration of Growth
4.2
Response t o Environmental S t r e s s
4.3
Height Growth
DISCUSSION
REFERENCES
IMPUCATIONS OF THE PIPE MODEL THEORY
ON DRY MATTER PARTITIONING
AND HEIGHT GROWTH IN TREES
Annikki Makela
1. INTRODUCTION
The pipe model t h e o r y by Shinozaki et al. (1964a) f i r s t theoretically
formulated t h e observation t h a t in t r e e s , sapwood a r e a is proportional t o
foliage biomass. The t h e o r y reasoned t h a t each unit of foliage r e q u i r e s a
unit pipeline of wood t o conduct water from t h e roots and t o provide physical support. Although t h e argument may be disputable, r e c e n t empirical
observation strongly s u p p o r t s t h e existence of such a relationship. Cons t a n t r a t i o s have been established both between sapwood and foliage a r e a
( o r 'oiomass) (Rogers and Hinckley, 1979; Waring, 1980; Kaufmann and
Troenale, 1981), and sapwood a r e a s of successive p a r t s of t h e water conducting wood (Kaipiainen et al., 1985). The r a t i o s a p p e a r fairly constant
within species despite l a r g e environmental variation (e.g. Kaufmann and
Troendle, 1981; Kaipiainen et al., 1985).
While t h e applicability of such constants t o inventories of standing
biomass h a s been widely recognized ( S h i n o z a ~ iet al., 1964b; Waring et al.,
1982), less attention h a s been paid t o t h e implications t h e maintenance of
t h e s t r u c t u r a l balance may have t o growth dynamics. Two observations are
important from this viewpoint. First, since standing biomass i s cumulative
growth minus t u r n o v e r , a balance between t h e r a t e s of t h e s e two processes
is required f o r maintaining t h e constant r a t i o s between t h e standing
biomass compartments. Secondly, as t h e tree grows h i g h e r , t h e build-up of
new pipelines r e q u i r e s more and more growth r e s o u r c e s p e r unit leaf
biomass.
Varying t h e density of a tree stand causes variation in individual tree
basal area and leaf
biomass, but tree height remains more or less
unchanged (Whitehead, 1978). A r e a s o n f o r this may be t h a t t h e significance of t h e length variables f o r survival is not in t h e i r contribution to t h e
internal balance of growth, but r a t h e r to t h e competitive ability f o r light
and space.
A similar r a t i o h a s been observed between roots and leaves, with t h e
difference t h a t t h e r a t i o v a r i e s with t h e soil nutrient level. White (1935)
observed t h a t d r y matter partitioning between t h e roots and shoots of g r a s s
and clover were dependent upon t h e light and nitrogen levels applied in
laboratory experiments. This principle of functional balance w a s f u r t h e r
developed and formalised mathematically by Brouwer (1964) and Davidson
(1969), and a dynamic model of partitioning in non-woody plants w a s
developed by Reynolds and Thornley (1982).
A dynamic growth model i s developed f o r f o r e s t trees where t h e p a r t i -
tioning of growth between foliage a n d wood i s performed s o as t o maintain
t h e sapwood:foliage r a t i o constant. Partitioning of growth t o f e e d e r r o o t s
i s t r e a t e d using t h e principle of functional balance. The model u s e s t h e
time resolution of o n e y e a r a n d t h e time span of t h e life-time of t h e tree.
Using t h e model, t h e consequences of t h e p i p e model t h e o r y t o t h e o v e r a l l
growth dynamics of trees are examined by studying t h e life-time growth
dynamics of trees in d i f f e r e n t environments as functions of length growth
p a t t e r n s of t h e woody o r g a n s .
2. TREE MODEL WITH PARTITIONING OF GROWTH
2.1. M a s s Balance Model of Growth
When r e g a r d e d as assimilation of t o t a l growth r e s o u r c e s and t h e i r distribution t o d i f f e r e n t o r g a n s , plant growth c a n conveniently b e analyzed
with a mass balance model which i n c o r p o r a t e s t h e biomasses of t h e o r g a n s ,
W t , t h e t o t a l growth rate, G , t h e t u r n o v e r rates of t h e p a r t s . S i , a n d t h e
p a r t i t i o n i n g coefficients,
X i ( i E N = t h e set of indices of t h e p a r t s ) . The
latter are defined as t h e f r a c t i o n of t o t a l growth allocated t o p a r t i , a n d
t h e y satisfy
f o r all times t . The growth rates of t h e biomass compartments are
- wi- -
XiG-St,
foralli~N
alt
i.e. n e t growth i s t h e d i f f e r e n c e between g r o s s growth and t u r n o v e r .
Provided t h a t t h e variables h i , G , and Sf can be presented as functions of t h e s t a t e variables and identifiable parameters, Equations (2) constitute an operational dynamic model f o r t r e e growth. The following parag r a p h s a r e devoted t o t h e derivation of t h e required relationships.
Following a customary approach, growth can b e derived from t h e c a r bon metabolism of t h e t r e e . On a n annual basis, it can b e assumed t h a t t h e
carbon assimilated is totally consumed in growth and respiration. Denote
annual photosynthesis (kg C a -')
and
the
carbon
(kg d r y weight a -')
G
=
content of
by P , annual respiration (kg C a-')
dry
matter
by f C .
Annual
by R ,
growth
G
is t h e r e f o r e
(3)
JC~(P-R).
Photosynthesis i s assumed t o b e proportional t o foliage biomass, Wf,
and t o t h e specific photosynthetic activity, uc (kg C a
(kg d r y weight)-'),
which depends upon environmental and internal factors:
P
=
uc Wf
(4)
Following McCree (1970), respiration is divided into growth respiration, R,,
and maintenance respiration, Rm :
R = Rm+Rg
Growth respiration Rg is proportional t o growth r a t e G
Rg = rs G ,
and maintenance respiration R,,,
(6)
is proportional t o t h e size of t h e main-
tained biomass compartment. Hence
Above, r g and r,* are constant coefficients.
Turnover is customarily determined on t h e basis of specific t u r n o v e r ,
s i , which is t h e r e c i p r o c a l of t h e a v e r a g e life-time of t h e organ:
Equations (3)-(8) r e p r e s e n t what can b e called a standard p r o c e d u r e
f o r determining total growth and senescence in growth models (Thornley,
1976; d e Wit et al., 1978; Agren and Axelsson, 1980). For t h e partitioning
coefficients, such a standard does not exist. Section 2.2 treats t h e partitioning of growth in forest trees by using t h e pipe model t h e o r y and t h e
principle of s t r u c t u r a l balance as premises.
2.2. Partitioning of Growth
2.2.1. Description of tree structure
Figure 1 depicts tree s t r u c t u r e as a combination of five functionally
different p a r t s : foliage, f e e d e r roots, s t e m , b r a n c h e s and t r a n s p o r t roots.
Foliage and f e e d e r r o o t s are considered simply as biomasses. Wf and W,,
whereas t h e biomasses of t h e woody organs are derived from t h e geometric
dimensions of t h e organs. Denote sapwood area at crown base by A , , t h e
total sapwood area of primary b r a n c h e s at foliage base by Ab and t h e total
sapwood area of t r a n s p o r t r o o t s at t h e stump by A t . The length dimensions
incorporated a r e tree height, h , , crown radius, h b ,and t r a n s p o r t r o o t sys-
t e m radius, h t . I t is assumed t h a t t h e sapwood biomasses of s t e m , W , .
branches, W b , and t r a n s p o r t r o o t s , Wt , are obtainable from t h e s e variables
through a simple a l g e b r a i c operation. The assumption is then:
Figure 1.
Variables aescribing t r e e geometry. For f u r t h e r explanation, see Table 1.
W, = q i p i h i A f , i = s , b , t
(9)
where q t i s an empirical constant. This means t h a t variation in t h e shape of
t h e woody o r g a n s i s allowed only as r e g a r d s t h e r a t i o of sapwood area and
t h e height/length of t h e organ. The variables are summarised in Table 1.
2.2.2. Foliagemood ratio
The pipe model theory maintains t h a t t h e sapwood area at height x and
t h e foliage biomass above x a r e r e l a t e d through a constant r a t i o (Shinozaki
et al., 1964a). According t o Shinozaki et a1 (1964a), this applies to both
s t e m and branches. More r e c e n t empirical r e s u l t s indicate, t h a t (1) t h e
r a t i o may be different for s t e m and branches, and (2) t h e t r a n s p o r t roots
obey a similar relationship (Kaipiainen et al., 1985). With t h e s e supplement a r y notions, t h e basic observation can be elaborated t o yield t h e following
t h r e e relationships.
(1) Stem sapwood area at crown base, A,, i s proportional t o total folia g e biomass Wf
(2)
The total sapwood area of primary b r a n c h e s (at foliage base), A*,
i s proportional to total foliage biomass
v b A b = Wf
(11)
(3) The total sapwood area of t r a n s p o r t r o o t s a t t h e stump, At, i s proportional t o t o t a l foliage biomass
Table 1:Model variables
Name
Meaning
f
foliage
r
feeder roots
s
stem
b
branches
t
conducting r o o t s
Variables describing t h e s t a t e of t h e t r e e
i
4
biomass (in c a s e of wood:sapwood BM)
i =j',r ,s ,b ,t
sapwood a r e a
i =s ,b ,t
Other variables
Xi
partitioning coefficient
i =j' ,r ,s , b ,t
G
total growth
P
photosynthesis
R
respiration
Rm
maintenance respiration
RL3
growth respiration
si
senescence of biomass
i =j',r
Di
senescence of sapwood
i =s ,b ,t
ui
height growth r a t e
i =s ,b ,t
Units
The constants
v i are species specific parameters.
2.2.3. Foliage: feeder root ratio
According t o a n empirical relationship f i r s t established by White
(1935), and f u r t h e r developed mathematically by Brouwer (1964) and Davidson (1969), t h e metabolic activities of t h e r o o t s and t h e photosynthetic
organs a r e in balance. Denote t h e specific photosynthetic activity by ac
(kg C a-' (kgdry weight)-')
uptake) by
ON
(kg N a
and t h e specific r o o t activity (of nitrogen
-'(kgdry weight)-').
The relationship can b e
expressed in t h e form
' r rO
~ c
Wf =
ON
WT
where nN i s a n empirical parameter.
This requirement c a n b e understood in terms of a balanced carbon and
nitrogen r a t i o in t h e plant (Reynolds and Thornley, 1982). If t h e specific
activities v a r y , like between growing sites, t h e relationship produces diff e r e n t fo1iage:feeder r o o t r a t i o s , thus explaining t h e variation in t h i s r a t i o
between growing sites.
2.2.4. Pdriitioning coefficients
Partitioning of growth between foliage, r o o t s and wood follows from t h e
requirement t h a t t h e s t r u c t u r a l balances presented in Section 2.2 are satisfied. First, l e t us consider t h e fo1iage:root r a t i o constrained by Equation
(13). Assuming t h a t t h e balance holds initially, i t c a n b e maintained if and
only if
This equation r e l a t e s t h e balance requirement with t h e partitioning coefficients which e n t e r t h e equation through t h e time derivatives of Wf and W,.
If these a r e substituted from Equation ( 2 ) , t h e following relationship
between X f and A, i s obtained:
where
Similarly t o Equation ( 1 4 ) , t h e requirements of t h e pipe model t h e o r y
can b e maintained if and only if
dAi dt -
71i
dWf
i =s,b,t
dt '
(18)
In o r d e r to utilise this equation f o r t h e derivation of t h e partitioning
coefficients of t h e woody organs, X i (i =s , b . t ) , l e t us f i r s t relate t h e time
derivative of Ai t o t h a t of Wi
d Wi
= Pi 4'i
dt
(
d Ai
h
dt
.
i +
By Equation ( 9 ) ,
dhi
Ai)
dt
, i=s,b,t
(19)
On t h e o t h e r hand, dWi / d t i s defined in t e r m s of Xi by Equation ( 2 ) . Combining (2) and ( 1 9 ) allows us to solve f o r t h e time derivatives of t h e sapwood
areas, c L A i / d t , which thus become functions of t h e partitioning coefficients:
Above, w e have denoted:
and
Equation (2), with i =f , t o g e t h e r with Equation (20) c a n now b e substituted into Equation (18). This gives r i s e t o t h e following partitioning coefficients Xi:
where
where i = s ,b ,t .
This gives a g e n e r a l growth partitioning p a t t e r n between foliage and
wood as a function of t h e lengths of t h e woody organs.
2.3. Summary of the Model
I t follows from t h e above derivation t h a t tree growth c a n b e d e s c r i b e d
in t e r m s of length growth and one of t h e dependent variables Wf , W,,A,, Ab
and At only. If foliage biomass is s e l e c t e d as a r e f e r e n c e , t h e model r e a d s
as follows:
where
and a* and
are defined as in Equations (15), (17), (24) and (25) f o r
i = r , s , b , t . Additionally,
The variables of t h e model are listed in Table 1, and Table 2 gives a
summary of its parameters.
In o r d e r t o fully understand t h e partitioning of d r y m a t t e r in trees, t h e
determination of t h e length growth p a t t e r n should b e understood. Since,
f o r t h e reasons pointed out in Section 1, i t is difficult t o relate length
growth t o biomass growth directly, w e shall consider t h e growth rates of t h e
length variables as unknown "strategies" t h a t can vary.
Below, t h e
behaviour of t h e model i s analyzed as a function of t h e s e strategies.
2.4. Extension of the Model to Changing Environment
S o far i t h a s been assumed t h a t t h e s t r u c t u r a l r a t i o s between t h e state
variables, Equations (10)-(13), are constant o v e r time. Actually, t h e m e t a bolic activities ac, ON, and
- as
will b e discussed l a t e r
- also t h e parame-
ters q i ,may respond t o environmental changes such as shading and fertilization during t h e life-time of t h e tree. If t h e change i s f a s t e r than t h e
response time of t h e tree, i t i s impossible even approximately t o maintain
t h e balance. However, t h e tree may b e able t o change t h e partitioning coefficients s o as t o eventually r e a c h a new balance in t h e new situation. The
p r e s e n t model manifests this behaviour if w e r e d u c e t h e assumption t h a t t h e
balance equations (10)-(13) hold a t e v e r y moment of time, but maintain t h e
Table 2. Parameters
Names
Meaning
Unit
Qc
foliage specific activity
kg[Cl/kg[DWla
QN
r o o t specific activity
kg[Nl/kg[DWla
PC
carbon content of d r y weight
kg[Cl/kg[DWl
TN
N:C r a t i o of d r y weight
kg[Nl/kg[Cl
qs
foliage D W:stem sapwood r a t i o
kg[~~l/rn~[S~l
qb
foliage DW:branch sapwood r a t i o
~~[DWI/~~[SWI
qt
foliage D W:transport root sapwood r a t i o
~~[DwI/~~[swI
ps
stemwood density
kg/m3
Pb
branchwood density
kg/m3
Pt
rootwood density
kg/m3
qs
form f a c t o r of s t e m
unitless
qb
form f a c t o r of branch system
unitless
qt
form f a c t o r of root system
Sf
foliage specific turnover r a t e
ST
feeder root specific turnover rate
ds
s t e m sapwood specific turnover rate
db
branch sapwood specific turnover rate
4
r o o t sapwood specific turnover rate
rm
specific maintenance respiration
Q
specific growth respiration
partitioning coefficients derived from those equations. The dynamics of all
t h e s t a t e v a r i a b l e s are included using t h e original differential equations,
Equation (2). A partitioning model with similar adaptive behaviour h a s
a l r e a d y been p r e s e n t e d by Reynolds and Thornley ( 1 9 8 2 ) f o r t h e shoot-root
r a t i o s in g r a s s .
The analysis of t h e model in Section 3 i s c a r r i e d o u t using t h e time
invariant form, but some of t h e considerations in Section 4 make use of t h e
possibility t h a t t h e p a r a m e t e r s v a r y in time.
3. PROPERTIES OF THE MODEL
3.1. Root-Foliage R e l a t i o n s h i p s
Let us define t h e p r o d u c t i v e b i o m a s s , W p , as follows:
WP = Wf
+ w,
The partitioning coefficient of t h e productive p a r t i s
AP
= Af + A,
(32)
Let us define t h e specific growth rate, 5 , as t h e growth rate p e r productive
biomass, and denote t h e specific t u r n o v e r rate of productive biomass by s p .
The n e t growth rate of Wp i s t h e n
The quantities
g and s p c a n b e r e d u c e d to t h e original p a r a m e t e r s through
t h e differential equations of Wf and W , . This yields
sp = ( l + % ) - ' ( s f
+a, s,)
where t h e p a r a m e t e r s u, r g and
correspond t o uc, r,, and rmf and a i ,
respectively, with t h e difference t h a t they a r e defined as activities p e r unit
productive biomass instead of unit foliage o r r o o t . The relationships are
The specific activity of t h e productive p a r t becomes, when t h e original
p a r a m e t e r s are substituted f o r a, :
which i s a Michaelis-Menten type function in both uc and
dence i s such t h a t If TTN uc
>> O N , then
UN.
The depen-
r o o t activity is r e s t r i c t i n g growth
and small changes in uc have little impact on t h e t o t a l productivity.
TNUC
<< U N ,
If
then uc i s limiting. If both terms are of t h e same o r d e r of
magnitude t h e n r e s p e c t i v e changes in e i t h e r one have equal effects.
Environmental variation affecting r o o t and foliage activity can b e brought
into t h e model through t h i s relationship.
3.2. Lengh Variables and Growth
The relative growth r a t e of t h e productive p a r t . Rp,is defined a s follows
This is readily obtainable from Equation (33):
Rp = A p i - s p ,
Let us define h and u as
and l e t
Assume t h a t length growths are small in comparison with t h e lengths themselves, such t h a t
This means t h a t t h e t e r m ui / h i i s negligible in t h e p a r a m e t e r f i t , Equation
(25). This assumption is reasonable especially in t h e l a t e s t a g e of growth.
Assume, f u r t h e r , t h a t dt and rmi are independent of i. and d r o p t h e subs c r i p t s . Denote
Rp c a n then b e written as a function of
h as follows:
The r e l a t i v e growth rate is thus a function of t h e metabolic and s t r u c t u r a l p a r a m e t e r s and t h e generalized length h . The r e s u l t s c a n b e summarized as Lemma 1 below.
Lemmal.
cP
I. If h < 7
,
then Rp > O .
B
2. If h
= =a, then Rp =O.
B
3. If h
a ,
>r
then Rp
B
<O.
This means t h a t if t h e lengths of t h e woody organs continue t o grow
a f t e r Equation (46) h a s been fulfilled, t h e relative growth r a t e of foliage
and r o o t s goes negative and t h e tree s t a r t s t o die off. However, if length
growth ceases b e f o r e t h e state Rp < O has been r e a c h e d , growth will continue exponentially at t h e rate Rp which thus remains constant.
The.time development of t h e productive biomass depends exponentially
on t h e relative growth rate:
With a n exchange of variables, t h e integral c a n b e e x p r e s s e d in terms of h :
This expression contains information on t h e relationship between t h e
growth rates of t h e biomass and length variables. This can b e summarized
a s t h e following Lemma:
Given a fixed set of p a r a m e t e r values, t h e size of t h e productive p a r t a t any
length h i s a function of t h e rate at which t h a t length h a s been achieved:
t h e faster t h e preceding growth rate u , t h e smaller t h e size at t h e length
h.
The r e s u l t can also ise interpreted vice versa. If t h e height and
length growth rates a r e fixed, then variation in t h e parameter values
causes differences in t h e size of foliage and roots.
4. SOME Ib!IPLICATIONS OF THE MODEL
4.1. D e c e l e r a t i o n o f growth
When young, t r e e s grow exponentially in a l l dimensions, then gradually
siow down t o r e a c h a constant growth r a t e , and t h e growth finally ceases
slowly. Although all p a r t s of t h e t r e e seem t o follow t h e same basic p a t t e r n ,
t h e i r relative time scales vary. Dominant t r e e s c a n maintain considerable
basal a r e a growth long a f t e r t h e slowdown of height growth (cf. Koivisto,
1959).
I t is widely accepted t h a t t h e decline in growth in old trees is a wholeplant phenomenon r a t h e r than a consequence of t h e aging of individual tissues (NoodBn, 1980), but t h e mechanisms of decline are not fully understood. I t has been pointed out t h a t t h e proportion of stem t o crown gradually increases with a g e and size and, consequently, t h e r a t i o of carbohyd r a t e s produced t o t h a t consumed in respiration d e c r e a s e s (Jacobs, 1955).
There is also evidence of increasing difficulty of material translocation
between t h e r o o t s and t h e crown (Kramer and Kozlowski, 1979; p. 611).
Particularly, difficulties in water relations have been argued t o cause
s e v e r e water deficits and death of leaves and branches (Went, 1942).
The r e s u l t of
Lemma 1 provides some f u r t h e r insight into t h i s
behaviour. Since net growth of r o o t s and foliage cannot b e maintained if
t h e woody p a r t s exceed a c r i t i c a l size, survival r e q u i r e s t h a t t h e length
growth r a t e s a r e decelerated. If this action is postponed until very close t o
t h e critical limit
-
which would b e expected because rapid dimensional
growth is a n advantage in competition f o r light and nutrients
- only a s m a l l
growth potential will remain. Nevertheless, a fraction corresponding t o t h e
turnover of sapwood must continuously b e allocated t o t h e growth of t h e
woody organs.
The model thus explains t h e deceleration of growth as a consequence of
the f a c t t h a t maintenance and growth requirements of t h e woody p a r t s
increase faster than t h e productive potential. This is ultimately due t o t h e
assumption t h a t sapwood and foliage grow in constant proportions, which
makes t h e fraction of wood in total t r e e biomass increase whenever t h e r e is
length growth. In t h e absence of length growth, all t h e organs would continue t o grow in constant proportions.
A numerical example is worked out below s o as t o study t o what extent
t h e processes described by t h e present model can b e responsible f o r t h e
slowdown of
growth
in t r e e s .
Let us consider a Scots pine tree
( P i n u s sylvestris) under b o r e a l conditions.
Table 3 summarizes t h e
necessary p a r a m e t e r values f o r such a t r e e . The s o u r c e of t h e value is
indicated in t h e table. Maximum sustainable t r e e heights have been calcillated f o r a r a n g e of nutrient uptake r a t e s , corresponding t o different growing sites, and f o r a r a n g e of different trunk-branch-root configurations.
The results a r e summarized in Figures 2 and 3. This shows tree heights of
t h e c o r r e c t o r d e r of magnitude. However, t h e parameter values a r e rough
estimates only, and r e s u l t is sensitive especially t o t h e maintenance
requirement of living woody tissue.
Table 3. P a r a m e t e r value
Name
Value
Source
2
Agren and Axelsson 1980
0.04
guess
0.01
0.5
estimates based on elemental contents
according t o various s o u r c e s
0.25
Agren and Axelsson 1980
600.
Kaipiainen et al. 1985
450.
Kaipiainen et al. 1985
2400.
Kaipiainen et al. 1985
400
Karkkainen 1977
0.75
I
estimates based on tree from measurements
according t o various sources
0.70
1
corresponds to a v e r a g e life t i m e of
4 years
P e r s s o n 1980
guess
Agren and Axelsson 1980
Agren and Axelsson 1980
ROOT A C T I V I T Y
Figure 2.
Maximum sustainable a v e r a g e length of woody organs h
(Equation 42) as a function of r o o t activity a
,,.
4.2. Response to Environmental Stress
Many environmental s t r e s s f a c t o r s cause a slowdown of metabolic
activities and/or a c c e l e r a t e t h e turnover of plant tissue.
Drought, f o r
example, f i r s t d e c r e a s e s photosynthesis through stomata1 closure, and if
prolonged, may lead t o shedding of leaves s o as t o d e c r e a s e t h e transpiring
surface. The s a m e i s t r u e of t h e increasing anthropogenic stress load.
Atmospheric a i r pollution has been r e p o r t e d t o d e c r e a s e specific photosynthesis and a c c e l e r a t e foliage aging, and soil acidification increases fine
r o o t mortality and d e c r e a s e s specific nutrient uptake rate by leaching
nutrients.
Such s t r e s s f a c t o r s can ' b e incorporated in t h e dynamic extension of
p r e s e n t model (Section 2.4) as changes in t h e corresponding parameter
STEM HEIGHT VERSUS AV LENGTH
2
.-m
50
0)
C
41
E
X
0)
40
a
a
a
a
a
30
20
10
0
0
5
10
15
20
25
Average Length
Figure 3.
Stem height as a function of average length using different
stem: branch: root length ratios. All curves have hb =
ah,, ht = Zah,, and a varies as follows:
values.
Hence, a d e c r e a s e in t h e specific activities oc and o ~ o,r a n
i n c r e a s e in t h e specific t u r n o v e r rates of foliage and r o o t s , sf and s,, may
b e involved. All t h e s e changes have a similar impact on t h e r e l a t i v e growth
r a t e of t h e productive p a r t : Rp decreases. W e shall study in more detail t h e
conditions under which t h e t r e e will not survive t h e stress.
In t e r m s of t h e model, t h e t r e e survives as long as i t maintains a positive level of productive biomass. This condition will be violated if t h e long-
t e r m a v e r a g e of t h e r e l a t i v e growth r a t e is negative. By Equation (46),
relative growth rate i s positive only if
Q >
Jho
(50)
If t h e opposite o c c u r s frequenily t h e tree will die because of a decline in
r o o t and foliage biomass.
Equation (50) states t h a t t h e immediate reaction of a tree t o a
d e c r e a s e in growth potential depends on t h e size h of i t s woody organs.
Again, t h e r e s u l t c a n b e a t t r i b u t e d t o t h e fact t h a t in t h e model, maintenance and growth of woody p a r t s i n c r e a s e f a s t e r than growth potential.
Since h does not, normally, d e c r e a s e in time, t h e model implies t h a t a g e
increases t h e susceptibility of trees t o stress f a c t o r s t h a t s u p p r e s s productivity.
The model implies t h a t t h e long-term response of a young tree initially
'
satisfying condition (49) largely depends on i t s ability t o a d a p t i t s height
and length growth t o t h e new situation. If adaptation o c c u r s , t h e environmental change solely means a smooth d e c r e a s e in growth rates and maximum
sizes of t h e affected trees. Catastrophic behaviour will e n t e r , however, if
t h e tree insists in following i t s normal growth p a t t e r n s . This would suggest
t h a t species with different capacities of environmental flexibility of height
growth could manifest different reactions t o long-lasting environmental
stresses.
4.3. H e i g h t Growth
T r e e height, relative t o i t s neighbours, i s a n important indicator of
foliage productivity because i t a f f e c t s t h e amount of shade cast on t h e tree
by t h e rest of t h e canopy. The specific photosynthesis of t h e s h o r t e r trees
will b e reduced, and they will probably also have a higher t u r n o v e r rate of
foliage, both f a c t o r s decreasing t h e i r productivity r e l a t i v e t o t h e well-todo neighbours. I t t h e r e f o r e a p p e a r s t h a t t h e only means of survival in a
stand is t o keep up with t h e height growth of t h e rest of t h e canopy.
Indeed, Logan (1965,1966a,1966b) h a s shown by experiments t h a t t h e height
growth rates of seedlings shaded t o different extents v a r y considerably less
than t h e corresponding foliage growth rates (Logan, 1965,1966a,1966b).
The suppressed trees have nevertheless a lower productive potential
than t h e dominant ones. According t o t h e r e s u l t of Lemma 2, this means t h a t
they are allocating a considerably l a r g e r fraction of
t h e i r growth
r e s o u r c e s t o t h e t r u n k than t h e o t h e r trees. The f a c t t h a t suppressed
trees are smaller a s r e g a r d s crown, r o o t s and basal area can thus b e
readily i n t e r p r e t e d in terms of the p r e s e n t model.
According t o Lemma 1,t h e suppressed trees with lower potential productivity r e a c h t h e point where they can no longer grow t a l l e r , much earl i e r than t h e i r dominant neighbours. But slowing down height growth in a
suppressed situation means more shade and less photosynthetic production,
and t h e r e f o r e t h e r e l a t i v e growth rate is bound t o t u r n negative no matter
which height growth p a t t e r n they have in t h e end. The model h e n c e explains
t h e mortality of s u p p r e s s e d trees as a consequence of not being a b l e t o
avoid one of t h e two p r o c e s s e s reducing growth potential, i.e. shading and
r e l a t i v e i n c r e a s e of wood in tree biomass.
Lemma 2 also applies to a n opposite situation where t h e r e i s no light
competition at all, i.e. a tree growing in t h e open. If i t maintains t h e same
height and length growth p a t t e r n s as t h e densely growing t r e e s i t will h a v e
a l a r g e r s h a r e of growth to b e allocated to foliage and r o o t s and more massive individuals will r e s u l t .
This is consistent with t h e observation t h a t
open-grown trees maintain l a r g e crowns although t h e y d o not grow conside r a b l y h i g h e r t h a n within-stand trees.
5. DISCUSSION
The p r e s e n t r e s u l t s c a n b e summarized as follows. Provided t h a t (1)
t h e pipe-model t h e o r y i s adequate, and t h a t (2) sapwood compensation i s
r e q u i r e d due t o t u r n o v e r , t h e growth and maintenance requirements of t h e
woody p a r t s i n c r e a s e f a s t e r t h a n t h e photosynthetic potential and t h e r e f o r e s o o n e r or later start t o limit growth. Under d i s t u r b a n c e s in productivity, t h e s e requirements may even make t h e system unstable, a n important
r e s u l t with r e s p e c t to t h e consequences of possible environmental change.
The implications of t h e model on t h e height growth p a t t e r n s of trees give
some f u r t h e r insight into t h e competitive p r o c e s s e s in a tree stand.
I t is interesting t h a t t h e summarizing form of t h e model, Section 2.3,
essentially r e d u c e s t h e dynamics of tree growth to t h a t of diameter and
height, t h e s t a n d a r d v a r i a b l e s used f o r t h e description of individual tree
growth in a v a r i e t y of empirical-statistical stand growth models (cf. Botkin
e t al., 1972; S h u g a r t , 1984). The s t r u c t u r a l constraints thus provide a way
of connecting t h e mass balance approach (Equation (2)) with t h e more conventional tree growth models. Moreover, if development of biomass c a n be
derived from t h a t of geometric dimensions, l a r g e d a t a s o u r c e s become
available f o r t h e verification of models. That helps t o solve one of t h e
major problems of t h e m a s s balance approach (cf. Makela and Hari, 1985).
The t r e e model used in t h e so-called gap models of f o r e s t stands (cf.
Shugart, 1984) v e r y much resembles t h e p r e s e n t one, with t h e p r o p e r t i e s
t h a t (1) leaf a r e a i s proportional t o basal area, and (2) increasing height
provides a limitation t o growth. An important difference is, however, t h a t
in t h e gap models maximum height is a species-specific constant r a t h e r than
a variable depending on s i t e conditions. When applied t o environmental
change t h e gap models do not, t h e r e f o r e , show t h e unstable behaviour
described in Section 4.2. Instead, they account f o r t h e increased mortality
under environmental stress with t h e aid of a stochastic dependence of mortality on growth r a t e (West et a l . , 1980).
So as t o assess t h e applicability of t h e results, let us review t h e premises of t h e derivation. The key assumption is t h a t sapwood area and foliage
biomass o c c u r in constant ratios. Although many empirical studies seem t o
confirm this (see Section I ) , more detailed consideration may yield contradicting observations. Following the argument t h a t sapwood s e r v e s as a water
conducting medium, i t i s expected t h a t variation of those environmental fact o r s t h a t affect t h e availability, conductance and transpiration of water will
also cause variation in t h e p a r a m e t e r s
vi.
In o r d e r t o i n c o r p o r a t e this in
t h e model, w e could attempt t o define q i in terms of t h e environmental variables.
Let u s denote t h e water conductivity of sapwood by ow (kg
~ a t e r / ~ e a r / sapwood),
m~
and t h e efficiency of water use by
srw
(kg water
transpired/kg C assimilated). A balance requirement analogous t o t h e one
applied t o t h e foliage-root r a t i o (Equation (13)) would then become
aW A =
srw ac Wf
implying t h a t
S o far o u r empirical d a t a do not allow, however, a p r o p e r test of this relationship.
A s pointed out in Section 4, t h e turnover rates of t h e different organs
play a n important r o l e in t h e resulting tree dynamics. Especially if height
and foliage growth i s slow, i t is t h e t u r n o v e r rate of sapwood t h a t essentially determines t h e growth requirements of t h e woody organs. I t seems
plausible t h a t sapwood t u r n o v e r i s r e l a t e d t o t h e pruning of branches,
which slows down in t h e later s t a g e of canopy development. According t o
t h e model results, t h i s would increase net productivity and t h u s postpone
t h e attainment of equilibrium.
A b e t t e r understanding of t h e t u r n o v e r
processes of sapwood t h e r e f o r e seems crucial for t h e f u r t h e r development
of t h e model.
A s w a s pointed out e a r l i e r , many r e c e n t observations s u p p o r t t h e ideas
of t h e pipe model t h e o r y at least roughly, suggesting t h a t a n a c c u r a t e
correspondence with measurements can b e obtained merely with minor
improvements, such as t h e time dependence of c e r t a i n parameters, and
p e r h a p s t h e incorporation of t h e relationship in Equation (52). However,
since t h e applicability of t h e model totally depends upon t h e adequacy of
t h e pipe model s t r u c t u r e , l e t us critically review t h e underlying hypotheses
and compare them with a tentative alternative.
The pipe model t h e o r y i s consistent with t h e idea t h a t i t is t h e conductivity of t h e pipeline t h a t r e s t r i c t s t h e availability of water t o t h e foliage
(Shinozaki et al., 1964a). One could also assume t h a t s t o r a g e capacity is a
c r i t i c a l parameter. Since s t o r a g e capacity is proportional t o volume, sapwood volume instead of area then becomes t h e c r i t i c a l p a r a m e t e r to match
with t h e size of t h e foliage. This implies t h a t tree height no longer supplies
a n entirely negative feedback t o growth, but a steady height growth c a n be
maintained.
The requirement t h a t conductivity r a t h e r than s t o r a g e capacity is restricting presumes t h a t t h e trees e i t h e r (1) cannot s t o r e water or r e c h a r g e
t h e sapwood a f t e r exhaustion, or (2) do not o c c u r in environments where
long d r y and moist periods a l t e r n a t e , making s t o r a g e capacity profitable.
Condition (1) seems t o b e approximately t r u e of many hardwood s p e c i e s with
a ring porous conducting s t r u c t u r e , whereas t h e coniferous conducting
s t r u c t u r e would b e more a p p r o p r i a t e f o r t h e development of s t o r a g e (Waring and Franklin, 1979). However, in t h e dominant region of conifers, t h e
boreal and o r o b o r e a l zones, t h e main growth limiting f a c t o r i s t e m p e r a t u r e
instead of water (e.g. Kauppi and Posch, 1985), which means t h a t condition
(2) is fulfilled.
A s a n interesting exception to this p a t t e r n , Waring and Franklin (1979)
have discussed t h e e v e r g r e e n coniferous f o r e s t s of t h e north-western coast
of t h e American continent. They draw attention to both t h e climate with w e t
winters and d r y summers, and t h e r e c h a r g e ability of t h e conifers' sapwood.
They a r g u e t h a t t h e s e two f a c t o r s , t o g e t h e r with longevity and sustained
height growth, c o n t r i b u t e t o t h e long-lasting high productivity of t h e s e
f o r e s t s . The p r e s e n t analysis goes e v e n f u r t h e r by suggesting t h a t a l s o t h e
longevity and t h e ability of sustaining height growth c a n b e consequences of
t h e unusual water economy. If t h i s i s t h e case, t h e giant trees should not
have a constant fo1iage:sapwood r a t i o . F o r t h e time being, w e may only
speculate: P e r h a p s i t w a s t h e capability of overcoming t h e r e s t r i c t i o n s of
t h e pipe-model s t r u c t u r e t h a t enabled t h e coniferous f o r e s t s of t h e Pacific
Northwest to become one of t h e world's g r e a t e s t accumulators of biomass.
This study h a s given some insight i n t o t h e r o l e of length growth patt e r n s in t h e dynamics of t r e e growth. The length growth p a t t e r n itself w a s
not defined, however. This a p p r o a c h was chosen b e c a u s e length growth,
although f a i r l y well understood in g e n e r a l t e r m s , is still missing a dynamic
connection t o t h e environment and t o t a l growth. Lemmas 1 a n d 2 show t h a t
u n d e r t h e assumptions of t h e pipe model t h e o r y , t h e growth of a tree i s v e r y
sensitive t o i t s length growth p a t t e r n . In t h e highlight of t h e f a i r l y s t a b l e
and r e g u l a r growth p a t t e r n s t h a t o c c u r in r e a l i t y , t h i s r e s u l t indicates t h a t
w e are missing a n additional balancing mechanism between length a n d diame-
ter - or biomass
-
growth.
As indicated in Section 1, s u c h a balancing
mechanism seems likely to involve interactions between trees. If t h i s i s
t r u e , a n analysis of t h e whole s t a n d is r e q u i r e d in o r d e r to understand t h e
length growth of individual trees. In t h i s r e s p e c t , some new insight h a s
been gained by looking at height growth p a t t e r n s as evolutionary consequences of intra-specific competition, with t h e a i d of game t h e o r y (Makela,
1985). The p r e s e n t study f u r t h e r emphasizes t h e n e e d f o r such i n t e g r a t e d
analyses as means of increasing o u r understanding of t h e f a c t o r s t h a t r e g u l a t e t h e development of tree geometry.
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