Implications of the Pipe Model Theory on Dry Matter Partitioning and Height Growth in Trees

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

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 r e c e i v e d only lim- ited review. Views o r opinions e x p r e s s e d h e r e i n d o not neces- s a r i l y r e p r e s e n t t h o s e 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 r a i n t o forests. Like all submodels of RAINS, t h e f o r e s t impact submodel will b e 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 r e q u i r e s 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 t h e r e a l object in t h e model.

Our understanding of t h e response of f o r e s t s 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 t h e 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 s t r u c t u r e . Because of its generality, t h e model can b e applied t o ' various situations, also giving insight into t h e impacts of a changing environment. An interesting contribution of t h e p a p e r is t h a t i t provides a quantitative relationship between sensitivity t o environmental s t r e s s , eco- physiological parameters and tree size.

Leen Hordijk P r o j e c t Leader

ACKNOWLEDGEMENTS

The author wishes to acknowledge the contributions of Pertti Hari and Leo Kaipiainen to the ideas presented in this paper. Special thanks a r e 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 p a r t i - tioning 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 ^main- tain 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 o n e 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 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 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 o r g a n s , 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, r e s p o n s e 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 s p e c i e s in t h e Pacific Northwest region to maintain a n almost unlimited height growth pat- t e r n are discussed in r e l a t i o n to t h e pipe model t h e o r y .

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 2.2.2 Foliage: Wood Ratio

2.2.3 Foliage: F e e d e r Root Ratio 2.2.4 Partitioning Coefficients 2.3 Summary of t h e Model

2.4 Extension of t h e Model t o Changing Environment 3. PROPERTIES OF THE MODEL

3.1 Root-Foliage Relationships 3.2 Length Variables and Growth

4. 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

5. 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 a l . (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 e a c h 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 physi- c a l support. Although t h e argument may b e 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. Con- s 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 con- ducting wood (Kaipiainen et al., 1985). The r a t i o s a p p e a r f a i r l y constant within s p e c i e s 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 ~ i et al., 1964b; Waring et al., 1982), l e s s 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 t h i s viewpoint. F i r s t , 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 p r o c e s s e s is r e q u i r e d 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, b u t tree height remains more o r l e s s unchanged (Whitehead, 1978). A r e a s o n f o r t h i s may b e t h a t t h e signifi- c a n c e of t h e length v a r i a b l e s f o r survival is not in t h e i r contribution to t h e internal balance of growth, b u t r a t h e r to t h e competitive ability f o r light and s p a c e .

A similar r a t i o h a s been observed between roots and leaves, with t h e d i f f e r e n c e t h a t t h e r a t i o v a r i e s with t h e soil n u t r i e n t level. White (1935) observed t h a t d r y m a t t e r partitioning between t h e roots a n d shoots of g r a s s and c l o v e r were dependent upon t h e light and nitrogen levels applied in l a b o r a t o r y 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 a n d Thornley (1982).

A dynamic growth model i s developed f o r f o r e s t trees w h e r e t h e p a r t i - tioning of growth between foliage a n d wood i s p e r f o r m e d s o as t o maintain t h e sapwood:foliage r a t i o c o n s t a n t . P a r t i t i o n i n g 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 p r i n c i p l e of functional balance. The model u s e s t h e time r e s o l u t i o n of o n e y e a r a n d t h e time s p a n of t h e life-time of t h e tree.

Using t h e model, t h e c o n s e q u e n c e s 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 a n d t h e i r dis- t r i b u t i o n t o d i f f e r e n t o r g a n s , p l a n t growth c a n conveniently b e analyzed with a mass b a l a n c e 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 c o e f f i c i e n t s , 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 a l l o c a t e d t o p a r t i , a n d t h e y s a t i s f y

f o r all times t . The growth rates of t h e biomass compartments are

- - -

wi X i G - S t , f o r a l l i ~ 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 a n d t u r n o v e r .

Provided t h a t t h e variables h i , _{G , and} Sf c a n be p r e s e n t e d as func- tions of t h e s t a t e v a r i a b l e s and identifiable p a r a m e t e r s , Equations (2) con- s t i t u t e a n operational dynamic model f o r t r e e growth. The following para- g r a p h s a r e devoted t o t h e derivation of t h e r e q u i r e d relationships.

Following a customary a p p r o a c h , 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 c a n b e assumed t h a t t h e carbon assimilated is totally consumed in growth and r e s p i r a t i o n . Denote annual photosynthesis (kg C a -') by P , annual r e s p i r a t i o n (kg C a-') by R , and t h e c a r b o n content of d r y matter by f C . Annual growth G (kg d r y weight a -') is t h e r e f o r e

G

=

J C ~ ( P - R ) . (3)

Photosynthesis i s assumed t o b e proportional t o foliage biomass, W f , and t o t h e specific photosynthetic activity, uc (kg C a (kg d r y weight)-'), which depends upon environmental and internal f a c t o r s :

P

=

uc Wf (4)

Following McCree (1970), r e s p i r a t i o n i s divided into growth r e s p i r a t i o n , R,,, and maintenance r e s p i r a t i o n , Rm :

R

=

R m + R g

Growth r e s p i r a t i o n Rg i s proportional t o growth r a t e G

Rg

=

rs G , (6)

and maintenance r e s p i r a t i o n R,,, i s 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 o n 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 c a n b e called a s t a n d a r d p r o c e d u r e f o r determining t o t a l growth and senescence in growth models (Thornley, 1976; d e Wit et al., 1978; Agren and Axelsson, 1980). F o r t h e partitioning coefficients, such a s t a n d a r d does not exist. Section 2.2 treats t h e p a r t i - tioning of growth in forest trees by using t h e pipe model t h e o r y a n d 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 d e p i c t s tree s t r u c t u r e as a combination of five functionally d i f f e r e n t p a r t s : foliage, f e e d e r r o o t s , 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 o r g a n s are d e r i v e d from t h e geometric dimensions of t h e o r g a n s . Denote sapwood area at crown b a s e by A , , t h e t o t a l sapwood area of primary b r a n c h e s at foliage b a s e by Ab and t h e t o t a l 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 i n c o r p o r a t e d a r e tree height, h , , crown r a d i u s , 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 , . b r a n c h e s , 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 v a r i a b l e s 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 explana- tion, 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 s h a p e 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 o r g a n . The v a r i a b l e s are summarised in Table 1.

2.2.2. Foliagemood ratio

The pipe model t h e o r y 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), t h i s a p p l i e s to both s t e m and b r a n c h e s . 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 d i f f e r e n t for s t e m and b r a n c h e s , 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 supplemen- t a r y notions, t h e basic observation can b e e l a b o r a t e d 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 foli- a g e biomass Wf

(2) The t o t a l sapwood area of primary b r a n c h e s ( a t foliage base), A*, i s p r o p o r t i o n a l to t o t a l foliage biomass

v b A b

=

Wf (11)

(3) The t o t a l 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 p r o - portional t o t o t a l foliage biomass

Table 1: Model variables

Name Meaning

f foliage

r f e e d e r r o o t s

s stem

b b r a n c h e s

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 biomass (in c a s e of wood:sapwood BM)

i =j' ,r ,s ,b ,t

4

sapwood a r e a

i =s ,b ,t

Other variables

X i partitioning coefficient i =j' ,r ,s , b ,t

G t o t a l growth

P photosynthesis

R

r e s p i r a t i o n

Rm maintenance r e s p i r a t i o n

RL3 growth r e s p i r a t i o n

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 s p e c i e s specific p a r a m e t e r s .

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 David- son (1969), t h e metabolic activities of t h e r o o t s and t h e photosynthetic o r g a n s a r e in balance. Denote t h e specific photosynthetic activity by ac (kg C a-' ( k g d r y weight)-') and t h e specific r o o t activity (of nitrogen uptake) by O N (kg N a

-'

( k g d r y weight)-'). The relationship c a n b e e x p r e s s e d in t h e form

' r r ~ O c

Wf =

^{O N}

WT

where nN i s a n empirical p a r a m e t e r .

This requirement c a n b e understood in t e r m s of a balanced c a r b o n 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 s i t e s , t h e relationship p r o d u c e s dif- f e r e n t fo1iage:feeder r o o t r a t i o s , t h u s 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 a n d 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 p r e s e n t e d in Section 2.2 are satis- fied. F i r s t , l e t u s 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 coeffi- cients which e n t e r t h e equation through t h e time derivatives of Wf and W,.

If t h e s e 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 c a n b e maintained if and only if

dAi

-

^dWf

71i

dt -

d t ' i = s , b , t ( 1 8 )

In o r d e r to utilise t h i s equation f o r t h e derivation of t h e partitioning coefficients of t h e woody o r g a n s , X i (i =s , b . t ) , l e t u s f i r s t relate t h e time d e r i v a t i v e of Ai t o t h a t of Wi

.

By Equation ( 9 ) ,

d W i

-

^d

^Ai

^{d h i}

d t

=

Pi 4'i ( _{d t} h i +

-

_{d t} ^{A i ) ,} i = s , b , t ( 1 9 ) 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 ) . Com- bining (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 t h u s become functions of t h e partitioning coeffi- cients:

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 substi- t u t e d into Equation (18). This gives r i s e t o t h e following partitioning coeffi- c i e n t s Xi:

w h e r e

w h e r e 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 a n d wood as a function of t h e l e n g t h s of t h e woody o r g a n s .

2.3. Summary of the Model

I t follows from t h e a b o v e d e r i v a t i o n 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 o n e of t h e dependent v a r i a b l e s

Wf

,

W,, A,,

Ab a n d At only. If foliage biomass i s 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:

w h e r e

and a* and are defined as in Equations (15), (17), (24) and (25) f o r i = r , s , b , t . Additionally,

The v a r i a b l e s of t h e model are listed in Table 1, and Table 2 gives a summary of i t s p a r a m e t e r s .

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 r e a s o n s pointed o u t in Section 1, i t is difficult t o relate length growth t o biomass growth d i r e c t l y , w e shall consider t h e growth rates of t h e length v a r i a b l e s as unknown "strategies" t h a t c a n v a r y . Below, t h e behaviour of t h e model i s analyzed as a function of t h e s e s t r a t e g i e s .

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 v a r i a b l e s , Equations (10)-(13), are constant o v e r time. Actually, t h e m e t a - bolic activities ac, O N , and

-

as will b e discussed l a t e r

-

a l s o t h e parame- ters q i , may respond t o environmental changes such as shading and fertili- zation during t h e life-time of t h e tree. If t h e change i s f a s t e r t h a n t h e r e s p o n s e 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 a b l e t o change t h e partitioning coef- ficients 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 t h i s 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, b u t maintain t h e

Table 2. P a r a m e t e r s

Names

Qc

QN

PC

T N

q s q b q t p s P b P t q s q b q t

Sf

ST

d s

Meaning

foliage specific activity r o o t specific activity

carbon content of d r y weight N:C r a t i o of d r y weight

foliage D W:stem sapwood r a t i o foliage DW:branch sapwood r a t i o

foliage D W:transport r o o t sapwood r a t i o stemwood density

branchwood density rootwood density form f a c t o r of s t e m

form f a c t o r of branch system form f a c t o r of root system foliage specific t u r n o v e r r a t e f e e d e r root specific turnover rate s t e m sapwood specific t u r n o v e r rate

d b branch sapwood specific turnover rate

4

r o o t sapwood specific t u r n o v e r rate r ^m specific maintenance respiration

Q specific growth respiration

Unit

kg[Cl/kg[DWla kg[Nl/kg[DWla kg[Cl/kg[DWl kg[Nl/kg[Cl

k g [ ~ ~ l / r n ~ [ S ~ l

~ ~ [ D W I / ~ ~ [ S W I

~ ~ [ D w I / ~ ~ [ s w I

kg/m3 kg/m3 kg/m3 unitless unitless

partitioning c o e f f i c i e n t s d e r i v e d from t h o s e 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 o r i g i n a l d i f f e r e n t i a l equations, Equation (2). A partitioning model with similar a d a p t i v e b e h a v i o u r h a s a l r e a d y b e e n 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 i n v a r i a n t form, but some of t h e considerations in S e c t i o n 4 make u s e 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 u s 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 p r o d u c t i v e p a r t i s

AP

=

Af

+

A, ( 3 2 )

Let us define t h e s p e c i f i c growth rate,

5 ,

as t h e growth rate p e r p r o d u c t i v e biomass, and d e n o t e t h e s p e c i f i c t u r n o v e r rate of p r o d u c t i v e biomass by s p . The n e t growth rate of Wp i s t h e n

The quantities

g

a n d 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 t h r o u g h t h e d i f f e r e n t i a l 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 c o r r e s p o n d t o uc, r,, and rmf and a i , respectively, with t h e d i f f e r e n c e t h a t t h e y 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 U N . The depen- dence i s such t h a t If TTN uc

>>

^{O N ,} t h e n r o o t activity is r e s t r i c t i n g growth and small changes in uc have l i t t l e impact on t h e t o t a l productivity. If

T N U C

<<

^{U N , t h e n 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 o n e have equal effects.

Environmental variation affecting r o o t and foliage activity can b e brought into t h e model t h r o u g h t h i s relationship.

3.2. Lengh Variables and Growth

The r e l a t i v e growth r a t e of t h e productive p a r t .

Rp,

is defined a s fol- lows

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 them- selves, 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 r e a s o n a b l e 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 sub- s c r i p t s . Denote

Rp

c a n t h e n b e written as a function of h as follows:

The r e l a t i v e growth rate is t h u s 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 a n d t h e generalized length h . The r e s u l t s c a n b e summar- ized as Lemma 1 below.

Lemmal.

I.

If h

<

^{7 ,} cP ^{t h e n}

^Rp

^{> O .} B

2. If h

= = , a

then Rp = O .

B

3. If h

>

^{r ,}

a

^then Rp < O .

B

This means t h a t if t h e lengths of t h e woody o r g a n s continue t o grow a f t e r Equation (46) h a s been fulfilled, t h e r e l a t i v e 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 h a s been r e a c h e d , growth will con- tinue exponentially at t h e rate Rp which t h u s remains constant.

The. time development of t h e productive biomass depends exponentially on t h e r e l a t i v e growth rate:

With a n exchange of v a r i a b l e s , t h e i n t e g r a l 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 c a n 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 c a n a l s o ise i n t e r p r e t e d vice versa. If t h e height and length growth rates a r e fixed, then variation in t h e p a r a m e t e r values causes d i f f e r e n c e s 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, t h e n gradually siow down t o r e a c h a constant growth r a t e , and t h e growth finally c e a s e s 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 r e l a t i v e time s c a l e s v a r y . 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 a c c e p t e d t h a t t h e decline in growth in old trees is a whole- plant phenomenon r a t h e r t h a n a consequence of t h e aging of individual tis- sues (NoodBn, 1980), b u t t h e mechanisms of decline are not fully under- stood. I t h a s been pointed out t h a t t h e proportion of stem t o crown gradu- ally i n c r e a s e s with a g e and size and, consequently, t h e r a t i o of carbohy- d r a t e s produced t o t h a t consumed in r e s p i r a t i o n 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).

P a r t i c u l a r l y , difficulties in water relations have been a r g u e d t o cause s e v e r e water deficits and death of leaves and b r a n c h e s (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 n e t growth of r o o t s and foliage cannot b e maintained if t h e woody p a r t s e x c e e d 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 d e c e l e r a t e d . If t h i s action is postponed until v e r y 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 con- tinue 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 calcil- lated f o r a r a n g e of nutrient uptake r a t e s , corresponding t o different grow- ing 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 2 0.04 0.01 0.5 0.25 600.

450.

2400.

400 0.75 0.70

S o u r c e

Agren and Axelsson 1980 guess

estimates based on elemental contents according t o various s o u r c e s

Agren and Axelsson 1980 Kaipiainen et al. 1985 Kaipiainen et al. 1985 Kaipiainen et al. 1985 Karkkainen 1977

I

estimates based on tree from measurements according t o various sources

1

corresponds to a v e r a g e life t i m e of 4 y e a r s

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 a n d / o r a c c e l e r a t e t h e t u r n o v e r 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 c l o s u r e , 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 s u r f a c e . The s a m e i s t r u e of t h e increasing anthropogenic stress load.

Atmospheric a i r pollution h a s been r e p o r t e d t o d e c r e a s e specific photosyn- thesis and a c c e l e r a t e foliage aging, and soil acidification i n c r e a s e s fine r o o t mortality and d e c r e a s e s specific n u t r i e n t uptake rate by leaching nutrients.

Such s t r e s s f a c t o r s c a n ' 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 p a r a m e t e r

STEM HEIGHT VERSUS AV LENGTH

Figure 3. Stem height as a function of a v e r a g e length using different stem: branch: r o o t length ratios. All c u r v e s have hb

=

ah,, ht

=

Zah,, and a v a r i e s as follows:

2 50

.-

m _{0 )} C

E

0 )

X 40

41

a a a a a

30

20

10

0

0 5 10 15 20 25

Average Length

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

d e c r e a s e s . W e s h a l l study in more detail t h e conditions u n d e r which t h e t r e e will not s u r v i v e 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 posi- tive level of productive biomass. This condition will b e 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), r e l a t i v e growth rate i s positive only if

Q >

J h o (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 r e a c t i o n 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, mainte- nance and growth of woody p a r t s i n c r e a s e f a s t e r t h a n growth potential.

Since h does n o t , normally, d e c r e a s e in time, t h e model implies t h a t a g e i n c r e a s e s 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 produc- tivity.

The model implies t h a t t h e long-term r e s p o n s e 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 environ- mental change solely means a smooth d e c r e a s e in growth rates and maximum sizes of t h e a f f e c t e d 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 s p e c i e s with d i f f e r e n t c a p a c i t i e s of environmental flexibility of height growth could manifest d i f f e r e n t r e a c t i o n s t o long-lasting environmental stresses.

4.3. H e i g h t Growth

T r e e height, r e l a t i v e 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 s h a d e 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 r e d u c e d , and t h e y will probably also have a h i g h e r 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-to- d o 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 k e e p 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 e x t e n t s v a r y considerably less t h a n t h e corresponding foliage growth rates (Logan, 1965,1966a,1966b).

The s u p p r e s s e d trees have n e v e r t h e l e s s a lower productive potential t h a n t h e dominant ones. According t o t h e r e s u l t of Lemma 2, t h i s means t h a t they are allocating a considerably l a r g e r f r a c t i o n 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 t h a n t h e o t h e r trees. The f a c t t h a t s u p p r e s s e d trees are smaller a s r e g a r d s crown, r o o t s and basal area c a n t h u s b e readily i n t e r p r e t e d in terms of t h e p r e s e n t model.

According t o Lemma 1, th e suppressed trees with lower potential pro- ductivity r e a c h t h e point where t h e y can no longer grow t a l l e r , much ear- l i e r than t h e i r dominant neighbours. But slowing down height growth in a suppressed situation means more s h a d e and l e s s 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 h e i g h t growth p a t t e r n t h e y h a v e 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 n o t being a b l e t o avoid o n e of t h e two p r o c e s s e s reducing growth potential, i.e. shading a n d r e l a t i v e i n c r e a s e of wood in tree biomass.

Lemma 2 also a p p l i e s to a n o p p o s i t e situation where t h e r e i s n o light competition at all, i.e. a tree growing in t h e open. If i t maintains t h e same h e i g h t a n d 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 a n d r o o t s a n d more mas- s i v e individuals will r e s u l t . This i s consistent with t h e o b s e r v a t i o n t h a t open-grown trees maintain l a r g e crowns although t h e y d o n o t grow consid- e 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 a d e q u a t e , a n d 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 r e q u i r e m e n t s 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 a n d 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 produc- tivity, t h e s e r e q u i r e m e n t s may e v e n make t h e system u n s t a b l e , 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 i s i n t e r e s t i n g 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 d i a m e t e r a n d h e i g h t , 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 d e s c r i p t i o n 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 c o n s t r a i n t s t h u s provide a way of connecting t h e mass balance a p p r o a c h (Equation (2)) with t h e more con- ventional tree growth models. Moreover, if development of biomass c a n b e 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 o n e of t h e major problems of t h e m a s s balance a p p r o a c h (cf. Makela a n d 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.

S h u g a r t , 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 p r o p o r t i o n a l t o basal area, and (2) increasing height provides a limitation t o growth. An important d i f f e r e n c e 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 t h a n a v a r i a b l e depending o n s i t e conditions. When applied t o environmental change t h e gap models d o not, t h e r e f o r e , show t h e unstable behaviour d e s c r i b e d in Section 4.2. Instead, they account f o r t h e i n c r e a s e d mortality under environmental stress with t h e aid of a stochastic dependence of mor- tality on growth r a t e (West et a l . , 1980).

S o as t o a s s e s s t h e applicability of t h e r e s u l t s , let u s review t h e prem- i s e s 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 r a t i o s . Although many empirical studies seem t o confirm t h i s ( s e e Section I ) , more detailed consideration may yield contrad- icting observations. Following t h e argument t h a t sapwood s e r v e s as a water conducting medium, i t i s e x p e c t e d t h a t variation of those environmental fac- t o r s t h a t a f f e c t t h e availability, conductance and t r a n s p i r a t i o n of w a t e r will a l s o c a u s e variation in t h e p a r a m e t e r s

v i .

In o r d e r t o i n c o r p o r a t e t h i s in t h e model, w e could attempt t o define q i in terms of t h e environmental vari- ables. Let u s denote t h e water conductivity of sapwood by ow (kg

~ a t e r / ~ e a r / m ~ sapwood), and t h e efficiency of water use by

srw

(kg w a t e r t r a n s p i r e d / k g 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 t h e n become

aW A

= srw

ac Wf implying t h a t

S o far o u r empirical d a t a d o not allow, however, a p r o p e r test of t h i s rela- tionship.

A s pointed o u t in Section 4, t h e t u r n o v e r rates of t h e d i f f e r e n t o r g a n s 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 essen- tially 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 b r a n c h e s , which slows down in t h e later s t a g e of canopy development. According t o t h e model r e s u l t s , t h i s would i n c r e a s e n e t 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 p r o c e s s e s of sapwood t h e r e f o r e seems c r u c i a l for t h e f u r t h e r development of t h e model.

A s w a s pointed o u t 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 l e a s t roughly, suggesting t h a t a n a c c u r a t e c o r r e s p o n d e n c e with measurements c a n b e obtained merely with minor improvements, such as t h e time dependence of c e r t a i n p a r a m e t e r s , and p e r h a p s t h e incorporation of t h e relationship in Equation (52). However, s i n c e 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 t e n t a t i v e 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 conduc- tivity 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 a l s o assume t h a t s t o r a g e c a p a c i t y is a c r i t i c a l p a r a m e t e r . Since s t o r a g e capacity is p r o p o r t i o n a l t o volume, sap- wood volume instead of area t h e n 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 n o longer supplies a n e n t i r e l y negative feedback t o growth, but a s t e a d y height growth c a n b e maintained.

The requirement t h a t conductivity r a t h e r t h a n s t o r a g e capacity is res- t r i c t i n g 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 p e r i o d s 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 p o r o u s 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 (War- ing and Franklin, 1979). However, in t h e dominant region of c o n i f e r s , t h e b o r e a l 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 t h i s 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 a n d sustained

height growth, c o n t r i b u t e t o t h e long-lasting high p r o d u c t i v i t y of t h e s e f o r e s t s . The p r e s e n t a n a l y s i s g o e s 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 a n d t h e ability of sustaining h e i g h t growth c a n b e c o n s e q u e n c e s 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 h a v e a c o n s t a n t 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 e n a b l e d t h e c o n i f e r o u s f o r e s t s of t h e P a c i f i c Northwest to become one of t h e world's g r e a t e s t a c c u m u l a t o r s of biomass.

This s t u d y h a s given some insight i n t o t h e r o l e of length growth pat- t 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 n o t defined, however. This a p p r o a c h was c h o s e n b e c a u s e l e n g t h growth, although f a i r l y well u n d e r s t o o d in g e n e r a l t e r m s , is s t i l l missing a dynamic connection t o t h e environment a n d 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 p i p e model t h e o r y , t h e growth of a tree i s v e r y s e n s i t i v e 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 a n d 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 i n d i c a t e s 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 S e c t i o n 1, s u c h a balancing mechanism seems likely to involve i n t e r a c t i o n s 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 u n d e r s t a n d 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 b e e n gained by looking at h e i g h t growth p a t t e r n s as evolutionary conse- q u e n c e s of i n t r a - s p e c i f i c 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 s u c h i n t e g r a t e d a n a l y s e s as means of i n c r e a s i n g 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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Botkin, D.B., J .F. Janak and J.R. Wallis, 1972. Some ecological consequences of a computer model of f o r e s t growth. J. of Ecol. 60:849-873.

Brouwer, R. 1962. Distribution of Dry Matter in t h e Plant. Neth. J. Agric.

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Davidson, R.L. 1969. Effect of Root/Leaf Temperature Differentials on Root/Shoot Ratios in Some P a s t u r e Grasses and Clover. Ann. Bot.

33:561-569.

Jacobs, M.R. 1955. Growth h a b i t s of t h e eucalypts. For. Timber Bur., Aust.

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