Acid Precipitation and Catastrophes in Forest Dynamics: A Conceptual Framework

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ACID PRECIPJTATION AND

CATASTROPHES IN FOREST DYNAMICS:

A CONCEPTUAL FRAMFWORK

M. Gatto and S. Rinaldi

UP-85-6 January

1935

Working F t q e r s a r e interim reports on work of t h e International Insti- t u t e for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do c o t necessarily represent those of t h e Institute or of its National Member Organiza- tions.

IhPTERNATIONAL INSTITUTE FOR APPLIED STSTEMS ANALYSIS A-236 1 Laxenburg, Austria

TKE

AUTHORS

M.

Gatto is Associate Professsr of Applied Ecology a n d S. Rinaldi Prc- fessor of Systems Theory in t h e Engineering School of t h e Politecnico di Milano, Milan (Italy).

ACKNOWLEDGEMENTS

The a u t h o r s a r e grateful for support f r c n t h e I c t e r ~ a t i o n a l I n s t i t u t e for Applied Systems Analysis, L a x e ~ b u r g (Austria) a n d from t h e Cen- trodi Teoria dei Sistemi

-

CNR, Milar, (Italy).

INTRODUrnON

Acid precipitatior, h a s raised wide concern in t h e last decade bctk i r Europe ar,d ir, Ncrth America. Several internaticnal institntions (amor,g which a r e OECD, ChXP, WMO, IIASA) s t a r t e d a n d in some cases completed programs for moritcririg, modeling and evaluating causes and effects of this moderr, p I a g ~ e . It is widely r e c c ~ c i z e d t h a t orie major impact of acid precipitztion - bcth dry a r d wet - is o r fcrests. Recent episcdes, like t h e most pr;b!icized c c e in t h e Black Fcrest, and many o t h e r s in t h e USA, Canada, Germany, Poland, etc., have throwr, this issue into t h e political arena, while pnblic opir,ion in several c c u r - tries is beccmirg increasingly sensitive t o t h e problem of forest damage from

"acid rair,", a s t h e phenomenon is commonly t e r m e d .

A large body of scientific literature is ncw available with regard to t h e effects cf air pcllution on forests: just a s a n example we cite t h e bock by S n i t h ( l g e l ) , t h e papers by Abrahamser, a n d Dollard (1979), Abrahamsen (1979, 1981), Matziler ar,d U!rick (1981), Ulrich (1984), a c d n a r , y r e p o r t s of t h e Norwe~iar, Interdisciplinary Research Programme or, "Acid Precipitatior, - Effects cr_

Fcrest and Fish". This l i t e r a t u r e is mainly ccncerned with t h e carefu! a n d detailed identificatior of t h e processes t h a t directly o r indirectly a f e c t t h e vegetaticn. The emergir,g evidence is t h a t t h e r e a r e many ways in which p l z r t growth and sumival c a n depend npon acidic deposition and, qucting from Smith (19El). t h z t scmetimes "fcrests a r e ineuenced by a i r c o n t a n i n a c t s ir, a subtle manner". This paper's c c n t r i b u t i o r is quite difIerent frcm t h e above apprcach and may be coRsidered somewhat simple-minded a n d c r ~ d e . In fact. we COP.-

dense a variety of c c a p l e x phenomena into a simple mat5ematical model cf t r e e and nutrierit dynamics. This simplicity itself, hcwever, allows ns t o analyze t h e stability properties of a forest ecosystem subject t o acid precipita- tion and to pinpoint some possible key mechanisms of acid impact t h a t deter- mine its lorg-term fate a n d possibly its collapse. It should be c l e a r t h a t t h e aim of this work is basically conceptual a n d qualitative a n d t h a t o u r mcdel is riot meant t o rivz!, on quantitative groucd, t h e detailed sirnulatior, models cf forest z r d / c r soil dynamics t h a t have been and a r e c u r r e n t l y being built.

The focus of this paper will be on t h e intrinsic n o ~ l i c e a r n a t u r e cf t h e vegetaticr. respcnse t o acid precipitation, which plays a principal role in caus- ing catastrophes i n fcrest dynamics ( a very brief a c c o u n t on t h e coccept cf catastrcphe will be given in t h e ~ e x t section). We contemplate t h r e e possible mechanisms of forest disruption from acidic depositicn: (i) through direct effects upor vegetation (snch a s physical damage t o tissues), (ii) through increased scil acidity, which i n t e r alia entails t h e release cf tcxic a m c u t s cf a l u m i r u m a n d macganese, (iii) thrcugh accumulation in t h e t r e e biomass cf excessive amour-ts cf n u t r i e n t s , which n a y be harmful t c t h e plaxts. We analyze t h e t h r e e corresponding modes of forest decline a n d d e m o r s t r r t e . within t h e framewcrk of c a t a s t r o p h e theory, t h a t t h e s e c c r d a n d third m e c h a r - isms give rise t c a so-called fold catastrcphe.

THE

CONCEPT

OF

C A T B R O m

The description of catastrophes, snrprises, discontinuities c r whatever t h e y might be czlled h a s recently b e c o n e quite fashiccable in many fields cf natural, physical a n d social sciences. With regard to forest dynamics a remark- aSle e x a ~ p l e is t h e long study conducted by Holling a n d coIleagues or, t h e i n t e r a c t i c r between t h e s p r u c e buriworm a n d t h e c o n i f e r c ~ s fcrest of e a s t e r n Ncrth America (Clark e t al. 1978, Ludwig e t el. 1978, Casti 19e2j.

X t t c ~ g h a formal theory cf c a t a s t r c ~ h e s was develcpcd rncre t k a r a decade agc, maizly by Thcm (1972), we dc pot a t t e m p t t c give a c y k i ~ d cf

g e n e r a l review of t h e t h e c r y , b u t c ~ l y a v e r y simple a n d brief p r e s e r t a t i c n of what is m e a n t by c a t a s t r o p h e i n t h i s specific c o n t e x t .

A first basic f a c t o r t h a t n e e d s t o b e i n t r o d u c e d i s t h e inflow of n u t r i e x t from e x t e r ~ a l s o u r c e s which, for o u r p u r p c s e s , a c t s a s t h e main driving f c r c e t o t h e f c r e s t e c o s y s t e m . If t h i s i n p u t is c o n s t a n t over t i m e , t h e s t a t e cf t k e sys- t e n , i.e. t h e v a r i a b l e s c h a r a c t e r i z i n g t h e problem ( b i o m a s s e s of d i f I e r e ~ t t r e e s p e c i e s , r , ~ t r i e r _ t c c r i c e c t r a t i o n s i r t h e scil a n d in t h e v e g e t a t i o n , e t c . ) will con- v e r g e t o o n e of t h e e c o s y s t e m s t a b l e eqcilibria. If t h e n u t r i e n t inflow - call it W

-

i s s m o c t h l y a n d slcwly growing, e.g., d u e t o a n i n c r e a s e of a c i d precipita- t i o n , also t h e s e equilibria will in g e n e r a l vary a c c o r d i n g t c a s m o o t h patterr?.

S o m e t i m e s , however,

-

a s m a l l ( i n f i ~ i t e s i m a l ) p e r t u r b a t i o n cf W from a p a r t i c t l a r given v a l u e W e n t a i l s t h e a p p e a r a n c e o r d i s a p p e a r a n c e of ar, equilibrium. In o t h e r c a s e s , a n equilibrium-which i s s t a b l e for W

<

b e c c n e s c r i t i c a l for W =

?

a n d u n s t a b l e f c r W

> W ,

o r vice versa. Now, if o n e i m a g i n e s t h a t a f c r e s t is in a s t a b l e s t e a d y s t a t e for a n u t r i e n t ir,f?ow W

< F

arid t h a t t h i s equilibrium disappears o r b e c o m e s u n s t a b l e for W

> W ,

crie c a n s a y t h a t a c a t a s t r o p h e h a s o c c u r r e d . I r fact, a s s o o n a s t h e i n p u t e x c e e d s t h e t h r e s h c l d

F,

t h e s t a t e c: t h e ecosystem moves t c w a r d a different s t a b l e r e g i m e which c a n b e f a r from t h a t existing befcre t h e p e r t u r b a t i o n . This d y n a m i c t r a n s i t i c n f r c n o n e t c a ~ o t h e r equilibrium i s a c a t a s t r o p h e . The m o s t s p e c t a c u l a r c a t a s t r c p h e s a r e t h e co!- l a p s e of a forest a n d t h e s u d d e n blocm of a deforested a r e a .

0

W ' W"

Nutrient Inflow W

Figure 1. Hypothetical r e p r e s e n t a t i c r , of t h e t r e e b i c n a s s T ccrrespor,dizg t o s t a b l e equilibria, a s f u n c t i o n of t h e n u t r i e n t irAfiow W . The p z t t e r n is t h a t cf a typical c a t a s t r o p h e with h y s t e r e s i s .

Figure 1 gives a s i m p l e , p i c t o r a l a n d hjrpothetical r e p r e s e n t a t i o r of t h e o c c u r r e n c e of c a t a s t r o p h e s with r e f e r e n c e t o t h e b i o m a s s T of e t r e e species. If t h e n u t r i e n t icFcw i s below

W o ,

t h e sci! i s simply t o c poor f c r t h e species t o exist a n d t h e cnly s t a b l e equilibrium is T

=

0 . By i ~ c r e a s i r g W beyond

W 3 ,

t h e e x t i n c t i o n i s n o l o n g e r s t a b l e a n d t h e u n i q u e s t a b l e s t e a d y s t a t e is a positive s t a n d i c g biomass, b u t wheri t h e ~ u t r i e r - t i~ . p c t e x c e e d s

W:.

T = 0 b e c o m e s

stable again, s o t k z t two stable equi!ibria a r e simultaneously present. The forest, however, is not driven t c t r e e extincticn, unless a n o t h e r massive p e r t c r - bation, like fire, wind, pest, e t c . . brings t h e s t a t e of t h e ecosystem to t h e domain of a t t r a c t i o n of t h e extincticn eqeilibrium. When K increases beyond t h e level W", t h e or,ly possible steady s t z t e is T

=

0 ar,d t h e forest is inevitably driven t h e r e . This catastrophe can be reversed by reducing t h e nutrient inflow.

However, t h e d e c r e a s e in W m u s t be substantial, namely below W', so t h a t t h e only possible equilibrium is t h e productive c n e a n d t h e t r e e bicmass is a t t r a c t e d back t h e r e , giving rise t o another c a t a s t r o p h e , t o a bloom.

A MODEL

OF

TREE AND NUTRIENT DYNAMICS

This section is devoted t o the presentation of t h e mathematical model which is t h e basis of o u r analysis. We also a t t e m p t to make a c l e a r s t a t e m e n t of t h e a s s u m p t i o c s which underlie t h e e q c a t i c r s so a s t o let t h e reader appreciate t h e limitations of t h e theory set fortk.

The first basic hypothesis is t h a t t h e s t a t e of t h e t r e e species endangered by zcidic depositicn c a n be well represer-ted by t h e t c t a l b i o r a s s T ir- a given a r e a , without detailing t h e age s t r u c t u r e cf t h e plant population. This a m o u n t s t c asscrning e i t h e r an even-aged stand or a slow increase in biomass sc t h a t , for any T, a stable age distribution is reached (Begcr, a n d M c r t i n e r 1981). Further, no distinction is made betweex fcliage a n d wood, assuming t h a t both a r e a d e c t e d by acid precipitatior-.

The second basic assumption is t h a t t h e n u t r i e n t pool fcr t h e t r e e s c a n be described by t h e concentration N of one z u t r i e n t which is hcmogeneously dis- tributed in t h e soil. This hypothesis is c r u d e , althocgh i t is well krcwr, t h a t usually only one e l e m e n t limits productivity in forests. Most often in t e m - perate forests t h i s limiting n u t r i e n t is nitrogen (Cole a n d Rapp 1981, Agreri 1983, Vitousek a n d Matson 1984). Nitrogen is also one of t h e mcst important ccmponents of anthropogenic emissions i ~ t o t h e atmosphere (about one third of s u l p h u r emissions, Saderland 1977, OECD 1979).

We f u r t h e r a s s u m e t h a t t h e nutrient is taken up by t r e e s and used a s a catalyst for t h e processes of bicmass prcduction. The r a t e of synthesis cf new biomass is t h u s supposed t o depend, in a n increasing a n d s a t u r a t i c g fashion, cpon t h e concentration S of nutrient in t h e t r e e biomass. Or, t h e o t h e r h a z d , t h e n u t r i e n t contained in t h e dead biomzss is a t least ir,

art

r e t u r n e d , via deconpcsition, to t h e n u t r i e n t pool.

If we now i ~ t r o d u c e t h e total amocr,t Q

=

TS cf n u t r i e ~ t stored in t h e standing biomass, t h e basic n o d e l can be s t a t e d a s follows:

t =

t i m e

T

=

t r e e living biomass

s

= n u t r i e n t concer,tration (per unit biomass) in t h e t r e e s Q

=

a n o u ~ t of n u t r i e n t s t c r e d in t h e standing biomass = T S

N

= nutrierit c o n c e r i t r a t i o n in t h e scil

m =

t r e e m o r t a l i t y

h ( T )

=

riiaximum production of new biomass p e r u n i t biomass; a decreasing f u n c t i o n of T

e ( S )

=

e E c i e n c y of productior-, i.e. t h e p e r c e n t a g e of m a x i m u m p r c d u c t i c n which is a c t u a l l y achieved; a n i n c r e a s i n g f u ~ c t i o r , of S

fl

= r a t e of n u t r i e ~ t u p t a k e by o n e u n i t of t r e e biomass p e r u n i t cf n ~ t r i e n t c o n c e n t r a t i o n

W =

i n p u t t o t h e n u t r i e n t pool from s o u r c e s o t h e r t h a n t r e e decomposition a

=

c o e f i c i e n t of n u t r i e n t decay d u e t o leaching losses f r o m t h e forest

w a t e r s h e d o r t o u p t a k e by vegetation o t h e r t h a n t r e e s

7

=

p e r c e n t z g e of dying t r e e biomass which, t h r o u g h decompositioc, r e t u r n s t o t h e n u t r i e n t pool.

0 0

0 5 10 15

Tree Biomass [ kg m-2

I

Figure 2. Mzximnm productior, h cf new t r e e b i o m a s s p e r u n i t bicmass a s a functicn of t r e e biomass. The g r a p h i s a c t u a l l y o b t a i c e d from d a t a O R bzlsam fir r e p o r t e d by Sprugel 1984.

Model (1) c c n t a i n s m a z y f u r t h e r a s s c m p t i c ~ s t h a t z e e d explanation a n d c c m - m e n t . ^As far a s t h e p r o d u c t i c n cf new biomass i s c o n c e r n e d , we t a k e i c t o a c c c u n t s o m e form of d e n s i t y d e p e n d e n c e , n a m e l y t h e fact t h a t a n i n c r e a s i c g t c t a l t r e e biomass e n t a i l s a d e c r e a s i n g p r o d u c t i o n of new biomass by e a c h u n i t of s t a n d i n g biomass. This is obviously d u e t c i n c r e a s e d shading, r o o t competi- t i c n , e t c . , which limit t h e a m o u c t of e n e r g y c a p t u r e d f r c m t h e s u m s u n d i n g e n v i r o n m e n t by e a c h t r e e u ~ i t (fcr a d e t a i l e d a c c o u n t of t h e s e p h e ~ o m e r a s e e

Silvertown 1982). T h e r e f c r e t h e f u n c t i o n h ( T ) is a s s u m e d t o be d e c r e a s i n g with T ( a s for i n s t a n c e i n Figure 2 which is based on t h e e l a b o r e t i c ~ of d a t a o n bal- s a m fir r e p o r t e d by Sprugel 1984). However h ( T ) is t h e primary productivity p e r u n i t biomass u n d e r o p t i m a l coriditions of t h e n u t r i e n t c o n c e n t r a t i o n S i n t h e t r e e s . If S i s t o o low t h e productivity will be l e s s t h a n t h e m a x i m u m achievable a t t h a t given b i o m a s s T. Thus h ( T ) is multiplied by a n efficiency e ( S ) which is i n c r e a s i n g from 0 a n d s z t u r a t i n g t o 1 when t h e c o n c e n t r a t i o n S is above a c e r t a i n t h r e s h c l d .

The n u t r i e n t u p t a k e from t h e soil by e a c h t r e e b i o m a s s u n i t is c o n s i d e r e d t o i n c r e a s e linearly with t h e n u t r i e n t c o n c e n t r a t i o n (PN). This is equivalent t o a s s u m i n g t h a t t h e r o c t s y s t e m is proportional t o t h e aboveground s t a n d i n g biomass a n d t h a t e a c h r o o t p u m p s u p a c o n s t a n t a m o u n t of w a t e r w h e r e t h e n u t r i e n t is dissolved with c o n c e n t r a t i o n N. We t h e r e f o r e s u p p o s e t h a t a t r e e c a n n o t withstand a n excessive a n d potentially h a r m f u l q c a n t i t y of n u t r i e n t by limiting i t s u p t a k e . This is a t l e a s t partially u n r e a l i s t i c , s i n c e a t r e e car- develop r o o t s in a soil l a y e r which i s t e m p o r a r i l y f r e e from a n excessive n u t r i e n t c o n c e n t r a t i o n . However, with ever-increasing n u t r i e n t loads - a n d t h i s p a p e r is mainly c o n c e r n e d with t h e long-term c c n s e q c e n c e s a r i s i c g from t h i s o c c u r r e n c e

-

t h e n u t r i e n t is m o r e o r l e s s homogeneously d i s t r i b u t e d ir, t h e whole soil a n d t h e a s s u m p t i o n cf p u r e l y passive u p t a k e b e c o n e s l e s s c r u d e u n d e r t h i s perspective.

The dying t r e e biomass ( m T ) c o n t a i n s a certair, a m o u r , t of x ~ t r i e c t ( r n n ) . In t h e model, t h i s is s u b t r a c t e d frcm t h e n u t r i e n t s t o r a g e i r t h e living biomass a n d a fixed f r a c t i o n .rl is t r a n s f e r r e ? b a c k t o t h e soil by d e c c m p c s i t i o n . The r e m a i n d e r 1

-

7 i s c o t d e c o m p c s a b l e o r is washed off t h e forest e c o s y s t e m by m e t e o r i c a g e n t s . We m a k e t h e s t r o n g assumptior- t h a t decomposition is fast, s i n c e t h e flow of n u t r i e n t from t h e decomposable d e a d biomass (7rnTS) e n t e r s t h e b a l a n c e of t h e soil pool without a n y delay. I t s h c u l d be r e r n z r k e d , o n t h e o t h e r h a n d , t h a t t h e comporients of t h e d e a d biomass which a r e m o s t readily decomposable (foliage a n d b r a n c h e s ) a r e usually r i c h e s t in n c t r i e n t c o n t e n t ( s e e Sprugel 1984 f c r r e l e v a n t d a t a c n balsam fir). For t h e s e c o m p o n e n t s m i n e r a l i z a t i o n i s a c h i e v e d witk a delay which i s s m a l l when c o m p a r e d with t r e e lifetime.

In t h e balance of t h e soil n u t r i e n t pool, besides t h e positive c c n t r i b u t i o n of decomposition a n d t h e n e g a t i v e o n e of t r e e u p t a k e , t h e r e a r e two c t h e r t e r m s . One i s of c o u r s e t h e e x t e r n a l inflow

W ,

which c o m e s m a i r l y f r o m t h e z t m o s - p h e r e in t h e c a s e cf nitroger- a n d s u l p h u r (Abrahamsen 19@0); a c i d precipitz- tion goes t o g e t h e r with a n i n c r e a s e d supply W cf n u t r i e n t . The o t h e r t e r m ( - a N ) is negative a n d t a k e s i n t o a c c o i l n t both t h e l e a c h i n g losses (which a t l e a s t for n i t r o g e n s e e m t o b e proportional t o t h e n u t r i e n t c o n c e n t r a t i o n in t h e soil, Abrahamsen 19@0) a n d t h e u p t a k e by t h e r e m a i n i n g vegetaticn. -is vege- t a t i o n is a s s u m e d t o be m o r e o r l e s s c o n s t a n t so t h a t i t s u p t a k e is agzir, propcr- t i o n a l t o N a n d i t s decomposition is p z r t of

W.

The final a n d m c s t i n p c r t a n t c o m m e n t c o n c e r c s t r e e mortality r n . Sc f a r a n d purposely, we h z v e n o t specified a n y kind cf f u c c t i o n a l r e l a t i c r s k i p f c r t h i s key f a c t o r of o u r model. If t h e r e were no a d v e r s e effects of a c i d i c deposition, only t h e n a t u r a l m o r t a l i t y wculd be p r e s e x t a n d m i g h t b e a s s u m e d t o be 2 c c r - s t a n t coefficient. But, of c o u r s e , a c i d p r e c i p i t a t i o n d o e s infiuence t h e mortaIity r z t e of t r e e s . T h e r e a r e s e v e r a l ways in which a forest suffers from a c i d precipi- t a t i o c a c d t h e different s c u r c e s of d a m a g e a r e ir, g e z e r z l p r e s e n t a t t h e s a n e t h e . Therefore, m o r t a l i t y i s likely t o be a f ~ n c t i o r , cf a l l t h e variables r e l a t e d t o a c i d i c deposition, notably of W , N z n d S. In s p i t e of all t h i s , we will t z k e i n t o c o n s i d e r a t i o n t h r e e basic m e c h a n i s m s of f c r e s t d a m a g e t h a t l e a d t o t h r e e

different types cf functional dependence fcr mortality. a n d separately ar,alyze t h e consequences of e a c h mechanism on t h e t r e e dynamics a s though only one mechanism a t a t i m e were operating. This procedure is limiting on one hand, but on t h e o t h e r helps u s t o assess which of t h e t h r e e mechanisms is responsi- ble for catastrophic effects.

One type of impact is t h e direct, immediate efIect of a i r contaminants or, vegetztion. There c a n be foliar damage, icfluence on phctosynthesis and respiraticc, e n h a n c e m e n t of riicrobial pathogens, increased leaching of vita!

e l e m e n t s from t h e foliage, e t c . (Smith 1981). In some cases t h e r e is a positive correlation between t h e n u t r i e n t inflow W , t h e magnitude of t h i s direct dam- age, a n d t h e intensity of acid precipitation (notice t h a t also t h e formation of photooxidants, which, in addition t o acid rain, seem t o play a c e n t r a l role in damaging crops a n d forests (e.g., Skarby a n d Sellddn 1984) is r e l a t e d t o nitro- gen oxides, hence possibly t o n u t r i e n t inflow). Thus t h e first mechanism we a s s u m e is t h a t mortality is a functior, of iY only. This function is roughly con- s t a c t a n d equal t o t h e c a t u r a l mortality up t c a c e r t a i n point a n d t h e n sharply increasing.

The second mechanism is linked t o t h e increase i c scil acidity. There a r e several buffering reactions in t h e forest scils which c a n partially c o u n t e r e c t t h e effect of acidic depcsition (U!rich 1983): but when p H falls below 5 t h e solu- bility of aluminum increases sharply a n d if pH is less t h a n 4.2, which o c c u r s in many forest soils in Central Europe, t h e aluminum buffer range is reached:

aluminum ions a r e present in high concentrations a n d c a n be toxic t o bacteria s n d plant roots. Simultaneously t h e leaching of calcium, magnesium a n d pcssi- bly potassium, which a r e vital, though usually nonlim iting factors, i s enhanced.

With even lower p H v a l u e s most heavy metals a r e mobilized a n d c a n damage t h e t r e e s . As soil acidity is positively correlated with n u t r i e n t concentration in t h e scil, t h e assumption t h a t follows from this second mechanism is t h a t mortality is a function of N characterized by being constant up t c a c e r t a i n value znd t h e n very sharply increasing.

The last mechanism is a n o t h e r indirect effect. Acid precipitation, bringicg about a n increase in t h e inflow iY, c a n c a u s e a n accumulatioc of n u t r i e n t i n t h e t r e e s . Usually t h i s is not harmful; o n t h e contrary, i t e n h a n c e s t h e primary productivity, a s was already pointed out. However, wher, very high levels of concectration a r e reached, tissile injury c a p result. As for nitrogen, t h i s thres- hold coccentration is about 2% of dry weight for cocifers (Ingestad 1979) a n d about 4% fcr deciduous species (Ingestad 1981). As a consequence, we assume t h z t , if this mechanism of t r e e damage is operating, t h e mortality r a t e m is a f u c c t i c c cf t h e t r e e n u t r i e n t concentraticn S and, a s usual, is c c ~ s t a n t over a wide range a n d t h e n sharply increases.

THE

JU4CEfMEX-Y OF F0RF;ST INCREASE. DECLINE

AND

COLLAPSE

This section is devcted t c analyzing t h e t r e e dynzrnics a s predicted by mcaei ( I ) under different c n t r i e n t loads iY a n d in t h e t h r e e c a s e s above described. Befcre proceedir-g k r t h e r , i t is cccvenient to r e s t a t e equations (1)

d S 1 dQ S d T ^", by takicg ir-tc a c c c u n t t h a t

Q =

T S , h e n c e - = -

-

- - - i h u s

dt T dt T dt ^'

- 7 -

arid we z r e l e f t with t h e t h r e e s t a t e variables T , S s n d N only

Nutrient Concentration in the Trees S

F i g c r e 3. The two f a m i l i e s of c u r v e s o b t a i n e d from e q u a t i o n s (3) ar,d (4) by let- t i r g R v a r y . I n t e r s e c t i o n s of c u r v e s with e q u a l W ( d o t s labelled a s 2 , ..., 6) give v a l u e s of T a n d S at nor-trivial equilibrium.

b r e c t Cree damage - In t h i s czse t h e m o r t a l i t y r a t e is a s s u m e d to b e a k c c t i c n of t h e n u t r i e n t i n p u t V only: m = m ( V). It i s f c r t h e r h y p o t h e s i z e d t h a t m ( W) is p r a c t i c a l l y c o n s t z c t u p t o a c e r t a i n point, a f t e r which i t s h a r p l y i n c r e a s e s . Given a c o n s t a n t R , t h e c o r r e s p c n d i c g s t e a d y s t a t e s of model (2) car, b e found by imposing t h e t i m e c c c s t a n c y of T , S a n d N, o r e q u i v a l e n t l y t h e s i r n u l t a c e o u s

dT d S

v a n i s h i n g of

- -

a n d

-.

^{d N} There a r e two possibilities:

d t ' d t dt

i) T

=

0 a n d c o n s e q u e n t l y N

=

V / a , S =

5

w h e r e i s t h e u n i q u e s o l c t i o n cf t h e e q u a t i o n

This e q u i l i b r i c m correspcrids t c t h e e x t i n c t i o n of t r e e s .

ii) e ( S ) h ( T ) = m ( W) = @ N / S from t h e f i r s t a n d s e c o n d of e q u a t i c n s (2).

H e r c e

and

where h-I is t h e inverse fucctior, cf h . Since h(T) is a decreasing function of T and e ( S ) an increasing a n d saturating functioE of S, t h e right-hand side of equation (3) is ar, increasicg and satnrating function of S. From t h e last of equations (2) and substituting for N one gets

If t h e curves given by equations (3) and (4) i r t e r s e c t in t h e first quadrant, t h e r e is a nontrivial equilibrium characterized by T

>

0 , S

>

0 , N

>

0 . This occurrence does not obtain for all values of W , In view of t h e assumptions on rn ( W ) it follows t h a t

W /

rn ( W ) is z bell-shaped functior,, hence t h e situa- tion is a s portrayed in Figure 3 . When W is low ( W

=

W, in fignre) t h e r e is no intersecticn. As soon a s W takes on t h e critical valne W,, ixitersections s t a r t appearing and can be found for a range of intermediate W values ( W,, W,, W,). When

FY

equals W, t h e curve described by equaticn (4) begins mcving leftward, so t h a t intersections Enally disappear for W equal t c t h e critical value We. For very high nutrient loads ( W,) no intersection is pcs- sible. I t is worthwhile to remark t h e smooth petterx

-

i.e., without jumps in T or S - of intersection appearznce-disappearacce.

We can summarize by saying t h a t when W is tco lcw or too high only t r e e extinction is a feasible equilibrium, while for ictermediate nutrient inputs t h e r e exists also a nontrivial equilibrium.

As for stability, i t can be stndied via linearization (see Appendix). It is t h u s possible to show t h a t t r e e extinction is stable when it is t h e ucique steady s t a t e , whereas, for t h e W values which permit t h e existence of a viable equili- brium, this one becomes stable a n d extinction unstable.

In c o n c l ~ s i o n , wher, W is very low (insufficient nutrient inEow, hence poor forest soil) o r very high (heavy acid precipitation) t h e forest is driven t o t r e e extinction, while t r e e s can survive when W is intermediate. These two regimes a r e smoothly joized, without ar,y catastrcphic event, as shown in Figure 4 , which displays t h e qualitative behavior of T, S and N as functions cf W.

hcreased soil acidity

-

This mechanism implies that mortality is a function of t h e nutrient soil concentration N cnly: rn

=

m ( N ) . A s usual, mortality is sup- posedly constant up to a threshold, beyond which it rapidly increases.

The equilibria corresponding to a constant inflow W can be found by setticg - ^{d T}

-

^- - ^{dS =}

=

in e q u a t i o ~ s (2). Again. t h e r e a r e two pcssible outcomes

dt dt dt

i) t r e e extinetic?, namely T = C , N

=

W / . a , S =

3

with

5

being t h e unique solution of t h e equation

ii) e(S!h(T)

= m ( N )

=

-

BN frcm t h e first and seccnd cf equaticns (2). It fc!- S

Iows t h a t t h e nutrient concentraticn S in t h e t r e e s is relzted t c t h e one in t h e sci! N by

which, owing to the a s s u m p t i o ~ s on t h e mortality rate, is a unirnodal f u ~ c - tioK of N. Substituting fcr S one also obtains

' 4 ' 2

Nutrient Inflow W

Figure 4. P a t t e r ~ l of variatioc of t h e stable steady s t a t e z s a function of

FY

w h e ~ t h e first mechanism of t r e e damage is o p e r a t i ~ g . Points 2 , ..., 6 make refererice t o those in Figure 3.

Since e ( S ) is increasing a n d saturating, e (@N/ m (N)) is dorne-shaped. pcs- sibly w i t h a flat tcp. Therefore, a s

h(T)

is decreasing, t h e right-hand side of equaticn (5) is unimodal, pcsitive in t h e interval

(N,,

Ne) a n d may h:_zve a flat top (see Figure 5). Notice t h a t it does not depend upon

W .

From t h e l a s t of e q u a t i o ~ s ( 2 ) ar,d replacing S by i t s expressicn a s a func- tion of N it is e a s y t c obtain

1 N 1

~ 3 1 ~

N 2

Nutrient Concentration in the Soil N

Figure 5. The c u r v e described by e q u a t i o n (5) a n d t h e family of c u r v e s obtained f r c m e q u a t i o n (6) by l e t t i n g W vary. I n t e r s e c t i o n s yield v a l u e s of T a n d N a t nontrivial equilibrium. When double i n t e r s e c t i o n o c c u r s , distinction is m a d e between s t a b l e (4' a c d 5 ' ) a n d unsta.b!e (4" a n d 5") equilibria.

Thus t h e r e exist nontrivial equilibria if t h e r e a r e i n t e r s e c t i o n s between t h e c u r v e s described by e q u a t i o n s (5) a n d (6). Figure 5 &splays t h e m c s t i n t e r e s t - ing s i t u a t i o n which a r i s e s w h e ~ t h e c u r v e s given by equation (6) a r e n o t t o o s t e e p ( t h i s o c c u r s when t h e leaching c o e f f c i e n t a is n o t too large). When t h e n u t r i e n t inflow i s low

C

W

=

W1 in figure) n o i n t e r s e c t i o n o c c u r s . At W equal t o t h e c r i t i c a l v a l u e W2 = a N l i n t e r s e c t i o x s begin showing up. For ir-termedizte v a l u e s cf W , or-ly o c e i n t e r s e c t i o n i s p r e s e n t ( 3 i n Figure 5), b u t as s o c n a s t h e n u t r i e n t load r e a c h e s t h e c r i t i c a l value W4

=

a N I ! a s e c c n d i n t e r s e c t i o n a p p e z r s a n d s t a y s o n for a rar-ge of high W v a l u e s ( W = WS). W h e ~ W e q u a l s t h e c r i t i c a l v a l u e

W 6

( t h e o n e for which t h e c u r v e d e s c r i b e d by e a u a t i c n (6) i s t a n g e n t t o t h e c u r v e d e s c r i b e d by equation (5)) t h e two i n t e r s e c t i c n s collapse i n t o o n e . At even h i g h e r n u t r i e n t i n p u t s n c i n t e r s e c t i o n o c c u r s .

The complex a n d a r t i c u l a t e p a t t e r n e m e r g i n g f r c n t h e above anzlysis c a n b e s u m m e d u p as follows. When t h e n u t r i e n t i n p u t is very low o r very h i g h

( W

<

W2 o r

W >

W6) only t r e e e x t i n c t i o n is a feasible s t e a d y s t a t e .

intermediate-low n u t r i e n t loads (

W E <

W

<

W,) imply t h e e x i s t e z c e of t h e trivia!

a c d of o n e nor-trivia! equilibrium. For i n t e r m e d i a t e - h i g h W ' s (We

<

W

<

We) t h e r e exist t h e trivia! ar-d two nontrivial e ~ d i l i b r i a .

it is very important to assess the stability of all these equilibria. When this is accomplished via linearizaticr? (see Appendix), the fcllcwing results a r e obtained:

W

<

WE o r W

>

W, t r e e exticctior, is stable

W2

<

W

<

W4 t h e unique viable equilibrium is stable; t r e e extinc-

tion is unstable

W, <

W

<

W, t h e viable equilibrium with higher t r e e biomass (5' ^il;_

Figure 5 ) is stable; the other one (5" i c Figure 5) is unstable; tree extinction is stable.

Therefore, when t h e r e is insuficient nutrient inflow or heavy acidic depositioc, t r e e s a r e doomed to extinction. For intermediate-low nctrient inpcts t h e forest is healthy and trees can survive, but under t h e burden of intermediate-high loads the trees can be a t t r a c t e d either to viability or to extinction depending on the initial conditions: if t h e t r e e bicmass is impoverished, but not toc much, the forest can regrow; if it is severely depleted, t h e t r e e s beccme extinct.

I t is just this f?ip-flopping pattern of equilibria which occurs ir. a certain W range t h a t determines t h e essentially catastrophic impact of increased soil aci- dity. This is clarified in Figure 6 which qualitatively shows how W aEects t h e T, S and

N

values corresponding t o stable steady states. It should be remarked t h z t t h e curve of t h e t r e e biomass has exactly t h e same shape as in Figure 1.

Thns all t h e relevant discussion on catastrcphes also pertains here. When acid precipitation beccmes so intense t h a t t h e nutrient load ever, slightly exceeds t h e threshold W6, trees inevitably collapse from an equilibrium characterized by a normal standing biomass to extinction. Moreover, t h e mere decrement cf the nutrient load below W6 is not sufficient to get t h e t r e e s out of t h e extincticn trap. Recolonization can occur only after a substantial decrease (below W,) of nutrient inflow. In t h i s case a reversed catastrophe, a Slcorn, takes place. This hysteresis pattern is typical of a so-called fold catastropke (Thorn 1972).

Excessive

nutrient accumulation -

This last impact is translzted icto a mcrtal- ity which is a funcitor, of S only

( m =

m ( S ) ) , is constant up t o a threshold and thereafter rapidly increases. The analysis cf equilibria and their stability very much resembles t h e orie performed with t h e previous mechanism. Sc, we dc not go icto much detail.

Besides t h e nsual trivial extinction equilibrium, t h e nontrivial ones satisfy t h e fcllowirig relationships

If t h e r e is any intersection in the first ~ u a d r a n t of t h e plane S-T between t h e ccrves giver? by ecilaticns (7) and

(e),

t h e n viable steady s t a t e s exist. Sirce t h e threshold a t which m ( S ) sharply increases is ~ o r ~ c e i v a b l y higher than t h e threshold: a t which t h e e s c i e n c y e (S) saturates, the fuzcticn m (S)/ e ( S ) is bowl-shaped with a fiat bottom. Consequently t h e right-hand side of e a ~ a t i o r r (7) is unimodal, positive in t h e interval (S1? S2) and has z fiat top (see Figure 7).

Moreoever it is independent of W. The curve giver by equatior? (@) cbviously decreases with S and depends on I Y , so t h z t t h e situation is as represected ir,

w2 W4 Nutrient Inflow W

Figure 6. P a t t e r n of variation of the s t a b l e s t e a d y s t a t e s v e r s u s n u t r i e n t inficw when t h e s e c c n d m e c h a n i s m of t r e e darnage i s operatirig. Pcir,ts labelled 2, ..., 6 c c r r e s p o n d t o t h c s e in Figure 5 .

Figure 7 . The re!evar,t discussion i s n o t given. b e c z u s e i t is t c t a l l y a n z I c g o c s t c t h e t fcr i h e previous m e c h a n i s m (Figure 5 ) . Also i h e stability a n a l y s i s lea& t c v e r y sirniiar r e s u l t s , which a r e s u m m a r i z e d in Figure 8. When t h e r i u t r i e n t i ~ f l o w is too low, only t r e e extir-ction i s s t a b l e ; by ir,creasir,g

W

o n e o b t a i n s a

S1

S2

Nutrient Concentration in the Trees S

Figure 7 . The curve described by equation ( 7 ) and t h e family of curves obtained frcm equation (8) by letting

W

vary. I ~ t e r s e c t i o ~ s give values of T and S a t nontrivial equilibrium. When double intersection occxrs, distinction is made between stable (4' and 5') and u ~ s t a b l e (4" and 5") equilibria.

pcsitive ar,d increasing t r e e biomass. Further i n c r e c e n t s of t h e nutrient load, leading to a harmful accumulaticr, in t h e t r e e s , i n c i ~ c e Erst a decrease of t h e standir-g biomass and t h e r a dramatic colIapse. wl-ie~ th e threshold

W 6

is exceeded. A s in t h e previous case recovery is pcssible only if the nutrient infiow is substantially decreased (below

W4).

The r e m a r k t h a t is worth making with reference to t h e last case is t h a t , eveE if trees a r e capable of successfully withstanding t h e direct damage from soil aci- dity and only t h e long term and very indirect impact from excessive nutrient accumulation is operating, catastrophes still occur.

CONCLUSION

We have illustrated hcw three possible impacts cf acid precipitaticn on t r e e survival and grcwth affect t k e ternpcral dynamics of t r e e biomass. Frcm t h e analysis cf equilibria and their stability twc dicerent patterns heve emerged.

The direct injuries to v e g e t a t i o ~ , which a r e not on!y t h e most conspicuous, but also t h e mcst important impact of zcic;ic depositioc, dc nct give rise, fcrmally speakixg, to a catastrophe, while the other two mechanisms do. This fact should cause no surprise since iccreased soil acidity and excessive n u t r i e c t buildup iii :he trees entail t h e accumulation in the fo-est ecosystem of a s t r e s s t h a t does n c t show up immediately, bnt explodes when a breakpoint is reacked.

This result should not be taken a s an advice to neglect t h e direct impact on vegetztion. This rnechznism, zs we have shcwn, causes t h e d e c I i ~ e ar,d

Nutrient Inflow W

Figure 8. P a t t e r n of variation of t h e s t a b l e equilibria a s f u n c t i c n s of

W

wher: t h e t h i r d m e c h a n i s m of t r e e d a m a g e is operating. P o i n t s labelled 2, ..., 6 c o r r e s p o n d t o t h o s e in Figure 7.

e x t i n c t i o n of t h e t r e e s s u b j e c t t c i n c r e a s e d a c i d precipitation in ar,y e v e n t . The c o u r s e of t h e s e u n p l e a s a n t e v e n t s c a n be q u i t e fast, t h u s becoming a c a t z s - t r o p h e i n p r a c t i c e , though n o t o n a m a t h e m a t i c a l basis. Therefore. if t h e r e i s a s m a l l lesson which mzy e m e r g e from t h i s p a p e r , is r a t h e r not t o n e g l e c t t h e

o t h e r two m e c h a n i s m s w h i c h , t h o u g h i n d i r e c t a n d t h u s given l e s s a t t e r i i o n , carr in t h e l o ~ g r u n s u r p r i s e by c a u s i n g t h e u n e x p e c t e d collapse of f o r e s t e c o s y s t e m s .

h o t h e r c a v e a t t h a t we would like t o give t h e r e a d e r i s t h a t t h i s p a p e r i s n o t m e a n t t o explain a l l t h e e p i s o d e s of f o r e s t d a m a g e f r o m a i r c o n t a m i ~ a n t s . Attentior? h a s been r e s t r i c t e d t o a c i d p r e c i p i t a t i o n a n d , e v e n w i t h i n t h i s f r a m e - work, p h e n o m e n a h a v e beer, g r e a t l y simplified, b u t t h e r e i s m o u n t i n g e v i d e n c e t h a t o t h e r c a u s e s of d a m a g e , s u c h a s o z o n e f o r m a t i o n , m a y be equally i m p o r - t a n t . An i n t e r e s t i n g d i r e c t i o n f o r f u t u r e r e s e a r c h m i g h t i n d e e d b e t h e con- s t r u c t i o r i of a model of tree d y n a m i c s w h i c h t a k e s t h e s e f u r t h e r i m p a c t s of a i r p o l l u t i c n o n f o r e s t s i n t o a c c o u n t .

REFERENCES

Abrahamsen, G. (1979) "Impact of a t m o s p h e r i c su!pkur depcsition on f c r e s t e c o s y s t e m s " i n

Atmospheric S u l f u r Lkwosition. Efivironmental i m p a c t a n d Health E f f e c t s

( S h r i n e r , D.S., Richmond, C.R. ar,d Lindberg, S.E e d s . ) , Ann Arbor S c i e ~ c e .

Abrahamser,, G. (1980) "Acid p r e c i p i t a t i o n , p l a n t n u t r i e n t s a n d f o r e s t grow5" i n

Ecologicd h p a c t of Acid Precipitation

(Drabls, D . a n d Tollan, k eds.), SXSF p r o j e c t , Norway.

Abraharnsen, G. (1981) "Effects cf a i r poliution o n forests" in

Beyond the

f i ~ g y Ctisis.

q3porCLtniCy a n d Chcrllenge

(Fazzolare, R.A. a c d S m i t h , C.B. eds.), Pergarnon P r e s s , Oxford.

Abraharnsec, G. a n d G.J. Dollard (1979) "Effects cf a c i d precipitatior, o c forest v e g e t a t i o n a n d soil" i n

Ecologkal E,qects of Acid R e c i p G t a t w n

r e p a r e d by Central Electricity R e s e a r c h Laboratories, L e a t h e r h e a d , England, EPRI SOA 77-403.

Agren, G.I. (1983) "The c o z c e p t cf citrogeri productivity i n f c r e s t grcwth model- ling"

hfiiteilungen

der

Forstlichen & n & s v e r s u c h s a n s t a l t ,

147: 1 9 9 2 1 0.

Begon, M. a n d Ed. M c r t i m e r (1981)

Population Ecology.

Blackwell, Oxford.

Casti, J. (1982) " C a t a s t r o p h e s , c o n t r c l a n d t h e inevitability of s p r u c e budworrn outbreaks"

Ecological Modelling,

14: 293-300.

Clark, W.C., D.D. Jones a n d C.S. Hc!ling (1979) "Lessons for ecological policy design: a c a s e s t u d y of e c c s y s t e m m a n a g e m e n t " .

Ecological -Modelling,

7:

1-53

Cole, D.W. ar,d

M.

Rapp (1981) "EIernectal cycling i n forest ecosystems" i n

-am* Properties of Forest E c o s y s t e m s

(Reichle, D.E. ed.), Cambridge University P r e s s .

I n g e s t a d , T. (1979) "Mineral n u t r i e n t r e q u i r e m e n t s of

Pin= s y l v e s t r i s

a n d

R c e a d i e s

seedlicgs" .

Physialogia P e a t m u m

45: 373-380.

h g e s t a d , T. (1979) "Nutrition a r d growth cf birch a n d grey a l d e r seedjings i n low conductivity s o l u t i o n s ar,d a t v a r i e d r a t e s of n u t r i e n t a d d i t i o ~ s " .

f i y - siologia P l a n t a r u m

52: 454-556

Ludwig, D., D.D. J o n e s a n d C.S. Holling (1978) "Qualitative analysis of insect out- b r e a k s y s t e m s : t h e s p r u c e budworm a n d t h e forest".

Journal of Animal Ecclogy,

47: 315-332.

Matzner. E, a n d B. Ulrich ( 1 9 e l ) "Effect of a c i d p r e c i p i t a t i o n O R scil" ir,

Beyond t h e k e r g y

^{M e .}

q 3 p o r t v n z t y a n d M l e n g e

(Fazzclare R.A. a n d S m i t h C.B. e d s . ) , P e r g a m o n P r e s s . Oxford.

OECD (1979) Th,e

OECD P r o g r a m m e o n Long Range Transport of Ai? Pollution.

OECD, P a r i s .

Siivertcwz. 2 . W . (1982)

Entroductwn t o Plant Population Ecolo*gy.

L c r g z a r , , Lcc- dcn.

Skarby, L. a n d G. Selldez ((1984) "The effects cf c z c n e c n c r o p s a n d iorests".

h 5 ~13: 68-72. ,

S m i t h , W.H. (1981) Air

Polluticm a n d Forests.

Springer-Verlag, New Ycrk.

S o d e r l u ~ d , R. (1977). "NO, p c l l u t a n t s a n d a m m o n i a e m i s s i o c s . A m a s s b a l z z c e for t h e a t m o s p h e r e o v e r Europe."

A m b w ,

6: 118-122.

S p r u g e l , D.G. (1984) "Density, b i o m a s s , p r o d u c t i v i t y a n d n u t r i e n t - c y c l i z g char-ges d u r i n g s t a n d d e v e l o p m e n t i n w a v e - r e g e z e r a t e d b a l s a m fir f o r e s ~ s . "

E c o l o g i c d M o n o g ~ a p h s , 54: 165- 186.

7 -

i h o r n , R. (1 972) S a b i l i t d s t r u c t u r e l l e e t r n o r p h o g e n ~ s e . W . A. Benjamir,. New Ycrk.

Ulrich. B. (1983) "Soil a c i d i t y a n d i t s r e l a t i o n t o a c i d deposition" in E1Sfectc of A c c u r n u l a t w n o f Atr P o l l u t a n t s tn f i r e s t E c o s y s t e m s ( U l r i c h , B . a n d

Pan-

k r a t h , J . e & . ) D. Reidel P u b l . Comp., N e t h e r l a n d s

Ulrich, B. (1984) "EfTects of a i r pollutioE o n f o r e s t e c c s y s t e m s a n d w a t e r s . The p r i n c i p i e s d e m o n s t r a t e d a t a c a s e s t u d y i n C e n t r a l Europe" A t m o s p h e r i c E h v i r o n m e n t , 18: 62 1-628.

Vitcusek, P.M. a n d P.A. Matson (1984) "Mechanisms of n i t r o g e n r e t e n t i o r - i n f c r e s t e c c s y s t e r n s : a field e x p e r i m e n t . " S c i e n c e , 225: 51-52.

APPENDIX

This appendix i s devoted t o explaining kcw t h e s t a b i l i t y a n a l y s i s of t h e equilibria of model ( 2 ) c z n b e performed. The m e t h o d we u s e is c a l l e d lineari- z a t i c c . which c o n s i s t s cf replacing t h e d y n a m i c s of e q u a t i c n s (2) i n t h e neigh- borhood of e a c h e q ~ i l i b r i u m by t h e a p p r c x i m a t e d y n a m i c s of a n eql;iva!ect l i ~ e a r model. More precisely, if we d e n o t e by

?, 3, N

t h e v a l u e s of t h e s t a t e v a r i a b l ~ s a t equilibrium a_nd by

& ( t )

t h e v e c t o r whose c o m p o n e n t s a r e T ( t ) - T , S ( t )

- 5,

^{N ( t )} - N , t h e n t h e t i n e evolution of & is a p p r o x i m a t e l y given by t h e v e c t o r l i n e a r differential equatior,

- - -

where

- 5

is t h e Jacobian m a t r i x of model ( 2 ) eva!uated at T , S , N . If t h e m a t r i x J h a s eiger-values with n e g a t i v e r e e l p a r t s , t h e correspondirig equilibrium of model ( 2 ) i s stable.

We c a n apply t h i s p r o c e d u r e t o t h e a ~ a l y s i s of t h e first m e c h a n i s m where t h e m o r t a l i t y is given by m = m ( W ) . In t h i s c a s e t h e J a c o b i a n m a t r i x is

-e ( S ) S- dh d T

By e v a l u a t i z g J a t t h e trivial equilibrium

o n e g e t s

S i r c e t h e eigenvalues of

7

a r e , i n t h i s c a s e , t h e e l e m e x t s on t h e d i a g c n a l , t h e trivial equilibrium Is s t a b l e if m ( W )

> e

( S ) h ( 0 ) . It is easy t o u c d e r s t a n d t h a t t h i s last condition is equivalent t o t h e nor,-existence cf t h e viable e c u i l i b r i u m . In f a c t , n o r t r i v i a l steady s t a t e s e x i s t o ~ l y wheri t h e i n t e r s e c t i o n S S of t h e c u r v e g i v e c by e q u a t i o n (3) with t h e S a x i s lies on t h e r i g h t of t h e i n t e r s e c t i o ~ S., cf t h e c u r v e giver, by e q u a t i o n ( 4 ) ( s e e Figure 3). S i c c e

i t t u r n s o u t t h a t

Therefore, f r o m t h e f a c t t h a t e ( S ) i s i n c r e a s i n g , i t fo!lows t h a t e i t h e r S , I

3 5

S4 o r

s4

⁵

31

S,.

On t h e other hand, t h e inequslity m (

w> >

e ( 3 ) h ( 0 )

is equivalent to e ( S 4 )

>

e ( S ) ^, h e n c e to

s, > S > s , .

The procedure is more complex when dealing with t h e nontrivial equili- brium. In this case t h e Jacobian matrix is

a n d t h e eigenvalues cannot explicitly be computed. However, t h e r e exists a condition on t h e elements of

5

which guarantees t h e negativity of t h e eigen- values' real parts. If we indicate t h e t r a c e of

1

^by C t J , t h e determinant by d e t J and t h e sum of t h e second order leading mincrs by C . t h e n t h i s conditicn is

ct5 <

0 , d e t J

<

0 , C t r ?

<

d e t J

.

As for t h e t r a c e , it results

and, since h ( T ) is a decreasing functicn a n d e ( S ) a n increasing one, t h e first inequality is satisfied. As for t h e determinant, one has

Sir,ce h ( T ) is decreasing, e ( S ) ircreasing a n d

<

I , zlso t h e negativity of t k e d e t e r m i n a c t is veriEed. The proof of t h e t h i r d inequality is straightforward b ~ t cumbersome ar,d is not reported here.

The linearization procedure can also be applied t c t h e stability a ~ a l y s i s fcr t h e second a n d third mechariisins of t r e e demage. The relevant computations a r e standard algebra, thongh lcng and boring, a r d a r e n c t given