Model of the Optimal Development of a Plant Taking Into Account Defence and Competition

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

MODEL OF THE OPTIMAL DEVELOPMENT OF A PLANT TAKING INTO ACCOUNT DEFENCE AND COMPETITION

M.Ya. Antonovski M.D. Korzukhin

M.T. Ter-Mikhaelian

November 1987 WF-87-106

I;=1 IIASA

l n l e r n a l ~ o n a l I n s t ~ t u l e for A p p l ~ e d Systems Analys~s

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MODEL OF THE OPTMAL DEWEZDPMENT OF

A

PLANT TAKING INTO ACCOUNT DEFENCE AND COKPEIlTION

M.Ya. Antonovski,

M.D.

K o r z u k h i n , M. T. Ter-Mikhaelian

November 1987 WP-87-106

Working P a p e r s are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o not necessarily 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 A-2361 Laxenburg, Austria

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Foreword

This p a p e r p r e s e n t s a v e r y simple model of t h e development of a species, taking into account t h e energy expended on growth and on competition. Using probability distributions on growth rates and population densities, some v e r y general r e s u l t s are obtained. The model predictions of c o u r s e need t o b e tested with field data, and t h i s may provide a conceptual framework f o r new measurement pro- grams.

M.T. Ter-Mikhaeljan was a YSSPer (Young Scientists' Summer Program), working with P r o f e s s o r Antonovski during t h e summer of 1987. The t h i r d a u t h o r , D r . Korzukhin, attended t h e IIASA conference on "Impacts of Changes in Climate and Atmospheric Chemistry on Northern Forests and Their Boundaries" in August. 1987.

Both are from t h e Natural Environment and Climate Monitoring Laboratory GOSKOMGIDROMET and USSR Academy of Sciences.

R.E. Munn Leader

Environment Program

-

iii

-

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HODEL OF THE OPTIMAL DEVELOPMENT OF A PLANT

TAKING

INTO ACCOUNT

DEFENCE

A N D

COMPETITION

M.Ya. A n t o n o v s k i ,

M.D.

K o r z u k h i n , *

M.T.

T e r - M i k h a e l i a n *

1. LNTRODUCTION

In t h i s p a p e r w e formulate and investigate a p a r t i c u l a r development model of a plant, whose growth maximizes r e p r o d u c t i o n during t h e life cycle. In comparison with available models (a brief review follows), w e h a v e included in o u r model t h e competition between plants and t h e costs of a n individual's defence. This connects t h e development problem with t h e problem of population dynamics.

A plant s p e c i e s w e see now i s t h e r e s u l t of a n evolutionarily formed genotype.

I t i s doubtful whether i t would b e possible or d e s i r a b l e to restore t h e l i s t of factors (with t h e i r densities and intensities), which had a significant influence o n t h e p r e s e n t s t a g e of development. A m o r e r e a l i s t i c way of investigating development and i t s changes u n d e r d i f f e r e n t ecological conditions i s to make hypotheses about t h e s e factors, to formulate models, and to test them with field d a t a . The important f a c t o r s in o u r model a r e t h e s i t e quality (generalized r e s o u r c e ) and population density.

2.

LITERATURE

R E Y r m

T h e r e i s a voluminous l i t e r a t u r e on t h e problem of optimal development ( o r life s t r a t e g i e s ) of p l a n t s and animals. W e shall dwell on p a p e r s containing models m o s t similar to o u r s .

W e study a perennial (plant or t r e e ) , which divides available r e s o u r c e s amongst a number of functions ( t h e growth of various plant o r g a n s , adaptation t o varying ecological conditions, d e f e n c e from mortality f a c t o r s ) . The problem of r e s o u r c e division i s discussed in detail from t h e biological point of view in Pianka (1981). T h e r e e x i s t t w o quite d i f f e r e n t a p p r o a c h e s to tree growth modelling, namely, optimal a n d non-optimal models. The basic merit of t h e second class of model i s t h a t t h e y are simpler f r o m t h e experimental point of view, viz., comparing simulation r e s u l t s with field d a t a . The merit of t h e f i r s t c l a s s of models i s t h e i r g r e a t e r biological validity and t h e i r consideration of t h e e n t i r e organism, which given a n opportunity to d e s c r i b e o n e of i t s m o s t important p r o p e r t i e s , namely, i t s ability to adapt. Two optimization principles (in d i f f e r e n t modifications) are com- monly used

-

productivity optimization ( i n c r e a s e in biomass) (Kibzun, 1983; Menju- lin and S a w a t e e v , 1981; Nilson, 1968; Oya, 1985; Oya, 1986; Racsko, 1979; Racsko, 1987; T a r k o and Sadulloev, 1985) and maximization of t h e number of s e e d s or population growth rate (Antonovski a n d Semenov, 1978; Antonovski and Korzukhin, 1983; Vorotintsev, 1985; Insarov, 1975; Korzukhin, 1985; Semevski and Semyonov,

*

Natural Environment and Climate Monitoring Laboratory COSKOMCIDROMET and Academy o f S c i - e n c e s , USSR.

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1982; T a r k o a n d Sadulloev, 1985; Hanin and Dorfman, 1973; Caswell, 1982). The second principle seems to b e more valid from a n evolutionary point of view (Pian- k a , 1981; Semevski and Semyonov, 1982; Hanin and Dorfman, 1980; Holden, 1935).

The main f a c t o r s to b e considered in a problem statement for optimal development are t h e following. (1) The partitioning of e n e r g y between growth a n d r e p r o - duction f o r a plant was considered in Vorotintsev, 1985; Insarov, 1975; K o n u k h i n , 1975; T a r k o a n d Sadulloev, 1985. (2) The growth c u r v e maximizing a n animal's fer- tility taking into account i t s defence expenses w a s investigated in Hanin and Dorf- man (1973); in f a c t i t was t h e balance between growth and defence e x p e n s e s t h a t was investigated. (3) Plant competition was t a k e n into a c c o u n t only in Korzukhin (1985), but population density w a s considered as a n e x t e r n a l p a r a m e t e r not as a dynamic variable. In Semevski and Semyonov (1982) f o r t h e f i r s t time a non- s t e p p e d r e p r o d u c t i v e c u r v e was obtained; t h i s r e s u l t w a s achieved by including a s t o c h a s t i c mechanism in t h e model, namely, t h e probability of s e e d germination at e a c h s t e p of development. A similar c u r v e w a s obtained in Tarko and Sadulloev (1985), but with simultaneous u s e of two nonintegral principles of optimization. In a l l o t h e r p a p e r s , only s t e p p e d r e p r o d u c t i v e c u r v e s were obtained. In Oya (1985), Oya (1986), Racsko (1979) and Racsko (1987), t h e p u r p o s e w a s t o find a c o r r e l a t i o n between d i f f e r e n t plant o r g a n s , maximizing t h e i n c r e a s e in biomass on t h e next s t e p ; smooth growth c u r v e s were obtained.

To o u r knowledge a s e a r c h w a s n e v e r made f o r a survival c u r v e by means of optimal development models. The optimal c o r r e l a t i o n of growth and defence f o r a n annual plant b u t f o r a n a r b i t r a r y number of mortality factors w a s obtained in Semevski and Semyonov (1982).

T h e r e i s a l s o a number of models of a different t y p e ; in t h e s e models t h e f e r - tility of vegetable c o v e r as a whole i s optimized, i.e., without considering s e p a r a t e individuals.

3. CONSTRUCTION OF THE MODEL

When attempting to c o n s t r u c t a model of optimal growth of a tree, w e would like f i r s t of a l l to obtain at l e a s t t h e main qualitative f e a t u r e s of t h i s growth ( s e e fig.1): smooth growth of tree biomass m ( t ) . attaining some maximal value; r e p r o - ductive c u r v e q ( t ) having no fruiting until some nonzero a g e t o , increasing t h e r e a f t e r up to t h e end of life; d e c r e a s i n g t o with d e t e r i o r a t i n g ecological conditions; increasing t o with increasing population density; estimates in even-aged plantations of probability p ( t ) of attaining a g e t (this probability is p r o p o r t i o n a l to c u r r e n t density n ( t ).

Since by means of v a r i a b l e n ( t ) w e in f a c t consider t h e population's dynamics, i t i s d e s i r a b l e a l s o to d e s c r i b e in t h e model t h e behavior of t h e main population variables. In o u r c a s e , m ( t ) a n d q ( t ) are v a r i a b l e s of t o t a l biomass and t o t a l number of s e e d s f o r unit s q u a r e of plantation

-

^M^{( t}⁾

=

m ( t ) n ( t ) and Q ( t )

=

q ( t ) n ( t ). The behaviour of M(t ) and Q(t ) i s usually nonmonotonous; for more detailed discussion of t h e v a r i a b l e s of a n even-aged population, see K o n u - khin (1986).

F o r m u l a t i o n of t h e g e n e r a l model.

Consider t h e population composed of n identical individuals, t h e i r life span being equal

to

N. Suppose t h a t individuals consume only one t y p e of r e s o u r c e ; i t s maximum quantity a c c e s s i b l e

to

o n e unit of plant's leaf s u r f a c e i s equal to amax, r e a l quantity is equal t o a

<

amax. The rate of n e t photosynthesis i s equal t o F.

Assume t h a t t h e s e e d s produced by individuals are k e p t during t h e i r life cycle;

t h u s t h e population i s always even-aged and i t s density i s monotonously decreasing.

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Figure 1: Usual behaviour of variables describing dynamics of uneven-aged t r e e population. t i s a g e of population; m ( t ) a n d M ( t ) a r e biomasses of individuals and t h e population respectively; q ( t ) and Q ( t ) a r e biomasses of t o t a l number of s e e d s produced by individuals and t h e population respectively; p ( t ) is an individual's probability of attaining a g e of t

.

W e shall t a k e t h e simplest method of describing a n individual's dynamics by one variable, namely, t h e biomass mi measured at d i s c r e t e moments of time t

=I,. ..

,N.

In o r d e r t o t a k e into account a n individual's defence expenses, w e include in t h e model dynamic v a r i a b l e pi denoting t h e probability of attaining a g e t ; i t i s obvious t h a t pi

=

ni / n l , where ni i s t h e population density at a g e i. Assume t h a t an individual spends f r a c t i o n s zi , yi , zi of photosynthetic product on t h e defence from mortality f a c t o r s , biomass growth and reproduction, respectively, zi

+

yi

+

zi

=

1 . These f r a c t i o n s according t o t h e optimization principle should

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b e found by maximizing t h e t o t a l number of s e e d s produced by a n individual during i t s life cycle

where q i s t h e number of s e e d s produced at s t e p i by a n individual with biomass mi in t h e case when i t s expenses o n r e p r o d u c t i o n are equal to z i F .

Let us discuss t h e equations describing t h e dynamics of biomass and population density (i.e., t h e probability of attaining a g e i). The f i r s t i s t h e well known equation of o r g a n i c s u b s t a n c e balance f o r a n individual

mi+l = m i + v i F ( m i . a ) - b ( m i ) , (2 where F i s t h e rate of photosynthesis a t given biomass mi and r e s o u r c e a , and b denotes t h e e x p e n s e of r e s p i r a t i o n . Equation (2) in d i f f e r e n t modifications h a s been used r e p e a t e d l y in growth models (Karev, 1985; Bugrovski et al., 1982; Oya, 1985; Racsko, 1979; Sirotenko, 1981).

The dependence of survival function V on t h e amount of r e s o u r c e a and t h e individual's p a r a m e t e r s in t h e equation describing t h e dynamics of population density

is known even more poorly t h a n t h e f o r m of F in (2). For example, w e cannot answer definitely whether V is dependent on an individual's c u r r e n t state or on some p a r t of i t s growth t r a j e c t o r y . The last assumption includes in t h e survival function something like "memory" and is quite possible from a physiological viewpoint. L a t e r w e s h a l l t a k e t h e simplest hypothesis about lack of "memory1'. I t i s a l s o possible t h a t V depends on a n individual's biomass but without a n y informa- tion on t h e possible form of V(m), w e shall not use it.

According t o t h e p r e s e n t state of survival t h e o r y (Semevski and Semyonov, 1982) t h e biomass depends on t h e p r e s s u r e of t h e mortality f a c t o r V and t h e amount of r e s o u r c e r s p e n t on defence. V

=

V(W,r) (we consider t h e one-factor c a s e ; by "resource" w e mean a n individual's i n t e r i o r r e s o u r c e , i.e., t h e p a r t of a n individual's e n e r g y s p e n t on defence); at t h e same time, V i s a concave function of r . Let u s u s e o n e of t h e possible ways to introduce r e s o u r c e (Antonovski e t al., 1984; Korzukhin, 1986), namely

where F i s photosynthesis;

z

i s t h e p a r t of i t s production expended on defence;

a ( n ) i s a n ecological r e s o u r c e (e.g. light) accessible to t h e individual t h a t depends o n density n of population; amax

=

a (0); Fmax

=

F [m ,amax] i s t h e maximum rate of photosynthesis. Assume t h a t in t h e case when a n individual expends maximum accessible r e s o u r c e ( a

=

1, F = p a x ) o n defence, t h e survival i s a maximum, i.e., t h e individual i s totally defended, V(r =1)

=

1.

Let us discuss possible ways of including population density in t h e list of factors t h a t h a v e b e e n influential in forming a n individual's genotype. Consider a n even-age population.

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Judging by t h e p r e s e n t dynamics of b o r e a l f o r e s t s , t h e conclusion c a n b e r e a c h e d t h a t m o s t exogeneous d i s t u r b a n c e s almost completely d e s t r o y a n initial stand (wildfires, windfalls, pests). In t h a t c a s e , a n intensive invasion of free t e r r i - t o r y begins with a "package" of pioneer individuals t h a t c a n b e considered a s even-aged (Kazimirov, 1971; Kirsanov, 1976); t h e s e individuals strongly i n t e r a c t with e a c h o t h e r and depend only slightly on younger individuals of lesser size. S o t h e following problem statements are possible.

I t c a n b e assumed t h a t individual trees were formed at d i f f e r e n t initial densi- t i e s of population ni (1) (each with i t s probability h i ) ; i i s a number of initial conditions. In t h a t case, t h e following quantity h a s to b e optimized

where qk i s t h e number of s e e d s produced by a n individual at a g e k .

I t can b e assumed t h a t a n individual tree was formed at one ( o r a group of closed values) initial density

n*

(1); t h i s assumption i s a p a r t i c u l a r case of (5). In t h i s case, a n individual's behaviour at initial densities d i f f e r e n t t o

n*

(1) i s suboptimal.

I t can b e assumed a l s o t h a t an individual was formed a t d i f f e r e n t initial densi- t i e s and w a s optimally adapted to e a c h of them. The formalization of t h i s viewpoint i s similar t o t h e previous one and l e a d s t o t h e maximization of t h e quantity

L a t e r on w e s h a l l u s e t h e simplest version (6).

Before writing t h e final version of t h e model, i t i s convenient t o introduce specific photosynthesis f' ( p e r sq.cm.) f o r a n individual instead of t h e t o t a l o n e F, and t h e individual's leaf s u r f a c e S, s o t h a t

F

=S.f'(S,a)

.

The dependence S ( m ) c a n b e t a k e n in t h e f o r m S

-

^{m d}^,d

^<

¹⁽Kuzmichev, 1977).

(Dependence f' ( S ) a p p e a r s through effects like crown self-shading).

The g e n e r a l f o r m of t h e model to b e suggested is:

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where a (mi , n i ) i s t h e amount of accessible ( p e r individual ecological r e s o u r c e in a n even-aged population of ni individuals with biomass m i , S ( m i ) being equal t o a n individual's leaf s u r f a c e . The p a r a m e t e r s to b e found are v e c t o r s ( x , y , z ) t h a t provide a maximum value f o r function (9).

Let us d e s c r i b e t h e p a r t i c u l a r case of model (7)-(9) investigated in t h e p r e s e n t p a p e r (one p a r t of t h e investigation w a s made by means of analytical tools and t h e o t h e r by computer simulation). I t w a s assumed t h a t :

-

t h e specific photosyntheses .f i s independent on leaf s u r f a c e S (i.e., t h e r e i s no crown self-shading);

-

t h e r e s o u r c e accessible t o a n individual depends exponentially on t h e p r o d u c t S.n (in f a c t h e r e w e used t h e competition model developed in Korzukhin and Ter-Mikhaelian (1982) t h a t contains an analytical deduction of t h i s formula);

-

leaf s u r f a c e i s proportional t o biomass m (a simplified version with d =1 in o r d e r t o make t h e investigation e a s i e r ) ;

-

t h e survival V, f i r s t l y , depends on i t s argument in a l i n e a r manner and second- ly assumes t h a t a n individual i s a b l e t o defend itself totally from exogenous in- fluences, i.e.,

t h e e n e r g y expended o n r e s p i r a t i o n i s proportional t o t o t a l photosynthesis;

i.e., in ( 8 ) b -S..f;

t h e number of s e e d s produced by an individual at one s t e p (during one y e a r ) i s proportional t o t o t a l photosynthesis, i.e., q in (9) depends on i t s argument in a l i n e a r manner;

specific photosynthesis 'J i s proportional t o t h e amount of accessible r e s o u r c e (in o r d e r t o d e c r e a s e t h e number of p a r a m e t e r s ) .

A s a r e s u l t , w e obtain

R = -

1

C

z i m i a e "nini

-.

m a r ; i = l

where 1 i s a n intensity of competition.

The p a r a m e t e r u i s equal to survival in t h e case when a n individual expends nothing on defence; otherwise i t c a n b e i n t e r p r e t e d as a measure of s i t e favoura- bility f o r a n individual.

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4. SOME RESULTS.

The p a r t i c u l a r case of t h i s model d e s c r i b i n g a n individual's development without competition h a s b e e n investigated by means of a n a l y t i c a l tools; in f a c t t h i s case d e s c r i b e s t h e growth of a single individual

ni +l

=

[u +(I- )zi

Ini

^; (13)

R

= -

1 zi a m i ni --, max ; n1 ^{i = l}

H e r e i t is possible to c o n s i d e r a as a n a b s t r a c t p a r a m e t e r being d e p e n d e n t o n some c o n c r e t e ecological f a c t o r (e.g., o n light in a n h y p e r b o l i c manner). The d i s c r e t e analogy of t h i s model, investigated in Semevski a n d Semyonov (1982), c a n b e o b t a i n e d in t h e case of a maximum f a v o u r a b l e environment, i.e., u

=

1 ( t h a t means ni =n l).

Let u s i n v e s t i g a t e t h e d e p e n d e n c e of a n individual's optimal s t r a t e g i e s on p a r a m e t e r s a a n d u , considering as usual R t o b e a function of ( z N , y N ) ,

. . .

, ( z l , y w h e r e k

=

1 ,

. . .

^,N i s t h e number of maximization s t e p s . W e m a r k t h e v a l u e s of z , y ,z, providing a maximum to functional R with u p p e r indices.

At e a c h maximization s t e p k , functional R i s a q u a d r a t i c function of v a r i a b l e s Z N - ~ : . Y N - ~ : .

where RN-k are items of R depending o n z , y with indices of l e s s e r v a l u e s t h a n N -k

.

I t follows from t h e r i g h t s i d e s of equations (13). (14). t h a t C3N -k

>

⁰for a l l k ; a c o n c r e t e form of c o e f f i c i e n t s C j l i s determined by values of z , y providing a maximum to R at t h e f o r e g o n g s t e p s of optimization.

S i n c e we c o n s i d e r function (16) within t h e t r i a n g l e

it follows f r o m C3,N +

>

⁰^{t h a t}a maximum c a n b e r e a c h e d e i t h e r at

z N *

=

0, y N + = O ,

or at some point of t h e segment

z N *

+

^yN*

=

1 .

Let u s d e n o t e t h i s l o c a l maximum R ~ O a n d R:', r e s p e c t i v e l y .

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I t i s obvious t h a t z N

=

1 ,

z =

y N

=

1. Consecutive maximization of functions of t y p e (16) f r o m k =1 to k =N-1 gives t h e following r e s u l t s (technical details of t h i s conclusion are obvious and t h e r e f o r e omitted).

Let

~ i - ~ ,

^{y i +} denote values of

z N + ,

y N d k at which derivatives dR / d y N + , dR / d z N + are equal to zero.

Let u s assume t h a t at maximization s t e p s t = I ,

. . .

, k -1, a maximum of R h a s been r e a c h e d in

z N =

y

-' ⁼

0. The equation

determines t h e c u r v e u:(a) on which z i * ( u , a ) at maximization s t e p k f o r t h e f i r s t time becomes equal to z e r o (being more than z e r o a t t h e foregoing steps). I t can b e shown t h a t at s t e p k , t h e value

vi+(u

, a ) i s always more t h a n 1.

On t h e c u r v e determined by equation

a local maximum R:' is located at t h e point

zN-k =

0 , yN-k

=

0. A t t h e points ( a ,u ) lying between c u r v e s u o ( a ) and u = l - a , t h i s maximum i s located in t h e po- sitive quadrant

(zN+ >

0 , yN +

>

0 ) ; at t h e points ( a , u ) lying above u o ( a ) , i t i s located in q u a d r a n t

( z N

^-k

<

0 , yN+

>

1 ) .

I t i s not difficult to show t h a t t h e problem of comparing R ~ O with R i l in t h e domain lying below u o ( a ) amounts t o defining t h e sign of t h e expression

where

Negative values Ak

<

0 c o r r e s p o n d to R:'

<

R#O and vice v e r s a . Curves u E ( a ) obtained f r o m Ak ( a ,u )

=

0 are shown in fig.2.

The optimal development s t r a t e g i e s constructed with t h e help of bifurcation c u r v e s ( 18 )

-

⁽²⁰) are t h e following ( see fig.2).

1. In domain w o o bounded with segments of c u r v e s u i -l ( a ) and u z -l ( a ) , th e individual r e a l i z e s a suboptimal s t r a t e g y

with t h e corresponding value of functional (15)

2. In t h e system of domains w l k bounded with segments of c u r v e s u:(a),

u:-l

( a ) and u o ( a ) , a n individual r e a l i z e s s t r a t e g i e s

z 1

= . . .

^{= z N} ^{= O ;}

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Figure 2: Separation of parameter's plane (a ,u ) f o r model (13-(15) into domains

W j r with different development strategies. For equations of bifurca-

tion curves and optimal strategies, see t e x t .

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with corresponding values of functional (15)

S t r a t e g i e s obtained in a "growth-reproduction" model (Semevski, Semyonov, 1982) c o r r e s p o n d t o t h e case u =1

.

3. In t h e system of domains w2k bounded with segments of c u r v e s u:(a ), ( a ) and u o ( a ), a n individual realizes s t r a t e g i e s

z1

= . . .

Z N *

,

Zmax Z N - k + l

= . . . =

Z N

=

0 ;

y l =

. . . =

Y ^N-k

=

ymax ^s Y N * + l =

. . .

= y N = O ;

where

Corresponding values of functional (15) a r e equal t o

1 1 ~ ~

R F a x

=

a m l [ q ( a , u ) l N *

S o all r e p r o d u c t i v e s t r a t e g i e s are s t e p p e d , i.e., zi=O i s changed immediately by z t

=

1 ( e x c e p t f o r t h e suboptimal s t r a t e g y (21)). Growth i s always exponential with e i t h e r y =1 o r y =ymax

<

1; dynamics of density i s e i t h e r exponential ( s t r a - t e g i e s (21, (23)) with ni +l=uni o r "biexponential" ( s t r a t e g i e s (25)) with

ni

=

[u +(1 -a )zmax]nt , i 5 N

-

k ;

The development s t r a t e g i e s w e obtained a r e c l e a r from a qualitative viewpoint. A t l a r g e values of u ( system of domains w lk) a n individual grows during some p a r t of i t s life and r e p r o d u c e s during t h e remainder, i t s defence expenses being equal t o zero. The most a p p r o p r i a t e domain i s w l l , where a n individual grows during t h e f i r s t N-1 s t e p s and t h e n r e p r o d u c e s at t h e Nth one. For a smaller u (systems of domains w 2 k ) an individual is r e q u i r e d t o expend r e s o u r c e s on defence a n d on growth during t h e e a r l y p a r t of i t s life b e f o r e s t a r t i n g t o r e p r o - duce. In conditions of r e s o u r c e deficiency (small a ) and s e v e r i t y of s i t e (small u ⁾ (domain woo), a n individual r e a l i z e s a s t r a t e g y of a n "ephemeral" t y p e , i.e., i t

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starts to r e p r o d u c e only in t h e f i r s t s t e p .

Thus t h e model (13)-(15) in s p i t e of i t s simplicity d e s c r i b e s plausibly t h e adaptation of optimal development s t r a t e g i e s to different ecological conditions.

Optimal s t r a t e g i e s in t h e model with competition (10)-(12) were investigated with t h e u s e of computer simulation. In c o n t r a s t to t h e l i n e a r model (13)-(15), two new p a r a m e t e r s a p p e a r , namely, intensity of competition 1 and initial density n However, s c a l e substitution I n i =si removes dependence o n 1 and w e have a r i g h t to v a r y , f o r example, I , with n l being constant. The simulations were made f o r

nl=lo4

and N=lO; just as in t h e l i n e a r c a s e , a s e a r c h f o r t h e s t r a t e g i e s w a s made on t h e p a r a m e t e r s ' plane ( a

, u )

f o r t h e following set of values of 1: 1

=

l o v 4 , 3.16 . l o 4 , 3.16 (these values were t a k e n from a model (Korzukhin et ^al., 1987) t h a t had been verified by field d a t a on birch-siberian pine succession; in t h a t model, t h e same e x p r e s s i o n w a s used f o r describing light competition).

The main e f f e c t found in model (10)-(12) consists of t h e p r e s e n c e of "smooth"

r e p r o d u c t i v e s t r a t e g i e s in c o n t r a s t to stepped s t r a t e g i e s of a l i n e a r model. The g e n e r a l p a t t e r n of behavior is t h e following: s t r a t e g i e s z k cease to b e stepped with increasing a ,

u ,

1, however. at t h e same time, t h e r e i s a n i n c r e a s e in t h e number of s t e p s with z k

#

0 , l . This p a t t e r n i s found only when t h e value of u i s big enough a n d a l l z k are equal t o zero. Thus t h e following s t r a t e g y i s added t o s t r a t e g i e s of t h e l i n e a r model:

= . . .

^{Z N}= O ;

An example of t h i s s t r a t e g y i s shown in fig.3a; t h e corresponding dynamics of a n individual's biomass and f e r t i l i t y a r e shown in fig.3b. The behaviour of population v a r i a b l e s M and Q is qualitatively similar t o t h a t shown o n fig.1.

Finally i t is n e c e s s a r y to mention one more e f f e c t d e r i v e d f r o m t h e mode!. I t is customary t o suppose (e.g., see Bugrovski et al., 1982) t h a t a n individual g e t s i t s maximum biomass through t h e f a c t t h a t r e s p i r a t i o n expenses i n c r e a s e f a s t e r t h a n t h e rate of photosynthesis F; as soon as t h e s e quantities become equal, growth ends. According to development s t r a t e g i e s obtained from t h e model, a n i n c r e a s e in biomass ends b e c a u s e t h e individual begins to spend all photosynthetic p r o d u c t s on s e e d production. I t seems t h a t both things t a k e place in r e a l life, i.e., t h e r e are both a n i n c r e a s e of r e p r o d u c t i o n expenses and nonproportional ( r e l a t i v e t o biomass of leaves) growth of organs, t h a t d o not produce b u t consume photosyn- t h e t i c products.

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Figure 3 Solution of model (10)-(12) (obtained by means of computer simulation) for following values of parameters: u

=

0.08, a

-

^{1.0, 1}

⁼

n l = 1 0 4 , N

=

10. Designations correspond t o those used in (10-(12) and in legend t o fig.1. Through z k being identical t o z e r o , t h e dynam- i c s of population density i s exponential and t h e r e f o r e i s omitted.

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The a u t h o r s wish t o e x p r e s s t h e i r g r a t i t u d e t o P r o f e s s o r Ted Munn f o r h i s h e l p .

Model of the Optimal Development of a Plant Taking Into Account Defence and Competition

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