Dynamical Systems and Environmental Models: Proceedings of an International Workshop cosponsored by IIASA and the Academy of Sciences of the GDR, held on the Wartburg, Eisenach (GDR), March 17—21, 1986 [Reprint 2021 ed.] 9783112484685, 9783112484678

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Mathematical Ecology

Mathematical Ecology edited by Peter Allen (Brussels), Werner Ebeling (Berlin), Manfred Peschel (Berlin), Peter Schuster (Vienna), Yuri M. Svirezhev (Moscow) Mathematical Ecology deals with mathematical models of evolution processes, in the biosphere as a unity of growth and structure-building, with spring-up of new species, their interaction and possible extinction, with the impacts of human activities on the environment and the corresponding consequences for the biosphere. The whole field seems to be a certain amalgam of suitable system philosophy, in which real world phenomena are considered through an ecological pair of spectacles with the help of system methodology and mathematics. All publications published herein are of interdisciplinary interest for ecology, biology, economy and technical engineering.

Dynamical Systems and Environmental Models Proceedings of an International Workshop cosponsored by IIASA and the Academy of Sciences of the GDR held on the Wartburg, Eisenach (GDR), March 17—21, 1986

Edited by Hans Günter Bothe Werner Ebeling Alexander B. Kurzhanski Manfred Peschel

Akademie-Verlag Berlin 1987

Editors: Dr. habil. Hans Günter Bothe Akademie der Wissenschaften der D D R Karl-Weierstraß-Institut für Mathematik Prof. Dr. Werner Ebeling Humboldt-Universität zu Berlin Sektion Physik Prof. Dr. Alexander B. Kurzhanski International Institute for Applied Systems Analysis (IIASA) Prof. Dr. Manfred Peschel Akademie der Wissenschaften der D D R Zentrum für wissenschaftlichen Gerätebau Reproduction of the original authors' manuscript.

ISBN 3-05-500334-9 Erschienen im Akademie-Verlag Berlin, DDR-1086 Berlin, Leipziger Straße 3—4 © Akademie-Verlag Berlin 1987 Lizenznummer: 202 • 100/521/87 Printed in the German Democratic Republic Gesamtherstellung: VEB Druckerei „Thomas Müntzer", 5820 Bad Langensalza Einbandgestaltung: Ralf Michaelis LSV 1095 Bestellnummer: 763 719 9 (9044) 03500

Introductory Remarks Advanced attempts on modelling the development of complex systems require to an increasing degree concepts and results from theoretical dynamics. This general fact applies especially to environmental models, and it is widely accepted that each of the both sides - the theoretical research and the work with realistic models - can gain a lot of ideas, fruitful motivations and improvements concerning the methods by knowing about the spirit, the problems and the methods of the other side. Nevertheless, in general there is a serious gap between the world of purely theoretical research and that of applications. This situation is rather stable, and it would not be realistic to expect that it will change drastically in the near future. But it seems to be an important task to look for already existing bridges and for developments which promise to lead to such bridges. This was the general aim of the workshop. So the scientific activities of the participants belong to different disciplines, and accordingly broad is the spectrum of their papers which are presented here. On the one end of this spectrum there are contributions to the mathematical theory of dynamical systems (Section I). They concern concepts and ideas which have proved to be (or promise to become) fruitful in connection with applications. These topics of pure mathematics are followed by the presentation of general mathematical models (in Section II) which are motivated by questions arising in attempts to get an insight into certain evolutions. Section III is devoted to mathematical models for more or less concrete evolution processes in ecology, while in Section IV mathematical methods are applied to describe facts which are known from experience or experiments. Here, in some cases, mathematics proves to be useful not only for a phenomenological description of processes in society and nature, but it helps to discover mechanisms which cause these phenomena and their laws.

5

List of Participants K.F. Albrecht, GOR

M. Kubiiek,

T. Aldenberg, The Netherlands

Z. Kubiikova, Czechoslovakia

E. Amann, Austria

E. Labos, Hungary

A. Arneodo, France

A. Lasota, Poland

Czechoslovakia

Th. Bley, GDR

R.W. Leven,'GDR

H.G. Bothe, GDR

L. Maistrenko, USSR

M. Bröhl-Kerner, FRG

H. Malchow, GDR

B. Bruhn, GDR

S. Markov, Bulgaria

P. Brunovsky, Czechoslovakia

V. Mazenko, USSR

G. Czajkowski, Poland

W. Mende, GOR

W. Ebeling, GDR

A. Molcanov, USSR

P. Erdi, Hungary

3.S. Nicolis, Greece

H.I. Freedman, Canadb

L.F. Olsen, Denmark

R. Funke, GDR

M. Peschel, GDR

K.P. Hadeler, FRG

B. Pompe, GDR

Z. Harnos, Hungary

K.R. Schneider, GDR

M. Heilig, Austria

E.E. Shnol, USSR

H.P. Herzel, GDR

K. Sigmund, IIASA

L. Kalriukstis, IIASA

U. Svedin, Sweden

P. Kindlmanh, Czechoslovakia

W.M. Terhorst, FRG

B.P. Koch, GDR

E. van de Vrie, The Netherlands

Organizing

Committee

F. Auert,

Karl-WeierstraB-Institute for Mathematics

E. Herbst,

IIASA

H. Pierau,

Karl-WeierstraQ-Institute

for Mathematics

D. Ruchhoft,

Karl-WeierstraB-Institute

for Mathematics

6

Contents Part I: Mathematical Theory of Dynamical Systems H.G. BOTHE Mausdorff dimension of attractors: An example

9

P. KINDLMANN K-stability as an ecologically more relevant stability concept

....

14

Stability, periodicity and chaos from the statistical point of view

24

K.R. SCHNEIDER Singularly perturbed autonomous differential systems A.N. SHARKOVSKY, YU.K. MAISTRENKO Difference equations with continuous time as mathematical models of the structure emergencies

32

A; LASOTA

40

E.E. SHNOL Stability of equilibria in critical cases

50

Part II: General Mathematical Models E. AMANN Permanence in population dynamics

58

U. EBELING, A. ENGEL, R. FEISTEL, V.G. MAZENKO Models of evolution processes including age structure

67

F. FELBER, A. HUNDING, H. MALCHOW Pattern formation in an ionic reaction-diffusion system: A contribution to the morphogene prepattern theory of mitosis

76

H. HERZEL, TH. SCHULMEISTER Chaotic dynamics and fluctuations in a biochemical system

85

R.W. LEVEN, B.P. KOCH, G.S. MARKMAN Periodic, quasiperiodic and chaotic motion in a forced predator-prey ecosystem

95

J.S. NICOL IS Chaotic dynamics of logical paradoxes

105

M. PESCHEL, M. VOIGT, W. MENDE Control of growth processes Part III. Mathematical Models for Concrete

114 Processes

A. F. ARGOUL, P. reaction: RICHETTI, A J.C. ROUX for The ARNE0D0, Belousov-Zhabotinskii paradigm studi es of dynamical systems

theoretical

122

P. BRUNOVSKY, T. KMET The nitrogen transformation cycle in water

132

G. CZAJKOVSKI Modelling of polymer growth

139

H.I. FREEDMAN, G.S.K. U0LK0WICZ A mathematical model of group defence in predator-prey systems K.P. HADELER Vector-transmitted

diseases in structured populations

....

149 154

7

Z. KUBÎÎKOVA, M. KUBÎCEK, M. MAREK Nonlinear behaviour in mathematical models of anaerobic digesters

«

162

E. LABOS, E. N0GRA0I Examples of computer-aided exploration and design of dynamical systems in neurosciences: Design of optimal nets

172

S. MARKOV, T. KOSTOVA-VASILEVSKA A dynamical model of synaptic transmission

182

K. SIGMUND Gradients for replicator systems

186

Part IV; Mathematical Description of Environmental Processes T. The ALDENBERG limiting factor concept in relation to stability, sensitivity and bifurcation in plankton models

196

G. BARNA, P. ERDI Effects of structured environment to dynamic .behaviour : Some illustrations

209

TH. BLEY, B. HEINRITZ Modelling and control of yeast growth in biotechnical systems

213

H. BOSSEL A dynamic simulation model of tree development under pollution stress

220

H. BRÜHL-KERNER A biota model to assess the influence of human impacts on the global carbon cycle

230

L. KAIRIUKSTIS An integrative approach to the solution of ecological problems with particular attention to the regional case

240

W. MENDE, K.-F. ALBRECHT Application of the evolon model on evolution and energy growth processes

253

L.F. OLSEN Non-linear dynamics of the epidemics of some childhood deseases in Copenhagen, Denmark

265

E.M. VAN DE VRIE Modelling an estimating transport and fate of heavy metals in water column and sediment layer in some enclosed branches of the sea in the S.W. Netherlands

273

POSTER SESSION T. CZIERZYNSKI Mathematical software developed by the Karl-WeierstraO-Institute of Mathematics

8

282

HAUSDORPP DIMENSION OF ATTRACTORS : AN EXAMPLE H. G. Bothe ")

The aim of this paper is to show by a simple example that the connection between the Ljapunov exponents and the Hausdorff dimension of an attractor is not so close as sometimes assumed. The mathematical background for the facts presented here (see [l]) covers a more general class of attractors (1-dimensional hyperbolic or higher dimensional expanding attractors of a diffeomorphism). Let S 1 be the unit circle which is regarded as the real line IR modulo 1, i.e. S1 is obtained from ® by identifying points whose difference is an integer. So for points t ^ t g C S 1 a sum t^+tg t S 1 is defined, and points t £ S 1 can be multiplied by real numbers. 2 2 By ID we denote the unit disk in the plane IR . The cartesian product V » S 1 x ID2 is a solid torus. For t £ S 1 the disk {tj x D 2 will be denoted by D^. We consider a (not necessarily continuous) embedding f: V —» Int V which has the following properties: (1) The restriction

f^

of

f

to

to a disk disk (i.e. f +

D^ is a linear embedding of D^ into the interior of the which contracts all distances by a factor A(t) > 0 is a similarity mapping). (2) There is a positive 6 1 such that for each t € S the distance between the two disks ^(D^), in D t is at least cf . By th-> *(t) the embedding f defines a (not necessarily continuous) positive function XsS 1 —f IR. If f is continuous, then to apply the mapping fsV—»Int V means to stretch V to a longer solid torus of twice the original length, where during the stretching each disk D^ is contracted to a smaller circular disk of radius ^.(t) (see Pig. 1). The set A «= O *" vi * 0 attracts all orbits (x, f(x), f2(x),...) (x e V): it is the attractor of f. As easily seen the intersection /\ r\ H of A with any disk D t is a Cantor set C + . We are interested in the *) Karl-Weierstrass-Institut ftir Mathematik DDR 1086 Berlin, Mohrenstr. 39

9

Hausdorff dimension dia^C^ of these sets. If these dimensions are independent of t, then the Hausdorff dimension dim^A is dim^C^ + 1. Under our assumptions (1), (2) for each t € S 1 the dimension dimgC^. is already determined by the function X . Some relations between X and dlm^C^. are stated in the following theorem. Theorem 1. (A) If for a positive real number equation X P (t)£(t) + ^ ( t + 1/2) | (t + 1/2) = with

p

p

pA the functional £ 0. The topology in £ is defined by the metric diet(X 1t X 2 ) " 8U eJ A,(t> - Aa|, i.e. we coneider the topology of uniform convergence. Then p^ (which for X e X is uniquely determined by L ) depends continuously on A , i.e. X p. is a continuous function on >C. (By (B) the space X 1 contains all Lipschits continuous functions on S .)

on

Corollary. If t.

A

belongs t o

X ,

then

dlmjjC.|. does not depend

For proofs of these facts see [1]. Now, for a continuous function A in X. , we try to reduce the functional equation (*) to an ordinary equation for p A , i.e. we look for a real function P(p) for which pA is the maximal solution of P(p) » 0. This can be done approximatively as follows: We first approximate \ (with respect to the topology of uniform convergence) by a step function A* which, for some n > 1, is constant on each subarc £(1-1 )2~n, i2" n ) of S 1 (i = 1,2,...,2n). If ^ denotes the value of X* on f(i-1)2"n, i2""n), then each solution fu'-'j^^n of the following 2 n linear equations (**) yields a solution of (*). 1

+

i+2*"1 " f 21 t 21 " S 21-1

(i - 1,2,...,2n-1)

(**)

Indeed the solution of (W) corresponding to is the step function jn on S 1 whose values on the arcs [(i-1)2~ttt 12" w ) are The matrix A on the left hand side of (**) is non negative, and, by the theorem of Perron-Frobenius (see [2], XIII,2), we get a positive solution of (**) if and only if 1 is an eigenvalue of A and no eigenvalue is of modulus greater than 1. Let P., be det(A - E) regarded as a function of p. Since 0< 1 the powers are decreasing functions of p, and, applying the Perron-Probenius theorem once more, we find that (**•) has a positive solution if and only if p « p,„ is the maximal solution of P.»(p) = 0. Since P,*(0) « -1, A. X \ lim ^*(p) = 1 » this solution pA, exists and is positive. Reaee A belongs to £ , and by (C) ¡p^» - px| is small provided max |X*(t) - X(t)1 is small. teS-» A similar approximative determination of dim^C^ point of view (not using (*)) is done in £3]. For

n • 2

from a different

we have p ( P ) - (a*p - o a ; p - 1 ) -

11

In this case we consider the step function A ' M t ) - * ( t + J ) . Then Pr(p> -

- DCA^ - 1) -

and for almost a l l values of

).*»•••»

A'*

which i s defined by

AyAy. we get

p^,. /

I t can be proved that for a function A tS 1 -* IR which i s smooth and d i f f e r s from X*" only near the points t = i/4 ( i • 0,1,2,3) aa indicated in Fig. 2 the difference | p^ - px„| i s small (see [ 1 ] ) . Therefore, i f A'sS1 IR is defined by A ' ( t ) = X(t + the inequality p,„j* p,,. implies p v j p , and we have proved the following X A • A i theorem. Theorem 2« There.is a smooth function \ in and a point t —? 1 ° in S such that f o r the function X ' s S ® which i s defined by X ' ( t ) » A.(t + t 0 ) we have p^, + p^. Remark. By (C) this phenomenon i s stable in the sense that the set of a l l functions \ in jC. for which the assertion of the theorem holds true i s C° open in L .

1 4*0

4*0 Pig. 2 For

tQ

e

S1

let

be

O

the

rotation of

(t,x) = (t + tQ, x)

V given by

( ( t , x ) e s 1 * ©2 - v ) .

I f f « V - + V i s an embedding satisfying ( 1 ) , ( 2 ) and i f the corresponding function, then the mapping

i s an embedding which also s a t i s f i e s ( 1 ) , ( 2 ) . The function belonging to f ' i s given by V ( t ) » X(t + t Q ) . 12

> IR

' «1 "X »S

is

This leads to the following corollary In which 7 " denotes the apace of all C 1 embeddings f:V—*• Int V satisfying (1),(2) with the C° topology. Corollary.

for each

j?"* contains a non empty open subset

t & T * there is an embedding

T * such that

f' » f_2t ' 9 t

e

for

o o which theHausdorff dimension of the corresponding attractor is different from the Hausdorff dimension of the attractor belonging to Remark. Probably

7*

is dense in

f.

T.

The Ljapunov exponents of an attractor A are defined with respect to an f-invariant ergodic probability measure on A . The measure of this kind which is most closely connected with the dynamics on /\ » i.e. with the restriction of f to A , is the so called Bowen-Ruelle measure ¡i which in our case can be characterized by the / a -I * condition that its projection to S is the Lebesgue measure on 8 . The ordinary Ljapunov exponents, i.e. those with respect to , for an attractor A of an embedding fsV—»Int V satisfying (1),(2) are (with multiplicity 1) and /logX(t)dt (with multiplicity 2). log 2 Hence, the Ljapunov exponents of the attractor belonging to an embedding f t ? do not change if we apply (as in the corollary) a rotation to f. This shows that the attractors of the mappings f, f' in the corollary have the same Ljapunov exponents but different Hausdorff dimensions. Besides the geometric Hausdorff dimension considered here another dimension of attractors A which dependes on the measure ju^ on ^ is defined as follows dim^

A formula for

= inf |dimjjT ; P a Borel set in A , y u A ( P ) * 1 ]• .

dim^A

proved in

dim

A

A

Corollary 5.1 implies

- 1 - log 2 //log A(t)dt .

>I

Therefore our corollary to Theorem 2 shows that in many simple cases dim /\ differs from the (geometric) Hausdorff dimension of A. References fl] H.G.Bothex On the Hausdorff dimension of expanding attractors. (Preprint 1986)

[2] F.R.Gantmacher: Matrieenrechnung.

Berlin 1959

[3J M.Schuls: Hausdorff-dimension von Cantormengen mit Anwendungen auf Attraktoren. (Dissertation Berlin 1986) [4] L.-S. Young: Dimension, entropy and Igrapunov exponents. Ergodic Theory and Dynam. Syst. 2 (1982), 109-124

13

K-STABILITY

AS AN E C O L O G I C A L L Y

MORE R E L E V A N T

Pave L K i n d I m a n n

1.

1

STABILITY )

Introduction The term s t a b i l i t y

mathematics. Whereas

is f r e q u e n t l y

in m a t h e m a t i c s

stability systems

concept

commonly

is the L i a p u n o v

u s e d b o t h in e c o l o g y a n d

u s e d in m a t h e m a t i c a l

stability.

the -needs of e c o l o g i s t s .

In t h i s

m o d e l s of

H o w e v e r , it d o e s not c o m p l e t e l y better

stability

concept

In m o s t m a t h e m a t i c a l

stability

concepts

t i m e . In the

field ecology

the c o n t i n u o u s

the e x c e p t i o n of a few o n e s ) is a l m o s t systems This

the v a r i a b l e s vertebrates

is p a r t i c u l a r l y

generations.

in p a r t i c u l a r

intervals

u s u a l l y o n c e a year for e c o l o g i c a l

the

some

motivation

measuring

continuously of v a r i a b l e s

changes

important

Moreover, (circadian,

o n l y , in t e r r e s t r i a l

plants

c o n c e p t s are

H o w e v e r , the e n v i r o n m e n t a l

a n d e v e n the e n d o g e n e o u s

fluctuations

in v a l u e s

more it

average

over

some

realistic approach of a p o i n t .

c o n d i t i o n s are n e v e r

d y n a m i c s of e a c h c o m m u n i t y

of p a r t i c u l a r

of e q u i l i b r i u m

s t u d i e s . One p o s s i b i l i t y

variables.

The

is

this equilibrium

p e r i o d w h i c h is c o n s i d e r e d to be is to c o n s i d e r

an i n t e r v a l

are a b l e to e x i s t

in

constant causes

quantitative

usually causes difficulties

to a p p r o x i m a t e

Some e c o s y s t e m s

and the

application.

In m a t h e m a t i c a l m o d e l s of e c o s y s t e m p o p u l a t i o n d y n a m i c s

determination

with

measure

u s u a l l y s u p p o s e d t h a t u n d e r n o r m a l c o n d i t i o n s the s y s t e m r e m a i n s "equilibrium".

(with

11]. C o n s e q u e n t l y , in many c a s e s stability

It in

the

in o r g a n i s m s

H e n c e , it is o f t e n m e a n i n g f u l to

m o d e l s a n d the c o r r e s p o n d i n g

appropriate

in e c o l o g y

e q u a t i o n s are u s e d .

impracticable.

are s u b j e c t e d to p e r i o d i c

phenomenon

non-overlapping

in nature

of

definition:

models, differential

that the v a r i a b l e s are s u p p o s e d to be c h a n g i n g

discrete

proposed

[4]. Here we s h a l l s t r e s s o n l y

p o i n t s of t h i s d i s c u s s i o n , in o r d e r to e l u c i d a t e

ecological

is

m o d e l s of p o p u l a t i o n d y n a m i c s

A t h o r o u g h d i s c u s s i o n of d i f f e r e n t

diurnal).

fit

definition

species:

was g i v e n by K i n d l m a n n a n d Lep§ of the new

The

ecological

p a p e r , a new m a t h e m a t i c a l

a n d a p p l i e d to s e v e r a l m a t h e m a t i c a l

major

defini-

is o f t e n v a g u e .

of s t a b i l i t y , w h i c h f i t s the d e m a n d s of e c o l o g i s t s one a n d t w o

in

t h e r e e x i s t a lot of r i g i d

of t h i s t e r m in e c o l o g y

t i o n s , the u n d e r s t a n d i n g

means

CONCEPT

in the

" n o r m a l " . The

(range

field

is to t a k e

the

more

of v a l u e s )

instead

only far f r o m the

equili-

1) P a v e l K i n d l m a n n , D e p a r t m e n t of B i o m a t h e m a t i c s , B i o l o g i c a l R e s e a r c h C e n t r e , C z e c h o s l o v a k A c a d e m y of S c i e n c e s , B r a n i S o v s k i 3 1 , 3 7 0 0 5 C e s k 6 Bud£jovice, Czechoslovakia

14

b r i u m . As a n i c e example may s e r v e snow a v a l a n c h e great

species

diversity

tem from time t o t i m e . would d i e

is If

these avalanches

The t y p e of L i a p u n o v s t a b i l i t y It

the e q u i l i b r i u m , about the s i z e interested

assures

commonly u s e d i s a l o c a l

of t h i s

neighbourhood.

great

s y s t e m s we meet f a i r l y

prey-predator

environmental

systems)

this

is

2 . The d e f i n i t i o n Bearing

of

more r e l e v a n t

counterparts.

is

invariant

systems

laboratory

w e l l f o u n d e d . But even

interpretation.

stability

concept

[4].

periodical

set w i l l

t h e n be s t u d i e d .

shall

t o us - i t

cases.

differential

i n some

than i n e q u i l i b r i u m .

or even a c h a o t i c

we

i n many

d e f i n i t i o n c o u l d be s t a t e d f o r t h e i r rather

ecologi-

I n the d e f i n i t i o n

w i l l not be of any i n t e r e s t

invariant

The

behaviour

may be a

perio-

m o t i o n . The s t a b i l i t y

Local s t a b i l i t y

will

be

the

system i s

dis-

s t a b l e . T h i s means t h a t the s y s t e m , a f t e r

p l a c e d by some e x t e r n a l acting

force

t o some p o i n t x < H - G , n e v e r

r e a c h e s the s e t G. The e x i s t e n c e

perturbations

leads e i t h e r

perturbed d i f f e r e n t i a l

to s t o c h a s t i c

or d i f f e r e n c e

equations.

stability

c o n c e p t s of p e r t u r b e d e q u a t i o n s

[2,3])

satisfactory

is

o f them assume t h a t the sense:

that

its

h a v i n g been of

leaves

m o d e l s , or t o some None o f the

f o r the use i n e c o l o g i c a l

e.g.,

applications.

"perturbation function'is

small

norm a p p r o a c h e s z e r o a s the s t a t e of the

in a

is

smaller

All certain

system

some c o n s t a n t

But i n e c o l o g y the norm of the

various

i n mathematics ( s e e ,

s m a l l ) , s o t h a t the norm of the p e r t u r b a t i o n fairly

the

permanently

approaches e q u i l i b r i u m , or that there e x i s t s constant.

(maybe

than

"normal"perturbations

very

this may be

h i g h . T h e r e f o r e , t h e r e s h o u l d be some p r e d e t e r m i n e d c o n s t a n t

s e r v i n g a s an upper bound f o r the norm of

of

re-

p l a c e d by the g l o b a l o n e , i . e . , we d e f i n e a s e t H , w i t h i n w h i c h

s e t H and f i n a l l y

in

K-stability

s e t G under n o r m a l c o n d i t i o n s set

always

in simple

We s h a l l assume t h a t the s y s t e m r e m a i n s

d i c a l , an a l m o s t

act

this

either.

concept

e q u a t i o n s , a s t h e y a r e more a p p r o p r i a t e

Obviously, a similar

within this

which

i n mind the c r i t i c i s m a b o v e , we s h a l l d e f i n e an

use d i f f e r e n c e

of

anything

and

- e . g . , i n the

quite

case one must be c a r e f u l w i t h the

etc.)

concept

(especially

conditions

concept

saying

perturbations

variability

i n the L i a p u n o v s t a b i l i t y

Under c e r t a i n c i r c u m s t a n c e s

and under c o n s t a n t

property

perturbations.

great

changes, b i o l o g i c a l

not i n c o r p o r a t e d

unrealistic.

this

species

I n e c o l o g y , h o w e v e r , we a r e

T h i s does not mean t h a t the L i a p u n o v s t a b i l i t y

cally

the sys-

of some n e i g h b o u r h o o d

s t a b l e , without

i n the c o n s e q u e n c e s of f a i r l y

In biological

this

the

were not t h e r e , many

o n l y the e x i s t e n c e

i n w h i c h the s y s t e m i s

permanently ( c l i m a t i c is

[ 5 ] , where

out.

o f the s y s t e m .

fact

paths

c a u s e d By a v a l a n c h e s w h i c h d i s t u r b

"normal"perturbations.

K,

All 15

the subsets of the set H may be d i v i d e d

into three

categories:

D E F I N I T I O N 1. Let N be the set of n a t u r a l n u m b e r s , R NxR

m - d i m e n s i o n a I E u c l i d e a n vector s p a c e , f u n c t i o n f: m"* ve n u m b e r , G e H e R . We shall denote the set P = / g > g: m * supn#N

y

T h e n we d e f i n e : the set G c H a) K - a t t r a c t i v e

is

in H for the system (1), if for all the s o l u t i o n s

the s y s t e m (1) it holds: 1) for all m c N , e x i s t s an n e N , geP

R

and n C N ,

n » m so that

y m « H and for all g « P

y n c G> 2) if m € N ,

n » m , it holds

y m « G , then for all

y n 6 G-

b) K - t r a n s i e n t for the system ( 1 ) , if for all m « N ,

y^e G and for

g C P there e x i s t s such n e N , n > m so that it is not

yn*G.

c) K - s e n i a t t r a c t i v e

of there

all

in H for the s y s t e m ( 1 ) , if it is neither

K - a t t r a e t i v e , nor K - t r a n s i e n t in H. D E F I N I T I O N 2. The set M c H

is c a l l e d the minimal K - a t t r a c t i v e

set

i n H for the system ( 1 ) , if and only if it is K - a t t r a c t i v e and if for all the K - a t t r a c t i v e sets S C H The d i f f e r e n c e

of the s y s t e m (1) it holds

b e t w e e n this s t a b i l i t y concept and the c o m m o n

is rooted in the fact that the constant K is a priori therefore

defined

are b o u n d e d and p e r m a n e n t l y acting "extreme"- catastrophes

ones

(and

it may be fairly great) and does not d e p e n d on x. This

lity concept d i s t i n g u i s h e s two types of p e r t u r b a t i o n s :

displace

MCG.

stabi-

"normal", which

(performed by the f u n c t i o n g) and

like f i r e , human a c t i v i t i e s e t c . , w h i c h

the system outside the regions of its

may

"normal"motion - outside

the K - a t t r a c t i v e set G. There are s e v e r a l e c o l o g i c a l l y

interesting

q u e s t i o n s , which can be a n s w e r e d only by means of this s t a b i l i t y

concept

[4]. We shall state some simple used in the a p p l i c a t i o n s .

lemmas on K - s t a b i l i t y , which will be

Their simple proofs will be

left to the

reader. LEMMA 1. D e c o m p o s i t i o n of exp(H) into K - a t t r a c t i v e , K - s e m i a t t r a c t i ve and K - t r a n s i e n t sets is a d e c o m p o s i t i o n of e x p ( H ) into pairwise joint

LEMMA 2. Let f: N x H - » G . If G'= { x « R m ; .

3

c H , then G ' i s K - a t t r a c t i v e in H for the system

16

dis-

sets. y«G: (1).

| Ix - y ] I - K }c

L E M M A 3 . Let t h e r e e x i s t an x'fiG, so t h a t h o l d s at a)

least one of the

llf(n',x')|| + K > sup

for s o m e n e N

there

conditions

y e G

lly||,

b) 0 < I I f ( n ' , x * ) | | - K < i n f y , G l | y | | . T h e n G is not K - a t t r a c t i v e LEMMA

4. If t h e r e e x i s t s an x ' « G , so t h a t

||f(n,x*)

a) s u p x i G ^

)

inf

(1). for a n y n € N

it

x.G,

h o l d one a n d o n l y one of the

n€N(llf(n,x)||

n 6 N

(

| | f

(

n

'

x

T h e n G is K - t r a n s i e n t

)

-

M

||x 11) = S < - K < 0

"

l,X,l

For K - s t a b i l i t y t h e r e

conditions

,

= l > K >

(1).

a l s o can be p r o v e d t h e o r e m s

of L i a p u n o v

let for e a c h n ( N ,

it h o l d s f ( n , y ) + g ( n , y ) « H .

for e a c h g e P

Furthemore,

let

some f u n c t i o n V: NxR —* R a n d for a l l n « N , for a l l x « H it m V ( n , x ) Si o a n d V ( n + 1 , f ( n , x ) ) - V ( n , x ) < 0. We s h a l l d e n o t e =

For some

sup

t> 0

x€H>V

L

x«G;

sup

type.

and for

holds

" V(n,x)||.

(2)

let

G'={

=

I I x-y I I < K, n c N ' l ^ " ' ^

x , y e H,

(1).

illustration:

T H E O R E M . Let K > 0 , N c R ^ , let for e a c h y € H

>

in H for the s y s t e m

We s h a l l b r i n g one for

"

holds

- x * | | - K , t h e n G is not K - t r a n s i e n t in H for the s y s t e m

L E M M A 5. Let t h e r e

b

in H for the s y s t e m

G = { x€H>

n

«N:

nCN

v

V(n+1, f(n,x))

- V(n,x)

>

- M - £},

(3)

("'x)'

CM

V n-0: V(n,x) - M + L).

T h e n G is K - a t t r a c t i v e case of a u t o n o m o u s

(5)

in H for the s y s t e m (1). (The s i t u a t i o n

systems

is i l l u s t r a t e d

in Fig.

for

the

1.)

P r o o f : We s h a l l v e r i f y the c o n d i t i o n s of the d e f i n i t i o n 1 a ) . 1) Let Y M € H - G. T h e n for e a c h g C G y ^ C H - G , t h e n it is

a n d for a l l n € N ,

n - M , if

satisfied

17

Fig.

1. C o n s t r u c t i o n of the set G i n the Theorem i n the case of a u t o n o mous systems. F u l l l i n e s : V(x) = c o n s t . , broken l i n e : V(x) = = V ( f ( x ) ) - V(x) = c o n s t . V(n+1, f ( n , y n )

+ 9(n,yn))

« V(n+1, f ( n , y n ) ) (as G'C G). so t h a t 2) Let ycG

Therefore

*

there

exists

£

(6)

an n « N ,

n

- (V(m,y m )

- M -

su

and M + L -

PneNv(n^y) > L-

- G* and from (2) and (3)

all

n - 0,

supniNV(n,y) holds

3.

Application

f(n,y)

V(n+1, f ( n , y )

We s h a l l models",

i.e.,

of

analyze

i.e.,

of

published

functions

f

[6],

[8],

[3],

1. The f u n c t i o n 2.

f * ( 0 ) > 1.

3.

There e x i s t s

L7], f

is

y«G

and

I.e.,

of

the

- V(n,y)

a wide c l a s s

of

models *one-species

R^-* r | i n the system ( 1 ) .

following

set

of

conditions

There e x i s t s

convex

18

for

for

Ma'ny

(see,

[9]):

continuous

and f ( 0 )

= 0.

1

decreasing

an x > 0 such t h a t

1

f(x)

is

increasing

for

xx^. an x^> x^, such t h a t

x > x^.

i

again f ( n , y ) +g(n,y)

t o nne- and t w o - s p e c i e s

where f , g :

satisfy

follows

(7)

€ G. I f

- h + L.

the b e h a v i o u r

e.g.

i*

(2) and as V(n +1 , f ( n , y ) )

+ g(n,y))

the case

'from ( M

- H + L

+ g(n,y)

K-stability

Then

we have

+ g(n,y)) 0

it h o l d s f ( x ) -

In this case any set G is e i t h e r dependence

of w h e t h e r

or w h e t h e r

number

from this

according

c a s e s for g i v e n f a n d K

K - t r a n s i e n t or not K - t r a n s i e n t

it d o e s not c o n t a i n a n u m b e r

it d o e s

(Fig.2):

K .

set

d i t i o n c H is not s a t i s f i e d , t h e n there d o e s not e x i s t K-attractive

set G C H

for

e x i s t s a u n i q u e n u m b e r x ® , sutfh t h a t f ( x 5 )

The

h o l d s in t h i s case as in the p r e v i o u s

i n f

y«H

l y l

=

i n f

y«Glyl

=

*5'

S U p

= f ( x 5 ) , t h e n G is K - a t t r a c t i v e

y6G

l y l

~ *3

a n d

if

inf

x > . A g a i n the

interval

in H , if it is c o n t a i n e d any K - a t t r a c t i v e c) T h e r e 5

exist 6

3

( 0 > x > is the

in H. For a n y o t h e r

substan-

set in H is the

=

inter-

minimal K-attractive

set H t h e r e d o e s not

set

exist

set in H for. (1).

two d i f f e r e n t

n u m b e r s x^ a n d x 6 , s a t i s f y i n g

in t h i s case

is much more

complicated.

one e x a m p l e , u s e f u l for b i o l o g i c a l a p p l i c a t i o n s . n u m b e r , for w h i c h f ( x 7 )

f(x^) - K =

We s h a l l s h o w

Let x 7 be the

= f ( x ^ ) . If x 3 < x 7 , t h e n from L e m m a 2.

t h a t G =

with some constants

>

> 1,

IS"(*)I

[S'(x)]2 i K < oo

(1.3)

X ,M and S(a l _ 1 ,a i ) -

(0,1)

for

i=1,...,q,

(1.4)

Under these conditions it is easy to calculate P g . In fact denoting by S^ the restriction of S to the interval (a i _ 1 ,a i ) we have s"1(o,x) = U C v i ' f d 3

( x ) )

o

(J k

(f^x),^)

where vj/^ - S~1 and the first union is over all Intervals in which S^ is increasing whereas the second is over intervals in which Sj is decreasing. Thus according to (1.2) q P s f(x) l^tolfi+^x)) . (1.5) i=1 25

Using (1.3) i t i s easy to v e r i f y (see [3] ) that f o r every C1 function f « D the logarithmic derivative of Pgf s a t i s f i e s K*go

V

,

for

n > nQ(f)

(1.6)

with some constant E independent on f . 2. Properties of the Frobeniua-Perron operator An operator P: IL11 - > I 1 1 i s called a Markov operator i f i t satisfies the following two conditions Pf

>, 0

| P f | - IfII

for

f » 0,

f « L1,

(2.1)

for

f

f t l

(2.2)

0,

1

.

Conditions (2.1) and (2.2) imply that Hpf 1 4 Ufll for

f « I1.

(2.3)

Using formula (1.1) i t i s easy to show that the Probenius-Perron operator P s i s a Markov operator. The behaviour of the iterates £ J allows to determine many propert i e s of the transformation S such as preservation of measure, ergodlclty, mixing and exactness. Here we make only a few remarks concerning these problems. Let a density f * be given. Define the corresponding measure nig(A) •

£f„.(x)m(dx)

for

A ac> kio £ for every m^- lntegrable function g and the convergence in (2.6) holds almost everywhere. Thus from the statistical properties of S we may derive corollaries concerning the behaviour of individual trajectories. 3. Asymptotical stability Let P be a Markov operator. We say that P is asymptotically stable if there exists a unique density t^ such that 11m jp^f - f#|| - 0

for every

i f D.

(3-1)

From (3.1) it follows that PJ^ - t^ and that is the unique fixed point of P. Thus, In particular, when the Probenius-Perron operator is asymptotically stable, then m^ given by (2.4) is invariant and ergodic . A convenient criterion for the asymptotical stability may be formulated by the use of lower bound functions. A function h « I 1 is called a lower bound function if Pnf »

h + £nf

for f € D

(3.2)

where the remainder £ Q f satisfies |(fnffj -9 0 as n —*ao. a lower bound function h is non-trivial if h ^ O and ||hg >0. We have the following £3} Theorem 3.1. A Markov operator P is asymptotically stable if and only if it has a nontrivial lower bound function. Observe that in applications it is not necessary to verify condition (3.2) for all possible f€ D. Sue to the fact that Markov operators are contractive (Inequality (2.3)) it is sufficient to verify (3.2) for all f belonging to a dense subset of D. Example 3.1. Consider the mapping S: to, 1] -*[0,1] described In the Example 1.1. Prom inequality (1.6) It follows that P£f(y) $

pgf (x) e ^ " yl

for x,y£ [0,1] .

Since E^f is a density there is a point y* to, 1} such that P^f(y) >, 1 and concequently p|f(x) >, e~K

for xt[0,1j ,

n >, n 0 (f) .

The last inequality shows that the constant h = e~K is a nontrivial lower bound function for p_ and that P_ is asymptotically stable.

27

The fact that In Examples 1.1 and 3.1 the transformation S maps the unit Interval £0,1] into itself is not essential. Analogous results may be obtained for expanding transformations on the real line and on manifolds (see [3] ). 4. Asymptotical decomposition Let P: L1(X) I 1 (X) be a given Markov operator. We say that P has an asymptotical decomposition if its iterates can be written in the form P°f(x) = where:

*i ( f ) g «, > < x )

+

f

nf(3c)

f

°r

fiI>1

(4

'1)

r is a fixed Integer, of is the n-th iterate of a permation oc of the sequence of numbers (1,...,r), g^-'-igj, are densities with mutually disjoint supports (gjg^-0 for ifj) and such that Pg

i

" srt(i)

for

1 = 1

»"'' r »

,..., are linear functional s on I 1 , the remainder ¿ Q f satisfies U6nf|( —> 0 a s n ^ « Observe that the summation part in (4.1) which does not exceeds r!. Thus if P has an then for every f € I 1 the sequence {P n f} i.e. (P n f) is a sum of a periodic sequence ges to zero.

is periodic in n with period asymptotical decomposition, is asymptotically periodic, and a sequence which conver-

A necessary and sufficient condition for the existence of an asymptotical decomposition may be formulated by the use of upper bound functions. A function h f L1 is called an upper bound function if P n f ^ h + £nf for f « D (4.3) where the remainder f Q f satisfies H£nf II -5> 0 as n -> eo . Again it is sufficient to verify (4.3) on a dense subset of D. Prom the results of J. Komomik [2 3 it is easy to derive the following Theorem 4.1. A Markov operator P has an asymptotical decomposition if and only if P has an upper bound function. Thus we may say shortly that fot Markov operators boundness implies periodicity. This fact has many interesting applications. In this section we will discuss two of them. Example 4.1. Consider a mapping S: [0,1] —> [0,1] which satisfies all the assumptions described in Example 1.1 except (1.4). In this case the operator P has the form

28

Pgf(x) -

IJ/^lfi^^x))!

(x)

(4.4)

1

1-1

where 1/^(x) denotes the characteristic function of the interval ^ " Si^ai-1'ai^ * I't 0811 1,6 verified L 3] that there exists a constant K, independent on on f, such that for every f t D of bounded variation Var P^f 4 K

for n >, no(f).

Since Pgf(y) i 1 for some yf[0,1] , this implies P°f(x) ^

K

+ 1

for x€[0,1],

n^nQ(f)

and shows that the constant h « K + 1 is an upper function. Thus, according to Theorem 4.1, the operator P„ has an asymptotical decomposition . u This fact was first observed by G.Keller [1J . Example 4«2. Consider the family of transformations S^ : [0,1J ->[0,1] given by the formula SA(x) = Ax(1 - x) with A € [°»4]. ?rom the resultB of M. Misiurewicz C 6 3 it follows that there exists an uncountable set /\ c. [0,4J

such that for every A 6 A

the operator P has an upper bound function and admits an asymptotical s» decomposition. 5. The Influence of noise In this section*X - Rd (d-dimensional Euclidean real space) or X = Td (d-dimensional torus). Consider a dynamical system with random perturbations given by the formula x

n+1

"

S(x

n^

+

In '

n

"

where S: X -> X is a Borel measurable transformation sequence of independent random variables with the same tion function g. We also assume that the initial value vector Independent o n { } .

and { f n } is a density distribux_ o is a random

Denote by f the density of in . It is well known that in the recu n rrence formula fn+1

= Pfn

,

n = 0,1,...

(5.2)

the operator P may be written in the form Pf(x) =

5 f(y)g(x - S(y))dy. X

(5.3)

It is difficult to verify directly that P has an upper bound function. 29

Thus In order to understand the behaviour of ll^f} we will ase another version of Theorem 4.1• Assume X « T d and denote by N the set of nonnegative integers. Consider all possible sequences I^f for f tD. This family will be shortly denoted by D • We say that the family {l^fJjj D is asymptotically uniformly integraand ble if there exist a positive valued function o: (0, w ) —s> (0, Integer valued function n Q : D N such that for every L > 0 ^ ^ f (x)dx Í £ A

whenever

m(A) ^ S(£)

and n >f nQ(f). (5.4)

From the results of J. Komomik £2 J it follows immediately Theorem 5.1. If the family ¿ ^ f } N D is asymptotically uniformly Integrable, then F has an asymptotical decomposition. In order to apply Theorem 5.1 to the operator F given by (5.3) we will use the fact that g is an integrable function. This implies the existence of a function (0, oo ) (0, CO ) such that $g(x)dx $ t A

"

"»(A) Í Sg(E)

Now consider P n f for given f € D and n have Jp^xjdx A

.

(5.5)

Setting f Q _ 1 « P n ~ 1 f we

-

$fn_., 0 for x tX, then F Is asymptotically stable.

30

References [1] Keller, G.,Stochastic stability in some chaotic dynamical systems, Monatsh. Math. 94(1982), 313 - 333. [2l Komornik, j., Asymptotic periodicity of the iterates of weakly constrictive Markov operators,TBhoku Hath. J. (In press). [3] Lasota, A. and Mackey, H. C., Probabilistic Properties of Deterministic systems, Cambridge Univ. Press, Cambridge 1985. [4] Lasota, A. and Mackey, M. C., Statistical periodicity and noise, Physica I) (to be published). [51 Li, T. Y. and Torke, J. A., Period three implie chaos, Amer. Math. Monthly 82(1975), 985-992. [6] MisiurewiCE, M., Absolutely continuous measures for certain maps of an interval, Publ. Math. IHES 53(1981), 17 - 51. [7l Sharkovsky, A.N., Coexistence of cycles of a continuous map of a line into itself, Dkr. Mat. Zh. 16(1964), 61 - 71.

31

SIHOULARLY PERTORBED AOTOHOMOUS DIFPEREHTIAL SYSTEMS Klaus R. Schneider 1 ^ 1. Introduction To motivate the following considerations we recall the famous BelousovZhabotinskii reaction [13, 153* When four or five chemical compounds (for example sodium bromate, malonic acid, sulfuric acid and ferroin) are mixed in the appropriate concentration ranges and at the appropriate temperature, the Belousov - Zhabotinskii system organizes itself into temporal dlssipative structures of macroscopio dimension: synchronous oscillations, I.e. spatially homogeneous time periodic oscillations with the same phase can be observed in a well-stirred reactor. The Belousov - Zhabotinskii reaction has played a stimulating role for the developpment of the theory of dlssipative structures and has caught the attention both of chemical engineers and of mathematicians. ?ield and Noyes Invented a kinetic model which they called the "Oregonator". It bases on five important steps of the underlying FKH-mechanism and reads C3] -

- x + 2h? -

p»-1-

u

' ,

(1.1)

* •

+4 Here we have ^ > [ H Br 02J> \ • TBrl! , z - [Ce J, 0 < £,q,p « 1. The system (1.1) is a special case of the following class of differential systems

jf - f(x,y,c,ot) , t §f - g(x,y,£ ,tx) ,

(1.2)

x e R n , j c e " , 1 hold, then the existence of a transversal homoclinic orbit of the degenerated system (1.4) implies the existence of a transversal homoclinic orbit of the system (1.2) for sufficiently small I . 35

It is well-known (Smale-Blrkhoff Homocllnlc Bieorem see C51) that the existence of a transversal homocllnlc orbit 7- Implies the existence of a hyperbolic invariant Cantor set near y such that the dynamics on J c is topologically conjugate to the shift automorphism on a closed subset of the set of bi-lnfinite sequences on a finite set of symbols (for definitions see [53, [8]). Therefore, the set J Q contains a countable set of closed orbits of arbitrarily long primitive periods, an uncountable set of bounded nonclosed orbits, a dense orbit. In case of existence of hyperbolic invariant sets one expects long chaotic transients before the orbits behave like their a -limit sets« Although, in general, we cant say that J c is an attracting set, there are examples with the property that the existence of J c is an indicator for the existence of a strange attractor for systems which are close to the considered one. In what follows we formulate conditions ensureing the existence of an invariant Cantor set J^ with the properties described above for the system (1.2). Let d(x,£,ec)

f(x,>(x,£,«c),£,