The Influence of Pests on Forest Age Structure Dynamics: The Simplest Mathematical Models

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THE

INFLUENCE

OF

PESlX

ON FOR=

AGE

STRUCTURE DYNAMICS:

THE

SIMPLEST MA-TICAL MODEX3

M.Ya. A n t o n o v s k y , Yu.A. K u z n e t s o v , a n d W. C l a r k

August 1987 WP-87-70

PUBLICATION NUMBER 37 of t h e project:

Ecologically S u s t a i n a b l e m v e l o p m e n t of t h e B i o s p h e r e

Working P a p e r s a r e interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only limited review. Views o r opinions 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 those of t h e Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

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Some of t h e most exciting c u r r e n t work in t h e environmental sciences involves unprecedentedly close interplay among field observations, realistic but complex simulation models, and simplified but analytically tractable versions of a f e w basic equations. IIASA1s Environment Program i s developing such parallel and comple- mentary approaches in its analysis of t h e impact of environmental change on t h e world's forest systems. In this paper, Antonovsky, Kuznetsov and Clark provide a n elegant global analysis of t h e kinds of complex behavior latent in even t h e simplest models of multiple-aged forests, t h e i r predators, and t h e i r abiotic environment.

Subsequent papers will apply these analytical results in t h e investigation of case studies and more detailed simulation models.

I a m especially pleased t o acknowledge t h e important contribution made to t h e paper by Yuri Kuznetsov, a participant in IIASAss 1986 Young Scientist Summer Program and one of t h e 1986 Peccei award winners.

R.E. Munn Leader

Environment Program

-

ⁱⁱⁱ

-

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ABSTRACT

This paper i s devoted to t h e investigation of t h e simplest mathematical models of non-even-age f o r e s t s affected by insect pests. Two extremely simple situations are aonsidered: 1) t h e pest feeds only on young trees; 2) t h e pest feeds only on old trees. I t i s shown t h a t a n invasion of a s m a l l number of pests into a steady-state f o r e s t ecosystem aould r e s u l t in intensive oscillations of i t s a g e s t r u c t u r e . Possi- ble implications of environmental changes on f o r e s t . ecosystems are also considered.

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Software is available to allow interactive exploration of t h e models described in this paper. The software consists of plotting routines and models of t h e systems described h e r e . I t can b e r u n on a n IBM-PC/AT with t h e Enhanced Graphics Display Adapter and 256K graphics memory.

For f u r t h e r information o r copies of t h e software, contact t h e Environment Program, International Institute f o r Applied Systems Analysis, A-2361 Laxenburg, Austria.

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vii

-

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TABLE

OF

CONTENTS

Introduction

1 . Results of the investigation of model (A.1) 2. Results of the investigation of model (A.2) 3. Discussion of the results

4. Surnmar).

Appendix: Numerical procedures for the bifurcation lines R and P 1. Andronov-Hopf bifurcation line R

2. Separatrix cycle line P References

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THE I N ~ C OF E

P E S ~ S

ON

mmsr

AGE

m m m

DYNAYIICS:

THE

WTHEMATICAL MODEIS M.Ya Antonovsw, Yu.A. Kuznetsov, and W. Clark

Introduction

The influence of insect pests on t h e age structure dynamics of forest systems has not been extensively studied in mathematical ecology.

Several papers (Antonovsky and Konukhin. 1983; Konukhin, 1980) have been devoted to modelling t h e age struoture dynamics of a forest not affected by pests.

Dynamical properties of insect-forest systems under t h e assumption of age and species homogeneity can be derived from t h e theoretical works on predator-prey system dynamics (May, 1981; Bazykin, 1985). In t h e present paper w e attempt to combine these two approaches to investigate t h e simplest models of non-even-age forests affected by insect pests.

The model from Antonovsky and Konukhin (1983) seems to be t h e simplest model of a g e s t r u c t u r e dynamics of a one-species system. I t describes t h e time evolution of only two a g e classes ("young" and "old1' trees). The model has t h e following form:

where t and y are densities of "young1' and "old" trees, p i s fertility of t h e species, h and f are death and aging rates. The function y(y ) represents a dependence of "young1' trees mortality on t h e density of "old" trees. Following Antonov- sky and Konukhin (1983) w e suppose that t h e r e exists s o m e optimal value of "old1' trees density under which t h e development of "young1' trees goes on most success-

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fully. In t h i s case i t is possible to chose y ( y )

=

a ( y

-

^b12

⁺

c (Figure 1). Let s = j + C .

Model (A.0) serves as t h e basis f o r o u r analysis. Let u s t h e r e f o r e recall i t s properties. By scaling variables ( z , y ), parameters ( a ,b ,c , p , j , h , s ) and t h e time, system (A.0) c a n b e transformed into "dimensionless" form:

I

²

⁼

^{= z}^py^{- h y ,}

^-

^{( y}

^-

¹⁾²²

^-

^S2

where w e have p r e s e r v e d t h e old notations.

The parametric p o r t r a i t of system (0.1) on t h e (p,h)-plane f o r a fixed s value i s shown in Figure 2, where t h e relevant phase p o r t r a i t s are also presented.

Thus, if parameters ( p , h ) belong to region 2, system (0.1) a p p r o a c h e s a sta- tionary state with constant a g e classes densities (equilibrium E 2 ) from a l l initial conditions. In region 1 between lines Dl and D 2 t h e system demonstrates a low density threshold: a sufficient d e c r e a s e of each a g e class leads to degeneration of t h e system (equilibrium Eo). The boundary of initial densities t h a t r e s u l t in t h e degradation i s formed by s e p a r a t r i c e s of saddle El. Finally, in region 0 t h e stationary existence of t h e system becomes impossible.

Let u s now introduce a n insect pest into model (A.0). The two extremely simple situations seem to b e possible:

1 ) t h e p e s t s feed only on t h e "young" trees (undergrowth);

2) t h e p e s t s feed only on t h e "old" (adult) trees.

Assume t h a t in t h e absence of food t h e pest density exponentially declines and t h a t forest-insect interactions c a n b e described by bilinear t e r n as in t h e c a s e of predator-prey system models (e.g

.,

May, 1981; Bazykin, 1985).

Thus, f o r t h e case where t h e pest feeds on undergrowth w e obtain t h e following equations:

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I.

²

⁼

py - y ( y ) z -12 -Azz

= f z ^{- h y}

Z

=

-ez + b z z ,

while for t h e case where t h e pest feeds on adult trees

1:

^z^{= P Y}- 7 ( v ) z -12

i

= f z - h y -Ayz (A.2)

Z

=

^-&Z

+

h z .

Here z i s insect density, e i s mortality rate of insect, and terms with z z and yz r e p r e s e n t t h e insect-forest interaction.

The goal of t h i s p a p e r i s t h e comparative analysis of m o d e l s (A.O), (A.1) and (A.2). In t h e final p a r t of t h e p a p e r w e consider biological implications of t h e ob- tained r e s u l t s and outline possible directions f o r elaborating the model. The main tools f o r o u r investigation are t h e bifurcation theory of d y n m i c a l systems and t h e numerical methods of t h i s theory.

1. M t . of the investigation of model (kl)

By a linear change of variables, parameters and time t h e system (A.1) c a n b e transformed into t h e form:

I

²

^{= #}

- ( y -I)% -sz - 2 2

i

= z - h y (1.1)

z

=

^-EZ

+

^bzz,

where t h e previous notations are p r e s e r v e d f o r new variables and parameters which have t h e same sense as in system (0.1).

In t h e f i r s t o c t a n t

system (1.1) can have from one to f o u r equilibria. The origin E o

=

(0,0,0) is always an equilibrium point. On t h e invariant plane z

=

0 at which t h e system coincides with system (0.1) t h e r e may exist e i t h e r one o r two equilibria with nonzero coordi-

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nates. As in system ( 0 . 1 ) , t h e t w o equilibria El

=

( z l , y l , O ) and E 2

=

( Z ~ , Y ~ ~ O ) where

a p p e a r in system ( 1 . 1 ) on t h e line:

On t h e line

equilibrium El coalesces with equilibrium Eo and disappears f r o m

: . R

Besides t h e equilibria E, , j =0,1,2, system ( 1 . 1 ) could have an additional equilibrium

c c p - s h E3

= ( -

b ' b h '

-

h

This equilibrium a p p e a r s in

B :

to t h e right of t h e line:

passing through t h e plane z =O and coalescing on this plane with e i t h e r equilibrium El or E 2 . Line S i s tangent to line Dl at point

and lies under it. Line S i s divided by point M into t w o parts: S 1 and S 2 . Equilibri- um E3 collides on S l with E l and on S 2 with E2.

The parametric p o r t r a i t of system ( 1 . 1 ) i s shown in Figure 3, while t h e corresponding phase p o r t r a i t s are presented in Figure 4. In addition to t h e described bifurcations of t h e equilibria, autooscillations can "emerge" and "van- ish" in system (1.1). These events t a k e place on lines R and P on t h e parameter plane, while t h e autooscillations exist in regions 5 and 6.

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Equilibrium E g loses i t s stability on line R due to t h e transition of two complex conjugated eigenvalues from t h e left to t h e right half-plane of t h e complex plane. This stability change results in t h e a p p e a r a n c e of a stable limit cycle in system (1.1) (Andronov-Hopf bifurcation).

There i s also a line corresponding to destruation of t h e limit cyales: line P on t h e (p,h)-plane. On line P a s e p a r a t r i x cycle formed by outgoing s e p a r a t r i c e s of saddles E l and E 2 does exist (Figure 5). While moving to t h e s e p a r a t r i x line t h e period of t h e cycle inareases to infinity and at t h e c r i t i c a l p a r a m e t e r value i t coalesces with t h e s e p a r a t r i x cycle and disappears.

The point M plays a key role in t h e parametric plane. This point i s a common point f o r all bifurcation lines: S 1 , S 2 , D l P 2 , R and P. I t corresponds to t h e existence of a n equilibrium with two z e r o eigenvalues in t h e phase s p a c e of t h e sys- t e m . This f a c t allows us to p r e d i c t t h e existence of lines R and P.

For parameter values close to t h e point M t h e r e is a two-dimensional stable- c e n t e r manifold in t h e phase s p a c e of system (1.1) on which all essential bifurcations t a k e place. The c e n t e r manifold i n t e r s e c t s with invariant plane z =O along a curve. Thus w e have a dynamical system on t h e two-dimensional manifold with t h e structurally unstable equilibrium with two z e r o eigenvalues and t h e invariant curve. This bifurcation h a s been treated in general form by Gavrilov (1978) in con- nection with a n o t h e r problem. I t w a s shown t h a t t h e only lines originating in point M are t h e mentioned bifurcation lines.

The locations of t h e R and P lines were found numerically on a n IBM-PC/XT compatible aomputer with t h e help of standard programs f o r computation of c u r v e s developed in Research Computing Center of t h e USSR Academy of Sciences by Bala-

baev and Lunevskaya (1978). Corresponding numeriaal procedures are described in t h e Appendix. W e have also used a n interactive program f o r t h e integration of ordinary differential equations

-

PHASER (Kocak, 1986). On Figures 6, 7, and 8 t h e

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changes in system behavior are visible.

2. Besulta of the investigation of model ( k 2 )

Model ( A . 2 ) can b e transformed by scaling into t h e following form:

I:

²^y

⁼

^{= z}^py^{- h y}

^-

^{( y}^{- y z}

^-

^112z

^-

^sz ( 2 . 1 )

2

=

^-LZ

+

b z *

where t h e meaning of variables and parameters is t h e same as in system ( 1 . 1 ) . System ( 2 . 1 ) can have from one to f o u r equilibrium points in t h e f i r s t octant

BQ

: E,,

=

( 0 , 0 , 0 ) , E l

=

( z l , y l , O ) , E 2

=

( z 2 , y 2 , 0 ) and E 3

=

( z 3 , p 3 , z 3 ) . Equilibria E l and E 2 on t h e invariant plane z

=

0 have t h e same coordinates as in system ( 1 . 1 ) ; they also bifurcate t h e same manner on lines Dl and D2. AS in system ( 1 . 1 ) t h e r e is a n equilibrium point of system ( 2 . 1 ) in

RQ

:

~ b

-

^L ² h

I ^.

= 1

⁽ ^- + s b 2 b * ( E

-$

^{+ s b 2}

This equilibrium a p p e a r s in

R :

below t h e line

S = [ ( ~ . h ) : pb -h = O

.

( E

-

b12

+

s b 2

1

But equilibrium E 3 does not lose its stability. Autooscillations in system ( 2 . 1 ) are t h e r e f o r e not possible. That i s why t h e parametric p o r t r a i t s of system ( 2 . 1 ) look Hke Figure 9. Numbers of t h e regions in Figure 9 correspond to Figure 4.

3. Dimcussion of the resalt.

The basic model ( 0 . 1 ) with t w o a g e classes describes e i t h e r a f o r e s t approach- ing an equilibrium state with a constant r a t i o of "young" and "old" trees

( z

=

h y ), o r t h e complete degradation of t h e ecosystem (and presumably, re- placement by t h e o t h e r species).

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Models (1.1) and (2.1) have regions on t h e parameter plane (0,l and 2) in which t h e i r behavior is completely analogous to t h e behavior of system (0.1). In these regions t h e system e i t h e r degenerates or tends to t h e stationary state with z e r o pest density. In this case t h e pest is "poorly adapted" to t h e tree species and oan not survive in t h e ecosystem.

In systems (1.1) and (2.1) t h e r e are also regions (4 and 3) where t h e station- a r y f o r e s t state with z e r o pest density exists, but i s not stable to s m a l l pest "invasions". After a small invasion of pests, t h e ecosystem approaches a new stationary state with nonzero pest density. The pest survives in t h e f o r e s t ecosystem.

The main qualitative difference in t h e behavior of models (1.1) and (2.1) i s in t h e existence of density oscillations in t h e f i r s t system but not in t h e second one.

This means t h a t a small invasion of pests adapted to feeding upon young trees in a t w ~ g e olass system could cause periodical oscillations in t h e f o r e s t a g e structure and r e p e a t e d outbreaks in t h e number of pests (i.e., z,y , z / y and z become periodic functions of time). I t should b e mentioned t h a t t h e existence of such oscillations is usual f o r simple, even-aged predator-prey systems.

In our case, however, t h e "prey" i s divided into interacting a g e classes and t h e "predator" feeds only on one of them. It i s important t h a t t h e pest invasions induce t h e oscillations in r a t i o z / y of t h e age classes densities. I t should b e mentioned also t h a t in t h e case of model (2.1) t h e pest invasion oan include damping os- cillations in t h e a g e structure.

When w e move on t h e parameter plane towards s e p a r a t r i x cycle line P , t h e amplitude of t h e oscillations increases and t h e i r period tends to infinity. The oscillations develop a strong relaxation c h a r a c t e r with intervals of s l o w and rapid variable change. For example, in t h e dynamios of t h e pest density z ( t ) t h e r e ap- p e a r periodic long intervals of almost z e r o density followed by rapid density outbreaks. Line P is a boundary of oscillation existence and a b o r d e r above which a

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small invasion of pests leads to complete degradation of t h e system. In regions 7 and 8 a small addition of insects to a forest system, which was in equilibrium without pests, results in a pest outbreak and then tree and pest death.

It can be seen t h a t t h e introduction of pests feeding only upon t h e "young"

trees dramatically reduces t h e region of stable ecosystem existence. The existence becomes impossible in regions 7 and &

W e have considered t h e main dynamical regimes possible in models (1.1) and (2.1). Before proceeding, however, let u s disauss a very important topic of time scufes of t h e processes under investigation. It is w e l l known t h a t insect pest dynamics reflect a much more rapid proaess than the response in tree density. It seems t h a t this difference in t h e time scales should be modeled by introduction of a s m a l l parameter p<U into t h e equations f o r pest density in systems (1.1) and (2.1):

2

+b.

But it can be shown t h a t t h e parametric p o r t r a i t s of t h e systems are robust to this modification. The relative positions of lines D1,D2 and S as w e l l as t h e coordinates of t h e key point M depend on r a t i o E / b which is invariant under substitutions E + B / p, b +b / IL. The topology of t h e phase p o r t r a i t s is not affected by introduction of a s m a l l parameter p, but in t h e variable dynamics t h e r e a p p e a r intervals of slow and rapid motions. Recall that in model (1.1) t h e similar relaxation c h a r a c t e r of oscillations w a s demonstrated n e a r line P of separatrix cycle without additional s m a l l parameter IL. So w e could say t h a t w e have an "implicit small parameter" in system (1.1).

To demonstrate t h e potential f o r extensions of this approach, let us now consider t h e qualitative implications of imposing on model (1.1) a n effect of a t m o s - pheric changes on t h e forest ecosystems. A s it w a s suggested in Antonovsky and Korzukhin (1983), a n increase in t h e amount of SO2 or o t h e r pollutants in t h e a t m o - s p h e r e could lead to a dearease of t h e growth rate p and a n increase of t h e mortality rate A . Thus, a n increase of pollution could result in a slow d r i f t along some

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curve on t h e (p,h)-plane (Figure 10).

Suppose t h a t parametric condition has been moved f r o m position 1 to position 2 on t h e plane but remains within t h e region where a stable equilibrium existence without pests is possible. But if t h e system i s exposed to invasions of t h e pest i t de- grades on line P. Therefore, slow atmospheric changes could induce vulnerability of t h e forest to pests, and forest death unexpected from t h e point of view of t h e forest's internal properties.

4. Snarnrrry

I t is obvious t h a t both models (A.l) and (A.2) are extremely schematic.

Nevertheless, they s e e m to be among t h e simplest models allowing t h e complete qualitative analysis of a system in which t h e predator differentially attacks various age classes of t h e prey.

The main qualitative implications from t h e present paper can be formulated in t h e following, to s a m e extent metaphorical, form:

1. The pest feeding t h e young trees destabilizes t h e forest ecosystem more than a pest feeding upon old trees. Based upon this implication, w e could t r y to ex- plain t h e well-known fact that in real ecosystems pests more frequently feed upon old trees than on young trees. It seems possible t h a t systems in which t h e pest feeds on young trees may be less stable and more vulnerable to external impacts than systems with t h e pest feeding on old trees. Perhaps this has led to t h e elimination of such systems by evolution.

2. An invasion of a s m a l l number of pests into an existing stationary forest eaosystem could result in intensive oscillations of its a g e structure.

3. The oscillations could be e i t h e r damping o r periodia.

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4. Slow changes of environmental parameters are able to induce a vulnerability of t h e forest to previously unimportant pests.

L e t us now outline possible directions f o r extending t h e model. It seems natur- al to take into account t h e following factors:

1) more than t w o age classes f o r the specified trees;

2) coexistence of more than one tree species affected by t h e pest;

3) introduction of more than one pest species having various interspecies relations;

4) the r o l e of variables like foliage area which a r e important f o r t h e description of defoliation effeot of t h e pest;

5) feedback relations between vegetation, landscape and microclimate.

Finally, w e express our belief that oareful analysis of simple nonlinear ecosystem models with t h e help of modern analytical and computer methods will lead to a b e t t e r understanding of r e a l ecosystem dynamics and to b e t t e r assess- ment of possible environmental impacts.

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Appendix: Numerical procedures for the bifurcation linemR and P

1. Andronov-Hopf bifurcation line R

.

On t h e (p,h)-plane t h e r e i s a bifurcation line R along which system (1.1) h a s a n equilibrium with a p a i r of purely imaginary eigenvalues All,

=

*i o (A,

<

0). I t i s convenient to calculate t h e curve R f o r fixed o t h e r parameter values as a p r o - jection on (p,h)-plane of a curve

r

in t h e d i r e c t p r o d u c t of t h e p a r a m e t e r plane by p h a s e space

R :

(Bazykin et al.. 1985). The c u r v e

r

in t h e 5-dimensional space with coordinates ( p , h , z , y , z ) i s determined by t h e following system of a l g e b r a i c equations:

py

-

^{( y}

-

^{1 1 2 2}

-

sz z z

=

0 z - h y = O

-ez

+

bzz

=

0 Q ( ~ ~ h , z , y , z )

=

0,

i

where G i s a corresponding Hurwtiz determinant of t h e linearization matrix

Each point o n c u r v e I' implies t h a t at p a r a m e t e r values (p, h ) a point ( z , y , z ) i s an equilibrium point of system (1.1) ( t h e f i r s t t h r e e equations of (8) are satisfied) with eigenvalues Al12

=

f i o ( t h e last equation of (8) i s satisfied).

One point on t h e c u r v e

r

i s known. I t c o r r e s p o n d s t o point M on t h e p a r a m e t e r plane at which system (1.1) h a s t h e equilibrium ( f . l . ~ ) with XI

=

A2

=

0 (9.g..

b

*i o

=

0). Thus, t h e point

t e e ( p ' , h ' , z ' , y ' , z ' )

=

(

-

b ' b ' b

-

^-,1,0 ⁾

lies on c u r v e

r

and c a n b e used as a beginning point f o r computations. The point- by-point computation of t h e c u r v e was done by Newton's method with t h e help of a s t a n d a r d EQRTRAN-program CURVE (Balabaev and Lunevskaya, 1978).

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2. Separatrix cycle line P

.

Bifurcation line P on t h e parameter plane w a s also aomputed with t h e help of program CURVE as a aurve where a "split" function F f o r t h e separatrix mnneat- ing saddles E2,1 vanishes:

F @ , h )

=

0.

For fixed parameter values this function can be defined following Kuznetsov (1983). Let

w2+

be t h e outgoing separatrix of saddle E 2 (the one-dimensional unstable manifold of equilibrium E2 in

R?).

Consider a plane z

=

6 , where 6 i s a small positive number; note t h e second intersection of

w2+

with this plane (Figure 11). Let t h e point of intersection be X . The two-dimensional stable manifold of sad- dle El interseats with plane z

=

6 along a curve. The distance between this curve and point X , measured in t h e direction of a tangent vector to t h e unstable manifold of E l , could be taken as t h e value of F f o r given parameter values. This funation i s w e l l defined n e a r its zero value and its vanishing implies t h e existence of a separa- t r i x cycle formed by the saddle El12 separatrices.

For numerical computations separatrix W; w a s approximated n e a r saddle E 2 by its eigenvector corresponding to X 1

>

0. The global p a r t of W $ w a s defined by t h e Runge-Kutta numerical method. Point X w a s calculated by a linear interpola- tion. The stable two-dimensional manifold of El w a s approximated n e a r saddle E l by a tangent plane, and a n affine coordinate of X in t h e eigenbasis of E l w a s taken f o r t h e value of split function F.

The initial point on t h e separatrix has z o

=

0.005. The plane z

=

6 was defined by 6

=

0.1 and t h e integration accuracy w a s

lo-'

p e r step. The initial point on P w a s found through computer experiments. A family of t h e s e p a r a t r i x cycles corresponding to points on curve P i s shown in Figure 12.

Figure 13 presents a n actual parametria portrait of sysbn (1.1) f o r s = b = l , t = 2 .

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Antonovsky ,

M

.Ya. and

M

.D. Korzukhin. 1983. Mathematical modelling of economic and ecological-economic processes. Pages 353-358 in

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1Q81. Leningrad: Gidromet

.

Bazykin, A.D. and F.S. Berezovskaya. 1979. Alleevs effect, low critical population density and dynamics of predator-prey system. Pages 161-175 in

Roblems of ecological monitoring and ecosystem modelling. u.2.

Leningrad: Gidromet (in Russian).

Bazykin, A.D. 1985.

Mathematicd Biophystcs of Interacting Populations.

Mos- cow: Nauka (in Russian).

Bazykin, A.D., Yu.A. Kuznetsov and A.I. Khibnlk. 1985.

m r c a t i o n diagrams of planar d y n a m i c d systems.

Research Computing Center of t h e USSR Academy of Sciences, Pushchino, Moscow region (in Russian).

Balabaev, N.K. and L.V. Lunevskaya. 1978.

Computation

@

a

cum

in

n-

d i m e n s i o n d space. FORTRRN Sonware Series, i.2.

Research Computing Center of t h e USSR Academy of Sciences, Pushchino, Moscow region (in Rus- sian).

Gavrilov, N.K. 1978. On bifurcations of a n equilibrium with one z e r o and p a i r of p u r e imaginary eigenvalues. Pages 33-40 in

Methods of q u d i t a t i u e theory of d w e r e n t t d equations.

Gorkii: State University (in Russian).

Kocak, H. 1986.

~ e r s n t i a r l and d w e r e n c e equations through computer ezper- iments.

New York: Springer-Verlag.

Konukhin, M.D. 1980. Age s t r u c t u r e dynamics of high edification ability tree population. Pages 162-178 in

Problems of ecologicd monitoring and ecosystem modelling, u.3.

(in Russian).

Kuznetsov, Yu. A. 1983.

Orre-dimensional i n v a r i a n t m a n t p l d s of ODE-systems depending u p o n parameters. FORTRAN Sonware Series,

i.8. Research Com- puting Center of t h e USSR Academy of Sciences (in Russian).

May, R.M. (ed.) 1981.

7heoreticta.L Ecology. Principles and Applications.

2nd Ed- ition. Oxford: Blackwell Scientific Publications.

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Figure 1. The dependence d "young" tree mortality on the density of "old" trees.

Figure 2. The parametric portrait of system (0.1) and relevant phase portraits.

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Figure 3. The parametric portrait of system (1.1).

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Figure 4 . The phase portraits of system (1.1).

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Figure 5. The separatrix c y c l e in system (1.1).

8

I

1 I

z i

¹

I

^I

I

i

3 ^I

x r

I

Figure 6. The behavior of system (1.1): s

=

b = 1, E

=

2, p

=

6 , h = 2 (region 3). The Y-axis extends vertically upward from the paper.

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Figure 7. The behavior of system (1.1): s = b = 1, c = 2, p = 6 , h = 3 (region 8)

-

Figure 8. The behavior of system (1.1): s = b = 1 , ^E = 2, p = 6 , h = 3.5 (region 7 )

-

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Figure 9. The parametric portraits of system (2.1).

Figure 10. The probable parameter drift under SOZ increase.

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Figure 11 . The separatrix split function.

1 1

Figure 12. The separatrix cycles in system (1.1).

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B I F U R C R T I O N CURVESs S = B = l E = 2

Figure 13. A computed parametric portrait of system (1.1).