The Influence of Pests on Forest
Age Structure Dynamics: The
Simplest Mathematical Models
Antonovsky, M.Y., Clark, W.C. and Kuznetsov, Y.A.
IIASA Working Paper
WP-87-070
August 1987
Antonovsky, M.Y., Clark, W.C. and Kuznetsov, Y.A. (1987) The Influence of Pests on Forest Age Structure Dynamics: The
Simplest Mathematical Models. IIASA Working Paper. IIASA, Laxenburg, Austria, WP-87-070 Copyright © 1987 by the
author(s). http://pure.iiasa.ac.at/2982/
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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. Clark
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 the Biosphere
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 expressed herein 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
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 complementary 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 -
ABSTRACT
This paper is 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 result in intensive oscillations of i t s a g e structure. Possible implications of environmental changes on f o r e s t . ecosystems are also considered.
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.
- vii -
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
THE I N ~ C OF
E PES~S
ON mmsr AGE m
m
mDYNAYIICS:
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 Antonovsky 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-
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
= py - ( y
=z -hy,
- 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 stationary state with constant a g e classes densities (equilibrium E 2 ) from a l l initial
conditions. In region 1between 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 station-
ary 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:
I.
2
Z
= py -y(y)z
=fz -hy
= -ez + b z z ,
-12
-Azz
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
Z
(A.2)
= -&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
=#
- ( y -I)%
i = z -hy
z = -EZ + bzz ,
2
-sz
-22
(1.1)
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-
nates. As in system (0.1), the t w o equilibria El
= ( z l , y l , O ) and
E2 = ( Z ~ , Y ~ ~ O )
where
appear 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
E3 = ( b ' b h '
This equilibrium appears in
p-sh
h
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 . Equilibrium 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 "vanish" in system (1.1). These events take place on lines R and P on t h e parameter
plane, while t h e autooscillations exist in regions 5 and 6.
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-
tem. 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 stablec 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 connection 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 Balabaev 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
changes in system behavior are visible.
2. Besulta of the investigation of model (k2)
Model (A.2) can b e transformed by scaling into t h e following form:
I:
2
= py
- ( y - 112z - sz
(2.1)
y = z -hy -yz
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 four equilibrium points in the 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
=
1
~b
L
- + s b 2 b * (E
(
This equilibrium a p p e a r s in
RQ
:
2
-$
h
+sb2
I
.
R
: below t h e line
S = [ ( ~ . h:)
pb
( E - b12
+ sb2
-h = O
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 portraits 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 forest approaching an equilibrium state with a constant r a t i o of "young" and "old" trees
(z
= hy
), 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).
Models (1.1) and (2.1) have regions on t h e parameter plane (0,land 2) in
which t h e i r behavior is completely analogous to the 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
zero 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 stationa r y forest 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 forest 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 olass
e
system could cause periodical oscillations in t h e forest a g e structure
and repeated outbreaks in t h e number of pests (i.e., z,y , z / y and z become
periodic functions of time). It should be mentioned that 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 age classes and
t h e "predator" feeds only on one of them. It i s important that t h e pest invasions induce the oscillations in r a t i o z / y of t h e age classes densities. It should b e mentioned also t h a t in the case of model (2.1) t h e pest invasion oan include damping oscillations in t h e a g e structure.
When w e move on t h e parameter plane towards separatrix 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 app 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 border above which a
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 us 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 atmospheric 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 atmos 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
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 1to 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 degrades 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 explain 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.
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 natural 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 assessment of possible environmental impacts.
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).
It
i s convenient to calculate t h e curve R f o r fixed o t h e r parameter values as a pro-
r in t h e d i r e c t product of t h e parameter plane
et al.. 1985). The c u r v e r in t h e 5-dimensional space
jection on (p,h)-plane of a curve
: (Bazykin
by phase space R
with coordinates ( p , h , z , y , z ) i s determined by t h e following system of algebraic
equations:
py
- ( y - 1 1 2 2 - sz z
z =0
z -hy = O
-ez + bzz = 0
Q ( ~ ~ h , z , y ,=
z 0,
)
i
where G is a corresponding Hurwtiz determinant of t h e linearization matrix
Each point on c u r v e I' implies t h a t at parameter values (p, h ) a point ( z ,y ,z ) i s an
equilibrium point of system (1.1) (the f i r s t t h r e e equations of (8) are satisfied) with
eigenvalues Al12 = f i o (the last equation of (8) is satisfied).
One point on t h e c u r v e
r i s known. It corresponds t o point M on t h e parameter
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
(p',h',z',y',z')
lies on c u r v e
r
t e e
=(- -,1,0
)
b'b'b
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
standard EQRTRAN-program CURVE (Balabaev and Lunevskaya, 1978).
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 mnneating 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
small positive number; note t h e second intersection of
= 6 , where 6 i s a
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 separat r i x cycle formed by the saddle El12 separatrices.
For numerical computations separatrix W; w a s approximated n e a r saddle E2
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 interpolation. The stable two-dimensional manifold of El w a s approximated n e a r saddle El
by a tangent plane, and a n affine coordinate of X in t h e eigenbasis of El 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-'
w a s found through computer experiments.
A family of t h e s e p a r a t r i x cycles
p e r step. The initial point on P
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.
Antonovsky , M .Ya. and M .D. Korzukhin. 1983. Mathematical modelling of economic
and ecological-economic processes. Pages 353-358 in Integrated globd mon-
itoring of environmentd pollution. R o c . of I1 Intern. a m p . , 'Ibilisi, U S R ,
.
1Q81.Leningrad: Gidromet
Bazykin, A.D. and F.S. Berezovskaya. 1979. Alleevseffect, 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. Moscow: 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 nd 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 Russian).
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 and
l
dwerence equations through computer ezperiments. 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 Computing Center of t h e USSR Academy of Sciences (in Russian).
May, R.M. (ed.) 1981. 7heoreticta.LEcology. Principles and Applications. 2nd Edition. Oxford: Blackwell Scientific Publications.
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.
Figure 3.
The parametric portrait of system (1.1).
Figure 4 .
The phase portraits of system (1.1).
Figure 5.
The separatrix cycle in system (1.1).
8
1
I
II
i1
z
I
i
I
I
Figure 6.
x
3
I
r
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.
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,
7 )-
E
= 2,
p
= 6 , h = 3.5 (region
Figure 9.
The parametric portraits of system (2.1).
Figure 10. The probable parameter drift under SOZ increase.
Figure 11. The separatrix split function.
1
Figure 12. The separatrix cycles in system (1.1).
1
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).