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THIS DISSERTATION
EXACTLY AS 9ECEIVED
M I C R ~ F I L M ~TELLE
E
QUE
NOUS L'AVONS REWE.
QUALITATIVE AND GUANTITATIVE ANALYSIS OF PERTURBED
LOTKA-VOLTERRA COMPETITION MODELS
by
*
Hyun-sook Kim
B.Sc., Simon Fraser University, 1975
4
THE hYQUIREAEXVTS FOR THE DEGREE OF
MASTER OF SCIENCE
in the Department
Mathematics and Statistics
SIMON FRASER UNIVERSITY
h
,
'", 4
4
April B 9 E
$.
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oT the author.
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ISBN
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'
'APPROVAL
Name :
Hyun-sook Kim
Degree :
Master of Scienee
Title of Thesis:
Qualitative and quantitative analysis of perturbed
Lotka-Volterra competition models.
Examining Committee:
Chairman :
Dr. A. R. Frec&nan
~r.(,k. Bojadziev
Senior Supervisor
,'
Dr. M. Singh
Dr. C.Y. Shen
-
.-
Ds. D. sharma
External Examiner
-
-
PARTIAL COPYRIGHT L ICENSE
'
1 h s r e b y xbrant t o Simon F r a s e r U n i v e r s I t y t h e r i g h t t o lend
my t h e s i s ,
p r o j e c t 4 3 r e x t e n d e d essay ( t h e t i t l e o f w h i c h i s shown b e l o w )
'to- u s e r s o f t h e Simon F r a s e r U n i v e r s i t y L i b r a r y , and t o make p a r t i a l o r
s i ng l e c o p i e s on 1 y f o r such u s e r s o r i n -response t o a r e q u e s t f r o m t h e
l i b r a r y o f any o t h e r u n i v e r s i t y , o r o t h e r e d u c a t i o n a l
i t s own b e h a l f o r f o r one o f i t s u s e r s .
i n s t i t u t i o n , on
I f u r t h e r agree t h a t permission
f o r m u l t i p l e c o p y i n g o f t h i s work f o r s c h o l a r l y purposes may be g r a n t e d
by me o r t h e Dean o f G r a d u a t e S t u d i e s .
I t i s understood t h a t copying
o r p u b i i c a t i o n o f t h i s work f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d
w i t h o u t my w r i t t e n p e r m i s s i o n .
T i t l e o f T h e s i s / P r o j e c t / E x t e n d e d Essay
Author:
(sYgnatu;e)
( name
ABSTRACT
h.
P
,
-
4-
b.
The Lotka-Volterra competition model>w.i t h c o n s t a n t r a t e
harvesting,
H
i s i n v e s t i g a t e d b o t h q u a l i t a t i v e l y and q u a n t i t a t i v e l y ,
i n . t h e neighbourhood o f t h e e q u i l i b r i a when small p e r t u r b a t i o n a l
terms o f t h e form
€fi (N1,~Z;
,i
= 1,2
, are
a p p l i e d t o t h e system.
-3
The e f f e c t of c o n s t a n t r a t e h a r v e s t i n g on t h e unperturbed system i s
analyzed.
For- a c e r t a i n v a l u e of
H
, ,the
unperturbed system h a s a
s t r u c t u r a l l y u n s t a b l e double e q u i l i b r i u m ' p o i n t (saddle-node),,
Lf-Y
t h e Jacobian i s z e r o which under small p e r t u r b a t i o n s b i f u r c a t e s
i n t o two s t r u c t u r a l l y s t a b l e simple e q u i l i b r i #,'of t h e p e r t u r b e d system.
P
AS an i l l u s t r a t i o n a numerical example is g i d n .
~ l s o ,a p a r t i c u l a r
c a s e o f t h e p e r t u r b e d system i s i c v e s t i g a t e d from t h e p o i n t o f view
of c o n t r o l .
Asymptotic s o l u t i o n s o f t h e p e r t u r b e d system w i t h no
h a r v e s t i n g is o b t a i n e d by u s i n g modified Krylov-Bogoliubov-Mitropolskii
method.
(iii)
DEDI CAT1ON
To my f a m i l y
I wish t o express my deepest g r a t i t u d e t o
3
Professor George N.
f o r giving me t h e
Bojadziev, my s e n i o r
t h i s study
i n i t i a l i n s p i r a t i o n and continued
would not have been p o s s i b l e .
\
J
I wish t o thank M.A.
S a t t a r and
Professor C.Y. Shen f o r t h e i r valuable crhIrents and suggestions.
1
I a l s o wish t o thank D r . Sharmal Bose, Chong-suh Chun, Richard Misiurka,
W i l l i a m McCuaig, Robert Townsend, P a t r i c k Vaugrante, Jeannie and
Kang Lee, C h r i s t i n e and S i e g f r i e d Schf f frnachgr and I3eS1'Carlson who
."
gave m e w a r m and u n f a i l i n g
moral support.
My s p e c i a l thanks goes t o
my l a t e f a t h e r Key-Dal K i m , my m t h e r Sung-Jeung Lee, my s i s t e r s :
Moon-Hye, Yeon-Koo,
my uncles:
existence
Dr.
ye-young;
my b r o t h e r s : Hwan-Koo, Hyung-koo;
Key-Hiuk K i m , D r . Ki-Hwan K i m ; who a r e my very
.
ina ally,
I wish t o thank Professor . ~ e c i lGraham,
'
Dep-ntal
C-
Chairman, P r o f e s s o r Choo-whan K i m , M s . Kathy Hammes and t h e rest of
t h e f a c u l t y and s t a f f members o f t h e Department o f Mathematics and
S t a t i s t i c s , S i m n F r a s e r M i y r s i t y , f o r t h e i r h e l p and kindness.
I
f
4
,
a l s o warmly thank Mrs. S y l v i a Holmss f o r h e r e x c e l l e n t t y p i n g of t h e
manuscript.
--
/
,f---\.
i
.x<
2%
TABLE O F CONTENTS
<
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii)
. . .,L. ;. . . . . . . . . . . . . . . . . . . . . . . . . (iii)
5
Abstract
+. ,
,\
\
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .( i v )
~cknowledgement
. . . . . . . . . . . . . . . . . . . . . . . . . . . (v)
Tableofcontents . . . . . . . . . . . . . . . . . . . . . . . . .(vi)
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . ( v i i )
Dedication
4
i
,CKAPTER 1.
c=+
.
1
1 .
Asymptotic method o f Krylov-Bogoliubov-Mitropolskii
II
......1
\
. . . . . . . . . . . .6
92.
Qualitat.lve methods of dynamic systems
•˜ 3 .
Survey of Lotka-Volterra competition models
.........
.16
2
CHAPTER 2.
(
. . . . . .19
. . . . . . . . .2 6
1 .
Lotka-Volterra competition model without h a r v e s t i n g
•˜ 2.
Harvesting o f Lotka-Volterra cornpeti t i o n model. -
CHAPTER 3 .
5 1. The p e r t u r b e d Lotka-Volterra c m p e t i t i o n model without
. . . . . . . . . . . . . . . . .-. . . . . . . . . -36
c o n t r o l l e d Loma-Volterra corns t i t i o n model . . . . . . . . . 4 2
harvesting.
5 2.
The
33.
The perturbed Lotka-Volterra competition model with
C t
harvesting.
.
. . . . . . . . . . . . . . . . . . . . . . . . . .44
Asymptotic s o l u t i o n s of t h e perturbed Lotka-Volterra
competition model
52.
.......................
.57
. . . . . . . . . . . .67
. . . . . . . . . . . . . . . . . . . ,. . . . . . . . . . . . .68
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
c is cuss ion o f t h e asymptotic s o l u t i o n s .
Conclusion
,
,
.,.
References
s
INTRODUCTION
The e f f e c t of small perturbations on models of the type
di' =
N.,F. (N
1
1.1
1 2
popul&ions
N1
i =. 1,2, describing the behaviour of two i n t e r a c t i n g
and
N
,
matheqtical
2
, has
.been a topic of continuing inte-rest i n
ecology.
The purpose of t h i s t h e s i s i s t o study q u a l i t a t i v e l y the
e f f e c t of small perturbations on Lotka-Volterra competition models.
The q u a l i t a t i v e behaviour of the Lotka-Volterra competition m d e l
i
with harvesting i s compared t o t h a t of the nodel without harvesting.
We a l s o intend t o study the b i f u r c a t i o n s i n t h e i r c r i t i c a l cases f o r
:
he unperturbed and perturbed ~ o t k a - ~ o l t e r rcofipetition
a
model with
harvesting.
-
A p a r t i c u l a r case of t h e perturbed Lotka-Volterra
competition model without harvesting i s i n v e s t i g a t e d from the point of
view of control.
Also, t h e a i m of t h i s work i s t o study the asymptotic
behaviour of qolutions of t h e Lotka-Volterra competit$onLmodel without
The c c n t r i b u t i o n s of t h i s t h e s i s s t a r t from the second p a r t
of s e c t i o n 2,. Chapter 2.
chapter 1'reviews some well-known material of both
-.
q u a n t i t a t i v e and q u a l i t a t i v e nature of d i f f e r e n t i a l equations which
w i l l be used
Krylov-Bogoli
thesis.
-*
The asymptotic method based on works of
equations i s reviewed i n s e c t i o n 1.
In s e c t i o n 2 , we o u t l i n e t h e
c l a s s i c a l methods f o r i n v e s t i g a t i n g the q u a l i t a t i v e behaviour of
1
e q u i l i b r i a of a given dynami~system.
Section 3 gives a b r i e f
survey o f works done on Lotka-Volterra c o m p e t i t i ~ nmodels.
Chapter 2 c a r r i e s out t h e s t a b i l i t y a n a $ p i s of e q u i l i b r i p
of t h e unperturbed Lotka-Volterra competition model with constant
rate harvesting,
H
, in
t h e c r i t i c a l and u n c r i t i c a l cases.
The
e f f e c t of constant r a t e h a r v e s t i n g i s e*ned.
Chapter 3 c o n s i s t s of t h r e e s e c t i o n s , dealing primarily with
t h e i n v e s t i g a t i o n of q u a l i t a t i v e behaviour i n t h e neighbourhood of the
z.
-
perturbed e q u i l i b r i a of Lotka-Volterra
r a t e harvesting.
competition model with constant
The influence of small p e r t u r b a t i o n s i s analyzed i n
/=
I
I
both c r i t i c a l and u n c r i t i c a l c a s e s , and a numerical example i s given in
t h e c r i t i c a l case.
A p a r t i c u l a r case of the perturbed system
i n v e s t i g a t e d ' f r o m t h e p o i n t of view of control.
is
Then the cornpari
OG
is
b
made to t h a t of the case without c o n t r o l .
Chapter 4 i s devoted t o t h e study of t h e asymptotic s o l u t i o n s
of the p e r t u b e d L o t k a - V d t e r r a compe ition mdel .without harvesting.
h
F i n a l l y , T?h conclusion i s given.
(viii)
-
-1
ASYMFTOTIC METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKII
.
The a s y m p t o t i c method o f Krylov-Bogoliuboy-Mitropolskii [ l , 2 1
i s one o f t h e widely used methods f o r o b t a i n i n g a n a l y t i c s o l u t i o n s o f
nonlinear equations with s m a l l n m l i n s a r i t i e s ,
31
The method 1.2,
which
was developed o r i g i n a l l y f o r systems w i t h p e r i o d i c s o l u t i o n s w a s l a t e r
e x t e n d e d by Popov [ 4 ] , f o r n o n l i n e a r damped o s c i l l a t o r y systems.
The a s y m p t o t i c s o l u t i o n s o f d i f f e r e n t i a l e q u a t i o n s a r e o b t a i n e d
a s power s e r i e s i n a s m a l l p a r a m e t e r
E
.
The s e r i e s themselves a r e .not
convergent ,. b u t f o r a f i x e d number of t e r m s , t h e approximate s o l u t i o n
-
A
t e n d s t o t h e e x a c t s o l u t i o n a s t h e small p a r a m e t e r
-
tends t o zero.
ens in t h e form of a s y w t o t i c
Approximate s o l u t i o n s o f d i f f e r e n t i a l equa
9
E
,\
s e r i e s were f i r s t p r e s e n t e d by ~ i o u v i l l eE51.
D
Here w e . p r e s e n t some m o d i f i c a t i o n s o f Krylov-BogoliubovM i t r o p o l s k i i (K-B-M)
method given by Murty [ 6 ] and B o j a d z i e v and
-\
Edwards [7f.
Consider a second-order n o n l i n e a r d i f f e f e n t i a l e q u a t i o n
where
E
function.
is a s m a l l p a r a m e t e r and
Let
-AlI
-A
2
f(x,
dx
x)
is a given n o n l i n e a r
b e t h e two r o o t s o f t h e c h a r a c t e r i s t i c
e p a t i o n when
E
0, so t h a t
E =
ZK
1
=
h + h
1
2
(1.19
= 0,the solution of equation
and
K
11
.
= Xlh2
When
-
i s given by
A
where
and
A
a r e a r b i t r a r y c o n s t a n t s t o b e determined from the
B
[x(0),
$1
1. When E # 0 , we
t=0
assume a s o l u t i o n a c c o r d i n g t o t h e a s y m p t o t i c method i n t h e form
. ' g i v e ~ ~ e to f i n i t i a l c o n d i t i o n s
h
7
where
a
UJ
and
a e d e f i n e d by
d
-da
=
-
with
2K
2
=
dt
h - h2
1
-
K a
and
1
+
EA
ul,
L-
(a)
u2,
+
...
E 2 A2(a)
+
..i
.
are functions of
(1.4)
a
and,
to
b e determined.
-
T h e o r e t i c a l l y , t h e s o l u t i o n can be o b t a i n e d up t o t h e accuracy
-
o f any o r d e r o f approximation.
However, owing t o t h e , r a p i d l y growing
a l g e b r a i c complexity f o r t h e d e r i v a t i o n o f t h e formulae, t h e solution
i s i n general, confined t o t h e f i r s t order.
I n o r d e r t o determine t h e s o l u t i o n s o f e q u a t i o n (1.1),w e
f i r s t d i f f e r e n t i a t e t h e assumed s o l u t i o n i n e q u a t i o n (1.3) t w i c e w i t h
respect t o
t and s u b s t i t u t e t h e s e e x . r e s s i o n s i n t o " t h e o r i g i n a l
e q u a t i o n (1.1) u s i n g the r e l a t i o n s i n e q u a t i o n s (1.4) and (1.5)
(i. e.
, we
have t o s u b s t i t u t e e x r e s s i o n s f o r
5'
2
d a/dt
2
, da/dt,
2 2
d $/dt
,
4
d$/dt,
etc.,
a b t a i n e d from
f o r the derivatives
.of
E
on (1.4) and (1.5)
2
2
d x / d t , d x/dt 1 .
, in
t h e equations
C o e f f i c i e n t s o f e q u a l powers
a r e then compared i n o r d e r t o o h t a i n e q u a t i o n s f o r t h e unknown
For t h e f i r s t approximation we compare p w e r s o f
E
and
o b t a i n t h e following :
W e now assume t h a t t h e right-hand s i d e of (1.6) can be expanded
i n powers of
e'
and
e
-'
with
03
f =
2
r=o
[h (alerq
r
+
g r ( a ) e -r$]
,=-
.-'
f
'-.
~ u b s t i t u t i r &t h e expression f o r
%
i
f
in
(1.6)
, we - c b t a i n .the
e q u a t i o n s •’&,omcomparing c o e f f i c i e n t s of e q u a l powers o f
e
trJl
following
, e-
e
a d
(~zJJ
d
- -1
%
da
2
-
where
(where
hl,
,-
(1.7)
-
glr hr
r > l)
e9
+ K2 (A1 + dBl) - h ( a ) f o r
da
1
and
g'
are the coefficients of
X
and
e
e -+ r'4J
r e s p e c t i v e l y , o b t a i n e d i n t h e T a y l o r ' s s e r i e s expansion
of t h e given n o n l i n e a r f u n c t l m .
Let us n o t e t h a t t h e zxpansion o f t h e right-hand s i d e of (1.6)
is certainly valid i f
f
is
F.
polynomial i n
x
and
dx/dt.
A f t e r some a l g e b r a i c manipulation, e q u a t i o n
t o Euler-type e q u a t i o n s whose p a r t i c u l a r s o l u t i o n s give
( 1 . 6 ) reduce
A
1
and - B1
.
and g;
'k-&a&
of the b . s n quantities
hl
values of
A
and
1
B
1
.
Substitution of these
i n e q u a t i o n s (1.4) and (1.5) and i n t e g r a t i n g
g i v e s t h e f i r s t o r d c r approximation terms i n t h e assumed s o l u t i o n o f
equation (1.3).
We now d e t e r m i n e , from e q u a t i o n ( 1 . 9 ) , t h e f i r s t o r d e r
ul(a,$)
c o r r e c t i o n term
where f h e c o e f f i c i e n t s
unknown.
or
e
-9
The f u n c t i o n
, which
and
'r
'0'
is assumed i n t h e form of t h e s e r i e s ,
on t h e r i g h t - h a n d s i d e a r e y e t
Dr
u ( a r $ ) does n o t c o n t a i n t e r m s i n v o l v i n g
1
s i n c e t h e s e a r e i n c l u d e d i n t h e f i r s t two t e r m s i n t h e r i g h t -
hand s i d e o f
e q u a t i o n (1.3)
.
S u b s t i t u t i n g t h e series
e q u a t i o n (1.9) and e q u a t i n g t h e c o e f f i c i e n t s o f
s e t of equations o f t h e E u l e r type.
r e s u l t i n g equations give
quantities
e '4
h
and
r
i s determined.
'r
C o t Cr
'
and
. kr$
e
(1.10) i n t o
terms, w e g e t a
Particular solutions of these
Dr
i n t e r m s o f t h e known
Thus, t h e f i r s t o r d e r c o r r e c t i o n t e r m
u
1
The s e c o n d and h i g h e r o r d e r a ~ p r o x i m a t i o n scan be
determined i n a s i m i l a r manner.
I'
For e q u a t i o n (1.1), i t i s a l s o p o s s i b l e t o assume a s o l u t i o n
where
a
and
9
s a t i s f y (1.3) and ( 1 . 4 ) .
The c h o i c e o f t h e s o l u t i o n
*i.s dependent on the given i n i t i a l c o n d i t i o n s .
QUALITATIVE METHODS O F DYNAMIC SYSTEMS.
52.
'
The d e f i n i t i o n s , theorems ' and methods p r e s e n t e d i n t h i s
i
4-
s e c t i o n a r e a d a p t e d from the books o f Andronov, Leontovich, Gordon
and Maier [8, 91, Sansone and C o n t i [ l o ] , Nemytskii and S t e f a n o v [ l l ]
and B i r k h o f f and Rota [ 1 2 ] .
*
.
When a p h y s i c a l problem l e a d s t o a n o n l i n e a r s y s t e m , i t i s
f r e q u e n t l y e a s i e r and more c o n v e n i e n t t o s t u d y t h e q u a l i t a t i v e b e h a v i o u r
o f s o l u t i o n c u r v e s ( t r a j e c t o r i e s o r o r b i t s ) d i r e c t l y , by examination o f
t h e v e c t o r f i e l d d e f i n e d by t h e s y s t e m , t h a n by means o f a n a l y t i c a l
L
- '
e x p r e s s i o n s o b t a i n e d by i n t e g r a t i o n s .
We w i l l d e v o t e t h i s s e c t i o n t o t h e s t u d y o f t h e c l a s s i c a l
methods f o r q u a l i t a t i v e i n v e s t i g a t i a n o f a g i v e n dynamic s y s t e m .
We
are p a r t i c u l a r l y i n t e r e s t e d i n t h e n a t u r e o f t h e o r b i t s i n t h e
neighbourhood o f t h e e q u i l i b r i u m .
-.
.
!A%
C o n s i d e r a dynamic system
where
-
P
and
Q
a r e f u n c t i o n s o f class
2 , k z l ,
d e f i n e d i n some c l o s e d bounded p l a n e r e g i o n
open s u b r e g i o n o f
G
-
G
,
and
or
G*
.\
2-
d";
,--
a c l o s e d dr
.
M f i n i t i o n 1. A dynamic system (A) is said t o be s t r u c t t ' w
.. a l l y
s t a b l e i n G*
C G
i f t h e r e exists an open domain
H
containing
-
G*
,
-
1
1
,
\.
----
which s a t i s f i e s the following conditionsp-For
any
> 0 , there e x i s t s
&
N
6 > 0
such t h a t i f system
i s 6-dlose t o system (A) i n
(A)
N
f i n d a region
G , one
can
, it
is-
C
f o r which
H
@
I f syst.em (A) is not s t r u c t u r a l l y s t a b l e i n region G*
s a i d t o be s t r u c t u r a l l y unstable i n t h a t region,
The s t r u c t u r a l s t a b i l i t y
of the dynamic system i s generally r s f e r r e d t o t h e property t h a t
q u a l i t a t i v e nature of t h e ' o r b i t s does not change under. small changes i n
the system.
An equilibrium
M(xOIyO) of system
.
(A)
M €
, is
&
i n t e r s e c t i o n point of two curves
,
system (A)
Definition 2.
An equilibium s t a t e
is s a i d t o be of m u l t i p l i c i t y
r
M(xo,yo)
if
M
of a dynamic
i s a commn point
I
of m u l t i p l i c i t y
r
of curves
An equilibrium s t a t e
(1.12)
M
i s c a l l e d a simple equilibrium.
.
with m u l t i p l i c i t y
r > 1
If
, then
M
1
(1l
,
r = 1)
is called a
multiple equilibrium..
An equilibrium s t a t e
if
M
M
i s s a i d t o be of i n f i n i t e m u l t i p l i c i t y
i s an i n t e r s e c t i o n point of i n f i n i t e m u l t i p l i c i t y of curves (1.12)
(note t h a t i n any closed bounded plane region an a n a l y t i c dynamic system
has e l n e r a f i n i t e n m b e r of equilibrium s t a t e s o r s i n g u l a r curves a l l of
k h s e points axe equilibrium s t a t e s ) .
I
?
,
An e q u i l i b r i u m
M (xO,yo)
is simple i f t h e Jacabian
Then, by t h e i m p l i 4 t function theorem 18, see appendix, 5 4 . 3 ,
Theorem V I , ~ernzka
~simple e q u i l i b r i u m
e x i s t s a neighbodhood of
An i s o l a t e d e q u i l i b r i u m
M
M
Hence a multiple-eq*librium
M
i s i s o l a t e d , i .e.
,there
c o n t a i n i n g no e q u i l i b r i u m o t h e r than
M
.
i s s t r u c t u r a l l y s t a b l e . o n l y i f i t is simple.
is not structurally stable.
The following theorems 181 e s t a b l i s h t h e necessary and
s u f f i c i e n t conditions f o r the e x i s t e n c e of a double and a multiple
e q u i l i b r i u m o f the dynamic system (A)
Theorem 1.
A c o m n point
.
0(0,0)
a r e s.s&umed t o be o f c l a s ~ 2, i n
(P, Q
-
G)
-
o f t h e t w o curves (1.12)
i s a double i n t e r s e c t i o n
I
p o i n t i f and only i f t h e following conditions a r e s a t i s f i e d :
(b)
A t l e a s t one of the e l e m n t s i n the determinant
A
is
d i f f e r e n t from zero;
(c)
The number
Q(x,rp(x)) = 0
for
y
where
x = 0
i s a double r o o t of t h e equation
y = q(x)
i s t h e s o l u t i o n of the
- equation
i n some s u f f i c i e n t l y small r e c t a n g l e
-
1x1 5 a
,
IY 1
5
B
~ ( x , y )= 0
( t h i s s o l u t i o n e x i s t s and is uniqsbe by virtue of m a t i o n (b) and
the theorem of i m p l i c i t f u n c t i o n s ;
P (0,O) = 0
~f
Y
cp (0) = 0)
.
/
, but
A
some o t h e r element o f
does n o t
-=
vanish, c o n d i t i o n ( c ) should be a p p r o p r i a t e l y reworded.
.
-- - .
The%rem-2.
r
of c l a s s
Let
be an e,quilibrium s t a t e of a system
0(0;0)
( i n p a r t i c u l a r , a n a l y t i c a l system) and l e t a t l e a s t one of
the f i r s t derivatives of functions
vanish a t t h e p o i n t
the equation
-
hood of
0
,
O(O.0)
Q(x,y) = 0
Jl ( x )
and
.
number
0
and
Let f u r t h e r
for
y
r
Q
.
say
y = cp ( x )
Qy (0 , O )
,
not
be t h e s o l u t i o n of
i n some s u f f i c i e n t l y s m a l l neighbour-
= P ( x , i (x) )
equilibrium o f m u l t i p l i c i t y
P
.
Then t h e p o i n t
o f system
is a multiple
i f and only i f t h e
(A)
i s a multiple r o o t of m u l t i p l i c i t y
0
r
of t h e e q u a t i o n $ ( x ) = 0 .
Before we p r e s e n t t h e theorem we f i r s t d e f i n e t h e concept o f a
l i m i t cycle and a s e p a r a t r i x .
Limit c y c l e i s a c l o s e d t r a j e c t o r y such t h a t no t r a j e c t o r y
2
s u f f i c i e n t l y n e a r t o i t is closed.
r
Separatrix is a path
" (generally a saddle point) a s
do n o t t e n d t o
t
-t
tending t o a s i n g u l a r point
+"J (-03)
such t h a t neighbouring p a t h s
under t h e s m c o n d i t i o n s and s o p a r t
A
A
r
as t
-+
+a
( - "J)
.
S t r u c t u r a l l y s t a b l e systems a r e c h a r a c t e r i z e d by t h e following
Theorem [ 8 ] .
b
-G*
Theorem 3.
The dynamic system
w i t h a normal boundary
IG* c G)
A
is structurally stable i n
i f and only i f t h e f o l l o w i n g necessary
and s u f f i c i h t c o n d i t i o n s are s a t i s f i e d :
(a) System (A) has only a f i n i t e number of e q u i l i b r i a
which are a l l simple and hyperbolic (no eigenvalue
matrix, i.e.
, ~ a c o b i & nmatrix,
in
-G*
.
o f t h e variationak
of (A) has zero r e a l p a r t )
(b) There a r e no saddle-to-saddle s e p a r a t r i c e s i n
.
*>
* *
-G* .
( c ) System (A) has only a f i n i t e number of c l o s e d t r a j e c t o r i e s
id G*
which a r e a l l simple l i m i t c y c l e s .
We now review t h e c l a s s i c a l methods f o r i n v e s t i g a t i o n of the
t o p o l o g i c a l s t r u c t u r e of an e q u i l i b r i u m
M(xO,yO) of system ( A ) .
We
d i s t i n g u i s h between two c a s e s : a simple ,equilibrium and a m u l t i p l e
equilibrium.
(i) Let
M
be. an i s o l a t e d simple equilibrium.
Without l o s s
of g e n e r a l i t y , w e a s s u m t h a t t h e e q u i l i b r i u m under consideration i s a t
,at
t h e o r i g i n , i .e.
'
change of v a r i a b l e s
the point
=
P ( O r @ ) , Q,(O,O),
Px(O,O),
Y
+
0 ( 0 , 0 ) ( t h i s can be ensured by t h e
X
, y = yo + Y) .@
Denote t h e values
*
Q (0,o)
Y
by
a , b, c, d l
respectively.
System (A) then can be w r i t t e n i n t h e form
P
dx
=
dt
where
0 (0,O)
ax + by + P2 (x,y) ,
and
p2
, the
dt = cx
+ dy +
22 a r e f u n c t i o n s of c l a s s
functions
P2
Q2
Q2 (x,y)
k 2 1 in
,
-G
(1.14)
;
a t the point
and t h e i r p a r t i a l d e r i v a t i v e s a l l vanish:
The ~ a c o b i a nmatrix of t h e l i n e a r p a r t of system (1.14) a t t h e
equilibrium
0
,
denoted by
J
0
,
is
L.
,
Since
0
i s a simple equilibrium, the Jaco@ian
The c h a r a c t e r i s t i c equation of (1.16)
has solutions
where
The numbers
A1
and
h2
eigenvalues of the matrix
are called the c h a r a c t e r i s t i c roots o r
J
0 .
The c h a r a c t e r i s t i c equation and i t s
roots play a major r o l e i n investigation of the topological s t r u c t u r e
of an equilibrium.
discriminant of
Several cases a r i s e depending on the value of the
(1.19)
X
The general c l a s s i f i c a t i o n c r i t e r i a are surmnariz~~d
in
Table 1.
,
Case
Equilibrium
Stability,
Node
Node i f
-
A > O
-
ll'
Stable .
if U < O
Real,
positive
Unstable
if- a > 0
Real,
negative
Stable i f
if a<O
Real, Distinct,
Negative
Unstable
if
0>O
Real, Distinct,
Positive
Saddle i f
A < 0
Real, Distinct,
Opposite sign
<
Stable
if
O <
Focus i f
o # o
Complex, Re A
O
~om~lex
R e, h
Unstable i f
if a>O
I
Center
I
1,2
112
< 0
> 0
Pure imaginary
Table 1
(ii) Let
be an i s o l a t e d multiple. equilibrium of an
0(0,0)
analytical dynamic sys tern
Two cases are possible:
(A)
0 =
0
which s a t i s f i e s the condition
,
0
# 0
.
-
Here we consider only the
I
case
where
# 0 , Then t h e system
U
P2
.
Q2
(A1
becomes
,
a r e a n a l y t i c i n t h e 'neighbourhood of t h e o r i g i n and
t h e i r s e r i e s expansions involve only terms of a t l e a s t second o r d e r ;
7
Under t h e s e assumptions, t h e r e e x i s t s a nonsingular l i n e a r transformation
reducing the system (1.23) t o t h e form
f
where
I
d
---\
T = ot
,
c
i s some c o n s t a n t , and
k o n d i t i o n s as t h e function
'.
,
P2
and
Q;
in
-
-
, Q2
P2
s a t i s f y t h e same
(1.23).
By t h e i m p l i c i t function,theorem, t h e r e i s a s o l u t i o n
where
cp
(S)
is q
i a n a l y t i c function such that
of the equation
i n a s m a l l neighbourhood of
0(0,0)
and l e t t h e expansion o f t h e
7
'
-
function
i n powers o f
5
have t h e form
where
m L 2
,
and
(The e x i s t e n c e of t h e s e numbers
m
and
A
m
follows from t h e f a c t
>
t h a t t h e equilibrium i s i s o l a t e d ) .
The following proposit$ons apply [9, see 5 21.2, Theorem 651 :
1.
If
m
i s odd, and
Am > 0
, the
equilibrium
0
of system (1.23)
i s a t o p o l o g i c a l node.
A
< 0
,0
2.
If
m
i s odd, and
3.
If
m
i s even, t h e e q u i l i b r i u m
m
is a topo&ical
0
saddle p o i n t .
i s a saddle-node, i.e.,
its
canonical neighbourhood c o n s i s t s o f a p a r a b o l i c and two hyperbolic
sectors.
,
?ccording t o Andronov, Leontovich, Gordon and Maier [ 81, t h e
theory o f b i f u r c a t i o n s i s concerned with t h e changes which occur i n t h e
system i n a p a r t i c u l a r region when t h e
t o p l o g i c a l s t r u c t u r e of&dynamic
/
'
-
\
-1
r i g h t - s i d e s o f tfie system a r e a l t e r e d .
I f system (A) i s s t r u c t u r a l l y
unstable, dynamic systems o f d i f f e r e n t t o p o l o g i c a l s t r u c t u r e s always
e x i s t i n any a r b i t r a r i l y small neighbourhood o f
(A)
.
Therefore,
only s t r u c t u r a l l y unstable systems should be considered.
A
m u l t i p l e e q u i l i b r i u m s t a t e of m u l t i p l i c i t y
r Z 2 , O(.O,O)
,
of an a n a l y t i c system (A) should s a . t i s f y the following conditions
( s e e D e f i n i t i o n 5, 52.1 [8] and D e f i n i t i o n 1):
(a) There e x i s t numbers
system
(ff)
bo-close t o rank
E
0
r
> 0
and
t o system
6O > 0
(A)
'
such t h a t any
has a t most
r
e q u i l i b r i u m s t a t e s i n t h e E -neighbourhood o f , 0 ;
0
(b)
For any
6 - d o s e t o rank
E
r
>
and
Eo
to
, there
0
€
0
-neighbourhood o f
0
,
k
..
O1 ,02,. ,Ok
0
equilibrium s t a t e s
6 0 - c l ~ s et o rank
O1,O2,...
,Ok
r
to
i n the
a n d - t h e y a r e a l l s t r u c t u r a l l y s t a b l e , w e say
stable e q u i l i b r i u m s t a t e s
equilibrium
n
(A)
equilibrium s t a t e s
t h a t the multiple equilibrium s t a t e
of t h e p o i n t s
r
.
I f an a n a l y t i o a l dynamic system
system ( A ) has p r e c i s e l y
n
e x i s t s a system (A)
which has a t l e a s t
!A)
i n t h e €-neigfibourhood of
6 > 0
i n t h e case
01,02,.
0
..,Ok
decomposes i n t o s t r u c t u r a l l y
n
on p a s s i n g t o system (A)
Each
which are obtained from an i s o l a t e a m u l t i p l e
o = px(Q,O)?+ Q (0.0) # 0
is either a
Y
s t r u c t u r a l l y s t a b l e node o r a s t r u c t u r a l l y s t a b l e saddle p o i n t
(see 923, Theorem 35 l81).
.
,
.
3.
SURYEY OF LOTKA-MLTERRA COMPETITION EIODELS
.
~ o s mathematical
t
models f o r the populations o f two species
competing for the same food supply are formulated i n terms of a system
of 'two ordinary d i f f e r e n t i a l equations.
I t is usually assumed t h a t the
growth r a t e of each species which measures the r a t e of increase of
population s i z e per member of the population i n u n i t time, depends
l i n e a r l y on the two population s i z e s .
A widely used model of i n t e r -
species competition describing the competitive interaction o f , two
species coexisting i n the same ecological environment w i t h f i n i t e
resources was proposed independently by Lotka (1924) and Volterra ( 1931) ;
---
\
-1-4
where
t
;
N
ai
\
%
.'
i s the number of i n d i v i d ~ l sof species i a t a given time
i
\
i s the i n t r a s p e c i f i c coeffidie;;h(i$nate capacity of increase
per individual ,of species
i); $
i
i s the i n t e r s p e c i f i c coefficient .
j
(competition coefficient of an individual of species
of s p e c i e s , i);
0i
positive constants.
i s the carrying capacities of
N
i
on an individual
;
they q e a l l
I n 1935, Gauss and W i t t [I31 i n v e s t i g a t e d (1.31) and gave t h e
necessary and s u f f i c i e n t conditions f o r s t a b l e coexistence of-two
competing s p e c i e s .
'
Abdelkader [14] found exact s o l u t i o n s of model (1.31)
A
under c e r t a i n conditions.
In 1967, Rescigno and Richardson [151 re-examined t h e Kolmogorov
mode 1
and analyzed t h e behaviour of s o l u t i o n s i n case of competition and
cooperat ion.
In 1974, Brauer and Sanchez [16] extended t h e Lotka-Volterra
model (1.31) t o include harvesting:
-2
where
H
is a p o s i t i v e parameter.
They i n v e s t i g a t e d t h e e f f e c t of
constant r a t e h a r v e s t i n g on e q u i l i b r i a and t h e i r s t a b i l i t i e s .
i
I n 1980, Freedman [17, s e e p. 156, 7.21 proposed an open
problem t o analyze the behaviour of sol'utions of t h e perturbed ~otka-
p
Volterra competition model
where
E
i s a small p o s i t i v e parameter.
Following the suggestion of
Freedman we now propose t o study the two-dimensional perturbed LotkaVolterra competition model.
h-=vesting.
Also, we extend our study t o include
CHAPTER 2
LQTKA-VOLTERRA COMPETITION MODEL WITHOT3 HARVESTING.
1 .
..
4
L e t us examine t h e two-dimensional L o t k a - V o l t e r r a c o m p e t i t i o n
model.
Here i t i s assumed t h a t b o t h competing s p e c i e s have c a r r y i n g
capacities (self-saturation levels).
The model i s g i v e n by t h e system
of nonlinear d i f f e r s n t i a l equations
where
8
i
is the carrying capacity o f
positive constants, f o r
i = 1,2
Ni
and
a . ' s , Bi's
--
1
are
.
We w i l l f i r s t determine t h e e q u i l i b r i a o f s y s t e m ( 2 . 1 ) .
a r e a t l e a s t t h r e e e q u i l i b r i a i n t h e f i r s t quadrant:
and
~ ~ ( 0 . 8 ~F)c r. t h e r a f o u r t h e q u i l i b r i u m
by s o l v i n g t h e system o f e q u a t i o n s
E (0,O)
4
E (q .q
4
1
2
There
, E2 (el,O)
i s determined
?
~
T h i s system h a s a unique s o l u t i o n i f and o n l y i f
Then assuming (2.3) t o h o l d , t h e s o l u t i o n s o f (2.2) a r e
,
'
Here we a r e i n t e r e s t e d o n l y i n nonzero e q u i l i b r i u m p o p u l a t i o n s i n t h e
i n t e r i o r of t h e f i r s t quadrant.
(ii) a
1
-
B 1 82 <
0
,
The c o n d i t i o n s f o r t h i s a r e :
a
2
- B2el
< 0
when
y
Y 0
.
B
To determine t h e s t a b i l i t y o f t h i s e q u i l i b r i u m p o s i t i o n ( t h i s
g i v e s only t h e l o c a l s t a b i l i t y o f e q u i l i b r i u m ) we f i r s t make t h e change
of variables
r
where
xl ( t ) and
x2 ( t ) a r e t h e two new v a r i a b l e s , which w i l l have t h e
e f f e c t of s h i f t i n g the equilibrium
Substituting
--
we g e t
(2.5)
(q1,q2)
into
(2.1)
t o the origin.
and
simplifying,
where
e
The c h a r a c t e r i s t i c eqda: iori of the l i n e a r i z e d system (2.6) i s
whose roots are
We first look a t the discriminant of (2.9).
Since
(A1
-
B
~ > )0
by u s i n g (2.7) , the discriminant of (2.9) i s always p o s i t i v e , which
~and
implies t h a t t h e o r i g i n of system (2.6) is e i t h e r . a saddle o r a node
and using ( 2 . 7 )
,
and
From these i t i s c l e a r t h a t
a a
1 2
>
BlB2
Re
A
a a
,
do < 0
if
1 2 <
-
A.
i s always negative and
.
B1B2
> 0
if
Hence the equilibrium E4(q1,q2)
0102
( i ) an asymptotically
2
- d2e1 > 0
and
s t a b l e node i f
y > 0
a
1
- B102 >
0
I
,
Thus, the s t a b l e coexistence can occur when
e,e,
1 L
BIB2
(i.e.,
the
species control t h e i r own growth stronger than they control t h e i r
c o q e t it o r s )
.-
One can e a s i l y see t h a t
El
i s a s t a b l e node since each
species grows i n the absence of the other.
a 2 - 2 ~1 e< o , saddle i f
if
cr1 - 8 18 2 < O
I
a2
saddle i f
E2
- B2e1 > 0 . E3
a 1 - Blez > 0 .
i s a s t a b l e node i f
i s a l s o a s t a b l e node
W e summarize the various cases as following;
r
a1 -
case^.
6 8 > O ,
1 2
a2 - 6 28 1 > O
and
y > O .
Thisisa
s i t u a t i o n where both species coexist i n stable equilibrium with p o s i t i v e
l i m i t i n g populations (see Figure
case
n.
al
- file2
< 0
a2
a . la) .
- B2e1
< 0
and
y < 0
.
I n L h i case
~
the
survival of the species depends on the i n i t i a l conditions (see Figure
Case C.
a1
species
N1
B102 < 0
as
a 1 - 6 18 2 >
case^.
CX2
- B2e1 > .
0
dies out while species
e2
saturation
N1
-
t
-+
0
N2
(see Figure
a2
-
b201 < 0
approaches i t s level of s a t u r a t i o n
out as
t
+
This i s a s i t u a t i o n where
approaches i t s l e v e l of
2 . 1 ~ ).'
.
O1
This i s a case where species
while species
N2
dies
(see Figure 2.1d).
--W
e now study the existence of l i m i t cycles by using DulacBendixson c r i t e r i a [9, see 5 1 2 , Theorem 311.
respective right-hand s i d e s of ( 2 . 1 )
where
G
Let
P, Q
denote the
and choose the Dulac function
y # 0
(see ( 2 . 3 ) ) .
Then
Since
a k + a2h #
, this
1
curves
N1 = 0
vanishes on the
quadrant.
0
and
$
!
2
= 0
function vanishes only along the i n t e g r a l
.
N -and N -axes,
1
2
a (PB)
a~ ,
--
Though the expression
+
a (QB)
aN,
it does not change sign i n any
.
.
Thus there can be no l i m i t cycles.
From the above analysis
we observe t h a t :
(i) A l l e q u i l i b r i a a r e hyperbolic.
(ii) There a r e no l i m i t cycle.
(iii) There a r e no saddle-to-saddle s e p a r a t r i x .
Thus, by theorem 3 i n Chapter 1, 8 2 , system
stable.
[2-11
i s structurally
Fig. 2.lc
52.
HARVESTING OF L-Y COMPETITION MODEL.
For populations of two species, i n competition, we model t h e
I
1
e f f e c t of h a r v e s t i n g one s p e c i e s by \the' introduct'ion of a p a r m t e r .
I f the s p e c i e s
N1
i s h a r v e s t e d a t a constant r a t e
while t h e second s p e c i e s
N2
H
y
p e r u n i t time,
i s undisturbed, then t h e governing
equations become
where
H
i s a p o s i t i v e constant.
--.-
-
The e q u i l i b r i u m p o i n t s of (2.16) a r e given by t h e i n t e r s e c t i o n
p o i n t s of the. curves (a hyperbola and a p a i r of l i n e s )
From t h e second equation of (2.17) we g e t
Substituting N2
•’ram (2.18) into (2.17) we obtain
correspondingly the follswing equations
where y
be
N1 (1)
is given in (2.3) .
N2
Let the solutions of (2.19) and (2.20)
(2) respectively, we get
where
In order to ensure real equilibrium solutions we require that
the discriminants of
(2.21) and (2.22) be nonnegative which leads to
the following conditions for H
respectively.
Since H > 0 , we also require that
Y >
0
.,
.
t-
Assuming (2.24) t o h o l d , we s u b s t i t u t e ( 2 . 2 2 ) i n t o t h e
second e q u a t i o n of (2.18) t o g e t
Also, t h e values of
(2.22) and - (2.25)
have p o s i t i v e r e a l p a r t s i f t h e
following. c o n d i t i o n s a r e s a t i s f i e d :
and
H >
-
a1 a2 (a2 - B 2 9 4
.
Note t h a t c o n d i t i o n s i n
( 2.26) a r e
a
independent o f
H
and t h e r e f o r e must be s a t i s f i e d by t h e unharvested
system i f s t a b l e c o e x i s t e n c e i s t o occur under h a r v e s t i n g .
We c o n s i d e r
only t h e system f o r
Assuming t h a t (2.26) holds t h e r e a r e two p o s s i b i l i t i e s .
I n t h i s c a s e , t h e system (2.16) g i v e s four s i w l e e q u i l i b r i a i n t h e
f i r s t quadrant.
y,
6
a r e given i n (2.3)
venience we l e t
and ( 2 . 3 3 ) , r e s p e c t i v e l y .
For con-
( P ~ ~ ~
andP (~ P~ ~ ) ~ I Pdenote
~ ~ )t h e c o o r d i n a t e s
>-.
of
I? 3
and
I?4 ,
respectiveAy.
Here we s t u d y t h e s t a b i l i t y i n t h e neighbourhood o f
equilibrium
J(Nl,
N2)
,
E4 (P12
the
F i r s t we compute t h e J a c o b i a n - m a t r i x ,
of (2.16)
We again apply a l i n e a r t r a n s l a t i o n
i
which reduces the system (2.16) t o the form
where
The c h a r a c t e r i s t i c equation of the l i n e a r p a r t of system (2 - 3 2 )
has roots
S t is c l e a r from (2.33) t h a t the discriminant of
Since from (2.30) the Jacobian a t
E4
(2.35) i s always positive.
and
-
is always negative ; hence E4 i s an asymptotically s t a b l e node (Table 1-11.
Following the same procedure we f i n d t h a t
E2' E3
a r e saddle p o i n t s .
-
E
1
i s an unstable node and
.-
The dynami,cs a r e shown i n Figure 2 . L
/
/
./
,,j
(ii)
H =
a2 0 1 (a1 4y
B1e2)
--'.
2
-.
,/-
.--,,
//
r Y > O :
J'
.
,-
'I
/-i
p
(2.38f
%/'
\
Then t h e system (2.16) has two simple e q u i l i b r i a
and a s t r u c t u r a l l y unstable double equillbriurn (by Theorem 1, 82,
A
A
~ e tpl,p2
denote t h e r e s p e c t i v e coordinates of
the n a t u r e of t h e double equilibrium E
E
.
W
e wish t o study
(61 ,; 2) .
A
n
We f i r s t s h i f t t h e equilibrium E (pl ,p2) t o t h e o r i g i n by
the change of v a r i a b l e s
i
--
Then o u r s y s t e m becomes
where
--
by u s i n g ( 2 . 3 9 ) ,and ( 2 - 4 2 )
f o l l o w i n g Andronov e t a1 [9], w e i n t r o d u c e
a nonsingular l i n e a r transformation
a
which r e d u c e s t h e s y s t e m (2.41) t o t h e form
where
with
Solving equation
we o b t a i n
Define a f u n c t i o n
$(S)
by
Substituting
the expression
cp
(c)
from (2.48) w i t h (2.46) i n t o (2.49)
t h e s e r i e s e x p a n s i o n o f the f u n c t i o n )<()I
Thlu s , t h e double eqiA l i b r i u m
'
(see c h a p t e r 1, 92)
1
(2
- k,
.
See
0(0,0)
Fig. 2 . 3 .
,
t a k e s t h e form
(2.41) is a saddle-node
of
I t can e a s i l y b e ' shown t h a t
0 ) i s a n u n s t a b l e node and (-
2
+
k , 0) i s a s a d d l e .
By the same analogy as i n system (2.11, w e f i n d t h a t t h e
s y s t e m (2.16)
s a t i s f y i n g t h e c o n d i t i o n s (2.28) and
(2.26) i s
s t r u c t u r a l l y s t a b l e , and t h a t s a t i s f y i n g ( 2 . 3 8 ) i s s t r u c t u r a l l y u n s t a b l e .
Fig. 2 . 2
Fig. 2 . 3
CHAPTER 3
1
THE PEHTURBED LOTKA-VOLTERRA COMPETITION MODEL WITHOUT HARVESTING.
We now consider t h e perturbed Lotka-Volterra sys tern, when
s m a l l p e r t u r b a t i o n a l terms
Efi (N1 , N 2 )
i = 1 ,2
are applied t o
I
sys tern ( 2 . 1 )
where
E
<< 1 i s a s m a l l p o s i t i v e parameter and
f. ( N ,N )
1
1
2
,i
= 1,2,
a r e a n a l y t i c function's of t h e i r arguments.
Here we i n v e s t i g a t e t h e q u a l i t a t i v e behaviour of (3.1) i n
t h e neighbourhood of the perturbed e q u i l i b r i a .
W
e f i r s t compute t h e
perturbed e q u i l i b r i a using t h e techniques of Freedman and Waltman [lo].
The equilibrium positic&
dN./ d t = 0 , i = 1 , 2
1
.
of (3.1) can be found by s e t t i n g
They a r e t h e s o l u t i o n s of t h e .system
Here we a r e only i n t e r e s t e d i n nonzero equilibrium populations i n the
f i r s t quadrant.
(q1,q2)
in
Note t h a t f o r
(2.4)
f o r (3.1) , denoted
Hence
T h i s gives
with
J (N
&
(2.3)
N ,€ )
1' 2
.
= 0
, the
equilibrium i s given by
f i r s t , we compute t h e Jacobian matrix
, and
obtain
by (2.3)
, and s o by t h e i m p l i c i t f u n c t i o n theorem, (3.2) can be s o l v e d
1.
i
f o r N1
and N2 a s f u n c t i o n s o f E f o r s u f f i c F e n t l y small &
Let
.
t h e s e s o l u t i o n s be
compute
q;
q;(€)
and
q;
N1
- q;
Substituting
f . (N ,N ) , i = 1 , 2 ,
1 1 2
(3.5)
and
t o order
and
q; (€1
E
.
*
N2 = q2
, respectively.
W e wish t o
Hence we s e t
into
i n Taylor s e r i e s about
(3.2)
, expanding
€ = 0
g i v e s , a f t e r comparing t h e c o e f f i c i e n t s o f
,
and u t i l i g i n g
€
To s t u d y t h e s t a b i l i t y i n t h e neighbourhood o f t h e p e r t u r b e d
equilibrium
where
E* f q f ,q*)
1 1 2
x 's-are
i
w e introduce a l i n e a r transformation
t h e two new v a r i a b l e s whose e q u i l i b r i u m p o s i t i o n i s
I
A
-
-
--
>
(0,O).
Without l o s s o f g e n e r a l i t y , we again use t h e same
S u b s t i t u t i n g (3.7) i n t o (3.1)
*
f. (ql
1
+ EX1,
*
q2 +
u t i l i z i n g (3:2)
&rld
q
A1,
b
€x2) , i =' 1 , 2 , w i t K (3.5)' i n Taylor s e r i e s i n
, dividing
B1, A 2 # B2
111q21
.
, expanding
by
E
, and
keeping terms up t o o r d e r
gives
where
E
a r e given i n (2.7) and
a r e given by (3.6).
The l i n e a r p a r t of system ( 3 . 8 ) i s
E
,
E
,
The c h a r a c t e r i s t i c equation of (3.10)
,
where
A = A B
1 2
-
A B
2 1
+.- E ( A 1b 2 + alBZ -
has r o o t s
W
e f i r s t compute t h e discriminant of (3.13)
A2bl
-
a B )
2 1
+
o(E'),
by w?ng
(2.7)-:
Hence f o r
s u f f i c i e n t l y small t h e d i s c r i m i n a n t
€
(3.14) 'is always p o s i t i v e which i m p l i e s t h a t t h e r o o t s
h1
h2
and
From tha Table 1.1 ( s e e p. 1 2 ) we conclude t h a t
are r e a l and d i s t i n c t .
the o r i g i n o f system (3.10) i s e i t h e r a node o r a s a d d l e .
Using (2.7) we compute
+&2)
a
0 = (A,
+
B2)
+
O(E1
=
-
+ Olt).
(3.15)
O2
d
L
?
7
-
.
-r
i
,which i s always n e g a t i v e .
Thus t h e n a t u r e o f t h e o r i g i n i s determined
:
as following :
(i) an a s y m p t o t i c a l l y s t a b l e node i f
(ii) a s a d d l e i f
a1 a 2
-
f3 f3 8 8
1 2 1 2
a1 a 2 - 81820182 >
0
I
< 0
foy system (3.10) and hence f o r system (3.8)
.
Again t h e s t a b l e
co-existence o c c u r s when t h e c o m p e t i t i o n w i t h i n ( i n t e r n a l grawthl i s
s t r o n g e r t h a n o u t s i d e ( e x t e r n a l growth) a s i n t h e unpe'rturbed system
( 2 .
Thusf t h e s t a b l e e q u i l i b r i u m
E
4
of t h e unperturbed system
(2.1) p e r s i s t s under t h e i n f l u e n c e of s m a l l p e r t u r b a t i o n s .
5 2.
THE CONTROLLED LOTKA-VOLTERRA COMPETITION MODEL.
'
The small terms
€fi(N1,N2)
for
i = 1 , 2 , p e r t u r b i n g the
s i t u a t i o n d e s c r i b e d by t h e Lotka-Volterra e q u a t i o n s s t u d i e d i n S e c t i o n 1
o f t h i s Chapqer can be t h o u g h t of a s c o n t r o l s .
As
an example l e t us
c o n s i d e r t h e c o n t r o l l e d system
which i s a p a r t i c u l a r c a s e o f (3.1) w i t h
W e assum t h a t t h e c o n t r o l l i n g f a c t o r
belong t o t h e s e t
functions.
For
f-1,1},
E = 0
hence
fl
f l = $lN1
@ ( t ) and
1
and
f2
2
and , f 2 =
02NZ 2
@2(,
t )t E 0
,
a r e discontinuous
, o u r system reduces t o ' l o t k a - ~ o l t e r r amodel
From (3.5) with ( 3 . 6 ) w e o b t a i n t h e nonzero e q u i l i b r i u m
.
where
of
G1
are given i n (2.4) with (2.3) . Depending on t h e values
91' 92
G2 , t h e
and
system generates f o u r systems with f o u r e q u i l i b r i u m
4
points.
According t o Section 1, Chapter 3 , t h e s t a b i l i t y of t h e
e q u i l i b r i a of t h e c o n t r o l l e d system remains t h e same a s i n t h e
uncontrolled system.
population s i z e s .
of s p e c i e s
N
1
i n c r e a s e s while
However, t h e r e can be small changes i n t h e
For example, when
decreases while
N
2
decreases i f
N2
= -1
and
increases i f
Y<
0
.
2
Y >
= 1
0
the size
and
N
1
53.
THE PERTURBED LO--VOLTERRA
COMPETITION
MODEL WI'TH HARVESTING.
I n t h i s s e c t i o n we c o n s i d e r t h e . L o t k a - V o l t e r r a system w i t h .
h a r v e s t i n g ' s t u d i e d i n Chapter 2 when p e r t u r b e d by small terms
€ f i (N1,N2)
where
ai
,
i = 1.2
, Bi , H
p a r a m e t e r and
arguments.
For
f . (N
1
.
-
-
Then t h e governing model becomes
a r e p o s i t i v e c o n s t a n t s , E < < 1 i s a small p o s i t i v e
N )
1' 2
€ = 0
,
i = 1,2
,
and . H = 0
The e q u i l i b r i u m p o s i t i o n s
a r e a n a l y t i c f l m c t i o n s of t h e i r
,
t h e s y s t e m . (3.16)
*
(pl
I
P;)
reduces t o
of t h e p e r t u r b e d s y s t e m
(3.16) a r e t h e s o l u t i o n s o f t h e system
We a r e a g a i n i n t e r e s t e d i n nonzero e q u i l i b r i a i n the f i r s t q u a d r a n t .
The J a c o b i a n m a t r i x f o r (3.16) i s g i v e n by (3.3)
.
H e r e , a s i n Chapter 2, w e c o n s i d e r s e p a r a t e l y two cases.
a e (a - 6 9 1 ~
< H <
2 1
.-
1 2
1
4~
-
does n o t vanish a t . E
case, t h e J a c o b i a n
3
.
d
I n This
-
and
E4
.
Hence by
i m p l i c i t f u n c t i o n theorem we a g a i n f i n d the p e r t u r b e d e q u i l i b r i a
-
E* (p* tp*
3 11 21
and
E*(p* ,p* )
4 12 22
i n t h e form o f power series i n
E
as
i n t h e u n h a r v e s t e d c a s e i n S e c t i o n 1 of t h e p r e s e n t c h a p t e r
f o r i = 1,2, respectively.
Hence
(p11'p21 1
and
%2
e q u i l i b r i u m p o s i t i o n s o f t h e u n p e r t u r b e d system (2.16)
coefficients of
E
)
sre the
(E = 0).
The
B i , A;,
2
in
-
B2
a r e g i v e n i n ( 2 . 3 3 ) . and
c a n be o b t a i n e d fzom (2.33) by r e p l a c i n g
respectively.
P22
pll,
p21
A;,
with
p
1 2 ' P22r
W e f i r s t study the s o l u t i o n s of (3.16) i n the neighbourhood
ii (P;2
of
,P;2)
.
* +
-$*
t o t h e o r i g i n (0,O)
--
\i
',.
(pY2
22
1
+
( l ) ,+ E 2
pi2
,
expanding
-
€ x i ( t ) = Pi2 +
shifts
I
T-
N. ( t ) = pi2
1
i
The l i n e a r t r a n s l a t i o n
(xi
.
S u b s t i t u t i n g (3.20) i n t o (3.16)
fi
--% 6
E
* +
( P ; ~ + &xl, p22
.
&x2)
,i
with
= 1,2
u t i l i z i n g ( 3 . 1 7 ) , d i v i d i n g by
E
,
(3.18) i n Taylor s e r i e s i n
and c o F l w o t h e f i r s t o r d e r
I'
terms of
E
,
gives
,/
where
...
and
xl,
B1,
x2, -B2
a r e given i n ( 2 . 3 3 ) .
The c h a r a c t e r i s t i c e q u a t i o n of LLe l i n e a r part of (3.21) i s
where
1
Hence
b
'1
S i n c e f o r s u f f i c i e n t l y small
E
,
G(E)
is always p o s i t i v e ( s e e Chapter 2 , 5 2 )
%@lZ
,p*
22 )
,
i s always n e g a t i v e and t h e
the perturbed equilibrium
is an asynptotically stable node.
A
-
Following t h e same p r o c e d u r e as
t o be a saddle.
equilibrium
Ei
Thus, t h e s t a b l e e q u i l i b r i u m
-
we f i n d
-
E4
E*3 (p*11&
and the unstable
o f t h e u n p e r t u r b e d system (2.16) p e r s i s t a g a i n under
E3
t h e i n f l u e n c e o f small p e r t u r b a t i o n s .
(ii) H =
a 20 1 (a1 -
B102)
2
-
I n t h i s c a s e t h e nonzero
+
4Y
A
equilibrium
A
E (pl I P ~ ) of t h e system (3.16) when
E = 0
is a
s t r u c t u r a l l y u n s t a b l e double e q u i l i b r i u m , which i r n p l i e s t h a t t h e
A
A
Jacobian
variishes a t
A
E (pl IP,)
which is a saddle-node
(see 2 43)
.
A
We seek a p e r t u r b e d e q u i l i b r i u m
form of series i n
&
(3.26)
, i
= 1,2
of t h e system (3.16) i n t h e
by p e r t u r b a t i o n t e c h n i q u e .
Substituting
' i ( pA*f;*)
1
2
(p;,p;)
N1 - pl*
A
and
N2 -
s;
, i n Taylor s e r i e s about
Hence we s e t
i n t o ( 3 . 1 7 ) , expanding
E = 0
g i v e s , a f t e r comparing t h e c o e f f i c i e n t s o f
, and
&
utilizing
where
C
All
I
A
B1,
f
i
)
I
A2, E 2
a r e given i n
(2 - 4 2 )
From t h e f i r s t e q u a t i o n s
of ( 3 - 2 7 ) w e g e t
/-
A f t e r some s i m p l i f i c a t i o n on s u b s t i t u t i n g ( 3 . 2 8 ) *o
e q u a t i o n of ( 3 . 2 7 ) and k e e p i n g terms
quadratic equation
where
order of
th& second
E
,
d
From t h i s it i s c l e a r t h a t (3.29) h a s s o l u t i o n s i f and only i f
Assuming t h a t (3.31) h o l d s it f o l l o w s t h a t there e x i s t p r e c i s e l y two
v a l u e s of
and
11
s a t i s f y i n g e q u a t i o n (3.29)
.
L e t t h e s e s o l u t i o n s be
A (2)
Pll :
Expanding t h e r i g h t - h a n d s i d e of ( 3 . 2 8 ) . we g e t
.
S u b s t i t u t i n g (3.32) i n t o ( 3.33) we o b t a i n r e s p e c t i v e l y
p
(1)
11
W
e d e n o t e two simple e q u i l i b r i a (see Chapter 1, 32) o b t a i n e d as
'
A
where
A
A
p iI p i l
and
Pi2
a r e given i n (2.391, (3.32)
and (3,341.
-
To s t u d y t h e s o l u t i o n s i n t h e neighbourhood of t h e s e e q u i l i b r i a
A
n
A
and
E2
o b t a i n e d from t h e double e q u i l i b r i u m
A
E(p1,p2)
we make
linear translations
w e i c h w i l l have t h e
of s h i f t i n g t h e e q u i l i b r i u m o f i n t e r e s t t o
the o r i g i n .
same p r o c e d u r e as i n S e c t i o n 1, w e f i n d t h a t
A
h
i s an a s y m p t o t i c a l l y s t a b l e node and E2 i s a s a d d l e . Hence
1
h
A
saddle-&
E (pl,p2) which i s the double e q u i l i b r i u m o f (2.16) under
E
-
small p e r t u r b a t i o n s s a t i s f y i n g (,3.j.-) b i f u r c a t e s i n t o two s t r u c t u r a l l y
stable equilibria
-
an asymptotically s t a b l e node and a saddle.
When system (3.16) does n o t s a t i s f y (3.31) , E(p1,p2)
'
_-
A
A
disappears.
As
an i l l u s t r a t i o n we give t h e following example.
-
A-
Consider the system
Ex.
,
For
E = 0
and
1
(z
, 65 ) .
t h e system (1) has s o l u t i o n s
Here we study equilibrium
1
(5
,
5
)
.
me can e a s i l y
check t h a t t h e conditions s t a t e d i n Theorem 1 ( s e e Chapter 1, 5 2 ) . a r e
fulfilled.
(ii) noEe of t h e e l e m n t i n t h e determinant i n (i)
i s zero;
(iii) let P, 2
1
?;
2
= -I
denote r e s p e c t i v e right-hand sides of
i s a double r o o t of the equation P (Nl,q (N1) = 0
3
= q(N1) = 1
1
Hence
(7,
\
5
)
1
-N
2 1
i s the s o l u t i o n of the equation
(3.37)
.
, where
Q (Nl.,N2)
i s t h e double equilibrium which i s unstable.
= 0
.
TO
of ($, $1,
s t u d y the behaviour o f t h e s o l u t i o n s i n t h e neighbourhood
we f i r s t make t h e change Of v a r i a b l e s
A f t e r some s i m p l i f i c a t i o n on s u b s t i t u t i n g i n t o system (3.37)
g e t the system
We then make a nonsingular linear transformation-
which reduces the system (3.39) t o t h e form
N o w consider
--
(E
= 0 ) ' we
BY the i m p l i c i t function theorem, the equation ( 3 . 4 2 ) has a s o l u t i o n
i n a s m a l l neighbourhood of o r i g i n .
Define
Then t h e s e r i e s expansion of t h e function
Hence by the theorem
(1F 5, c )i s
6
$(c)
becomes
( s e e Chapter 1, 8 2 ) ' t h e double equilibrium
a t o p o l o g i c a l saddle-node.
W
e now consider t h e perturbed system
(3.37).
One can e a s i l y
see t h a t t h e p e r t u r b a t i o n a l terms of ( 3 . 3 7 ) s a t i s f y the condition (3.31)
t"
W
e n w seek t h e perturbed e q u i l i b r i a i n t h e form of power s e r i e s i n
Substituting
( 3 . 4 7 ) i n t o (3.37) and expanding t e n & of o r d e r
5iviiing by
E
,
gives
E
and
€
From (3.48) w e get
Substituting (3.49) into the second equation of (3.48) we get
Equation (3.50) has solutions
gubstitute (3.51) into (3.849) we obtain
Hence t w o perturbed equilibria of system (3.37) are
h
F i r s t we consider
E2
neighbourhood of the e q u i l i b r i u m
where
.
To study t h e s o l u t i o n s i n the
E2
w e make a l i n e a r t r a n s l a t i o n
a r e two new v a r i a b l e s and whose e q u i l i b r i u m i s o r i g i n .
xl, x2
S u b s t i t u t i n g (3.53) i n t o (3.37) and expanding, keeping up t o o r d e r
d i v i d i n g by
E
,
E
,
gives
The c h a r a c t e r i s t i c e q u a t i o n o f t h e l i n e a r p a r t o f system
From this i t f o l l o w s t h a t f o r s u f f i c i e n t l y small
h
(3.55) a r e n e g a t i v e d i s t i n c t and r e a l , Hence
2
E
,
(3.54) is
roots satisfying
is a s t a b l e node.
h
S i m i l a r l y , we f i n d
(17 , -)65
E
1
t o be a s a d d l e .
Thus, t h e saddle-node
which i s t h e double e q u i l i b r i u m a o f t h e system 13.37)
JE
'.
= 0)
b i f u r c a t e s i n t o a s t r u c t u r a l l y s t a b l e s a d d l e and an a s y m p t o t i c a l l y
%
'
-,
s t a b l e node under s m a l l p e r t u r b a t i o n s .
\
I
I
- -. -
CHAPTER 4
-
1.
ASYMPTOTIC SOLUTIONS OF THE PERTURBED
LO'I'KA-VOLTERRA MODEL.
Here we are interested in obtaining the asymptotic solutions
-.
of the nonlinear system ( 3 . 8 ) by using the modified K-B-M method
presented in Chapter 1, 51, and also papers by G. Bojadziev [ 2 0 ] and
G. Bojadziev and M. Bojadziev
When
system
-.E
= 0
C191.
, the nonlinear system
( 3 . 8 ) reduces to the linear
'
where AII BII A2
and
B2
are given in (2.7).
We first seek the solutions of the linear system (4.1). -The
~har~cteristic
equation of
(4.1) is
-.
where
2K1 =
-
(Al + B2)
and
Kll = A B
1 2
-
A2Bl
.
I t i s assumed t h a t
K
11
> 0
.
E q u a t i o n (4.2) h a s two n e g a t i v e real
distinct roots
where
Then t h e , - a o l u t i o n o f e q u a t i o n s (4.1) becomes
+
,=
A. + K, + K,
whe re
aol bo
-(K,
+
A, + K,
K,) t
- K,
L
- -4K 1
- K,
are a r b i t r a r y c o n s t a n t s t o b e d e t e r m i n e d from the g i v e n
set o f i n i t i a l conditions
Following t h e
xl ( 0 )
and
x2 ( 0 ) .
K-B-M method w e s e e k a s o l u t i o n o f t h e non-
l i n e a r s y s t e m ( 3 . 8 ) i n the form o f
-
\
\
..'
where
ui (a,$)
,i
= 1.2,
are functims i n
a
and
t o be determined by
$J
the d i f ferentiax equation3
&e f u n c t i o n s
that
(4.7)
u
, A,
i
B ,
satisfies
8
1
..
have ' t o be determined from t h e c o n d i t i o n
t o each o r d e r o f
',
E
.
For
E = 0
the
%
s o l f i t i o n (4.7) with (4.9i9'f
l i n b system (4.1).
up t o .the o r d e r of
( 3 . 8 ) reduces t o t h e s o l u t i o n (4.61 o f t h e
Toqfind t h e f i r s t o r d e r approximation, only terms
E
have t o be considered.
,
I n x d e r t o determine t h e unknowns
A
t h e assumed s o l u t i o n (4.7) u s i n g (4.9) t o o b t a i n
These expressions' t o g e t h e r w i #
*.
B
dxl/dt
we d i f f e r e n t i a t e
and
dx2/dt
(4.7) a r e s u b s t i t u t e d i n t o t h e
o r i g i n a l d i f f e f - e n t i a l ' e q u a t i o n s (3.8) and t h e term o f e q u a l powers o f
f
a
and
&
are compapd.
.
The z e r o o r d e r terms c a n c e l i d e n t i c a l l y while t h e
I
,
i.\'
terms. of o r d e r
E
give t h e following e q u a t i o n s - f o r u1 and
u
2
.
Equations (4.10) and (4.11)
where
p i
if
r
i'
s
i
and
v
i
are functions of
the coefficients of equal p i e r s of
equations
suggest solutions in the form of
a
to be determined.
eiU terms we have the following
dpl
Ka1 da
'
dw
1
Ka1 da
-6
+
11 2
+ B p '= -a.
AIPl
1-2
2B,
1
+ B o = - ( A + B ~ - a a - b c a)
1
1 2
2
1
11
+
(A1
-
+
(A1
+ 2K2)v1 +
K )u
2
-
do
2.
Ka1 da
dr
1
Ka1 da
dr
2
Ka1 da
ds
1
Ka1 da
ds
2
Ka1 da
dvl
K a 1 da
dv
Ka1 da
+ A2v 1 +
fB2
v
1 2
a2
a
= p($
1
a2
+ 2K2)v2 = -
a
+ 61 c 2 )
'
L
In order to avoid arbitrariness in determining
ui
, as is
customary in the K-B-M method, we introduce additional conditions.
For such conditions we chTose w 1 = r1 = 0
(4.16), (4.17) and
.
(4.18) reduce to'
We assume particular solutions of A(a) . and
for some constants
%
Substituting
simplifying we obtain
and
v2
Then equations
.-
B(a)
in the form of
to be determined.
(4.25) into
(4.15),
(4.23) and
(4.24)
and
Solving equations (4.26) simultaneously and simplifying using relations
(4.8) we get
Hence
Substituting (4.28) intd
function
u
i
.
Equations (4.13)
the form of polynomials.
-
(4.15)
(4.22)
Substituting
-
(4.18) ' we now deternine the
suggest particular solutions i n
x
1
i
1I
into
(4.13)
(4.1,4)
and
,-"
( 4 . 5 ) a n d ( 4 . 8 ) -we
~ i m i l a r l ~we, f i n d
where
obtain
and simplifying using relations (2.7)
<
,
Substituting
A(a)
and
B(a)
n e g l e c t i n g t h e terms o f t h e o r d e r o f
cla
--
dt
-
where
a
from
2
O(E )
(4.28) i n t o (4.9) and
g i v e s the e q u a t i o n s
a +b
= [ - K1
0
=a(O)
+
E
,
2,2 l a
(
$
0
= )I ( 0 )
,
.
The s o l u t i o n s o f
(4.33)
The f i r s t improved approximate s o l u t i o n of
from (4.7) by t r u n c a t i n g t h e t e r m s of o r d e r o f
here
pl, sl, vl,
p 2 , w 2 ' s 2 ' r2 ' v 2
the terms w i t h f a c t o r '
E
.
is obtained
2
O(E )
are g i v e n i n
The f i r s t a ~ r o x i r n a t es o l u t i o n o f
(3.8)
are
(4.30) and (4.31).
(3.8) i s (4.35) w i t h o u t
The second approximate s o l u t i o n of
s i m i l a r l y keeping terms of order of
r a t h e r Labourious
.
c2
, but
(3.8) can be found
the c a l c u l a t i o n s are
DISCUSSIm OF THE ASYHPTOTIC SOLUTIONS.
52.
According t o (3.7) and (4.35) t h e populations
N1
and
e x h i b i t small v i b r a t i o n s i n t h e neighbourhood o f the e q u i l i b r i u m
(ql , q2)
.
'Since t h e dominant p a r t s of
(4.35)
and
x
2
Also, s i n c e
w i l l approach t h e equilibrium p o s i t i o n a s
t
K1 > K ~ ,
.
+
$
Formula (4.33) show t h a t i n t h e f i r s t approximation t h e phase
linearly increasing function in
t
changing exponential f u n c t i o n i n
behaviour of amplitude
-
K1 =
-21
(A1
+
B2)
Since
-
a (see
Ni,
K1
I
.
4.34)
x
i
,
a
is a
is a f a s t
For s u f f i c i e n t l y s m a l l
E
the
depends on t h e expression
(2.7)
is
i s always n e g a t i v e , t h e amplitude
i = 1,2
given by
i = 1.2, approach t h e e q u i l i b r i u m
which i s a l o c a l a t t r a c t o r .
$4
.
which according t o
exponentially decaying and
(0.0), hence
while t h e amplitude
t
-
c o n s i s t s of r e a l
exponential terms, t h e r e w i l l be no o s c i l l a t i o n s .
x1
*L
a. is
(4.35)
* q;)
(ql,
approach
of
(3.1)
CONCLUSION
We have proved t h a t t h e Lotka-Volterra system (2.1 ) f o r two
s p e c i e s i n competition w i t h o u t h a r v e s t i n g i s s t r u c t u r a l l y s t a b z e .
In
o t h e r words, t h e t o p o l o g i c a l s t r u c t u r e of system (2.1) does n o t change
s i g n i f i c a n t l y under s m a l l changes i n t h e system.
We have observed t h i s
p r o p e r t y i n S e c t i o n 1 and s e c t i o n 3 of Chapter 3 when small p e r t u r b a t i o n s
:*
o r controls
~ f(N. ,N )
1 1 2
,
i = 1 , 2 , a p p l i e d t o modify t h e Lotka-Volterra
.
-
.
system (2.1) do n o t change s i g n i f i c a n t l y ' t h e behaviour o f t h e s o l u t i o n s .
For i n s t a n c e , t h e e q u i l i b r i u m
E4
o f t h e unperturbed system ( 2 .l) i s an
I
a s y m p t o t i c a l l y s t a b l e node which, under t h e i n f l u e n c e of small
p e r t u r b a t i o n s , remains an a s y m p t o t i c a l l y s t a b l e node.
The Lotka-Yolterra system with harvesting (2.16) s a t i s f y i n g
( 2 . 2 8 ) and ( 2 . 2 6 ) i s s t r u c t u r a l l y s t a b l e , and t h a t s a t i s f y i n g (2.38) i s
s t r u c t u r a l l y unstable.
E
Its s t r u c t u r a l l y u n s t a b l e double e q u i l i b r i u m
which i s a saddle-node,
under c e r t a i n small p e r t u r b a t i o n s , b i f u r c a t e s
into e i t h e r two s t r u c t u r a l l y s t a b l e simple e q u i l i b r i a , a s a d d l e p o i n t
and an a s y m p t o t i c a l l y s t a b l e nade, o r d i s a p p e a r s .
111
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the theory of nonlinear oscillations", New Yor, Gordon and
-Breach (1961)
.
[3] Minorsky, N., "I
duction to nonlinear mechanics", J.E. Edwards,
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3
Bojadziev, G. and Edwards, F., "On some asymptotic methods for
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.
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r
Gauss, G.F. and Witt, A.A., "Behaviour of Mixed populations
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Rescigno, A. and Richardson, I.W., "The Struggle for life: I.
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A
[17] Freedman, H.E. , "~ete-ministic
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P
Freedman, H.I. and Waltman, Per "Perturbation of
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i
i
i
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c
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