LIMIT CYCLES IN A KOLMOGOROV-TYPE MODEL

13 NO. 3

(1990)

555-566

LIMIT CYCLES IN A KOLMOGOROV-TYPE MODEL

XUN-CHENG HUANG Department

of Mathematics New

Jersey

Institute ofTechnology Newark, New

Jersey

07102

U.S.A.

(Received June 2, 1989 and in revised form February 2, 1990)

ABSTRACT. In this paper, a Kolmogorov-type model, which includes the

Gause-type

model (Kuang and Freedman,

1988),

thegeneral predator-prey model (Huang 1988,

Huang

and Merrill

1989),

^and manyother specialized models,isstudied. The stabilityofequilibrium points, the existence and uniqueness oflimitcyclesinthe model areproved.

KEY WORDS AND

PHRASES.

Kolmogorovmodel,predator-preymodel, limitcycles.

1980AMS SUBJECT CLASSIFICATION CODE. 92A15.

1. INTRODUCTION.

Thetopicoflimitcyclesisinterestingbothinmathematics and inscience. This concept firstappearedinprint inthe famouspaper byPoincare

(1881,

^1882,1885,

1886).

Then in 1926, van der Polproposedanequationinthe studyofa self-sustained oscillation occurringina vacuumtubecircuit which showed that the closed orbitⁱⁿthe phaseplane ofthe equationis alimit cycle as consideredbyPoincare. After thisobservation, the existence, non-existence, uniqueness and otherproperties of limitcycleswere studiedextensively bymathematicians and physicists.

Bythe 1950’s,manymodelsfrom physics, engineering, chemistry, biology,economics, etc.weredisplayed as plane^autonomoussystems withlimitcycles. Since then, more and more mathematicians and scientists have been attractedtothe topic. Eveninthe renowned 23 Hilbertproblems, youwillfindaplaceforlimitcycles,specificallyin the 16th problem

(see [9]

^for

example).

Inmathematical modelingofecological systems,sincethepapersof

May (1972

8

]),

and Albrecht, Gatzke and Wax

(1973 1])

finding conditionsthatguarantee theuniqueness of alimitcycleinpredator-preymodel has been considered an outstandingproblem. Recently, severalresults have beenpublished

(see,

e.g. Cheng 1981

[2], Kuang

andFreedman

198817], Huang 198815], Huang

^andMerrill

198916]).

Inthispaper,ageneralmodelofKolmogorov-typeisinvestigated. Thismodeltakesinto accountall of the above models as special cases. Weare goingtoprovethe stability of the equilibrium points,the existence and the uniquenessconditionsoflimitcycles. Several known theorems will be easily derived again as anillustrationof our theorem.

Themethodused in thispapercanbeemployed foruse in thestudy of general Kolmogorovsystemsandwillbe published elsewhere for^furtherstudy.

2. THE MODEL.

Consider themodel

dx

d--- ^{(x) (F(x) (3)}

dy

a---: ^{o(3) (V,(x)}

⁺

⁽³⁾⁾

(2.1)

wherex is thepreydensity,yisthepredatordensity,

(x)F(x)

isthe intrinsicgrowthrateof the preyintheabsence ofpredators,and

(y) (3)

istheintrinsicrateof the increasing

(or

decreasing)of thepredator. Theterm

(x)zt (3)

represents the functionalresponse ofthepredator,i.e.

#(x)(3)

x

istherateofpreyconsumptionper predator. Mostof the authors simply takezt(y)= y,but a functionzt(y) that increasesslower than the linear couldbeusedtomodel interference amongpredatorswitheachother’s hunting, or fasterthanthe linear could beusedtomodelpredatorcooperation

[3].

^Theterm0(y)p(x)istheresponseof the predator, which means the difference of the actualrateof increase and the intrinsicrateofincreaseof thepredator.

Inparticular,

efy)(0) + (3)

y isthe deathrate ofthepredatorinthe absence ofprey.

Forthefollowingdiscussion, we needto assume:

(H,): ,p,

st,

O,

e C

([0, m));F CI(0, o),F(0) (0, oo,](0) r(0) t)(0) =(0)

=0,

’>0

^{for x > 0,t’>0,}

’

^>0,

’

^{0 for y > 0" there}

exists

-

^>0 such thatp() 0, p’(x) >0 for x

-.

Moreover,

(x)

isbounded bysomelinearfunction forO<xK.

(HI*) (H2)’

^Thecurve

t(3) F(x)

0 is defined for allx> 0, and

p(x)

+ (y) 0

isdefinedfor 0<x ^_< K.

(Ha)"

There existsK>" suchthat

F(K)

O,

F’(K)

^<0,

F(x)

>0 forall0<x <K, and forany

-

^>K,

^F’(-)

⁰

^ifF(-)

^0.

Moreover, theseexists aK* < such that

F(K*)

^{0 and}

F(x)

^{0 for} any x > K*.

(H4)"

There exist positive numbers

M

and

s

^suchthat

^t(y)> M0(y

^for Y

>-es,

and also there exist positive N and eN ^suchthat

0(Y)

^>

Ny

for y ^> eN

The constantKin

(H1)

and

(H2)

isthe same as in

(Ha).

Itispossibletohave

F(0)

in mostofthisdiscussion.

In

that case

(0,0)

isnolongeran equilibrium point.

This discussion isinthe interiorofthe firstquadrant.

Clearly, system

(2.1)

consistsof those studiedbyLotka and Volterra,Gause, Rosenzwieg and MacArthur

(1963),

GeneralizedGause

(Freedman, 1980),

Hsu

(1978),

Hsu, HubbellandWaltman

(1978),

Kazarinnoff and Driessche

(1978),

Cheng

(1981),

LiouandCheng

(1988), Kuang

^andFreedman

(1988), Huang (1988), Huang

and Merrill

(1989).

For example,

Kuang

andFreedmanmodel

(1988)

isa special case of system

(2.1)

with

F(0)

<

, F(x)

< 0 for all x> K,and

(3)

0. Also,theassumptions

(H*), (H)

^and

(H4)

employed here butnot employedin

Kuang

and Freedman webelieveshould be required there, also.

3. THEOREMS AND PROOFS.

Clearly,system

(2.1)

withassumptions

(H1) (H4)

has a positive equilibrium(x’,y’),where

" ^x"

^{< K and}

one or more saddles, forexample,

(K,0),

and

(0,0)

(if

F(0)

<

Forthe stability of(x’,y’) we have:

THEOREM3.1. Let

n(x,

y)

(x)F’ (x)

+

e(Y)’ (Y).

H(x*,y*) <0, the equilibrium (x*,y*) is stable, whileH(x*,y*) >0 unstable.

PROOF. The Jacobianofsystem

(2.1)

^at (x*,y*)is

j(x

.,y ,) ( ^{(x ")F’(x ")’} -(x ")’(Y ")

I

e(y

*)’(x *),

e(y *)’(y

*)

and the signs ofthereal parts oftheeigenvalues are determinedby

t(x ",y ’) =(x ")’(x ")

+e(y

")’(y ").

(3.1)

But

(2.1)

implies that

Hence, for all

>-

0, whichisa contradiction.

If

(xo,

yo)

fl

{(x,y): p(x)

+@(y) O,F(x) -t(y)

< 0,y >0

},

^then

(2.1)

^and

(Ha)

implythat either

x(t)

decreasestosomeconstant,orthereexists t2 >0 such that

x(t2)

< K.

By

the same argument as the case

x(0)

< K,it isnecessarythat

x(t)<K

^{for all >_}

t.

x(t)

^_<

TI=

^max

^{K,x(0)}.

If

(xo, Yo)

{(x,y):

p(x)+(y)>_O,F(x)-t(y)>O,y>O},

^{by the phase} portrait analysis, the trajectorystartingat

(x0, Y0)

willcross theboundaryof into

fl.

Hencethere exists t3>0, such that

x(t)<K

^{for all}

e

t3, and

x(t)<K

for all

>-

0.

Therefore,

x(t)

^isbounded.

x’ (t 1) #(K)t(K)

<0 Itiseasyto seethat Theorem 3.1 isvalid.

THEOREM 3.2.

Suppose

(i)

^{there exists}a

co

^> such that

M/co’(x) ’(x)

^<_ 0 for allx _> 0 and lim

(--M-M (x)-(x))

^C(R)>0,

(ii)

F(x)

<0forx> K*, whereK* isasin

(Ha).

Let

x(t),

y(t)be the solution of system

(2.1)

with a positive initial condition

x(0)=

x0>0, y(0)= y0>0. Then thereexists T>0 suchthat

O<x(t)<T

and O<y(t)<Tfor all t->0, and exists

to >-

^{0 such that 0}

<x(t)

<K for all >_

to.

PROOF. Bythephaseportrait analysis,itisclearthatx

(t),

y

(t)

>0 for all 0. If x

(0)

< K,thenx

(t)

< K for all >_ 0. Otherwise, there exists tl >0 such that

x(tl)

=K and

x’(tO

O.

558

To showy(t) is bounded, byusing (H4), we estimate the following +

-k(x) (F(x)

zt(y) +

O(Y) (P(x)

+

(Y)

Co dt dt Co

<_--qb(x)F(x)

O(Y)

(--qb(x)M p(x)

Co Co

<_

mO(x)F(x) -p(y)C(R)

Co

(qb(x)F(x) +

C(R)Nx(t)

C(R)(e(y Ny(t) C(R)N(¹

--- ^x(t)

^{+ y(t))}

< (q(x)F(x)+

C(R)Nx(t)) C(R)N(--x(t)

+ y(t))

Co Co

Mo ^No(-x(t)

1 ⁺

^y(t)),

Co

1

where

M0

^max

(-qCx)FCx)

+

NxCt)), No

^C(R)N.

CO Co

Since

x(t)

^isbounded, so are

(x)

^and

F(x).

^Thus

M0

^isaconstant.

Now, let

z(t)

satisfy

dz

Mo Noz(t)

dt

z(0) x(0)

+y(0).

(3.2)

Then

Z(t) z(O)e

^-lqOt

+’o Mo ^{(1 e-nOr). (3.3)}

Since

z(t) -,x(t)

+ y(t) >^{0 [or}all 0 and

z(t), x(t)

are bounded,y(t)isbounded. Let

to

^max

{tl

,t2

},

then 0<

x(t)

< Kfor all ^_>

to.

The proof of Theorem 3.2iscompleted.

Fromtheproof ofTheorem 3.2, we have

THEOREM 3.3. Under the assumptions asinTheorem 3.2,if(x*,y*) isstable,it isasymptotically global stable.

Inthe case when

(x*,

y’) isunstable, we have

THEOREM 3.4. There existsatleast onelimitcyclearound

(x"

,y’)if

(x"

,y’)is aunstableequilibrium point of system

(2.1)

PROOF. Let /’1 be the curve p(x) + (y)=0. If gl intersects theray x K,y >0at

Pa(xp,yp),

^then

F1 AP--’-U "’U ^B-’-U

whereA

(Xp1,0),

^B

(0,ypl),

^O

(0,0),

^isthe boundary and any trajectory whichintersects it eithercrossesfrom exteriortointerior orremainsonit. Therefore,by thePoincare-Bendixson annular region theorem, thereexists at least one limitcycle around

(x*,y*).

If

t’l

^doesnot intersecttherayx K, y >0 atall, letting m0= max

{F(x)}>0,

thenby

(H2),

there exists Yl suchthat

:(yl) m0.

Consider the auxiliarysystem dx

d" ^{ok(x) (mo r(y)}

ely O(Y) 0P(K)

+

(2y,))

dt

andthe trajectory startingatpoint

PI’(K,2y

will intersect the curve

t

since

(Hi*).

(3.5)

P1

X

Fig.

rt AP"-’

^LI

P-’tJ ^BO

^{00A is}

theboundaryof the Poincare-Bendixson annularregion.

Fig.2 If

/tdoes

not intersecttheray x=K, y>0,

r’2 APt OPt P2UPzB’UB-’r-O’U ’"

is theboundary ofthePoincare-Bendixson annular region.

Suppose

the intersectionis

P2(x,yp2 ).

Let

B’ (0,y&).

^Then

Fa AP---’ro P1 ’P

⁰

PB’

^IJ

^B’O

⁰

isaboundaryof anannularregion. Since

F(x)

^(y)

mo

^{(y) <0}

(x)

+

(y)

<

(/c)

+ (2yl),

(3.6)

anytrajectoryintersects

rz

will eithercross fromexterior tointerior orremainonit. The Poincare-Bendixson Theorem guarantees that thereis atleast onelimitcycleinside

rz.

Therefore, inany casesthere existsatleastone limitcyclearound

(x*,y*).

Now, for theprovingof the uniqueness theorem oflimit cycles,wedefine"

F

0 int

F

*,

F

*isthe Poincare-Bendixson’souterboundary, f

{(x, y)[(x,

y) e

f"-, x>0},

fj= (x,

y)

(x,

y)

,

sgn(W(x) +(y)

(- 1)J},

j= 1,2,

e0 rYn6,

and let

IC(x,

y) H(x,y)

(x)

+(y)

(x,

y)

fl

^0f2,

560

w(x,

y)

(#’ (x)F’ (x)

+

(x)F" (x))

((y) +

V:(x)) V:’ (x)H(x,

y), (x, y)

,

whereH(x,y) isdefined as in

(3.1).

Assume

(Hs): H(x,y)lt

> 0 and

W(x,y)ln

^< O, and

(3.7)

’ ^(y)

^{0 or}

’ ^(y)

^{;t 0 a.e. on}

^{6. (3.8)}

SinceW(x,y

")

<0 impliesH(x,y

)

> 0,Theorem 3.4 guarantees that system

(2.1)

haslimitcyclesin

.

Suppose C1

^and

C2

aretwolimitcycles around

(x

,y

)

such that

C1

C

C2

and, without lossofgenerality,suppose

C:

isstable from inside.

Let Q

be the point on

C

^suchthat

xo=

^rain

^{{xl(x,y) C}.}

Since (y) isstrictly increasing,

Q

isunique,

we

claimthat LEMMA 3.5

xo . ^(3.9)

PROOF. If

’(y)

0

(or

(y)

0)

on10,then

(3.9)

isclearlytrue. If

’(y)

0.

Suppose

(see

Fgure

3),

Xo.

^>t’.

Let Q1

(xol

,YOl ^{be the}intersectionof theray x xo y >^{0 with}

it0.

^Since

Yo

^{>0 and}

^(3.8),

there exists

Qz (Xoa,yo)

^on

l0

^{such that}

’<xo

^{<xo, 0}<yo2 <yol Let

Define an auxiliary function

or, inthe other form,

By (3.7),

Wehave

Xo- Xo Xo " ^{Yoz }.}

th=min {-,

2 2

L(x,

y)

H(x,

y)

K(x

_O,y)

(W(x)

+

!(Y) ),

L(x,

y) (p(x) + (y))(K(x, y)

K(xo,

y)).

dK(x,

y)

<O.

dx

L(x,

y) 0 for

(x, y)

f]fl

{(x, y)10

<x<

xo. ^}.

Onthe other hand, L(x,y)iscontinuous inthe 1-neighborhood ofQ2"

Nq (Q2),

and

Nq ^(Q2)

^fl

(x, y)Ix xo}

(.

Thus,

H(xo, Yo2)

L(x, y)1(22 H(x02, YO2) W(--@) --@) ^(P(x’2)

⁺

^(Yo2))

H(xo, Yo)

^>0.

Hence, thereexists

(0,)

such that

L(x,

y) >0 for

(x,

y)N,(2)

n {(x, y)I-

<x<

xo),

which isa contradictionto

(3.13).

Therefore, theclaim

(3.9)

holds.

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

Wealso claim that on the periodic orbits

C,

we have

LEMMA

3.6

div((x)(F(x) t(y), O(y)Op(x)+l(y))dt

L(x,y)dt,

for i= 1,2.

PROOF: BytheGreenformula,

div

((x)(F(x)-t(y),e(y)(P(x)+(y))dt

ok’ ^{(x) (F(x)}

^{r(y)) +}

^{(x)F’ (x)}

⁺

e’ (Y) (7,(x)

+ (y)) +

O(Y)!’ (Y))dt

(V:(x) +l(y))

K(x,y)

+/j(y---’(q’(x)((y)-F(x))

o’

^(y)

((x)

^{/(y))l &}

OP(x)

+(y))(X(x, y)

K(xo,

y)dt L(x,y)dt, 1,2,

since,

((x)

+(y))

K(xo,

y)

+(y--- (O’(x)((y) -F(x))

O’

^{(y)(p(x)+(y)) dt}

=0.

Now, we areinthe position toprovethe following uniqueness theorem.

THEOREM 3.7. Inadditiontoassumption (Hs), if

(F(x)

^zr(y))

Ly(x,y) ^>-

^{0 for (x,y)efl,}

then there exists at mostone limitcycle in system

(2.1).

PROOF. As inFig. 4, let

t0

^intersect

C1

^at At, Az,

Cz

^{at B1,}

Bz.

Then

(3.16)

(3.17)

C1 =A1A4

^{U A4Q O QAI}

C2 =B3B2

^LI

B2B4

^U

^B4B5

^U

^BsB3.

562

Hence,

L(x,

y)at

t.(x,

y)at

cl c2

(3.18)

y:

xo

^{X 0 x}o ^Y:

Fig. 3 The assumption x

o

^>

.

results inacontradiction. Fig. 4 Itis impossiblethatthe system

(2.1)

hastwolimitcycles.

Itis nothardtoseethat

K.(x,

y)

dxdy +

f ^K(x,

^y)

^K(x

^O,y)

+

3 ^K(x,

^y)

^{o(y) K(x}

^O,

^{y) ay}

>0,

since

Kx

^{(x,y) _< 0 and}

Kx

^{(x,y) 0 for}

_{xal -<}

^x<K.

Suppose

A4Q’ y=y(x),

xo.

^{<_ x<}XA4, and

(3.19)

Then

B2B+"

y=y2(x), x₀^<_x^<_ x/t+.

[xA+ ^L(x,

^y)

" ^{xo (x) ((x)}

^n’(y2))dx

^f+A+

^+"o

^{(x) (3 L(x, Yt) :(y,))}

f’+ ^(F(x)

^(y)

^(t.(x,

^y)

^t.Cx,

^{yl) +}

^t.(x,

^{y2) ((y2) (y,)} _dx

’"o ^(x)(F(x)

^(y2))

^(.(x)

^(y,))

>0,

(3.20)

since

(3.17)

and (p(x)

+

(y2))(K(x,y2)

-K(xo.,y2)) >-

0 for x0 <_ x <_

xa+

Similarly,

Finally,

(3.21)

y)dt |^f"(p(x)+(y))(K(x, y)

K(x

O,y))dt JB4B

5 ^{(X(x, Y)ofy)- r(xo,}

^y)dy

f’+ ^{K(x, y) K(x}

^O,y)

(3.22)

sinceforx < xo, K(x,y)

K(

x

o,y)

^>_ 0.

From

(3.19)

^to

(3.22),

wehave

L(x,

y)dt ^f

L(x,

y)dt>

o.

C JC2

(3.23)

f

Since

L(x,y)dt

<O, so

L(x,y)dt<0.

JC1 JC2

NOW, ifwe canprove

C1

is nota semi-stablelimitcycle, then

C2

mustbe internally unstable. Thatis

L(x, y)dt

0, c2

whichcontradictsthe fact

L(x,y)dt < O.

c2

Nowconsiderthe followingsystem containingaparameter dx

d--: ^{(x) (l(x) (y) t}

^r

d--: ^{Ors) ((x)}

^{+ (y)}

(3.24)

where

Let

and

//’(x,

y)

(x)iV’ (x) +

o(y)’

0’) X’(x,

y)

/7(x, Y.__2__)

_{(x, y),}

_{v 2}

(x) +@)

W(x,

y)

(’ (x)1’ (x)

+

(x)" (x) ((y)

+

(x)) ’(x)/7(x,

y).

Clearly,ifyissmallenough,thenallthe assumptions for system

(2.1)

aresatisfied forsystem

(3.24).

Thus,if system

(3.24)

hastwo limitcycles and

L’’z,

C

L’z,

^{then wehave}

div(1 y, Q)dt

<

div(1 r, Q)dt

< O.

J

(3.25)

Furthermore, let

Then

Q(x,y,,)

(3.26) fl(x,

y,

,) tan-’/Y(x,

y,

)

> 0

(3.27)

forall ordinary points

(x,y)

ofsystem

(3.24).

Also, the equilibrium points ofsystem

(3.24)

^are not dependent on

,.

Thus, system

(3.24)

^{forms a} generalized rotatedvectorfield in

Accordingtothetheory ofgeneralizedrotatedvectorfield

(see [9],

for

example),

for sufficiently small y, >

0, system

(3.24)

producesa generalized limit cycle

Ca

^C

Ca

^{whichis atleaststable}internally and ageneralizedlimit cycle

Cz

^D

^C,

^{whichis at}least unstable on oneside. This isacontradiction to

(3.25).

^We,thus,completethe proofof Theorem3.7.

4. EXAMPLES AND

DISCUSSION.

EXAMPLE 1. (Huang 1988

[5], Huang

and Merrill, 1989

[6])

dx

(x) (F(x)

(y) dt

dt

(4.1)

withthe assumptions

(H+)-(H4)

in

[6].

Itiseasytoseethat system

(4.1)

^isaspecial^casethat (y) 0 in

(2.1);

andthe assumptions

(H;)-(H,0

in Chapter 2 ofthispaperare satisfied. Therefore, Theorem 3.1 implies that if

F’(x )

>0 the equilibrium point (x*,y*)of

(4.1)

is unstable,andif

F’(x*)

< 0it isstable. Theorem 3.4tellsus thatwhen

(x*,y*)

isunstablethere exists atleastonelimitcyclein

(4.1).

Forthe uniquenessoflimitcycles, byTheorem 3.7, we have:

THEOREM 4.1. (Huang andMerrill

[6])

Inadditiontothe assumptions

(H;)-(H4)

ⁱⁿ

[6],

if

F’(x’)>O

and

(’(x)F’(x)) ^O’(x)

thenthereis auniquelimit cycle aroundtheequilibrium point (x’,y

)

in

(4.1).

PROOF. Since (y) 0,

(Hs)

ishold. Furthermore, if (y)=O,

F’(x *)

^{>0 and}

(q(x)F’(x) , ^,(x) ),

⁰

L(x,

y)

#(x)F’ (x)

p(x)

(4.2)

(xo)

which isonlya functionofx. Hence

Lr(x,y

^0,and consequently

(F(x)

:t(y))

Ly(x,y)

^{0 for} (x,y) Ef.

Therefore,the conditions ofTheorem 3.7are allsatisfied. Employingtheorem 3.7 will end theproofofTheorem 4.1.

REMARK. In

[6],

when weprovedthe uniquenessoflimitcyclesin

(4.1),

weemployedthe Zhang theorem

(or

equivalent, Cherkas and Zhilevich

theorem). But

inthispaperwedonotneedtouseit. Wealsomay simplify ourassumptionsfor the existence anduniqueness^oflimitcyclesbecause someoftheassumptions^aremadeforthe globalstability such as Theorems 3.2 and 3.3.

The ZhangtheoremandCherkas and Zhilevichtheoremcanbe foundin

[9].

EXAMPLE2. (Kuangand Freedman, 1988

[7])

dx

a-T ^xs(x)

dY _tl(y)

(_ ,+ q(x))

(4.3)

dt withthe assumptions

(H1)-(Hs)

in

[7].

Clearly, system

(4.3)

isa special case of

(4.1)

^{andhence ifall}the assumptionsinthispaperare satisfied, Theorem3.7isapplicable.

Theoriginalproof ofthe uniquenessoflimitcyclesin

[7]

isbasedonZhang’s^theorem. However,since the assumptions

(H1 *), (Ha)

^and

(H4)

inthispaperarenotassumedthere,the existence oflimitcyclesis notguaranteed and someargumentsneedtobemodified.

Asanexample,let us consider the following system:

dx x

(1 +

2x x

2) yx

at (4.4)

=(-+x),

dt

whichsatisfiesall therequired hypotheses by

Kuang

andFreedman

[7].

Unfortunately, since

(4.4)

ⁱⁿ

[7],

"=

e(s’) (s+x’)

2 2

---’ ^(4.5)

-x+

By (4.1)

in

[7],

X+x* =x. Hence 2

(4.6)

Therefore, u has no definition on ,+

566

and consequently the hypothesis that (v)/ F(u) 0is definedforall ue

(-

^m,+m)hasnotbeen sattsfed. Thus, Zhang’stheoremis notapplicable and the uniquenessoflimitcyclescannotbe obtainedbytheargumentm

[7].

Clearly,system

(4.4)

does satisfy^{all the}requirements of Theorem^4.1. Soifourtheoremisemployedwecan stillhave the uniqueness result for system

(4.3).

The idea usedinthispaperispossible for use in determining the uniquenessconditionsoflimitcyclesinthe general Kolmogorovsystem. Thiswork will be publishedina separatepaper.

ACKNOWLEDGMENT. Theauthor extendshisthankstoProfessorStephen J.Merrillforhisvaluable discussions.

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