Conditions for uniqueness of limit cycles in general predator-prey systems

Conditions for Uniqueness of Limit Cycles in General Predator-Prey Systems XUN-CHENG HUANG Department of Mathematics,

New Jersey Institute

of Technology, Newark,

NJ 07102

AND

STEPHEN

J. MERRILL

Department Milwaukee,

of Mathematics, WI 53233

Received

9 August

Statistics and Computer Science, Marquette

Umversity,

1988; revised I5 December 1988

ABSTRACT Using a transformation to a generalized Lienard system, theorems are presented that give conditions under which unique limit cycles for generalized ecological systems, including those of predator-prey form, exist. The generalized systems contain those studied by Rosenzweig and MacArthur (1963); Hsu, Hubbell, and Waltman (1978); Kazarinnoff and van den Driessche (1978); Cheng (1981); Liou and Cheng (1987); and Kuang and Freedman (1988). Although very similar and Freedman, the conditions presented original the prey allowable.

1.

(untransformed)

functions.

as shown

in the examples

in approach to the result presented by Kuang here are of simpler form and in terms of the

The results also apply to more general provided.

In particular,

growth

an immigration

terms for term

is

INTRODUCTION

Finding models that display a stable limit cycle, an attracting stable self-sustained oscillation, is a primary problem in mathematical ecology. If a model is to describe some particular ecological system, structurally stable features, like limit cycles, should be visible in both if displayed in either. For that reason, finding conditions that guarantee the uniqueness of a limit cycle is an important problem. Unfortunately, most of the classical results concerning periodicity in planar systems (Bendixson’s criterion, PoincarbBendixson theorem, Dulac’s criterion, and topological methods, for example) deal only with the existence or nonexistence of periodic solutions. Bifurcation results can generally produce only “small” solutions and guarantee uniqueness locally. Recently, there have been a number of results concerning the uniqueness of limit cycles for generalized Lienard equations. These results grew from a MA THEMA

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Co., Inc., 1989 New York, NY 10010

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48

XUN-CHENG

HUANG AND STEPHEN J. MERRILL

theorem of Zhang in 1958 [l]. Cherkas and Zhilevich in 1970 [2] gave a modified version and provided the first proof of the generalized result. In addition, Zhang in 1986 [3] gave a proof of her original theorem. By way of transformations, generalized predator-prey models can be put in the form of a generalized Lienard equation. For uniqueness of limit cycle results, this was done independently by Kuang and Freedman [4] and by Huang [5]. In this paper, transformations essentially identical to those employed by Kuang and Freedman are used to obtain conditions guaranteeing uniqueness of limit cycles. The condition obtained is more easily applied (being in terms of the untransformed functions) and can be applied to a wider range of situations. There are many examples in the literature where the uniqueness of limit cycles can be proved by application of the theorems discussed here. For example, those studied by Rosenzweig and MacArthur [6], Hsu et al. [7], Kazarinnoff and van den Driessche [8], Cheng [9], Liou and Cheng [lo], Kuang and Freedman [4], and Huang [5]. In Section 4 another two examples are presented that cannot be currently handled by the theorem of Kuang and Freedman [4]. In addition, in Theorem 3.2, explicit conditions guaranteeing the existence of a limit cycle for this general system is presented. Both in that proof and in the discussion of global stability, assumptions on the predator (y) functions not necessary in the linear case discussed in Freedman [ll] were employed. Other hypotheses used here (Hz) are sufficient for verification of several of the conditions required in the Lienard results. In Kuang and Freedman these were implicitly assumed to hold. 2.

THE MODEL We consider

the system g

=+(x)[F(x)- 4Y>l

(2.la)

$‘P(YM”)

(2.lb)

with the assumptions (I-I,): $7 $3 8, P E C’[O, co), (P(O)= a(O) = p(O)= 0, F E C'(0, co),

F(O)E (0,001; +'>O n’> 0,

p’>O

forxZ0;

fory>O;

there exists x* > 0 such that

#(x*1

=o,

$4 x*)

’

0,

and(x-x*)+(x)

>Oforxfx*

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CYCLES

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49

SYSTEMS

(H,): The curve a(y)F(x) = 0 is defined exist positive constants m, and m2 such that

+(x)6ml+m2x System (2.1) has at least and at least one equilibrium, limX,,+ F(x) = + 00 .] When in the y direction. Since the

J(x*,y*)

the eigenvalues

forx>O.

(HZ)

two equilibria, (0,O) and (x*, y*), if F(0)
G(x*) F’(x*)

=

for all x > 0; and there

I

-cp(x*)+(Y*)

P(Y*) JI’(x*)

’

0

are given by

The signs of the real parts of the eigenvalues and as a result.

are determined

( x*, y*) is stable

if F’(x*)

( x*, y*) is unstable

if F( x*) > 0.

by +(x*) F’( x*),

< 0,

while

The case where F’(x*) = 0 is undecided, as (x*, y*) could be either a center or a focus. We will prove a uniqueness theorem of limit cycles for system (2.1) under assumptions (Hi) and (H,). Note. Although (p, 4, rr, and p are given on [0, + co), it is easy to extend them to the whole real axis. One can consider, if necessary, +, #, rr, p E C’(-oo,+co). In order to guarantee the existence of limit cycles, we need two additional assumptions: (H,): F(K)

There exists a K > x* such that =O,F’(K)


Also,foranyK>KwithF(K)=O, such that F(K*)=O

and

F(x)

>O

forO
F’(K)#O;andthereexistsa

and

F(x)

+O

foranyx>

K*
K*

50

XUN-CHENG

HUANG

AND STEPHEN

J. MERRILL

(H4): There exist positive numbers M and E such that n(y) for y > c and such that there exists a number y, with p(yl)

>y+c

z Mp(y)

forallxE[x*,K].

System (2.1) with assumptions (H,), (H,), and (H4) now has a positive equilibrium (x*, y*) and one or more saddles; for example, ( K,O) and (0,O) [if F(0) < co]. The model studied by Kuang and Freedman [4] is a special case of system (2.1) with F(0) < 00 and F(x) < 0 for all x > K. The more general form of F(x) allows examination of prey growth equations displaying multiple equilibria, first suspected in the field by Voute [12]. See also Southwood [13]. In the case where F(0) is undefined, (0,O) is not an equilibrium point. As will be seen in Section 4, this case corresponds to having a source in the prey equation. 3.

THEOREMS

THEOREM

A (Zhang

AND REMARKS [l. 31)

Consider the generalized Lienard system dx ;~i.=-h(y)-A(x),

(3.la)

dy z =g(x).

(3.lb)

Assume that (1) g(x) satisfies a Lipschitz condition on every finite interval, xg(x) G(+oo)= +oo, where G(x)

=l’g(x)

> 0,

dx;

(2) A(0) = 0, A’(x) is continuous, and A’(x)/g(x) is nondecreasing for x in (- oo,O) and (0, + 00); A’(x)/g(x) f constant in a neighborhood of x = 0; (3) h(y) satisfies a Lipschitz condition on euery finite interval, yh( y) > 0 for y # 0, h(y) is nondecreasing; the curue h(y) + A(x) = 0 is defined on all XE(-CCI,+W); also h;(O).h’(O)#O in the case A’+(O).A’_(O)=O (see Notes 2 and 3 in [3]). Then system (3.1) has at most one limit cycle, and if it exists it is stable. The proof of Theorem A as well as some modification on the assumptions are given in [3]. Also, a modified theorem was proved by Cherkas and

LIMIT

CYCLES

Zhilevich theorem.

IN PREDATOR-PREY

([2], in English).

THEOREM

51

SYSTEMS

We summarize

these results

as the following

B

Assume that all the functions in system (3.1) are continuously differentiable and (1) xg( x) > 0 for x + 0; yh( y) (2) h(y) is nondecreasing, and XE(-co,+ccJ); (3) there exists an interval (x,, limit cycle for x < x1 and x 2 x2,

> 0 for y f 0; A(0) = 0, A’(0) < 0; the curve h(y) + A(x) = 0 is defined for all xz) with x1 < 0 < x2, such that there is no and A’( x)/g( x) is nondecreasing for x in

(x1,0) and (0, x2). Then system (3.1) has at most one limit cycle, and if it exists it is stable. Our result on uniqueness TffEOREM

is the following theorem.

3.1

In addition to assumptions (HI) and (H,),

assume that

(H,) - F(x)+(x)/+(x) is nondecreasingfor - 00 < x < x* and x* < x
Y=?(u)+Y*

(3.2)

system (2.1) to

du ;i~=~~(~(u)+x*)[F(C(u)+x*)-.(ll(u)+~*)l,

(3.3a)

do -= dt

(3.3b)

Here t(u)

~P(“(“)+y*)~(s(“)+“*)’

and n(v) satisfy the initial-value

problems

5’(u) =+(5(u)+x*)?

5(O) = 0,

(3.4)

n’(u) =P(?(u)+Y*)T

q(O) = 0.

(3.5)

and

[The initial conditions ensure that (x*, y*) will be transformed into (0,O) in the new variables.] Since (p and p E C’, there exist a unique functions t

52

XUN-CHENG

HUANG

AND STEPHEN

J. MERRILL

and 11 satisfying (3.4) and (3.5). Clearly, (Hz) guarantees that E(u) is defined for all UE(-cc,+m). Moreover, Range{n(u)} =(-co,+co). In fact, for any ui > u2 > 0,

and thus n’(u) is strictly increasing 1)(u)-7)(O)

for u > 0. By the mean value theorem, for some u, E (0,~).

=9’(uo)(u-0)

Hence, 9( u> ’ 1)‘(O)u9 which approaches + cc. By the suitable (- oo,O), we will have n(u) 4 - 00. Now, system (2.1) is reduced to

extension

of p(y)

in the interval

du

-=-[~(~(U)+y*)-~(y*)l-C-~(Z(u)+x*)+~(~*)l dt =-h(u)-A(u),

(3.6a)

$=q(((u)+x*)=g(u). Checking the hypotheses of Theorem B: Clearly, (u, u) = (0,O) is the only equilibrium point of system (3.6). (1) Since 6(O) = 0 and t’(u) > 0, E(u) > 0 for u > 0, that is, u and t(u) have the same sign. Since ~(u)J/(~(u)+x*)=[~(u)+x*-x*]J,(.$(u)+x*) > 0, ulj/(s‘(u)+x*) = ug(u) > 0.

(2)

4 u> -=_ td u)

F’(-t(u)+x*)+(5(u)+x*)

+(t(u)+x*>

isnondecreasingfor -ccO, h(u) is nondecreasing and uh (u) > 0 for u f 0. Also, h(u) + A(u)= 0 is defined for all u E (- cc, + co) by (H,) and the fact that Domain{ [( u)} = ( - w , + 00) and Range{ q( u)} = (- 00, + co). Thus, Theorem

B implies Theorem 3.1.

Remark I. For the other equilibria of system (2.1) for example, (0,O) and (K,O), Theorems A and B are not applicable. That is because after the change of variables, the second equation of system (3.6) becomes

$=g(u)=+(t(u)+x*),

(3.7)

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CYCLES

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53

SYSTEMS

in which the right-hand function g(u) = 0 only at u = 0 (i.e., only at x = x*); hence (x, y) = (0,O) and (K,O), etc. are no longer equilibria of system (3.6). Remark 2. According to the proof of Zhang’s theorem [3], equilibrium point (0,O) is unstable or system (3.6) does not nontrivial periodic orbits. Therefore, for system (2.1), if (x*, y*) system (2.1) does not have any nontrivial periodic solution around [Note that the Jacobian of (3.6) about (0,O) is also J(x*, y*).]

either the have any is stable, (x*, v*).

Remark 3. Several authors simply consider Zhang’s theorem (i.e., Theorem A) as a special case of Cherkas and Zhilevich’s modified theorem [2] where /3 = 0 and f, (x)/g( x) is nondecreasing for x in ( - o3,O) and (0, + 00). (See, for example, Kuang and Freedman [4].) But, actually, it is not. Remark 2 pointed out that under the assumptions of Theorem A, the equilibrium (0,O) may be stable, but in the Cherkas-Zhilevich theorem condition (3) guarantees that (0,O) is unstable. For example, the system

(3.8) which has a stable focus (O,O), satisfies all the assumptions in Theorem A but fails to satisfy the condition (3) of the Cherkas-Zhilevich theorem (see Kuang and Freedman [4], Theorem 3.1). Remark 4. Corresponding 3.1 could be reduced to (Hi)

- F’(x)+(x)/+(x)

to Theorem

B, the assumption

of Theorem

is nondecreasing

for a < x < x* and x* < x < b, where u, b are some constants x, and x2, respectively.

depending

on

Remark 5. In order to use the results of Zhang, it is necessary that h(r) + A(x) = 0 be defined for all x. This is actually a statement concerning the domain of A and the range of h. Both limit the nonlinearities that may be examined using these theorems. On the existence theorem. THEOREM

3.2 (Huang

of limit cycles of system (2.1) we present

the following

[5])

System (2.1) with assumptions (Hi), (H,), equilibrium or a limit cycle, or both.

and (H4) u/ways has u stable

Proof. Suppose (x*, v*) is unstable. We want to construct an outer boundary of an invariant region R that contains (x*, v*). Let L, be the line

54

XUN-CHENG

HUANG

AND STEPHEN

J. MERRILL

x = K. Since dx

co.

=-$(K)n(y)

Jf

I.,

the AB of L, is a transversal intersects it moves from right to left. Let M, E, and yr be numbers such that

of (2.1), and any trajectory

that

and NC

-sup i*<.rsz

K

(1

Let L, be the Line segment slope N, i.e., L,:

G(x)

~bmxMYl)BC passing

Ml

through

y-y,=N(x-K),

x*
Ii

point

B( K, y,) with

K.

Since

BC is a transversal of (2.1), and any trajectory right to left as shown in Figure 1. Let CD be a segment of the line L, :

y=N(x*-K)+y,,

intersecting

it

moves from

o
Since

CD is also a transversal from above to below.

of (2.1), and any trajectory

intersecting

it moves

LIMIT

CYCLES

IN PREDATOR-PREY

55

SYSTEMS

B I/‘%

.

.

‘I ’

--

\

rc . \

I

\ t I

\l

*r

\ *

\R

I I

0

I

0

K\

X*

/

\ \

FIG. 1. Any trajectory which intersects exterior to interior or remains on it.

My

/

\A

L -

6

LC

/

\’

I

-c

the boundary

.’ OAKDO

either

crosses

from

Obviously, the x and y axes are trajectories of system (2.1). So OABCDO is the desired boundary, and any trajectory that intersects it either crosses from the exterior to the interior or remains on it. Therefore, by a generalized version of the Poincart-Bendixson annular region theorem (see [14], there exists at least one limit cycle surrounding (x*, y*). Note. Hypothesis (H4) is necessary in this proof of the existence of limit cycles in system (2.1) to ensure that the trajectories cross as required. From Theorem 3.1, Remark 4, and Theorem 3.2, we have the following. THEOREM 3.3 (Huang In

[5])

uddition to assumptions (HI)-(H,),

if

(i) F’(x*) > 0 and (ii) - p(x)+(x)/#(x) is nondecreasingfor 0 < x < x* and x* < x < K, then system (2.1) has a unique limit cycle surrounding the equilibrium (x*, y*). Moreover, (iii) If 3K* such that F(x) < 0 for x > K*, then the limit cycle is stable and globally attracting in the first quadrant. (iv) If 3K* such that F(x) > 0 for x > K*, then the limit cycle is stable and at least locully attracting.

56

XUN-CHENG

HUANG AND STEPHEN J. MERRILL

Proofi Theorems 3.1 and 3.2 guarantee there exists a unique limit cycle around (x*, y*>_ Now let Q, be the region {(x, y)]y > 0, x > K, F(x)- n(y) < 0}, and let r, be any trajectory initiating in Q, [say, at (x,,, A,)]. If (x,,,.~y,,)~~~={(~,y)~y~O,K~-,~~~-,~tF(~~~O, and F(x,f= F( x1) = 0) c Sit,, then since

T, will cross line x = x, and move to the left of x = x1 in finite time. We now suppose (x,,, yO) E fJ[’ = ((x, y)]y b 0, K < x1 < x < x7, F(x)r(y) < 0, F(x) > 0, and Ff x, ) = F(x2) = 0} c f2,. Since +/dx c 0, r, goes upward to the left. Without losing generality, assume

and

we have

So I?, will cross x = xi to the left too. Hence, any r, initiating in Q2, will eventually intersect line x = K and move into the left of x = K. Let Q, be the region ((x, y)]y z 0, x > K, F(x) - m(y) > 0} and let I, be any trajectory initiating in !A, [say, at (Xol ?,,)I. Since &/dx > 0, r1 moves upward to the right. If there exists X, < -K < K* such that F(E) = 0, and F(x)>0 for Yo 0. K < x < K, F(X)n(y) < 0) c 0,. Thus, if 3K* such that F(x) < 0 for x > K*, all trajectories initiating in 0, u 52, = {(x, y)jy > 0, x > K } will eventually cross line x = K from right to left, and hence the limit cycle is globally attracting. (See Figure 2.)

LIMIT

CYCLES

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57

SYSTEMS

Y

FIG. 2. All trajectories initiating in Q; ally cross line x = K from right to left.

u a;

u 8,

= ((x,

.v)ly

>

0,x > K) will eventu-

If there isn’t such a K, i.e., 3K* such that F(x) > 0 for some trajectories initiating (X,,, &) in 52’ = ((x, y)ly F(x) - a(y) > 0} might remain in a’. In that case the limit locally attracting. Note. Hypothesis (H4) is also necessary for this proof property. 4.

x > K*, then > 0, K* < x, cycle is only of the global

APPLICATIONS

There are many examples in the literature whose uniqueness of limit cycles can be proved by the theorems in Section 3. Here we just provide two simple examples that have special growth terms for the prey. Example

I.

Consider

the system

$=x[F(x)-Y(Y+~)],

(4.1)

where F(x)

=

sinx+sin7r/l2, - 1 + sin 7r/12

x d IT/2 x > 7n/2,

-1-E

44x)=

I

sinx-2e 13

2

X6-z

2 ’

_1<,<1 2

2 2

X$

(4.2)

2

(4.3)

58

XUN-CHENG

Clearly, rium,

assumptions

(H,)-(H4)

HUANG

AND STEPHEN

J. MERRILL

are satisfied with the only positive equilib-

x* = ?! 4’

_v*G 0.4806508,

and K=13n/12,K*=3~r+?r/12 such that F(K)=F(K*)=O,F(x)>O for O< x K*. Since F’(x) =cosx for - 71/2 Q x < 7r/2, F’( n/4) = a/2 > 0. Thus, the equilibrium (x*, y*) is unstable. Also, for 0 < x < 7r/2,

Jz

6

X x-sinxcosx+Tcosx-Txsmx t

\ I

Let H(x)

fi sinxcosx+-cosx--xsmx. 2

=x-

fi

(4.4)

For 0 < x i 71/4, H(x)

> 0,

(4.5)

which is because

Jz.

1-Tsmx

>O

and

cosx(q

-sinxj

> 0

j For n/4
1

Q

>L--+Tcosl-Tsml 4 2

Jz.

Jz L 0.2853982 $ +0.841471A 0.0724377 > 0.

0.5403023) (4.6)

For 1 G x < ?r/2, H(x)

=x/L-qsinxj-cosx(dnx-$j

(4.7)

LIMIT CYCLES IN PREDATOR-PREY

59

SYSTEMS

d dx

= l-&2

l

( xsinx-cosx)

> 0.

(4.8)

(xsinx-cosx)

> 0

(4.9)

For n < x c: 13n/12, d dx

1

-F’(x)+(x) 4(x)

= l-J?/2

since x < cot x for R < x < 13~r/12. x ) is increasing on 0 < x < 7r/4 and From (4.5)-(4.9), - F’(x)$(x)/$( lr/4 < x < 137r/12. Theorem 3.3 implies that system (4.1) has a unique limit cycle that is globally attracting in the first quadrant. Example 2.

Consider

the system dx x=xg(x)-6(y)p(x)+r, (4.10)

x(0)

2

0,

y(0) > 0.

which is obtained by adding a positive source term r to the prey equation of the general Gause-type model studied recently by Kuang and Freedman [4]. We can prove THEOREM

4.1

Under the same assumptions (Hz)-(H7) (H, )‘I for O
of Kuang and Freedman [4] and

There exist K* z K > 0 such that Kg(K) Kandxg(x)+y K*;

+ y = 0, xg( x) + y > 0

if also

is nondecreasing for 0 < x < x* and x* < x < K, then system (4.10) has u unique limit cycle surrounding the positive equilibrium (x*, y*) that is globully attracting in the first quadrant.

60

XUN-CHENG Proof.

HUANG

AND STEPHEN

Rewrite system (4.10) into the standard dx

x

=Ax>

4 x) + Y P(X)

J. MERRILL

form

-2(_vJ]

(4.11a) (4.11b)

It is clear that assumptions (HI)-(H4) as well as conditions (i)-(iii) of Theorem 3.3 are satisfied. Thus, the conclusion follows immediately from Theorem 3.3. REFERENCES 1 2

Zhifen Zhang, Dokl. Akad Nouk USSR 119:659-662 (1958). L. A. Cherkas and L. I. Zhilevich, Some criteria for the absence

3

the existence of a single limit cycle, DI# Eq 6:891-X97 (1970). Zhifen Zhang, Proof of the uniqueness theorem of limit cycles of generalized

4

equations, Y. Kuang

5 6 1 8 9 10 11 12 13

14

Appl. At&. 23161-16 (1986). and H. I. Freedman, Uniqueness

predator-prey X. C. Huang,

of limit cycles and for

of limit cycles in Game-type

systems, Math. Biosci. 88:67-X4 (1988). Ph.D. Thesis, Marquette University, Milwaukee.

Wisconsin,

Lienard models

of

19X8.

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability condition of predator-prey interactions, Am. Nor. 47:209-223 (1963). S. B. Hsu, S. P. Hubbell, and P. Waltman. Competing predators, SIAM J. Appl. Muth. 35:617-625 (1978). N. Kazarinnoff and P. van den Driessche, A model predator-prey system with functional response, Murh. Biosci. 39:125-134 (1978). K. S. Cheng, Uniqueness of a limit cycle for a predator-prey Anul. 12:541-548 (1981).

system.

SIAM J. Mrrrh.

L. P. Liou and K. S. Cheng, On the uniqueness of a limit cycle for a predator-prey system, 1987 preprint. H. I. Freedman, Determinisnc Mathematicul Models in Populution Ecologv. Marcel Dekker, New York, 1980. A. D. Voute. Regulation of the density of the insect population in virgin forests and cultivated woods, Arch. Neer. Zool. 7:435-470 (1946). T. R. E. Southwood, Stability in field populations of insects, in The Muthemutrcul Theon, of the D_ynamicsof B~ologud Populutlons. II, R. W. Hiorns and D. Cooke, cds.. Academic Press, London, 1981 pp. 31-45 F. Albrecht et al.. The dynamics of two interacting 46:65X-670

(1974).

populations,

J. Math. ilnul. Appl.