Multiple Limit Cycles for Predator-Prey Models

W O R K I N G P A P E R

MULTIPLE LIMIT CYCLES FOR PREDATOR-PREY MODELS

Josef Hofiauer J . Wai Hung S o

September 1989 W P-89-069

l n t e r n a t ~ o n a l l n s t ~ t u t e for Appl~ed Systems Analysis

MULTIPLE LIMIT CYCLES FOR PREDATOR-PREY MODELS

Joaef Hofbauer J . W a i Hung So

September 1989 WP-89069

Working Papera are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein d o not necessarily represent those of the Institute or of its National Member Organizations.

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

Foreword

One of the classical objects of study in mathematical ecology is the predator-prey in- teraction. In particular, the well-known model by Gause exhibits a rich dynarnical struc- ture. In this paper, a Gause type predator-prey model with concave prey isocline and (at least) two limit cycles is constructed. This serves as a counterexample to a global stabili- ty criterion of Hsu 131.

Alexander B. Kurzhanski Chairman System and Decision Sciences Program.

M u l t i p l e L i m i t C y c l e s f o r P r e d a t o r - P r e y M o d e l s ' Josef HOFBAUER

Instilut fiir Malhemalik, Universilat Wien Strudlhofgasse 4 , A-1090 Vienna, Austria

Joseph W.-H. SO

Department of Mathematics, University of Alberta Edmonton, Canada T6G 2G1

1. I n t r o d u c t i o n .

We consider the classical Gause predator prey model

where x represents the density of the prey and y t h a t of the predator. T h e death rate J of the predator and the conversion factor c are positive numbers.

T h e growth rate g ( x ) of the prey and the predator response function p ( x ) are assumed t o satisfy

( x - K ) g ( x )

<

O for x

2

O , x

#

K , ( 1 . 2 ) for some K

>

0 and

T h e prey isocline is given by y = h ( x ) :=

#

and is assumed to be concave down, i.e.

h f l ( x )

<

O for x 2 0. ( 1 . 4 )

Under these assumptions, the interior equilibrium E* = ( x * , y * ) exists and is unique. Let 5 be the unique point where h ( x ) attains its maximum. Then 0

<

^ai:

<

K. E* is locally stable if h l ( x * )

<

0 or equivalently x*

>

ai: and it is

Research was done while the second author was visiting IIASA, Laxenburg, Austria. Re- search partially supported by F W F of Austria, NSERC of Canada and CRF of the University of Alberta.

unstable if h f ( x * )

>

0, t h a t is z *

<

2. It is known t h a t if 3i: = 0, i.e. the prey isocline is always decreasing, then E* is globally stable, see Hsu [3, Theorem 3.21.

In 13, Theorem 3.31, it was claimed that (1.4) together with local stability implies the global stability of the interior equilibrium. Later it was pointed out in [l] th a t the proof was 'not rigorously correct'. In spite of this, this condition seems to be still believed by some as a criterion for global stability. T h e purpose of this note is t o construct counterexamples which satisfy the above conditions, but with the (locally stable) interior equilibrium being surrounded by ( a t least) two limit cycles.

2. M u l t i p l e L i m i t C y c l e s .

T h e idea for constructing an example with multiple limit cycles is as fol- lows. We will use ³ as a bifurcation parameter. For ³

<

^{i :=} p ( i ) , the interior equilibrium E,* = (x:, y,*) is unstable and hence, by boundedness of solutions, there is an attracting limit cycle or an attracting invariant annulus surrounding E f . When ³ increases beyond i, xz passes i, so that E,* becomes stable and there is a Hopf bifurcation a t ^{3 =} i . If we can ma.ke this Hopf bifurcation sub- critical, there will be an unstable limit cycle bifurcating from E,* for s slightly larger than i. Hence there will be a t least two limit cycles.

Multiplying the vector field (1.1) by the positive function p(x)-1y8-' (where the real number

p

will be fixed later) we get

- 3

y = yo(-

+

^c),

^.

P(X)

Clearly (1.1) and (2.1) have identical phase portraits. The divergence of the vector field (2.1) is given by

Evaluating D,(x) a t x = x: we have D,(x:) = hf(x:), in particular D i ( S ) = 0.

Hence hf(x:) equals the real part of the eigenvalues a t

Ei,

up to a positive constant. Since & h f ( x i ) = hff(x:)E58 8

<

0, the transversality condition for a Hopf bifurcation is satisfied.

Differentiating with respect to x we obtain D:(x) = h"(x)

+ 8%.

Now we choose

p

such that D \ ( i ) = 0 i.e.

p

⁼

-

i p l ( % )

-

~h~~

and

We will show in section 3 that it is ~ o s s i b l e t o find functions h and p obeying the assumptions (1.2) - (1.4) and satisfying

Then D;(x)

>

0 for x close to (but different from) 2. Hence, by a theorem of Bendixson, Ef is repelling. Consequently the Hopf bifurcation is subcritical and there are small limit cycles for s slightly larger than i.

Alternatively, one could also follow the procedure as described in [2]. In their notation (see p. 90 of [2]) Re c1 (0) leads to the same expression as given in (2.2) for our D;(2), up to a positive factor, and hence is positive by assumption (2.3). Consequently, their pz and

p2

are both positive. Therefore there is a unique unstable limit cycle bifurcating from El for ³ slightly larger than i .

3. A n E x a m p l e .

We conclude by giving a concrete example satisfying all the assumptions in sections 1 and 2. Let

c = 1, g(x) = (1

+

^{x)(1 - -)(3} x ^{- 4x}

+

2 s 2 ) and ~ ( x ) = x(3 ^- 4x

+

^2x2)

3

Then g and p satisfy (1.2) and (1.3) with K = 3. Also

satisfies (1.4). Moreover, i = 1 and = 1. Using (2.2), D y ( i ) = and (2.3) is also satisfied.

References.

1. K.-S. Cheng, S.-B. Hsu and S.-S. Lin, Some results on global stability of a predator-prey system, J. h4ath. Biol. 1 2 , 115-126 (1981).

2. B.D. Hassard, N.D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981.

3. S.-B. Hsu, On global stability of a predator-prey system, Math. Biosci.

3 9 , 1-10 (1978).