Multiple Limit Cycles for Predator- Prey Models

Transcription

1 Multiple Limit Cycles for Predator- Prey Models Hofbauer, J. and So, J.W.H. IIASA Working Paper WP September 1989

2 Hofbauer, J. and So, J.W.H. (1989) Multiple Limit Cycles for Predator-Prey Models. IIASA Working Paper. IIASA, Laxenburg, Austria, WP Copyright 1989 by the author(s). Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting

3 WORKING PAPER MULTIPLE LIMIT CYCLES FOR PREDATOR-PREY MODELS Josef Hofiauer J. Wai Hung So September 1989 W P lnternat~onal lnst~tute for Appl~ed Systems Analysis

4 MULTIPLE LIMIT CYCLES FOR PREDATOR-PREY MODELS Joaef Hofbauer J. Wai Hung So September 1989 WP 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 do not necessarily represent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

5 Foreword One of the classical objects of study in mathematical ecology is the predator-prey interaction. In particular, the well-known model by Gause exhibits a rich dynarnical structure. 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 stability criterion of Hsu 131. Alexander B. Kurzhanski Chairman System and Decision Sciences Program.

6 Multiple Limit Cycles for Predator-Prey Models' 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. Introduction. We consider the classical Gause predator prey model where x represents the density of the prey and y that of the predator. The death rate J of the predator and the conversion factor c are positive numbers. The growth rate g(x) of the prey and the predator response function p(x) are assumed to satisfy for some K > 0 and (x - K )g(x) < O for x 2 O,x # K, (1.2) The prey isocline is given by y = h(x) := # and is assumed to be concave down, i.e. hfl(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 hl(x*) < 0 or equivalently x* > ai: and it is Research was done while the second author was visiting IIASA, Laxenburg, Austria. Research partially supported by FWF of Austria, NSERC of Canada and CRF of the University of Alberta.

7 unstable if hf(x*) > 0, that is z* < 2. It is known that if 3i: = 0, i.e. the prey isocline is always decreasing, then E* is globally stable, see Hsu [3, Theorem 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] that 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. The purpose of this note is to construct counterexamples which satisfy the above conditions, but with the (locally stable) interior equilibrium being surrounded by (at least) two limit cycles. 2. Multiple Limit Cycles. The idea for constructing an example with multiple limit cycles is as follows. 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 Ef. When 3 increases beyond i, xz passes i, so that E,* becomes stable and there is a Hopf bifurcation at 3 = i. If we can ma.ke this Hopf bifurcation subcritical, there will be an unstable limit cycle bifurcating from E,* for s slightly larger than i. Hence there will be at 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) at x = x: we have D,(x:) = hf(x:), in particular Di(S) = 0. Hence hf(x:) equals the real part of the eigenvalues at Ei, up to a positive 8 constant. Since &hf(xi) = hff(x:)e58 < 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 = - ipl(%) - ~h~~

8 and We will show in section 3 that it is ~ossible to 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. An Example. We conclude by giving a concrete example satisfying all the assumptions in sections 1 and 2. Let x c = 1, g(x) = (1 + x)(1 - -)(3-4x + 2s2) and ~(x) Then g and p satisfy (1.2) and (1.3) with K = 3. Also 3 = x(3-4x + 2x2) satisfies (1.4). Moreover, i = 1 and also satisfied. = 1. Using (2.2), Dy(i) = and (2.3) is 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. 12, (1981). 2. B.D. Hassard, N.D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, S.-B. Hsu, On global stability of a predator-prey system, Math. Biosci. 39, 1-10 (1978).