Multiple Limit Cycles for Predator-Prey Models^{1}

Josef HOFBAUER

Institut für Mathematik, Universität Wien Strudlhofgasse 4, A-1090 Vienna, Austria

Joseph W.-H. SO

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

Abstract. We construct a Gause type predator-prey model with concave prey isocline and (at least) two limit cycles. This serves as a counterexample to a global stability criterion of Hsu [3].

1. Introduction.

We consider the classical Gause predator prey model

\begin{aligned}ẋ & = xg(x) − yp(x), \\ ẏ & = y(−s + cp(x)), \end{aligned}  (1.1)

where x represents the density of the prey and y that of the predator. The death rate s 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

(x − K)g(x) < 0  for x ≥ 0, x ≠ K,  (1.2)

for some K > 0 and

p(0) = 0, p(K) > \frac{s}{c}, p′(x) > 0  for x ≥ 0.  (1.3)

The prey isocline is given by y = h(x) := \frac{xg(x)}{p(x)} and is assumed to be concave down, i.e.

h′′(x) < 0 for x ≥ 0. (1.4)

Under these assumptions, the interior equilibrium E^{∗} = (x^{∗}, y^{∗}) exists and is unique. Let x̂ be the unique point where h(x) attains its maximum. Then 0 ≤ x̂ < K. E^{∗} is locally stable if h′(x^{∗}) < 0 or equivalently x^{∗} > x̂ and it is unstable if h′(x^{∗}) > 0, that is x^{∗} < x̂. It is known that if x̂ = 0, i.e. the prey isocline is always decreasing, then E^{∗} is globally stable, see Hsu [3, Theorem 3.2].

In [3, Theorem 3.3], it was claimed that (1.4) together with local stability implies the global stability of the interior equilibrium. Later it was pointed out in [1] 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.

1Published (with a figure) in: Mathematical Biosciences 99 (1990) 71-75.

Research was done while the second author was visiting IIASA, Laxenburg, Austria, in the summer of 1989. Research partially supported by FWF of Austria, NSERC of Canada and CRF of the University of Alberta.

2. Multiple Limit Cycles.

The idea for constructing an example with multiple limit cycles is as follows. We will use s as a bifurcation parameter. For s < ŝ:= p(x̂), the interior equilibrium E_{s}^{∗} = (x_{s}^{∗}, y_{s}^{∗}) is unstable and hence, by boundedness of solutions, there is an attracting limit cycle or an attracting invariant annulus surrounding E_{s}^{∗}. When s increases beyond ŝ, x_{s}^{∗} passes x̂, so that E_{s}^{∗} becomes stable and there is a Hopf bifurcation at s = ŝ. If we can make this Hopf bifurcation subcritical, there will be an unstable limit cycle bifurcating from E_{s}^{∗} for s slightly larger than ŝ. Hence there will be at least two limit cycles.

Multiplying the vector field (1.1) by the positive function p(x)^{−1}y^{β−1} (where the real number β will be fixed later) we get

\begin{aligned}ẋ & = y^{β−1}h(x) − y^{β}, \\ ẏ & = y^{β}(\frac{−s}{p(x)} + c), \end{aligned} .  (2.1)

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

div = y^{β−1}h′(x) + βy^{β−1}(\frac{−s}{p(x)} + c) =: y^{β−1}D_{s}(x).

Evaluating D_{s}(x) at x = x_{s}^{∗} we have D_{s}(x_{s}^{∗}) = h′(x_{s}^{∗}), in particular Dŝ(x̂) = 0. Hence h′(x_{s}^{∗}) equals the real part of the eigenvalues at E_{s}^{∗}, up to a positive constant. Since \frac{∂}{∂s}h′(x_{s}^{∗}) = h′′(x_{s}^{∗})\frac{∂}{∂s}x_{s}^{∗} < 0, the transversality condition for a Hopf bifurcation is satisfied.

Differentiating with respect to x we obtain D′_{s}(x) = h′′(x) + β \frac{sp′(x)}{p^{2}(x)}. Now

we choose β such that D′ ; ŝ(x̂) = 0 i.e. β = −\frac{p(x̂)^{2}h′′(x̂)}{ŝp′(x̂)}. Then

D′′_{s}(x) = h′′′(x) + βs\frac{p(x)p′′(x) − 2p′(x)^{2}}{p(x)^{3}}

and

D′′ ; ŝ(x̂) = h′′′(x̂) − h′′(x̂) (\frac{p′′(x̂)}{p′(x̂)} − \frac{2p′(x̂)}{p(x̂)}) .  (2.2)

We will show in section 3 that it is possible to find functions h and p obeying the assumptions (1.2) – (1.4) and satisfying

D′′ ; ŝ(x̂) > 0.  (2.3)

Then Dŝ(x) > 0 for x close to (but different from) x̂. Hence, by a theorem of Bendixson, E∗ ; ŝ is repelling. Consequently the Hopf bifurcation is subcritical and there are small limit cycles for s slightly larger than ŝ.

Alternatively, one could also follow the procedure as described in [2]. In their notation (see p. 90 of [2]) Re c_{1}(0) leads to the same expression as given in (2.2) for our Dŝ(x̂), up to a positive factor, and hence is positive by assumption (2.3). Consequently, their μ_{2} and β_{2} are both positive. Therefore there is a unique unstable limit cycle bifurcating from E_{s}^{∗} for s slightly larger than ŝ.

3. An Example.

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

c = 1, g(x) = (1 + x)(1 − \frac{x}{3})(3 − 4x + 2x^{2}) and p(x) = x(3 − 4x + 2x^{2}).

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

h(x) = (1 + x)(1 − \frac{x}{3})

satisfies (1.4). Moreover, x̂ = 1 and ŝ = 1. Using (2.2), D′′ ; ŝ(x̂) = \frac{4}{3} 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. Math. Biol. 12, 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. 39, 1-10 (1978).