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On the uniqueness and nonexistence of limit cycles for predator-prey systems. (English) Zbl 1042.34060

Here, the generalized Liénard system \(x'=\phi(y)-F(x),\;y'=-g(x)\), is considered. The authors give new results on the uniqueness, stability and nonexistence of a limit cycle for this system, under conditions where monotonicity of \(\frac{f(x)}{g(x)}\) is not required (\(f=F'\)). As in the classical approaches, the method of proof uses integration of the differential of “energy” along some orbits. Then, the results are used in the study of a predator-prey system with an equilibrium \((x^*,y^*)\) in the interior of the first quadrant; this system is reduced to a Liénard system by a change in the time variable and a powerful change of dependent variables. There are results on global asymptotic stability of \((x^*,y^*)\) as well as on the existence and uniqueness of a limit cycle which has the complement of \((x^*,y^*)\) as the region of attraction. These general results are illustrated with a number of examples.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
34D20 Stability of solutions to ordinary differential equations
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