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On the number of limit cycles in a predator prey model with non-monotonic functional response. (English) Zbl 1092.92045
Summary: We analyze a Gause type predator-prey model with a non-monotonic functional response and we show that it has two limit cycles encircling a unique singularity at the interior of the first quadrant, the innermost unstable and the outermost stable, completing results obtained in previous papers [J. Huang and D. Xiao, Acta Math. Appl. Sin. Engl. Ser. 20, 167–178 (2004; Zbl 1062.92070); S. Ruan and D. Xiao, SIAM J. Appl. Math. 61, No. 4, 1445–1472 (2001; Zbl 0986.34045); D. Xiao and S. Ruan, Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, No. 8, 2123–2131 (2001; Zbl 1091.92504); H. Zhu et al., SIAM J. Appl. Math. 63, No. 2, 636–682 (2002; Zbl 1036.34049)].
Moreover, using Poisson brackets we give a proof, shorter than the ones found in the literature, for determining the type of a cusp point of a singularity at the first quadrant.

MSC:
92D40 Ecology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
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