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May  2006, 6(3): 525-534. doi: 10.3934/dcdsb.2006.6.525

On the number of limit cycles in a predator prey model with non-monotonic functional response

E. González-Olivares 1, , B. González-Yañez 1, , Eduardo Sáez 2, and  I. Szántó 3,

1. 

Instituto de Matemáticas, Grupo de Ecología Matemática, Pontificia Universidad Católica de Valparaíso. Casilla 4059, Valparaíso, Chile, Chile
2. 

Universidad Técnica Federico Santa María, Departamento de Matemática, Casilla 110-V, Valparaíso
3. 

Departamento de Matemáticas, Universidad Técnica Federico Santa María, Valparaíso, Chile

Received  August 2004 Revised  December 2005 Published  February 2006

In this work we analyze a Gause type predator-prey model with a non-monotonic functional response and we show that it has two limit cycles encircling an unique singularity at the interior of the first quadrant, the innermost unstable and the outermost stable, completing the results obtained in previous paper [12, 17, 26, 28].
    Moreover, using the Poisson bracket 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.
Keywords: stability., bifurcations, predator-prey models.
Mathematics Subject Classification: 92D25, 34C, 58F14, 58F2.
Citation: E. González-Olivares, B. González-Yañez, Eduardo Sáez, I. Szántó. On the number of limit cycles in a predator prey model with non-monotonic functional response. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 525-534. doi: 10.3934/dcdsb.2006.6.525
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