Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure

Jaume  Llibre; Claudio  Vidal

doi:10.3934/dcdsb.2016026

Article Contents

2016, Volume 21, Issue 6: 1859-1867. Doi: 10.3934/dcdsb.2016026

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Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure

Jaume Llibre^1, and
Claudio Vidal^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Universidad del Bio Bio, Concepción, Avda. Collao 1202

Received: October 2014

Revised: March 2016

Published: August 2016

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Abstract

A ratio--dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium point $E^*$ at the positive octant is unstable.
Here we characterize the existence of Hopf bifurcations for the systems. In particular we provide a positive answer to the above question showing for such models the existence of small--amplitude Hopf limit cycles being the equilibrium point $E^*$ unstable.

Keywords:

Mathematics Subject Classification: Primary: 34D23, 92D25.

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References

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