Rocky Mountain Journal of Mathematics

A predator-prey system involving five limit cycles

Eduardo Sáez, Eduardo Stange, and Iván Szántó

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Abstract

In this paper we consider the multiparameter system introduced in \cite{sch}, which corresponds to an extension of the classic minimal Daphnia-algae model. It is shown that there is a neighborhood in the parameter space where the system in the realistic quadrant has a unique equilibrium point which is a repelling weak focus of order four enclosed by a global attractor hyperbolic limit cycle. For a small enough change of the parameters in this neighborhood, bifurcation occurs from the weak focus four infinitesimal Hopf limit cycles (alternating the type of stability) such that the last bifurcated limit cycle is an attractor. Moreover, for certain values of parameters, we concluded that this applied model has five concentric limit cycles, three of them being stable hyperbolic limit cycles. This gives a positive answer to a question raised in \cite{col, Llo}.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 6 (2014), 2057-2073.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1422885107

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-2057

Mathematical Reviews number (MathSciNet)
MR3310961

Zentralblatt MATH identifier
1312.34088

Subjects
Primary: 34C 58F14 58F21 92D25: Population dynamics (general)

Keywords
Stability limit cycles bifurcations predator-prey models

Citation

Sáez, Eduardo; Stange, Eduardo; Szántó, Iván. A predator-prey system involving five limit cycles. Rocky Mountain J. Math. 44 (2014), no. 6, 2057--2073. doi:10.1216/RMJ-2014-44-6-2057. https://projecteuclid.org/euclid.rmjm/1422885107


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