Rocky Mountain Journal of Mathematics

A predator-prey system involving five limit cycles

Eduardo Sáez, Eduardo Stange, and Iván Szántó

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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

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

First available in Project Euclid: 2 February 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Stability limit cycles bifurcations predator-prey models


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.

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  • G. Caughley and J.H. Lawton, Plant-herbivore systems, in Theretical ecology, principles and applications, R.M. May, ed., Blackwell Scientific Publications, Oxford, UK, 1981.
  • C.S. Coleman, Hilbert's $16^{th}$ problem: How many cycles?, in Differential equations models, Vol. 1., W. Lucas, ed., Springer-Verlag, New York, 1978.
  • N.G. Lloyd, Limit cycles of polynomial systems-some recent developments, in New direction in dynamical systems, Cambridge University Press, Cambridge, 1988.
  • N.G. Lloyd, J.M. Pearson, E. Sáez and I. Szántó, Limit cycles of a cubic Kolmogorov system, Appl. Math. Lett. 9 (1996), 15–18.
  • E. McCauley and W.W. Murdoch, Cyclic and stable populations: plankton as paradigm, Amer. Nat. 129 (1987), 97–121.
  • E. McCauley, W.E. Murdoch and R.M. Nisbet, Growth, reproduction and mortality of Daphnia pule Leydig: Life at low food, Funct. Ecol. 4 (1990), 505–514.
  • J. Roughgarden, Competition and theory in community ecology, Amer. Nat. 122 (1983), 583–601.
  • E. Sáez,, 2000.
  • M. Scheffer, S. Rinaldi and Y.A. Kuznetsov, Effects of fish on plankton dynamics: A theoretical analysis, Canad. J. Fish. Aq. Sci. 57 (2000), 1208–1219.
  • Stephen Wolfram, The Mathematica book, 5th ed., Wolfram Media, 2003.