Singular perturbations of the Holling I predator-prey system with a focus

André Zegeling*, Robert E. Kooij

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

11 Citations (Scopus)
12 Downloads (Pure)


We study the occurrence of limit cyles in a model for two-species predator-prey interaction (referred to as Holling I). This model is based on the phenomenological fitting of the so-called functional response function to observed data in nature by a continuous, piecewise differentiable function. Of the original models presented by Holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet. We extend the work of Liu who showed that for certain values of parameters of the Holling I system at least two nested limit cycles will occur surrounding a focus. His case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities, i.e. a singular perturbation problem with slow-fast solutions. We prove that exactly two hyperbolic limit cycles occur after perturbation.

Original languageEnglish
Pages (from-to)5434-5462
Number of pages29
JournalJournal of Differential Equations
Issue number6
Publication statusPublished - 2020

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • Functional response
  • Generalized Gause model
  • Holling
  • Limit cycles
  • Singular perturbation


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