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Canards and mixed-mode oscillations in a forest pest model

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  • Brøns, Morten
  • Kaasen, Rune

Abstract

We consider a three-variable forest pest model, proposed by Rinaldi & Muratori (1992) [Rinaldi, S., Muratori, S., 1992. Limit cycles in slow–fast forest–pest models. Theor. Popul. Biol. 41, 26–43]. The model allows relaxation oscillations where long pest-free periods are interspersed with outbreaks of high pest concentration. For small values of the timescale of the young trees, the model can be reduced to a two-dimensional model. By a geometrical analysis we identify a canard explosion in the reduced model, that is, a change over a narrow parameter interval from outbreak dynamics to small oscillations around an endemic state. For larger values of the timescale of the young trees the two-dimensional approximation breaks down, and a broader parameter interval with mixed-mode oscillations appear, replacing the simple canard explosion. The analysis only relies on simple and generic properties of the model, and is expected to be applicable in a larger class of multiple timescale dynamical models.

Suggested Citation

  • Brøns, Morten & Kaasen, Rune, 2010. "Canards and mixed-mode oscillations in a forest pest model," Theoretical Population Biology, Elsevier, vol. 77(4), pages 238-242.
    • Handle: RePEc:eee:thpobi:v:77:y:2010:i:4:p:238-242
    DOI: 10.1016/j.tpb.2010.02.003
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    References listed on IDEAS

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    1. David Gray, 2008. "The relationship between climate and outbreak characteristics of the spruce budworm in eastern Canada," Climatic Change, Springer, vol. 89(3), pages 447-449, August.
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    Cited by:

    1. Morten Brøns & Mathieu Desroches & Martin Krupa, 2015. "Mixed-Mode Oscillations Due to a Singular Hopf Bifurcation in a Forest Pest Model," Mathematical Population Studies, Taylor & Francis Journals, vol. 22(2), pages 71-79, June.
    2. Feibiao Zhan & Yingteng Zhang & Jian Song & Shenquan Liu, 2023. "Canard Mechanism and Rhythm Dynamics of Neuron Models," Mathematics, MDPI, vol. 11(13), pages 1-22, June.
    3. Rinaldi, Sergio, 2012. "Recurrent and synchronous insect pest outbreaks in forests," Theoretical Population Biology, Elsevier, vol. 81(1), pages 1-8.
    4. Huang, Juanjuan & Bi, Qinsheng, 2023. "Mixed-mode bursting oscillations in the neighborhood of a triple Hopf bifurcation point induced by parametric low-frequency excitation," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    5. Wen, Zihao & Li, Zhijun & Li, Xiang, 2019. "Bursting oscillations and bifurcation mechanism in memristor-based Shimizu–Morioka system with two time scales," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 58-70.

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