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Nilpotent singularities and chaos: Tritrophic food chains

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  • Drubi, Fátima
  • Ibáñez, Santiago
  • Pilarczyk, Paweł

Abstract

Local bifurcation theory is used to prove the existence of chaotic dynamics in two well-known models of tritrophic food chains. To the best of our knowledge, the simplest technique to guarantee the emergence of strange attractors in a given family of vector fields consists of finding a 3-dimensional nilpotent singularity of codimension 3 and verifying some generic algebraic conditions. We provide the essential background regarding this method and describe the main steps to illustrate numerically the chaotic dynamics emerging near these nilpotent singularities. This is a general-purpose method and we hope it can be applied to a huge range of models.

Suggested Citation

  • Drubi, Fátima & Ibáñez, Santiago & Pilarczyk, Paweł, 2021. "Nilpotent singularities and chaos: Tritrophic food chains," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    • Handle: RePEc:eee:chsofr:v:142:y:2021:i:c:s0960077920307992
    DOI: 10.1016/j.chaos.2020.110406
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    References listed on IDEAS

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    1. Bernd Blasius & Amit Huppert & Lewi Stone, 1999. "Complex dynamics and phase synchronization in spatially extended ecological systems," Nature, Nature, vol. 399(6734), pages 354-359, May.
    2. Upadhyay, Ranjit Kumar & Naji, Raid Kamel, 2009. "Dynamics of a three species food chain model with Crowley–Martin type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1337-1346.
    3. Sahoo, Banshidhar & Poria, Swarup, 2014. "The chaos and control of a food chain model supplying additional food to top-predator," Chaos, Solitons & Fractals, Elsevier, vol. 58(C), pages 52-64.
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