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Existence of limit cycles in a three level trophic chain with Lotka–Volterra and Holling type II functional responses

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  • Castellanos, Víctor
  • Chan-López, Ramón E.

Abstract

In this paper we analyze a three level trophic chain model, considering a logistic growth for the lowest trophic level, a Lotka–Volterra and Holling type II functional responses for predators in the middle and in the cusp in the chain, respectively. The differential system is based on the Leslie–Gower scheme. We establish conditions on the parameters that guarantee the coexistence of populations in the habitat. We find that an Andronov–Hopf bifurcation takes place. The first Lyapunov coefficient is computed explicitly and we show the existence of a stable limit cycle. Numerically, we observe a strange attractor and there exist evidence of the model to exhibit chaotic dynamics.

Suggested Citation

  • Castellanos, Víctor & Chan-López, Ramón E., 2017. "Existence of limit cycles in a three level trophic chain with Lotka–Volterra and Holling type II functional responses," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 157-167.
    • Handle: RePEc:eee:chsofr:v:95:y:2017:i:c:p:157-167
    DOI: 10.1016/j.chaos.2016.12.011
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    Cited by:

    1. Ruiqing Shi & Ting Lu & Cuihong Wang, 2019. "Dynamic Analysis of a Fractional-Order Model for Hepatitis B Virus with Holling II Functional Response," Complexity, Hindawi, vol. 2019, pages 1-13, August.
    2. Nooteboom, Peter D. & Scutt Phillips, Joe & Kehl, Christian & Nicol, Simon & van Sebille, Erik, 2023. "Modelling of tuna around fish aggregating devices: The importance of ocean flow and prey," Ecological Modelling, Elsevier, vol. 475(C).
    3. Bashkirtseva, Irina & Ryashko, Lev & Ryazanova, Tatyana, 2019. "Stochastic variability and transitions to chaos in a hierarchical three-species population model," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 276-283.
    4. Bashkirtseva, Irina & Ryashko, Lev & Ryazanova, Tatyana, 2020. "Analysis of regular and chaotic dynamics in a stochastic eco-epidemiological model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    5. Lusardi, Léo & André, Eliot & Castañeda, Irene & Lemler, Sarah & Lafitte, Pauline & Zarzoso-Lacoste, Diane & Bonnaud, Elsa, 2024. "Methods for comparing theoretical models parameterized with field data using biological criteria and Sobol analysis," Ecological Modelling, Elsevier, vol. 493(C).

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