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Neimark–Sacker bifurcation in a tritrophic model with defense in the prey

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  • Blé, Gamaliel
  • Dela-Rosa, Miguel Angel

Abstract

We analyze a tritrophic discrete system. The analysis focuses in two cases, the two dimensional one in which the absence of super-predator lead to a predator-prey discrete system; secondly, we consider a model constructed using the average grow method in each density of the species. In both cases, we prove that the coexistence of species takes place by means of a supercritical Neimark–Sacker bifurcation at one of the fixed points that the system could have. We assume that the lowest trophic level has logistic grow and the functional responses for predators-prey and superpredator-mesopredator are Holling type IV.

Suggested Citation

  • Blé, Gamaliel & Dela-Rosa, Miguel Angel, 2019. "Neimark–Sacker bifurcation in a tritrophic model with defense in the prey," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 124-139.
    • Handle: RePEc:eee:chsofr:v:123:y:2019:i:c:p:124-139
    DOI: 10.1016/j.chaos.2019.03.034
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    References listed on IDEAS

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    1. Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    2. Baogui Xin & Tong Chen & Junhai Ma, 2010. "Neimark-Sacker Bifurcation in a Discrete-Time Financial System," Discrete Dynamics in Nature and Society, Hindawi, vol. 2010, pages 1-12, September.
    3. Banerjee, Ritwick & Das, Pritha & Mukherjee, Debasis, 2018. "Stability and permanence of a discrete-time two-prey one-predator system with Holling Type-III functional response," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 240-248.
    4. Xia Liu & Yanwei Liu & Qiaoping Li, 2015. "Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-10, February.
    5. Cui, Qianqian & Zhang, Qiang & Qiu, Zhipeng & Hu, Zengyun, 2016. "Complex dynamics of a discrete-time predator-prey system with Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 158-171.
    6. S. M. Sohel Rana & Umme Kulsum, 2017. "Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-11, July.
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