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Indirect Allee effect, bistability and chaotic oscillations in a predator–prey discrete model of logistic type

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  • López-Ruiz, Ricardo
  • Fournier-Prunaret, Danièle

Abstract

A cubic discrete coupled logistic equation is proposed to model the predator–prey problem. The coupling depends on the population size of both species and on a positive constant λ, which could depend on the prey reproduction rate and on the predator hunting strategy. Different dynamical regimes are obtained when λ is modified. For small λ, the species become extinct. For a bigger λ, the preys survive but the predators extinguish. Only when the prey population reaches a critical value then predators can coexist with preys. For increasing λ, a bistable regime appears where the populations apart of being stabilized in fixed quantities can present periodic, quasiperiodic and chaotic oscillations. Finally, bistability is lost and the system settles down in a steady state, or, for the biggest permitted λ, in an invariant curve. We also present the basins for the different regimes. The use of the critical curves lets us determine the influence of the zones with different number of first rank preimages in the bifurcation mechanisms of those basins.

Suggested Citation

  • López-Ruiz, Ricardo & Fournier-Prunaret, Danièle, 2005. "Indirect Allee effect, bistability and chaotic oscillations in a predator–prey discrete model of logistic type," Chaos, Solitons & Fractals, Elsevier, vol. 24(1), pages 85-101.
    • Handle: RePEc:eee:chsofr:v:24:y:2005:i:1:p:85-101
    DOI: 10.1016/j.chaos.2004.07.018
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    Cited by:

    1. Marius-F. Danca & Michal Fečkan & Nikolay Kuznetsov & Guanrong Chen, 2021. "Coupled Discrete Fractional-Order Logistic Maps," Mathematics, MDPI, vol. 9(18), pages 1-14, September.
    2. Merdan, H. & Duman, O., 2009. "On the stability analysis of a general discrete-time population model involving predation and Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1169-1175.
    3. López-Ruiz, Ricardo & Fournier-Prunaret, Danièle, 2009. "Periodic and chaotic events in a discrete model of logistic type for the competitive interaction of two species," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 334-347.
    4. Duman, O. & Merdan, H., 2009. "Stability analysis of continuous population model involving predation and Allee effect," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1218-1222.
    5. Liu, Xiaoli & Xiao, Dongmei, 2007. "Complex dynamic behaviors of a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 80-94.
    6. Çelik, C. & Merdan, H. & Duman, O. & Akın, Ö., 2008. "Allee effects on population dynamics with delay," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 65-74.
    7. Xiongxiong Du & Xiaoling Han & Ceyu Lei, 2022. "Behavior Analysis of a Class of Discrete-Time Dynamical System with Capture Rate," Mathematics, MDPI, vol. 10(14), pages 1-15, July.
    8. Rogovchenko, Svitlana P. & Rogovchenko, Yuri V., 2009. "Effect of periodic environmental fluctuations on the Pearl–Verhulst model," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1169-1181.
    9. Merdan, H. & Duman, O. & Akın, Ö. & Çelik, C., 2009. "Allee effects on population dynamics in continuous (overlapping) case," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1994-2001.

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