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Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

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  • Zhang, Limin
  • Wang, Tao

Abstract

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Suggested Citation

  • Zhang, Limin & Wang, Tao, 2023. "Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    • Handle: RePEc:eee:chsofr:v:175:y:2023:i:p1:s0960077923008962
    DOI: 10.1016/j.chaos.2023.113995
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    References listed on IDEAS

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    1. Marotto, F.R., 2005. "On redefining a snap-back repeller," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 25-28.
    2. Zhang, Limin & Zhang, Chaofeng & Zhao, Min, 2014. "Dynamic complexities in a discrete predator–prey system with lower critical point for the prey," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 105(C), pages 119-131.
    3. Li, Xiaoshuang & Pang, Danfeng & Wallhead, Philip & Bellerby, Richard Garth James, 2023. "Dynamics of an aquatic diffusive predator–prey model with double Allee effect and pH-dependent capture rate," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    4. Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    5. Huang, Tousheng & Zhang, Huayong, 2016. "Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 92-107.
    6. Luo, Demou & Wang, Qiru, 2021. "Global dynamics of a Beddington–DeAngelis amensalism system with weak Allee effect on the first species," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    7. Han, Renji & Dey, Subrata & Banerjee, Malay, 2023. "Spatio-temporal pattern selection in a prey–predator model with hunting cooperation and Allee effect in prey," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    8. Huang, Tousheng & Yang, Hongju & Zhang, Huayong & Cong, Xuebing & Pan, Ge, 2018. "Diverse self-organized patterns and complex pattern transitions in a discrete ratio-dependent predator–prey system," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 141-158.
    9. Zhang, Limin & Zhang, Chaofeng & He, Zhirong, 2019. "Codimension-one and codimension-two bifurcations of a discrete predator–prey system with strong Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 162(C), pages 155-178.
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    1. Zhang, Huayong & Guo, Fenglu & Zou, Hengchao & Zhao, Lei & Wang, Zhongyu & Yuan, Xiaotong & Liu, Zhao, 2024. "Refuge-driven spatiotemporal chaos in a discrete predator-prey system," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).

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