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Dynamic complexities in a discrete predator–prey system with lower critical point for the prey

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  • Zhang, Limin
  • Zhang, Chaofeng
  • Zhao, Min

Abstract

In this paper, a discrete predator–prey system is proposed and analyzed. It is assumed that the prey population has a lower critical point, which is also referred to as extinction threshold. Such behavior has been reported for many flowering plants, many fishes, epidemiology, and so on. The existence and stability of nonnegative fixed points are explored. The conditions for the existence of flip bifurcation and Hopf bifurcation are obtained by using manifold theorem and bifurcation theory. Numerical simulations, including bifurcation diagrams, phase portraits and Maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit other complex dynamics and certain biological phenomena. Complex dynamics include quasi-periodicity, period-doubling bifurcations leading to chaos, chaotic bands with periodic windows, intermittent, supertransient, and so on. Simulations suggest that appropriate growth rate can stabilize the system, but the high growth rate may destabilize the stable system into more complex dynamics. As well, simulations suggest that the system is stable when the lower critical point parameter c is small, but when c increases beyond the critical values, the system changes from quasi-period to collapses. Furthermore, the simulated results are explained according to a biological point of view.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:105:y:2014:i:c:p:119-131
    DOI: 10.1016/j.matcom.2014.04.010
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    References listed on IDEAS

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    1. Zhao, Min & Yu, Hengguo & Zhu, Jun, 2009. "Effects of a population floor on the persistence of chaos in a mutual interference host–parasitoid model," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1245-1250.
    2. Abu-Saris, Raghib & AlSharawi, Ziyad & Rhouma, Mohamed Ben Haj, 2013. "The dynamics of some discrete models with delay under the effect of constant yield harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 54(C), pages 26-38.
    3. Barbaro, Alethea B.T. & Taylor, Kirk & Trethewey, Peterson F. & Youseff, Lamia & Birnir, Björn, 2009. "Discrete and continuous models of the dynamics of pelagic fish: Application to the capelin," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(12), pages 3397-3414.
    4. Zhu, Lili & Zhao, Min, 2009. "Dynamic complexity of a host–parasitoid ecological model with the Hassell growth function for the host," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1259-1269.
    5. Zhang, Limin & Zhao, Min, 2009. "Dynamic complexities in a hyperparasitic system with prolonged diapause for host," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1136-1142.
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    Cited by:

    1. 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.
    2. 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).

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