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Allee effect in a discrete-time predator–prey system

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  • Çelik, Canan
  • Duman, Oktay

Abstract

In this paper, we study the stability of a discrete-time predator–prey system with and without Allee effect. By analyzing both systems, we first obtain local stability conditions of the equilibrium points without the Allee effect and then exhibit the impact of the Allee effect on stability when it is imposed on prey population. We also show the stabilizing effect of Allee effect by numerical simulations and verify that when the prey population is subject to an Allee effect, the trajectory of the solutions approximates to the corresponding equilibrium point much faster. Furthermore, for some fixed parameter values satisfying necessary conditions, we show that the corresponding equilibrium point moves from instability to stability under the Allee effect on prey population.

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  • Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:4:p:1956-1962
    DOI: 10.1016/j.chaos.2007.09.077
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    References listed on IDEAS

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    1. Jiang, Guirong & Lu, Qishao & Qian, Linning, 2007. "Complex dynamics of a Holling type II prey–predator system with state feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 448-461.
    2. Hadjiavgousti, Despina & Ichtiaroglou, Simos, 2008. "Allee effect in a prey–predator system," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 334-342.
    3. Jing, Zhujun & Yang, Jianping, 2006. "Bifurcation and chaos in discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 259-277.
    4. Cheng, Zunshui & Lin, Yiping & Cao, Jinde, 2006. "Dynamical behaviors of a partial-dependent predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 67-75.
    5. Sun, Chengjun & Han, Maoan & Lin, Yiping & Chen, Yuanyuan, 2007. "Global qualitative analysis for a predator–prey system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1582-1596.
    6. Moghadas, S.M. & Corbett, B.D., 2008. "Limit cycles in a generalized Gause-type predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1343-1355.
    7. 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.
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    Cited by:

    1. 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.
    2. Hu, Zengyun & Teng, Zhidong & Zhang, Tailei & Zhou, Qiming & Chen, Xi, 2017. "Globally asymptotically stable analysis in a discrete time eco-epidemiological system," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 20-31.
    3. Pal, Pallav Jyoti & Saha, Tapan, 2015. "Qualitative analysis of a predator–prey system with double Allee effect in prey," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 36-63.
    4. Saifuddin, Md. & Biswas, Santanu & Samanta, Sudip & Sarkar, Susmita & Chattopadhyay, Joydev, 2016. "Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 270-285.
    5. Guangye Chen & Zhidong Teng & Zengyun Hu, 2011. "Analysis of stability for a discrete ratio-dependent predator-prey system," Indian Journal of Pure and Applied Mathematics, Springer, vol. 42(1), pages 1-26, February.
    6. Érika Diz-Pita & M. Victoria Otero-Espinar, 2021. "Predator–Prey Models: A Review of Some Recent Advances," Mathematics, MDPI, vol. 9(15), pages 1-34, July.
    7. 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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