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Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior

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  • Tang, Xiaosong
  • Song, Yongli

Abstract

In this paper, we consider a delayed diffusive predator–prey model with herd behavior. Firstly, by choosing the appropriate bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations, induced by diffusion and delay respectively, are investigated by analyzing the corresponding characteristic equation. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Furthermore, the instability of the Hopf bifurcation leads to the emergence of spatial patterns. Finally, some numerical simulations are also carried out to illustrate and expand the theoretical results.

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  • Tang, Xiaosong & Song, Yongli, 2015. "Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 375-391.
    • Handle: RePEc:eee:apmaco:v:254:y:2015:i:c:p:375-391
    DOI: 10.1016/j.amc.2014.12.143
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    1. Wang, Weiming & Zhang, Lei & Wang, Hailing & Li, Zhenqing, 2010. "Pattern formation of a predator–prey system with Ivlev-type functional response," Ecological Modelling, Elsevier, vol. 221(2), pages 131-140.
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    Cited by:

    1. Djilali, Salih, 2019. "Impact of prey herd shape on the predator-prey interaction," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 139-148.
    2. Gökçe, Aytül & Yazar, Samire & Sekerci, Yadigar, 2022. "Stability of spatial patterns in a diffusive oxygen–plankton model with time lag effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 109-123.
    3. Jiang, Xiaowei & Chen, Xiangyong & Chi, Ming & Chen, Jie, 2020. "On Hopf bifurcation and control for a delay systems," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    4. Lv, Yun-fei & Li, Tongtong & Pei, Yongzhen & Yuan, Rong, 2016. "A complete analysis of the global dynamics of a diffusive predator and toxic prey model," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 182-196.
    5. Tang, Xiaosong, 2022. "Periodic solutions and spatial patterns induced by mixed delays in a diffusive spruce budworm model with Holling II predation function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 420-429.
    6. Chen, Jianxin & Zhang, Tonghua & Zhou, Yongwu, 2020. "Dynamics of a risk-averse newsvendor model with continuous-time delay in supply chain financing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 169(C), pages 133-148.
    7. Han, Renji & Dai, Binxiang, 2017. "Spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 177-201.
    8. Ghanbari, Behzad & Djilali, Salih, 2020. "Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    9. Yang, Ruizhi & Ma, Jian, 2018. "Analysis of a diffusive predator-prey system with anti-predator behaviour and maturation delay," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 128-139.
    10. Yang, Ruizhi, 2017. "Bifurcation analysis of a diffusive predator–prey system with Crowley–Martin functional response and delay," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 131-139.

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