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Bifurcations in a diffusive predator–prey system with linear harvesting

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  • Wang, Yong
  • Zhou, Xu
  • Jiang, Weihua

Abstract

Complex spatiotemporal dynamical behaviors of a diffusive predator–prey system with Michaelis–Menten type functional response and linear harvesting are investigated. Firstly, the critical conditions for the occurrence of Turing instability, which are necessary and sufficient, are derived in a novel way. Then, the existence conditions of codimension-1 Turing bifurcation, Hopf bifurcation, and codimension-2 Turing–Turing bifurcation, Turing–Hopf bifurcation are established. Furthermore, the detailed bifurcation set is given by calculating the amplitude equation with the method of the multiple time scale near the Turing–Hopf bifurcation. We find that the system may exhibit nonconstant steady-state solutions, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions, which can be verified by a series of numerical simulations. These investigations not only explain the effect of diffusion and harvesting on the dynamic behavior of the system, but also reveal the mechanism of spatiotemporal complexity in the diffusive predator–prey system.

Suggested Citation

  • Wang, Yong & Zhou, Xu & Jiang, Weihua, 2023. "Bifurcations in a diffusive predator–prey system with linear harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    • Handle: RePEc:eee:chsofr:v:169:y:2023:i:c:s096007792300187x
    DOI: 10.1016/j.chaos.2023.113286
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    References listed on IDEAS

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    1. Hunki Baek, 2014. "Spatiotemporal Dynamics of a Predator-Prey System with Linear Harvesting Rate," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-9, April.
    2. Souna, Fethi & Lakmeche, Abdelkader & Djilali, Salih, 2020. "Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Hunki Baek & Dongseok Kim, 2014. "Dynamics of a Predator-Prey System with Mixed Functional Responses," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-10, September.
    4. Zhang, Lei & Wang, Wenjuan & Xue, Yakui, 2009. "Spatiotemporal complexity of a predator–prey system with constant harvest rate," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 38-46.
    5. Ghorai, Santu & Poria, Swarup, 2016. "Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food," Chaos, Solitons & Fractals, Elsevier, vol. 85(C), pages 57-67.
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    Cited by:

    1. Liang, Ziwei & Meng, Xinyou, 2023. "Stability and Hopf bifurcation of a multiple delayed predator–prey system with fear effect, prey refuge and Crowley–Martin function," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    2. Ming Liu & Linyi Ma & Dongpo Hu, 2024. "Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting," Mathematics, MDPI, vol. 12(18), pages 1-44, September.
    3. Wang, Fatao & Yang, Ruizhi, 2023. "Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    4. Chen, Mengxin & Ham, Seokjun & Choi, Yongho & Kim, Hyundong & Kim, Junseok, 2023. "Pattern dynamics of a harvested predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

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