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Steady states and spatiotemporal evolution of a diffusive predator–prey model

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  • Chen, Mengxin
  • Wu, Ranchao

Abstract

The existence of steady states, bifurcations and the spatiotemporal patterns are presented for the diffusive predator–prey model. First, the boundedness and positivity of solutions are justified, respectively. By employing the priori estimates, Poincare inequalities and Leray–Schauder degree, nonexistence and existence of nonconstant steady states are established, respectively. To further explore the pattern dynamics, the Hopf bifurcation and Turing instability are analyzed, the weakly nonlinear analysis is employed to establish the amplitude equations. It is found that the various complex pattern solutions can be identified from amplitude equations. The numerical results are in agreement with the theoretical analysis. We also find that the predator–prey model with Beddington–Deangelis (BD)-type functional response can admit various spatiotemporal patterns, such as labyrinthine-like patterns, the wave patterns near the Hopf–Turing bifurcation threshold, and so on. Such complex spatiotemporal patterns may be useful to help us understand the interaction among species.

Suggested Citation

  • Chen, Mengxin & Wu, Ranchao, 2023. "Steady states and spatiotemporal evolution of a diffusive predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    • Handle: RePEc:eee:chsofr:v:170:y:2023:i:c:s0960077923002989
    DOI: 10.1016/j.chaos.2023.113397
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    References listed on IDEAS

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    1. Li, Yilong & Xiao, Dongmei, 2007. "Bifurcations of a predator–prey system of Holling and Leslie types," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 606-620.
    2. Zhong, Shihong & Xia, Juandi & Liu, Biao, 2021. "Spatiotemporal dynamics analysis of a semi-discrete reaction-diffusion Mussel-Algae system with advection," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    3. Li, Qiang & Liu, Zhijun & Yuan, Sanling, 2019. "Cross-diffusion induced Turing instability for a competition model with saturation effect," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 64-77.
    4. Chen, Mengxin & Wu, Ranchao & Chen, Liping, 2020. "Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system," Applied Mathematics and Computation, Elsevier, vol. 380(C).
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    Cited by:

    1. Anita Triska & Agus Yodi Gunawan & Nuning Nuraini, 2023. "The Effects of the Susceptible and Infected Cross-Diffusion Terms on Pattern Formations in an SI Model," Mathematics, MDPI, vol. 11(17), pages 1-18, August.
    2. Owolabi, Kolade M. & Jain, Sonal, 2023. "Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    3. Chen, Mengxin & Zheng, Qianqian, 2023. "Diffusion-driven instability of a predator–prey model with interval biological coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 172(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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