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Spatial patterns through Turing instability in a reaction–diffusion predator–prey model

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  • Guin, Lakshmi Narayan

Abstract

Pattern formation in nonlinear complex systems is one of the central problems of the natural, social and technological sciences. In this paper, we consider a mathematical model of predator–prey interaction subject to self as well as cross-diffusion, arising in processes described by a system of reaction–diffusion equations (coupled to a system of ordinary differential equations) exhibiting diffusion-driven instability. Spatial patterns through Turing instability in a reaction–diffusion predator–prey model around the unique positive interior equilibrium of the model are discussed. Furthermore, we present numerical simulations of time evolution of patterns subject to self as well as cross-diffusion in the proposed spatial model and find that the model dynamics exhibits complex pattern replication in the two-dimensional space. The obtained results unveil that the effect of self as well as cross-diffusion plays an important role on the stationary pattern formation of the predator–prey model which concerns the influence of intra-species competition among predators.

Suggested Citation

  • Guin, Lakshmi Narayan, 2015. "Spatial patterns through Turing instability in a reaction–diffusion predator–prey model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 109(C), pages 174-185.
    • Handle: RePEc:eee:matcom:v:109:y:2015:i:c:p:174-185
    DOI: 10.1016/j.matcom.2014.10.002
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    References listed on IDEAS

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    1. Sun, Gui-Quan & Jin, Zhen & Liu, Quan-Xing & Li, Li, 2008. "Dynamical complexity of a spatial predator–prey model with migration," Ecological Modelling, Elsevier, vol. 219(1), pages 248-255.
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    Cited by:

    1. Huang, Tousheng & Yang, Hongju & Zhang, Huayong & Cong, Xuebing & Pan, Ge, 2018. "Diverse self-organized patterns and complex pattern transitions in a discrete ratio-dependent predator–prey system," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 141-158.
    2. Wen, Zijuan & Fu, Shengmao, 2016. "Turing instability for a competitor-competitor-mutualist model with nonlinear cross-diffusion effects," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 379-385.
    3. Huang, Tousheng & Zhang, Huayong, 2016. "Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 92-107.
    4. Zhu, Linhe & Tang, Yuxuan & Shen, Shuling, 2023. "Pattern study and parameter identification of a reaction-diffusion rumor propagation system with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    5. Xu, Li & Liu, Jiayi & Zhang, Guang, 2018. "Pattern formation and parameter inversion for a discrete Lotka–Volterra cooperative system," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 226-231.

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