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Cross-diffusion induced Turing instability for a competition model with saturation effect

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  • Li, Qiang
  • Liu, Zhijun
  • Yuan, Sanling

Abstract

Competitive behavior, like predation, widely exists in the world which has influences on the dynamics of ecosystems. Though a good knowledge for the patten formation and selection of predator–prey models with diffusion is known in the literature, little is known for that of a diffused competition model. As a result, a competition model with saturation effect and self- and cross-diffusion is proposed and analyzed in this paper. Firstly, by making use of nullcline analysis, we derive the condition of existence and stability of positive equilibrium and prove global stability. Then using the linear stability analysis, the Turing bifurcation critical value and the condition of the occurrence of Turing pattern are obtained when control parameter is selected. Finally, we deduce the amplitude equations around the Turing bifurcation point by using the standard multiple scale analysis. Meanwhile, a series of numerical simulations are given to expand our theoretical analysis. This work shows that cross-diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:64-77
    DOI: 10.1016/j.amc.2018.10.071
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    References listed on IDEAS

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    1. Gui-Quan Sun & Zhen Jin & Yi-Guo Zhao & Quan-Xing Liu & Li Li, 2009. "Spatial Pattern In A Predator-Prey System With Both Self- And Cross-Diffusion," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 71-84.
    2. Ghorai, Santu & Poria, Swarup, 2016. "Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 421-429.
    3. Gambino, G. & Lombardo, M.C. & Sammartino, M., 2012. "Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1112-1132.
    4. Lei Zhang, 2014. "Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, April.
    5. Peng, Yahong & Zhang, Tonghua, 2016. "Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 1-12.
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    Cited by:

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