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Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model

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  • Banerjee, Malay
  • Zhang, Lai

Abstract

In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey–predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large.

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  • Banerjee, Malay & Zhang, Lai, 2014. "Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 73-81.
    • Handle: RePEc:eee:chsofr:v:67:y:2014:i:c:p:73-81
    DOI: 10.1016/j.chaos.2014.06.012
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    References listed on IDEAS

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    1. Fasani, Stefano & Rinaldi, Sergio, 2011. "Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems," Ecological Modelling, Elsevier, vol. 222(18), pages 3449-3452.
    2. Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Exploring dynamical complexity in diffusion driven predator–prey systems: Effect of toxin producing phytoplankton and spatial heterogeneities," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 584-594.
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    Cited by:

    1. Tian, Canrong & Ling, Zhi & Zhang, Lai, 2017. "Nonlocal interaction driven pattern formation in a prey–predator model," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 73-83.
    2. Chang, Lili & Jin, Zhen, 2018. "Efficient numerical methods for spatially extended population and epidemic models with time delay," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 138-154.
    3. Chang, Lili & Sun, Gui-Quan & Wang, Zhen & Jin, Zhen, 2015. "Rich dynamics in a spatial predator–prey model with delay," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 540-550.

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