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Pattern Formation in Predator‐Prey Model with Delay and Cross Diffusion

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  • Xinze Lian
  • Shuling Yan
  • Hailing Wang

Abstract

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie‐Gower predator‐prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self‐replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

Suggested Citation

  • Xinze Lian & Shuling Yan & Hailing Wang, 2013. "Pattern Formation in Predator‐Prey Model with Delay and Cross Diffusion," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:147232
    DOI: 10.1155/2013/147232
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    References listed on IDEAS

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    1. Sheng Wang & Wenbin Liu & Zhengguang Guo & Weiming Wang, 2013. "Traveling Wave Solutions in a Reaction‐Diffusion Epidemic Model," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Wang, Weiming & Zhang, Lei & Wang, Hailing & Li, Zhenqing, 2010. "Pattern formation of a predator–prey system with Ivlev-type functional response," Ecological Modelling, Elsevier, vol. 221(2), pages 131-140.
    3. Sheng Wang & Wenbin Liu & Zhengguang Guo & Weiming Wang, 2013. "Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-13, April.
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    Cited by:

    1. Xinhong Pan & Min Zhao & Chuanjun Dai & Yapei Wang, 2015. "Dynamic Analysis of a Delayed Reaction‐Diffusion Predator‐Prey System with Modified Holling‐Tanner Functional Response," Abstract and Applied Analysis, John Wiley & Sons, vol. 2015(1).
    2. Xinhong Pan & Min Zhao & Chuanjun Dai & Yapei Wang, 2014. "Stability and Hopf Bifurcation Analysis of a Nutrient‐Phytoplankton Model with Delay Effect," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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