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Mathematical Biosciences and Engineering

2014, Volume 11, Issue 6: 1247-1274. doi: 10.3934/mbe.2014.11.1247
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Spatiotemporal complexity in a predator--prey model with weak Allee effects

  • 1.
    Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275
  • 2.
    Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh
  • 3.
    Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212
  • 4.
    College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035
  • Received: 01 March 2014 Accepted: 29 June 2018 Published: 01 September 2014

  • MSC : Primary: 35B36, 45M10; Secondary: 92C15.

  • In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.

    Citation: Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang. Spatiotemporal complexity in a predator--prey model with weak Allee effects[J]. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247

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  • In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.


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