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Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 6753-6768. doi: 10.3934/mbe.2019337
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Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

  • 1.
    School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
  • 2.
    Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA
  • Received: 25 January 2019 Accepted: 04 July 2019 Published: 26 July 2019

  • In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack's inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.

    Citation: Xiaoling Li, Guangping Hu, Xianpei Li, Zhaosheng Feng. Positive steady states of a ratio-dependent predator-prey system with cross-diffusion[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6753-6768. doi: 10.3934/mbe.2019337

    Related Papers:

  • In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack's inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.


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