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A note on a diffusive predator-prey model and its steady-state system

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Abstract

In this note, a diffusive predator-prey model subject to the homogeneous Neumann boundary condition is investigated and some qualitative analysis of solutions to this reaction-diffusion system and its corresponding steady-state problem is presented. In particular, by use of a Lyapunov function, the global stability of the constant positive steady state is discussed. For the associated steady state problem, a priori estimates for positive steady states are derived and some non-existence results for non-constant positive steady states are also established when one of the diffusion rates is large enough. Consequently, our results extend and complement the existing ones on this model.

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References

  1. Leung, A.: Limiting behaviour for a predator-prey model with diffusion and crowding effects. J. Math. Biol., 6, 87–93 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  2. Du, Y. H., Hsu, S. B.: A diffusive predator-prey model in heterogeneous environment. J. Differential Equations, 20, 331–364 (2004)

    Article  MathSciNet  Google Scholar 

  3. Du, Y. H., Wang, M. X.: Asymptotic behavior of positive steady-states to a predator-prey model. Proc. Roy. Soc. Edinburgh A, 136, 759–778 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Peng, R., Wang, M. X.: Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model. Appl. Math. Letters, 20, 664–670 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Peng, R., Wang, M. X.: Positive steady states of the Holling-Tanner prey-predator model with diffusion. Proc. Roy. Soc. Edinburgh A, 135, 149–164 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Pang, P. Y. H., Wang, M. X.: Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc. Roy. Soc. Edinburgh A, 133, 919–942 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wang, M. X.: Non-constant positive steady-states of the Sel’kov model. J. Differential Equations, 190, 600–620 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lou, Y., Ni, W. M.: Diffusion, self-diffusion and cross-diffusion. J. Differential Equations, 131, 79–131 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lin, C. S., Ni, W. M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations, 72, 1–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Nonlinear Complex Systems, College of Science, China Three Gorges University, Yichang, 443002, P. R. China

    Rui Peng

  2. Department of Mathematics, Southeast University, Nanjing, 210096, P. R. China

    Ming Xin Wang

  3. Department of Mathematics, Henan Polytechnic University, Jiaozuo, 454100, P. R. China

    Guo Ying Yang

Authors
  1. Rui Peng
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  2. Ming Xin Wang
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  3. Guo Ying Yang
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Corresponding author

Correspondence to Rui Peng.

Additional information

The first and third authors are partially supported by National Natural Science Foundation of China (Grant Nos. 10801090, 10871185, 10726016); the first author is also supported by the Scientific Research Projects of Hubei Provincial Department of Education (Grant No. Q200713001) and Scientific Innovation Team Project of Hubei Provincial Department of Education (Grant No. T200809); the second author is partially supported by National Natural Science Foundation of China (Grant No. 10771032)

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Peng, R., Wang, M.X. & Yang, G.Y. A note on a diffusive predator-prey model and its steady-state system. Acta. Math. Sin.-English Ser. 26, 963–974 (2010). https://doi.org/10.1007/s10114-010-6565-5

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  • DOI: https://doi.org/10.1007/s10114-010-6565-5

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