American Journal of Applied Mathematics

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A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment

Received: 14 September 2020    Accepted: 12 October 2020    Published: 26 October 2020
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Abstract

This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.

DOI 10.11648/j.ajam.20200805.17
Published in American Journal of Applied Mathematics (Volume 8, Issue 5, October 2020)
Page(s) 284-292
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Free Boundary, Predator-Prey Model, Spreading-Vanishing Dichotomy, Spreading Speed, Heterogeneous Environment

References
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Author Information
  • School of Mathematics and Statistics, Shandong Normal University, Jinan, P. R. China

  • School of Mathematics and Statistics, Shandong Normal University, Jinan, P. R. China

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  • APA Style

    Shiwen Niu, Hongmei Cheng. (2020). A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment. American Journal of Applied Mathematics, 8(5), 284-292. https://doi.org/10.11648/j.ajam.20200805.17

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    ACS Style

    Shiwen Niu; Hongmei Cheng. A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment. Am. J. Appl. Math. 2020, 8(5), 284-292. doi: 10.11648/j.ajam.20200805.17

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    AMA Style

    Shiwen Niu, Hongmei Cheng. A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment. Am J Appl Math. 2020;8(5):284-292. doi: 10.11648/j.ajam.20200805.17

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  • @article{10.11648/j.ajam.20200805.17,
      author = {Shiwen Niu and Hongmei Cheng},
      title = {A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment},
      journal = {American Journal of Applied Mathematics},
      volume = {8},
      number = {5},
      pages = {284-292},
      doi = {10.11648/j.ajam.20200805.17},
      url = {https://doi.org/10.11648/j.ajam.20200805.17},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20200805.17},
      abstract = {This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment
    AU  - Shiwen Niu
    AU  - Hongmei Cheng
    Y1  - 2020/10/26
    PY  - 2020
    N1  - https://doi.org/10.11648/j.ajam.20200805.17
    DO  - 10.11648/j.ajam.20200805.17
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 284
    EP  - 292
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20200805.17
    AB  - This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.
    VL  - 8
    IS  - 5
    ER  - 

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