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Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration

Received: 25 October 2019     Accepted: 18 November 2019     Published: 30 December 2019
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Abstract

A delayed reaction Cdiffusion Rosenzweig-MacArthur model with a constant rate of prey immigration is considered. We derive the characteristic equation through partial differential equation theory, and by analyzing the distribution of the roots of the characteristic equation, the local stability of the positive equilibria is studied, and we get the conditions to determine the stability of the positive equilibria. Furthermore we find that Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values, and we get the conditions under which the Hopf bifurcation occurs and so periodic solutions appear near the positive equilibria. By using the center manifold theory and normal form method, we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, some numerical simulations are carried out to illustrate the analytic results of our study.

Published in American Journal of Applied Mathematics (Volume 7, Issue 6)
DOI 10.11648/j.ajam.20190706.14
Page(s) 164-176
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), 2019. Published by Science Publishing Group

Keywords

Delay, Stability, Bifurcation, Center Manifold, Normal Form

References
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  • APA Style

    Gang Zhu, Chunyan He. (2019). Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration. American Journal of Applied Mathematics, 7(6), 164-176. https://doi.org/10.11648/j.ajam.20190706.14

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    ACS Style

    Gang Zhu; Chunyan He. Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration. Am. J. Appl. Math. 2019, 7(6), 164-176. doi: 10.11648/j.ajam.20190706.14

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    AMA Style

    Gang Zhu, Chunyan He. Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration. Am J Appl Math. 2019;7(6):164-176. doi: 10.11648/j.ajam.20190706.14

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  • @article{10.11648/j.ajam.20190706.14,
      author = {Gang Zhu and Chunyan He},
      title = {Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration},
      journal = {American Journal of Applied Mathematics},
      volume = {7},
      number = {6},
      pages = {164-176},
      doi = {10.11648/j.ajam.20190706.14},
      url = {https://doi.org/10.11648/j.ajam.20190706.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190706.14},
      abstract = {A delayed reaction Cdiffusion Rosenzweig-MacArthur model with a constant rate of prey immigration is considered. We derive the characteristic equation through partial differential equation theory, and by analyzing the distribution of the roots of the characteristic equation, the local stability of the positive equilibria is studied, and we get the conditions to determine the stability of the positive equilibria. Furthermore we find that Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values, and we get the conditions under which the Hopf bifurcation occurs and so periodic solutions appear near the positive equilibria. By using the center manifold theory and normal form method, we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, some numerical simulations are carried out to illustrate the analytic results of our study.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration
    AU  - Gang Zhu
    AU  - Chunyan He
    Y1  - 2019/12/30
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ajam.20190706.14
    DO  - 10.11648/j.ajam.20190706.14
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 164
    EP  - 176
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20190706.14
    AB  - A delayed reaction Cdiffusion Rosenzweig-MacArthur model with a constant rate of prey immigration is considered. We derive the characteristic equation through partial differential equation theory, and by analyzing the distribution of the roots of the characteristic equation, the local stability of the positive equilibria is studied, and we get the conditions to determine the stability of the positive equilibria. Furthermore we find that Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values, and we get the conditions under which the Hopf bifurcation occurs and so periodic solutions appear near the positive equilibria. By using the center manifold theory and normal form method, we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, some numerical simulations are carried out to illustrate the analytic results of our study.
    VL  - 7
    IS  - 6
    ER  - 

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Author Information
  • School of Education Science, Harbin University, Harbin, P. R. China

  • School of Education Science, Harbin University, Harbin, P. R. China

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