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

Gang Zhu; Chunyan He

doi:doi:10.11648/j.ajam.20190706.14

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

Gang Zhu, Chunyan He

Published in American Journal of Applied Mathematics (Volume 7, Issue 6)

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

Gang Zhu

School of Education Science, Harbin University, Harbin, P. R. China
Chunyan He

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

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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}
}

Copy | Download

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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