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Volume 34, Issue 2
Hopf Bifurcation of Delayed Predator-Prey System with Reserve Area for Prey and in the Presence of Toxicity

Zizhen Zhang, Yugui Chu & Xin Zhang

DOI: 10.13447/j.1674-5647.2018.02.08

Commun. Math. Res., 34 (2018), pp. 161-170.

Published online: 2019-12

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  • Abstract

A kind of three species delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed in this paper. Local stability of the coexistence equilibrium of the system and the existence of a Hopf bifurcation is established by choosing the time delay as the bifurcation parameter. Explicit formulas to determine the direction and stability of the Hopf bifurcation are obtained by means of the normal form theory and the center manifold theorem. Finally, we give a numerical example to illustrate the obtained results.

  • Keywords

delay, Hopf bifurcation, predator-prey system, periodic solution, toxicity

  • AMS Subject Headings

34C15, 34C23, 37G15, 37N25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zzzhaida@163.com (Zizhen Zhang)

xinz15@mails.jlu.edu.cn (Xin Zhang)

  • BibTex
  • RIS
  • TXT
@Article{CMR-34-161, author = {Zhang , ZizhenChu , Yugui and Zhang , Xin}, title = {Hopf Bifurcation of Delayed Predator-Prey System with Reserve Area for Prey and in the Presence of Toxicity}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {34}, number = {2}, pages = {161--170}, abstract = {

A kind of three species delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed in this paper. Local stability of the coexistence equilibrium of the system and the existence of a Hopf bifurcation is established by choosing the time delay as the bifurcation parameter. Explicit formulas to determine the direction and stability of the Hopf bifurcation are obtained by means of the normal form theory and the center manifold theorem. Finally, we give a numerical example to illustrate the obtained results.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2018.02.08}, url = {http://global-sci.org/intro/article_detail/cmr/13523.html} }
TY - JOUR T1 - Hopf Bifurcation of Delayed Predator-Prey System with Reserve Area for Prey and in the Presence of Toxicity AU - Zhang , Zizhen AU - Chu , Yugui AU - Zhang , Xin JO - Communications in Mathematical Research VL - 2 SP - 161 EP - 170 PY - 2019 DA - 2019/12 SN - 34 DO - http://doi.org/10.13447/j.1674-5647.2018.02.08 UR - https://global-sci.org/intro/article_detail/cmr/13523.html KW - delay, Hopf bifurcation, predator-prey system, periodic solution, toxicity AB -

A kind of three species delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed in this paper. Local stability of the coexistence equilibrium of the system and the existence of a Hopf bifurcation is established by choosing the time delay as the bifurcation parameter. Explicit formulas to determine the direction and stability of the Hopf bifurcation are obtained by means of the normal form theory and the center manifold theorem. Finally, we give a numerical example to illustrate the obtained results.

Zhang , ZizhenChu , Yugui and Zhang , Xin. (2019). Hopf Bifurcation of Delayed Predator-Prey System with Reserve Area for Prey and in the Presence of Toxicity. Communications in Mathematical Research . 34 (2). 161-170. doi:10.13447/j.1674-5647.2018.02.08
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