Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient

Zhichao Jiang; Xiaohua Bi; Tongqian Zhang; B.G. Sampath Aruna Pradeep; Zhichao Jiang; Xiaohua Bi; Tongqian Zhang; B.G. Sampath Aruna Pradeep

doi:10.3934/mbe.2019188

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3807-3829. doi: 10.3934/mbe.2019188

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Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient

1.
Fundamental Science Department, North China Institute of Aerospace, 133 Aimin East Road, Langfang, 065000, P. R. China
2.
College of Mathematics and Systems Science, Shandong University of Science and Technology, 579 Qianwangang Road, Qingdao, 266590, P. R. China
3.
State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, 579 Qianwangang Road, Qingdao, 266590, P. R. China
4.
Department of Mathematics, University of Ruhuna, Wellmadama, Matara, 81000, Sri Lanka

Received: 09 February 2019 Accepted: 15 April 2019 Published: 28 April 2019

In this paper, a delayed phytoplankton-zooplankton system with the coefficient depending on delay is investigated. Firstly, it gives the nonnegative and boundedness of solutions of the delay differential equations. Secondly, it gives the asymptotical stability properties of equilibria in the absence of time delay. Then in the presence of time delay, the existence of local Hopf bifurcation is discussed when the delay changes. In addition to that, the stability of periodic solution and bifurcation direction are also obtained through the use of central manifold theory. Furthermore, he global continuity of the local Hopf bifurcation is discussed by using the global Hopf bifurcation result of FDE. At last, some numerical simulations are presented to show the rationality of theoretical analyses.
- phytoplankton-zooplankton system,
- delay,
- center manifold,
- stability,
- global Hopf bifurcation
Citation: Zhichao Jiang, Xiaohua Bi, Tongqian Zhang, B.G. Sampath Aruna Pradeep. Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3807-3829. doi: 10.3934/mbe.2019188

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Abstract

In this paper, a delayed phytoplankton-zooplankton system with the coefficient depending on delay is investigated. Firstly, it gives the nonnegative and boundedness of solutions of the delay differential equations. Secondly, it gives the asymptotical stability properties of equilibria in the absence of time delay. Then in the presence of time delay, the existence of local Hopf bifurcation is discussed when the delay changes. In addition to that, the stability of periodic solution and bifurcation direction are also obtained through the use of central manifold theory. Furthermore, he global continuity of the local Hopf bifurcation is discussed by using the global Hopf bifurcation result of FDE. At last, some numerical simulations are presented to show the rationality of theoretical analyses.

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