Hopf bifurcation analysis and its preliminary control in a Hasting-Powell food chain model with two different delays
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Authors
Jiangang Zhang
- School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China.
Jiarong Lu
- School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China.
Wenju Du
- School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China.
Yandong Chu
- School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China.
Hongwei Luo
- School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China.
- Department of Information Engineering, Gansu Forestry Technological College, Tianshui, Gansu 741020, China.
Abstract
Keeping the balance of nature is important, and it is very significant to effectively control the number of species for ecosystem stability. In this paper, we propose a tritrophic Hastings-Powell (HP) model with two different time delays, and the local stability of equilibrium, Hopf bifurcation, and the existence and uniqueness of the positive equilibrium are analyzed in detail. Besides, we obtain the stable conditions for the system and prove that Hopf bifurcation will occur when the delay pass through the critical value. And the stability and direction of the Hopf bifurcation are also investigated by using the center manifold theorem and normal form theorem. Finally, some numerical examples are given to illustrate the results.
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ISRP Style
Jiangang Zhang, Jiarong Lu, Wenju Du, Yandong Chu, Hongwei Luo, Hopf bifurcation analysis and its preliminary control in a Hasting-Powell food chain model with two different delays, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4181--4196
AMA Style
Zhang Jiangang, Lu Jiarong, Du Wenju, Chu Yandong, Luo Hongwei, Hopf bifurcation analysis and its preliminary control in a Hasting-Powell food chain model with two different delays. J. Nonlinear Sci. Appl. (2017); 10(8):4181--4196
Chicago/Turabian Style
Zhang, Jiangang, Lu, Jiarong, Du, Wenju, Chu, Yandong, Luo, Hongwei. "Hopf bifurcation analysis and its preliminary control in a Hasting-Powell food chain model with two different delays." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4181--4196
Keywords
- Tritrophic Hastings-Powell model
- local stability
- delays
- Hopf bifurcation.
MSC
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