# Hopf bifurcation analysis and its preliminary control in a Hasting-Powell food chain model with two different delays

Volume 10, Issue 8, pp 4181--4196
Publication Date: August 13, 2017 Submission Date: December 14, 2016
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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.

### Share and Cite

##### 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.

•  74H55
•  74H60

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