# Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays

Volume 9, Issue 6, pp 3479--3490 Publication Date: June 05, 2016
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### Authors

Dingyang Lv - Department of Mathematics, Hunan First Normal College, Changsha, 410205 Hunan, P. R. China. Wen Zhang - School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, P. R. China. - School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China. Yi Tang - Department of Mathematics, Hunan First Normal College, Changsha, 410205 Hunan, P. R. China.

### Abstract

In this paper, we consider a ratio-dependent predator-prey system with multiple delays where the dynamics are logistic with the carrying capacity proportional to prey population. By choosing the sum $\tau$ of two delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

### Keywords

• Ratio-dependent
• delay
• Hopf bifurcation
• center manifold
• periodic solutions.

•  34C25
•  37L10

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