文献类型：期刊文献

英文题名：Nonresonant Double Hopf Bifurcation in Toxic Phytoplankton-Zooplankton Model with Delay

作者：Yuan, Rui[1,2];Jiang, Weihua[2];Wang, Yong[3]

机构：[1]Guangdong Ocean Univ, Dept Math, Zhanjiang 524000, Peoples R China;[2]Harbin Inst Technol, Dept Math, Harbin 150001, Peoples R China;[3]Tianjin Univ Finance & Econ, Dept Informat Sci & Technol, Tianjin 300222, Peoples R China

年份：2017

卷号：27

期号：2

外文期刊名：INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS

收录：SCI-EXPANDED(收录号：WOS:000397284700018)、、EI(收录号：20171303499215)、Scopus(收录号：2-s2.0-85016021775)、WOS

基金：The authors wish to express their gratitude to the editors and the reviewers for the helpful comments. This work is supported in part by the National Natural Science Foundation of China (Nos. 11371112 and 11626166) and PhD Start-up Fund of Guangdong Ocean University.

语种：英文

外文关键词：Phytoplankton-zooplankton model; Michaelis-Menten type phytoplankton harvesting; stability switching; Hopf bifurcation; double Hopf bifurcation; quasiperiodic orbit

外文摘要：This paper investigates a toxic phytoplankton-zooplankton model with Michaelis-Menten type phytoplankton harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, fold, Hopf, fold-Hopf and double Hopf bifurcation, when the parameters change and go through some of the critical values, the dynamical properties of the system will change also, such as the stability, equilibrium points and the periodic orbit. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations, and the completion bifurcation set by calculating the universal unfoldings near the double Hopf bifurcation point are given by the normal form theory and center manifold theorem. We obtained that the stable coexistent equilibrium point and stable periodic orbit alternate regularly when the digestion time delay is within some finite value. That is, we derived the pattern for the occurrence, and disappearance of a stable periodic orbit. Furthermore, we calculated the approximation expression of the critical bifurcation curve using the digestion time delay and the harvesting rate as parameters, and determined a large range in terms of the harvesting rate for the phytoplankton and zooplankton to coexist in a long term.