Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response
Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response
复制标题
DOI：
10.1137/050623449
复制
发表时间：
2006-07
期刊：
影响因子：
--
通讯作者：
Dongmei Xiao;Huaiping Zhu
中科院分区：
文献类型：
--
作者：
Dongmei Xiao;Huaiping Zhu
在本文中，我们发展了一个准则来计算一般的捕食-食饵系统的多重焦点。作为该准则的应用，我们计算了Zhu，坎贝尔，和Wolkowicz [SIAM J. Appl. Math.，63（2002），pp. 636-682]，证明了退化Hopf分支是余维2分支。此外，我们还证明了存在一个参数值，使得该系统有唯一的正双曲稳定平衡点和两个极限环，即内部不稳定的极限环和外部稳定的极限环。数值模拟的存在性的两个极限环分叉的多焦点也给出了支持的标准。
In this paper, we develop a criterion to calculate the multiplicity of a multiple focus for general predator-prey systems. As applications of this criterion, we calculate the largest multiplicity of a multiple focus in a predator-prey system with nonmonotonic functional response $p(x)=\frac{x}{ax^2+bx+1}$ studied by Zhu, Campbell, and Wolkowicz [SIAM J. Appl. Math., 63 (2002), pp. 636-682] and prove that the degenerate Hopf bifurcation is of codimension two. Furthermore, we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles, the inner one unstable and outer one stable. Numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion.