具有时滞的珊瑚礁模型的Hopf分支分析

Hopf Bifurcation Analysis in the Coral Reef Delay Differential Equations (DDE) Model

摘要: 本文探讨具有时滞的珊瑚礁模型的内部平衡点产生的Hopf分支，李熊等人在文献[1]中得到了该模型内部平衡点的稳定性条件，并把时滞作为分支参数，得到了时滞界限，给出了Hopf分支存在的条件，但没有进一步讨论模型中内部平衡点的Hopf分支的分支方向及其周期解的稳定性。这篇文章中我们主要利用正规型方法和中心流形理论讨论内部平衡点的Hopf分支的分支方向以及周期解稳定性性质，并给出数值计算。

Abstract: The dynamics of the coral reef DDE model is investigated. Li et al. [1] proved that a sequence of Hopf bifurcations occured at the positive equilibrium as the delay increased. In this paper, by applying the center manifold theorem and the normal form theory, we provide a detailed analysis of the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions at the positive equilibrium. Finally, focused parameters are obtained which determine property of the Hopf bifurcation and numerical calculation are given to justify the valid of the theoretical analysis.