﻿ 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

Vol.05 No.01(2016), Article ID:16965,10 pages
10.12677/AAM.2016.51005

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

Qiuju Li, Weirui Zhao

Wuhan University of Technology, Wuhan Hubei

Received: Jan. 30th, 2016; accepted: Feb. 20th, 2016; published: Feb. 23rd, 2016

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.

Keywords:Coral Reef Models, Delay, Hopf Bifurcations, Periodic Solutions

1. 引言

2. 有关珊瑚礁模型的国内外研究现状

(2.1)

(2.2)

(2.3)

3. Hopf分支的分支方向及周期解的稳定性

，并把(2.3)式在(0,0)处线性化，则系统(2.3)可转化为

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

，我们可以得到

(3.18)

。 (3.19)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

(3.32)

4. 数值计算

5. 结论

Hopf Bifurcation Analysis in the Coral Reef Delay Differential Equations (DDE) Model[J]. 应用数学进展, 2016, 05(01): 31-40. http://dx.doi.org/10.12677/AAM.2016.51005

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