一类具有收获率竞争系统的稳定性及Hopf分岔

2012-07-05陈红兵何万生

纯粹数学与应用数学 2012年5期

关键词：天水计算公式时滞

陈红兵,何万生

(天水师范学院数学与统计学院,甘肃天水 741001)

一类具有收获率竞争系统的稳定性及Hopf分岔

陈红兵,何万生

(天水师范学院数学与统计学院,甘肃天水 741001)

首先建立了一类具有时滞的捕获率的竞争系统,该系统具有Holling II功能.接着应用特征方程,发现当τ穿过某些数时出现了Hopf分岔,并用规范型方法和中心流形定理得到Hopf分岔和分岔周期解的稳定性的计算公式.最后举例论证.

竞争;稳定性;平衡点;Hopf分岔

1 引言

受到文献[1-5]的启发,本文建立具有收获率的时滞Holling II功能反应系统:

2 有界性

3 Hopf分岔的存在性

4 Hopf分岔与分岔周期解的计算公式

5 举例

[1]陈兰荪.数学生态学建模与研究方法[M].北京:科学教育出版社,1988.

[2]马知恩.种群生态学的数学建模与研究[M].安徽:安徽教育出版社,1996.

[3]Faria T.Stability and bifurcation for a delayed predator-prey model and the e ff ect of di ff usion[J].J.Math. Appl.,2001,254:433-463.

[4]Yan X P,Li W T.Hopf bifurcation and global periodic solutions in a delayed predator–prey system[J]. Appl.Math.Comput.,2006,177:427-445.

[5]May R M.Time delay versus stability in population models with two and three trophic levels[J].Ecology, 1973,4:315-325.

[6]Hassard B,Kazarino ffD,Wan Y.Theory and Applications of Hopf Bifurcation[M].Cambridge:Cambridge University Press,1981.

[7]Meng X,Wei J.Stability and bifurcation of mutual system with time delay[J].Chaos,Solitons and Fractals, 2004,21:729-40.

[8]Hale J,Lunel S M.Introduction to Functional Di ff erential Equations[M].New York:Springer,1993.

[9]Ruan S,Wei J.On the zeros of transcendental functions with applications to stability of delay di ff erential equations with two delays[J].Dyn.Contin.Discrete Impuls.Syst.Ser.A Math.Anal.,2003,10:863-874.

[10]许丹丹,李艳玲,吴迪.一类带扩散项的 HIV系统的平衡解的稳定性分析 [J].纯粹数学与应用数学, 2011,27(3):369-375.

The Hopf bifurcation and stability of competitive system with rate harvesting

Chen Hongbing,He Wansheng
(School of Mathematics and Statistics Tianshui Normal University,Tianshui741001,China)

First,established a competitive mold with Holling II functional response.Further,by analyzing the associated characteristic equation,it is founded that Hopf bifurcation occurs when τ crosses some critical value. The direction of Hopf bifurcation as well as stability of periodic solution are studied.The method which we used is the normal form theory and center manifold method.An example showed the feasibility of results.

compete,stability,equilibrium point,Hopf bifurcation