Hopf Bifurcation for a Ecological Mathematical Model on Microbe Populations

Guo Ruihai; Yuan Xiaofeng

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2000 > 21(7): 693-700

Guo Ruihai, Yuan Xiaofeng. Hopf Bifurcation for a Ecological Mathematical Model on Microbe Populations[J]. Applied Mathematics and Mechanics, 2000, 21(7): 693-700.

Citation:

Guo Ruihai, Yuan Xiaofeng. Hopf Bifurcation for a Ecological Mathematical Model on Microbe Populations[J]. Applied Mathematics and Mechanics, 2000, 21(7): 693-700.

Citation:

Guo Ruihai, Yuan Xiaofeng. Hopf Bifurcation for a Ecological Mathematical Model on Microbe Populations[J]. Applied Mathematics and Mechanics, 2000, 21(7): 693-700.

PDF( 311 KB)

Hopf Bifurcation for a Ecological Mathematical Model on Microbe Populations

Guo Ruihai¹,
Yuan Xiaofeng²

1.
Department of Mathematics, Southwest Nationalities College, Chengdu 610041, P R China;
2.
Center for Mathematical Sciences, CICA, Academia Sinica, Chengdu 610041, P R China

Received Date: 1998-09-30
Rev Recd Date: 2000-04-13
Publish Date: 2000-07-15

Abstract

Abstract

The ecological Model of a class of the two microbe populations with second-order growth rate is studied.The methods of qualitative theory of ordinary differential equations are used in the four-dimension phase space.The qualitative property and stability of equilibrium points are analysed. The conditions under which the positive equilibrium point exists and becomes and O⁺ attractor are obtained.The problems on Hopf bifurcation are discussed in detail when small perturbation occurs.
- mathematical model,
- qualitative theory,
- equilibrium points,
- Hopf bifurcation

FullText(HTML)

References(7)

References

[1]	袁晓凤,刘世泽,郭瑞海等. 厌氧消化过程三种群微生物生态数学模型的定性分析[J]. 四川大学学报(自然科学版),1997,34(3):373～376.
[2]	梅茨基 B.B, 斯捷巴诺夫 B.B. 微分方程定性理论[M].(王柔怀译)北京: 科学出版社,1959,201～230.
[3]	刘世泽. n维空间奇点的拓扑分类[J]. 数学进展,1965,8(3):217～242.
[4]	Hartman P. Ordinary Differential Equation[M]. Baston: Birkhauser,1982,228～250.
[5]	李继彬,冯贝叶. 稳定性、分支与混沌[M]. 昆明: 云南科技出版社,1995,85～127.
[6]	Wiggins S. Introduction to Applied Nonlinear Dynamical System and Chaos[M]. New York: Springer-verlag,1990,193～284.
[7]	Liu Zhenrong, Jing Zhujun. Qualitative analysis for a third-order differential equation in a model of chemical systems[J]. Syst Scie Math Scie,1992,5(4):299～309.