Applied Mathematics and Mechanics

, Volume 21, Issue 7, pp 767–774 | Cite as

Hopf bifurcation for a ecological mathematical model on microbe populations

  • Guo Ruihai
  • Yuan Xiaofeng
Article

Abstract

The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used used in the four-dimension phase space. The qualitative property and stability of equilibrium points were 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.

Key words

mathematical model qualitative theory equilibrium points Hopf bifurcation 

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References

  1. [ 1 ]
    Yuan Xiaofeng, Liu Shize, Guo Rouhai, et al. Qualitative analysis on ecological models of three microbe populations in the anaerobic digestion process [J].J Sichuan University (Natural Science Edition),1997,34(3):373 ∼ 376. (in Chinese)MathSciNetGoogle Scholar
  2. [ 2 ]
    Nemenskii B. B, Sichepalov B. B.Qualitative Theory of Differential Equation [M]I. Wang Rouhai transl. Beijing: Science Press, 1959,201 ∼ 230. (Chinese version)Google Scholar
  3. [ 3 ]
    Liu Shize. Topologicl classification of singular points inn-dimensional space [J].Mathematical Advance, 1965,8(3):217 ∼ 242. (in Chinese)Google Scholar
  4. [ 4 ]
    Hartman P.Ordinary Differential Equation[M]. Boston: Birkhauser,1982,228 ∼ 250.Google Scholar
  5. [ 5 ]
    Li Jibin, Feng Beiye.Stability, Bifurcations and Chaos [M]. Kunmin: Yunnan Science and Technology Publishing House, 1995,85 ∼ 127. (in Chinese)Google Scholar
  6. [ 6 ]
    Wiggins S.Introduction to Applied Nonlinear Dynamical System and Chaos [M]. New York: Springer-Verlag,1990,193 ∼ 284.Google Scholar
  7. [ 7 ]
    Liu Zhenrong, Jing Zhujun. Qualitative analysis for a third-order differential equation in a model of chemical systems [J].Systems Sci Math Sci,1992,5(4):299 ∼ 309.MathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1980

Authors and Affiliations

  • Guo Ruihai
    • 1
  • Yuan Xiaofeng
    • 2
  1. 1.Department of MathematicsSouthwest Nationalities CollegeChengduP R China
  2. 2.Center for Mathematical SciencesCICA, Academia SinicaChengduP R China

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