International Journal of Applied Science and Engineering
Published by Chaoyang University of Technology

Xiaoning Xua and Xuncheng Huanga,b1

a Yangzhou Polytechnic University, Yangzhou, Jiangsu 225012, China.
b RDS Research Center, Infront Corp, Kearny NJ 07032, U.S.A.


Download Citation: |
Download PDF


A three-dimensional bio-reactor model of exploitative competition of two predator organisms with inhibition responses for the same renewable organism with reproductive properties is considered. By using a Lyapunov function and the center manifold theorem the global stability, the existence of Hopf bifurcation and limit cycles are proved. This result is useful in understanding the nonlinear oscillation phenomena in bio-engineering.

Keywords: bio-reactor; center manifold theorem; Lyapunov function; Hopf bifurcation; limit cycles; nonlinear oscillation.

Share this article with your colleagues



  1. [1] Smith, H. L. and Waltman, P. 1995. “The Theory of the Chemostat”, Cambridge. Cambridge Univ. Press.

  2. [2] Wolkowicz, G. and Global, Lu. Z. 1992. dynamics of a mathematical model of competition in the chemostat: general response function and different death rates, SIAM Journal of Applied Math, 52(1): 222-233.

  3. [3] Hsu, S. B., Hubbell, S. P. and Waltman, P. A. 1978. contribution to the theory of competing predators, Ecological Monograph, 38: 337-349.

  4. [4] Hsu, S. B., Hubbell, S. P., and Waltman, P. 1978. Competing predators, Siam Journal of Applied Mathematics, 35(4): 617-625.

  5. [5] McGehee, R., and Armstrong, R. A. 1977. Some mathematical problems concerning the ecological principle of competitive exclusion, Journal of Differential Equations, 21: 50-73.

  6. [6] Koch, A. L. 1974. Competitive coexistence of two predators utilizing the same prey under constant environment conditions, Journal of Theoretical Biology, 44: 373-386.

  7. [7] Yang, R. D., and Humphrey, A. E. 1975. Dynamics and steady state studies of phenol biodegradation in pure and mixed cultures, Biotechnology and Bioengineering, 17: 1211-1235.

  8. [8] Huang, X. C., and Zhu, L. M. 2005. Limit cycles in a general Kolmogorov model, Nonlinear Analysis: Theory, Method and Applications, 60(8): 1393-1414.

  9. [9] Hsu, S. B. 2005. A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese Journal of Mathematical, 9(2): 151-173.

  10. [10] Huang, X. C., Zhu, L. M., and Wang, Y. M. A note on competition in the bioreactor with toxin, Journal of Mathematical Chemistry, to appear in 2007.

  11. [11] D’Heedene, R. N. 1961. A third order autonomous differential equation with almost periodic solutions, Journal of Mathematical Analysis and Applications, 3: 344-350.

  12. [12] Schweitzer, P. A. 1974. Counterexample to the Serfert conjecture and opening closed leaves of foliations, American Journal of Mathematics, 100:2: 386-400.

  13. [13] Zhang, J. 1987. “The Geometric Theory and Bifurcation Problem of Ordinary Differential Equations, Peking University Press, Beijing.

  14. [14] Huang, X. C., Zhu, L. M., and Wang, Y. M. A note on competition in the bioreactor with toxin, Journal of Mathematical Chemistry, to appear in 2007.

  15. [15] Huang, X. C. 1988. “A Mathematical Analysis on Population Models”, PhD Dissertation, Milwaukee, Marquette University.

  16. [16] Wang, Y. Q., and Jing, Z. J. 2006. Global qualitative analysis of a food chain model, Acta Mathematica Scientia, 26A: 410-420.

  17. [17] Chiu, C. H., and Hsu, S. B. 1998. Extinction of top-predator in a three-level food-chain model, Journal of Mathematical Biology, 37: 372-380.

  18. [18] Chiu, C. H. 1999. Lyapunov functions for the global stability of competing predators, Journal of Mathematical Analysis and Applications, 230: 332-341.

  19. [19] Deng B. 2001. Food chain chaos due to junction-fold point, Chaos, 21(3): 514-525.

  20. [20] Deng B. 2004. Food chain chaos with canard explosion, Chaos, 24(4): 1083-1092.


Accepted: 2007-10-31
Publication Date: 2007-12-01

Cite this article:

Xu, X., Huang, X. 2007. Nonlinear oscillations in a three-dimensional competition with inhibition responses in a Bio-reactor. International Journal of Applied Science and Engineering, 5, 139–150.

We use cookies on this website to improve your user experience. By using this site you agree to its use of cookies.