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

Farzana Nasrina,1 and Sarker Md. Sohel Ranab

a Institute of Natural Sciences, United International University, Dhaka-1209, Bangladesh
b Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

Download Citation: |
Download PDF


In this paper, we consider a model describing predator prey interaction in a chemostat that incorporates general response functions and identical removal rates. The existence of steady states, their local stability and persistence of the model are presented. We construct a Lyapunov function in the study of the global stability of a predator free steady state. We show that a slight fluctuation in the maximal growth rates of prey and/or predator devastate the form of conservation principle. Numerical simulations are also presented to analyze the model equations and determine the effect of the operating parameters of the chemostat on its dynamics.

Keywords: Chemostat; food web; predator; prey; stability.

Share this article with your colleagues



  1. [1] Alhumazi, K. and Ajbar, A. 2005. Dynamics of Predator-Prey Interaction in Continuous Culture. Engineering in Life Sciences, 5-2:139-147.

  2. [2] Baiely, J. E. and Ollis, D. F. 1986. “Biochemical Engineering Fundamentals”. 2nd ed., Mc-Graw Hill, NY.

  3. [3] Butler, G. J. and Wolkowicz, G. S. K. 1985. A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM Journal on Applied Mathematics, 45: 138-151.

  4. [4] Butler, G. J. and Wolkowicz, G. S. K. 1986. Predator-mediated competition in the chemostat. Journal of Mathematical Biology, 24: 167-191.

  5. [5] El-Owaidy, H. M. and Moniem, A. A. 2003. On food chain in a chemostat with distinct removal rates. Applied Mathematics E-Notes, 3: 183-191.

  6. [6] Gause, G. F. 1934. “The struggle for existence”. Williams & Wilkins Co., Baltimore.

  7. [7] Grivet, J. P. 2001. Nonlinear Population Dynamics in the Chemostat. Computing in Science and Engineering, 3-1: 48-55.

  8. [8] Hale, J. K. 1980. Ordinary Differential Equations. Krieger, Malabar, FL.

  9. [9] Hsu, S. B., Hubbell S., and Waltman, P. 1977. A mathematical theory of single-nutrient competition in continuous cultures for microorganisms. SIAM Journal on Applied Mathematics, 32: 366-383.

  10. [10] Jost, J. L., Drake J. F., Fredrickson, A. G., and Tsuchiya H. M. 1973. Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandii and Glucose in a Minimal Medium. Journal Bacteriology, 113-2: 834-840.

  11. [11] LaSalle, J. 1960. Some extensions of Lyapunov’s second method. IRE Trans. Circuit CT., 7: 520-527.

  12. [12] Li, B. 1998. “Analysis of chemostat-related models with distinct removal rates”. Ph. D. thesis, Arizona State University.

  13. [13] Li, B. 1999. Global asymptotic behavior of the chemostat: General response functions and different removal rates. SIAM Journal on Applied Mathematics, 59: 411-422.

  14. [14] Li, B. and Kuang, Y. 2000. Simple food chain in a chemostat with distinct removal rates. Journal of Mathematical Analysis and Applications, 242: 75-92.

  15. [15] May, R. M. 1972. “Stability and Complexity in Model Ecosystems”. Princeton University Press, NJ (USA).

  16. [16] Moghadas, S. M. and Gumel, A. B. 2003. Dynamical and numerical analysis of a generalized food chain model. Applied Mathematics and Computation, 142: 35-49.

  17. [17] Smith, H. L. 1982. The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model. SIAM Journal on Applied Mathematics, 42: 27-43.

  18. [18] Smith, H. L. and Waltman, P. 1995. “The Theory of Chemostat”. Cambridge Univ. Press, Cambridge (UK).

  19. [19] Tsuchiya, H. M., Drake, J. K., Jost, J. L. and Fredrickson, A. G. 1972. Predator-prey interaction of Dictyostelium discoideum and Escherichia coli in continuous culture. Journal Bacteriology, 110: 1147-1153.

  20. [20] Turchin, P. 2003. “Complex Population Dynamics: A Theoretical/Empirical Synthesis”. Press Princeton University Press, Princeton, NJ (USA).

  21. [21] Vayenas, D. V. and Pavlou, S. 1999. Chaotic dynamics of a food web in a chemostat. Mathematical Biosciences, 162: 69-84.

  22. [22] Waltman, P. 1990. Coexistence in chemostat-like models. Rocky Mountain Journal of Mathematics, 20: 777-807.

  23. [23] Waltman, P., Hubbel, S. P., and Hsu, S. B. 1980. Theoretical and experimental investigation of microbial competition in continuous culture. Modeling and Differential Equations in Biology, Burton (ed.): 107-192.

  24. [24] Wolkowicz, G. S. K., and Lu, Z. 1992. Global Dynamics of a mathematical model of competition in the chemostat; General response function and differential death rates. SIAM Journal on Applied Mathematics, 52: 222-233.

  25. [25] Zhu, H., Campbell, S. A., and Wolkowicz, G. S. K. 2002. Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM Journal on Applied Mathematics, 63: 636-682.


Accepted: 2011-11-18
Available Online: 2011-12-01

Cite this article:

Nasrin, F., Rana, S.M.S. 2011. Three species food web in a chemostat. International Journal of Applied Science and Engineering, 9, 301–313.