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

Sarker Md. Sohel Rana*

Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh


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ABSTRACT


In this paper, we consider a classical food chain model describing predator-prey interaction in a chemostat. The Michaelis-Menten kinetics is used as the uptake for both predator and prey. We observe the dynamical behavior of the model around each of the equilibria and points out the exchange of stability. We use Lyapunov function in the study of the global stability of predator-free equilibrium. Using removal rate of prey as the bifurcation parameter, we prove that the model undergoes a Hopf bifurcation around interior equilibrium. It has been found that the dynamical behavior of the model is very sensitive to the parameter values. With the aid of numerical simulations we analyze the model equations and illustrate the key points of analytical findings, and determine the effects of operating parameters of the chemostat on the dynamics of the system.


Keywords: Chemostat; food chain; global stability; Hopf bifurcation; dissipative; Dulac criteria.


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REFERENCES


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

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

[3] Chiu, C.-H. and Hsu, S.-B. 1998. Extinction of top-predator in a three-level food-chain model, Journal of Mahematical Biology, 37, 4: 372-380.

[4] 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.

[5] El-Sheikh, M. M. A. and Mahrouf, S. A. A. 2005. Stability and bifurcation of a simple food chain in a chemostat with removal rates, Chaos, Solitons and Fractals, 23, 4: 1475-1489.

[6] Li, B. and Kuang, Y. 2000. Simple food chain in a chemostat with distinct removal rates. Journal of Mathematical Analysis and Applications, 242, 1: 75-92.

[7] Nasrin, F. and Rana, S. M. S. 2011, Three species food web in a chemostat, International Journal of Applied Science and Engineering, 9, 4: 301-313.

[8] Al-Sheikh, S. A. 2008. The Dynamics of a Tri-Trophic Food Chain in the Chemostat, International Journal of Pure and Applied Mathematics, 47, 1: 101-111.

[9] Alqahtani, R. T., and Nelson, M. I., and Worthy, A. L. 2012. Analysis of a chemostat model with variable yield coefficient: Contois kinetics. ANZIAM Journal (EMAC2011). 53: C155-C171.

[10] Alqahtani, R. T., Nelson, M. I., and Worthy, A. L. 2013. A fundamental analysis of continuous flow bioreactor models governed by Contois kinetics. IV. Recycle around the whole reactor cascade. Chemical Engineering Journal, 218: 99–107.

[11] Ws, M. S., Mohd, I. B., Mamat, M., and Salleh, Z., 2012. Mathematical model of three species food chain interaction with mixed functional response. International Journal of Modern Physics: Conference Series. Vol. 9: 334–340.

[12] Waryano Sunaryo, M. S., Salleh, Z., and Mamat, M., 2013. Mathematical model of three species food chain with Holling Type-III functional response. International Journal of Pure and Applied Mathematics, 89, 5: 647-657.

[13] Boonrangsiman, S. and Bunwong, K. 2012. Hopf bifurcation and dynamical behavior of a stage–structured predator sharing a prey. International Journal of Mathematical Models and Methods in Applied Sciences, 8, 6: 893-900.

[14] Upadhyay, R. K., and Raw, S. N. 2011. Complex dynamics of a three species food-chain model with Holling type IV functional response.  Nonlinear Analysis: Modelling and Control, 16, 3: 353–374.

[15] Butler, G. J., Hsu, S. B., and Waltman, P. 1983. Coexistence of competing predators in a chemostat, Journal of Mathematical Biology, 17, 2: 133-151.

[16] Boer, M. P., Kooi, B. W., and Koojiman, S. A. L. M. 1998. Food chain dynamics in the chemostat, Mathematical Biosciences, 150, 1: 43-62.

[17] Freedman, H. I. and Waltman, P. 1977. Mathematical analysis of some three-species food chain models, Mathematical Biosciences, 33, 3-4: 257-276.

[18] Hastings, A. and Powell, T. 1991. Chaos in a three-species food chain, Ecology, 72, 3: 896-903.

[19] Klebanoff, A. and Hastings, A. 1994. Chaos in three species food chain, Journal of Mathematical Biology, 32, 5: 427-451.

[20] Drake, J. F. and Tsuchiya, H. M. 1976. Predation of Escherichia coli by Colpoda stenii. Applied and Environmental Microbiology, 31, 6: 870-874.

[21] Jost, J. L., Drake, J. F., Tsuchiya, H. M., and Fredrickson, A. G. 1973. Microbial food chains and food webs. Journal of Theoretical Biology, 41, 3: 461-484.

[22] 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 of Bacteriology, 113, 2: 834-840.

[23] Tsuchiya, H. M., Drake, S. F., Jost J. L., and Fredrickson, A. G. 1972. Predator-prey interaction of Dictyostelium discoideum and Escherichia coli in continuous culture. Journal of Bacteriology, 110, 3: 1147-1153.

[24] Ali, E., Asif, M., and Ajbar, A. 2013. Study of chaotic behavior in predator–prey interactions in a chemostat. Ecological Modelling, 259: 10– 15.

[25] Freedman, H. I. and Ruan, S. 1992. Hopf bifurcation in three-species food chain models with group defense, Mathematical Biosciences, 111, 1: 73-87.

[26] Marin, A. M., Rubén, D. O. and Rodríguez, J. A. 2013. A Dulac function for a quadratic system. Theoretical Mathematics & Applications, 3, 2: 49-54.

[27] 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 of Applied Mathematics, 52, 1: 222-233.

[28] Marsden, J. E. and Mckracken, M. 1976. “The Hopf Bifurcation and its Applications”. Springer-Verlag, New York.

[29] Walter, E. and Pronzato, L. 1997. “Identification of Parametric Models from Experimental Data”. Springer, London.


ARTICLE INFORMATION


Received: 2013-12-29
Revised: 2015-05-07
Accepted: 2015-07-29
Available Online: 2015-09-01


Cite this article:

Rana, S.M.S. 2015. Bifurcation analysis of a food chain in a chemostat with distinct removal rates. International Journal of Applied Science and Engineering, 13, 217–232. https://doi.org/10.6703/IJASE.2015.13(3).217


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