A system of periodic coefficients functional differential equations is used to model the single microorganism in the chemostat environment with a periodic nutrient and antibiotic input. Furthermore, the total toxic action on the microorganism expressed by an integral term is considered in our system. Based on the technique of analysis, we obtain sufficient conditions which guarantee the permanence of the system and extinction of the microorganism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cushing J.M. (1977). SIAM J. Appl. Math. 32: 82–95
Krukonis G., Schaffer W.M. (1991). J. Theor. Biol. 148: 469–493
Veldkamp H. (1977). Adv. Microbiol Ecol. 1: 59–94
Hale J.K., Somolinos A.S. (1983). J. Math. Biol. 18: 255–280
Hsu S.B. (1980). J. Math. Biol. 18: 115–132
Smith H.L. (1981). SIAM J. Appl. Math. 18: 498–522
Butler G.J., Hsu S.B., Waltman P. (1985). SIAM J. Appl. Math. 45: 435–449
Tang B., Wolkowicz G.S. (1992). J. Math. Biol. 31: 1–23
N. MacDonald, in: Microbial Population Dynamics, ed. M.J. Bazin (Boca Raton, CRC Press, 1982), pp. 33–53.
Chen L.S., Chen J. (1993) Nonlinear Dynamic System in Biology. Science Press, Beijing (in Chinese).
Smith H.L., Waltman P. (1995) The Theory of the Chemostat. Cambridge University, Cambridge, UK
Pilyugin S.S., Waltman P. (1999). SIAM J. Appl. Math. 59(4): 1157–1177
Huang X.C., Zhu L.M. (2005). J. Math. Chem. 38: 575–588
Wang K.F., Fan A.J. (2004). Discrete Contin. Dyna. Syst.-Ser. B4: 789–795
Borsellino A., Torre V. (1974). Kybernetik 16: 113–118
Hallam T.G., Ma Z.E. (1986). J. Math. Biol. 24: 327–339
Gopalsamy K. (1991) Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer, Dordrecht, Boston, London
Teng Z. (2000). Nonlinear Anal.: RWA 42: 1221–1230
Lakshmikantham V., Matrosov V.M., Sivasundaram S. (1991) Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. Kluwer, Dordrecht, Boston, London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, H., Chen, L. Toxic action and antibiotic in the chemostat: permanence and extinction of a model with functional response. J Math Chem 43, 1256–1272 (2008). https://doi.org/10.1007/s10910-007-9278-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-007-9278-y