Abstract
In this paper, a chemostat model with delayed response in growth and pulse input in polluted environment is considered. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution. Further, it is globally attractive. The permanent condition of the investigated system is also obtained by the theory on impulsive delay differential equation. Our results reveal that the delayed response in growth plays an important role on the outcome of the chemostat.
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References
Smith H., Waltman P.: Theory of Chemostat. Combridge University, Combridge (1995)
Smith H., Waltman P.: Perturbation of a globally stable steady state. Proc. A.M.S. 127(2), 447–453 (1999)
Yang K.: Delay Differential Equation with Application in Population Dynamics. Academic Press, Boston (1993)
H.I. Freedman, J.W.H. So, P. Waltman, in Chemostat Competetition with Delays, ed. by J. Eisenfeld, D.S. Levine. Biomedicial Modelling and Simulation (Scientific Publishing Co., 1989), pp. 171–173
Ellermeyer S.F.: Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth. SIAM J. Appl. Math. 154, 456–465 (1994)
Caltagirone L.E., Doutt R.L.: Global behavior of an SEIRS epidemic model with delays, the history of the vedalia beetle importation to California and its impact on the development of biological control. Ann. Rev. Entomol. 34, 1–16 (1989)
Hsu S.B., Waltman P., Ellermeyer S.F.: A remark on the global asymptotic stability of a dynamical system modeling two species competition. Hiroshima Math. J. 24, 435–445 (1994)
Ellermeyer S., Hendrix J., Ghoochan N.: A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria. J. Theor. Biol. 222, 485–494 (2003)
Bush A.W., Cook A.E.: The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. Theor. Biol. 63, 385–395 (1975)
Hass C.N.: Application of predator-prey models to disinfection. J. Water Pollut. Contr. Fed. 53, 378–386 (1981)
Hsu S.B., Waltman P.: Competition in the chemostat when one competitor produces a toxin. Jpn. J. Ind. Appl. Math. 15, 471–490 (1998)
Jenson A.L., Marshall J.S.: Application of surplus production model to access environmental impacts in exploited populations of Daphnia pluex in the laboratory. Environ. Pollut. (Ser. A) 28, 273–280 (1982)
De Luna J.T., Hallam T.G.: Effects of toxicants on population: a qualitiative approach. IV. Resource-consumer-toxicants model, Ecol. Model. 35, 249–273 (1987)
Dubey B.: Modelling the effect of toxicant on forestry resources. Indian J. Pure Appl. Math. 28, 1–12 (1997)
Freedman H.I., Shukla J.B.: Models for the effect of toxicant in a single-species and predator-prey systems. J. Math. Biol. 30, 15–30 (1991)
Hallam T.G., Clark C.E., Jordan G.S.: Effects of toxicant on population: a qualitative approach. II. First order kinetics. J. Math. Biol. 18, 25–37 (1983)
Zhang B.: Population’s Ecological Mathematics Modeling. Publishing of Qingdao Marine University, Qingdao (1990)
Hallam T.G., Clark C.E., Lassider R.R.: Effects of toxicant on population: a qualitative approach. I. Equilibrium environmental exposure. Ecol. Model. 18, 291–340 (1983)
Liu B., Chen L.S., Zhang Y.J.: The effects of impulsive toxicant input on a population in a polluted environment. J. Biol. Syst. 11, 265–287 (2003)
Bulert G.L., Hsu S.B., Waltman P.: A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45, 435–449 (1985)
Hale J.K., Somolinas A.M.: Competition for fluctuating nutrent. J. Math. Biol. 18, 255–280 (1983)
Hsu S.B., Hubbell S.P., Waltman P.: A mathematical theory for single nutrent competition in continuous cultures of microorganisms. SIAM J. Appl. Math. 32, 366–383 (1977)
Wolkowicz G.S.K., Zhao X.Q.: N-species competition in a periodic chemostat. Diff. Integr. Eq. 11, 465–491 (1998)
Wang L., Wolkowicz G.S.K.: A delayed chemostat model with general nonmonotone response functions and differential removal rates. J. Math. Anal. Appl. 321, 452–468 (2006)
D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 (Longman, NY, 1993)
Jiao J., Chen L.: Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators. Int. J. Biomath. 1, 197–208 (2008)
Funasaki E., Kot M.: Invasion and chaos in a Lotka-Volterra system. Theor. Popul. Biol. 44, 203–224 (1993)
Smith R.J., Wolkowicz G.S.K.: Analysis of a model of the nutrent driven self-cycling fermentation process. Dyn. Contin. Disctete Impul. Syst. Ser. B 11, 239–265 (2004)
Jiao J., Chen L., Cai S.: An SEIRS epidemic model with two delays and pulse vaccination. J. Syst. Sci. Complex. 28(4), 385–394 (2008)
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Supported by National Natural Science Foundation of China (10771179). The Nomarch Fund of Guizhou Province, and the Science Technology Fund of Guizhou Province.
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Jiao, J., Chen, L. Dynamical analysis of a chemostat model with delayed response in growth and pulse input in polluted environment. J Math Chem 46, 502–513 (2009). https://doi.org/10.1007/s10910-008-9474-4
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DOI: https://doi.org/10.1007/s10910-008-9474-4