Abstract
In this paper, a mathematical model with the impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of order-1 and order-2 periodic solutions are obtained by the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state and the initial concentration of microorganism and substrate. Finally, some discussions and numerical simulations are given.
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Armstrong R.A., McGehee R.: Competitive exclusion. Am. Nat. 115, 151–170 (1980)
Butler G.J., Wolkowicz G.S.K.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J. Appl. Math. 45, 138–151 (1985)
Hsu S.B.: Limiting behavior for competing species. SIAM J. Appl. Math. 34, 760–763 (1978)
Hsu S.B., Hubbell S., Waltman P.: A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32, 366–383 (1977)
Levins R.: Coexistence in a variable environment. Am. Nat. 114, 765–783 (1979)
Wolkowicz G.S.K., Lu Z.: Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates. SIAM J. Appl. Math. 52, 222–233 (1992)
Grover J.P.: Resource Competition. Chapman and Hall, London (1997)
Hansen S.R., Hubbell S.P.: Single-nutrient microbial competition: agreement between experimental and theoretical forecast outcomes. Science 207, 1491–1493 (1980)
Tilman D.: Resource competition and Community Structure. Princeton U.P., Princeton, N.J (1982)
De Leenheer P., Smith H.L.: Feedback control for the chemostat. J. Math. Biol. 46, 48–70 (2003)
Flegr J.: Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems. J. Theor. Biol. 188, 121–126 (1997)
H. Guo, L. Chen, Periodic solution of a turbidostat system with impulsive state feedback control. J. Math. Chem. doi:10.1007/s10910-008-9492-2
Jiang G.R., Lu Q.S.: Impulsive state feedback control of a predator-prey model. J. Comput. Appl. Math. 34(2), 1135–1147 (2007)
Jiang G.R., Lu Q.S., Qian L.N.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals 31(2), 448–461 (2007)
Zeng G.Z., Chen L.S., Sun L.H.: Existence of periodic solution of order one of planar impulsive autonomous system. J. Comp. Appl. Math. 186, 466–481 (2006)
Yang T.: Impulsive Control Theory, pp. 307–333. Springer, Berlin Heidelberg (2001)
Kamel O.M., Soliman A.S.: On the optimization of the generalized coplanar Hohmann impulsive transfer adopting energy change concept. Acta Astronaut. 56, 431–438 (2005)
Lakshmikantham V., Bainov D.D., Simeonov P.: Theory of impulsive differential equations. World Scientific, Singapore (1989)
Zeng G.Z., Chen L.S., Sun L.H.: Existence of periodic solution of order one of planar impulsive autonomous system. J. Comp. Appl. Math. 186, 466–481 (2006)
Ayaaki Ishizaki, Tomomi Ohta: Batch culture kinetics of -lactate fermentation employing Strep-tococcus sp. IO-1. J. Ferment. Bioeng. 68, 123–130 (1989)
Simeonov P.E., Bainov D.D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Int. J. Syst. Sci. 19, 2562–2585 (1988)
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This work is supported by the National Natural Science Foundation of China (No.10771179) and Henan Science and Technology Department (No. 082102140025).
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Zhao, Z., Yang, L. & Chen, L. Impulsive state feedback control of the microorganism culture in a turbidostat. J Math Chem 47, 1224–1239 (2010). https://doi.org/10.1007/s10910-009-9644-z
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DOI: https://doi.org/10.1007/s10910-009-9644-z