Abstract
In this paper, we considered the non-autonomous Droop model for phytoplankton growth in a chemostat in which the nutrient input varies non-periodically. It is assumed that growth rate varies with the internal nutrient level of the cell and the uptake rate of phytoplankton depends on both the external and the internal nutrient concentrations. A series of new criteria on the positivity, boundedness, permanence and extinction of the population is established.
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References
Butler G.J., Waltman P.: SIAM J. Appl. Math. 45, 435–499 (1985)
Droop M.R.: UK: J. Mar. Biol. Assoc. 48, 689–733 (1968)
Droop M.R.: J Phycol. 9, 264–272 (1973)
El-Owaidy H.M., Ismail M.: Appl. Math. Comp. 13, 787–795 (2002)
El-Owaidy H.M., Moniem A.A.: Appl. Math. Comp. 147, 147–161 (2004)
Freedman H.I., Moson P.: Proc. Am. Math. Soc. 109, 1025–1033 (1990)
Grover J.P.: Am. Nat. 138, 811–835 (1991)
Grover J.P.: Oikos 62, 231–243 (1991)
Grover J.P.: J. Theor. Biol. 15, 409–428 (1992)
J. Monod, Recherches sur la Croissance des Cultures Bacteriennes (Herman, Paris, 1942)
Sommer U.: Limnol. Oceanogr. 30, 335–346 (1985)
Smith H.L., Waltman P.: The Theory of the Chemostat. University Press, Cambridge (1994)
H.L. Smith, Monotone Dynamical Systems (American Mathematical Society, Mathematical Surveys and Monographs, 1995)
Smith H.L.: J. Math. Biol. 35, 545–556 (1997)
Smith H.L.: SIAM. J. Appl. Math. 40, 498–521 (1981)
Stephanopolous G., Fredrickson A.G., Aris R.: Am. Inst. Chem. Eng. J. 25, 863–872 (1979)
Tilman D.: Resource Competition and Community Structure. University Press, Princeton (1982)
Teng Z.: Appl. Anal. 72, 339–352 (1999)
Teng Z.: Nonlinear Anal. 42, 1221–1230 (2000)
Teng Z., Chen L.: Nonlinear Anal. RWA 4, 335–364 (2003)
Teng Z., Li M., Jiang H.: Dyn. Syst. 19, 1–24 (2004)
Wolkowicz G.S.K., Zhao X.Q.: Diff. Integral Equ. 11, 465–491 (1998)
Wolkowicz G.S.K., Xia H., Ruan S.: SIAM. J. Appl. Math. 57, 1281–1310 (1997)
Xia H., Ruan S.: J. Math. Biol. 37, 253–271 (1998)
Xia H., Ruan S.: SIAM. J. Math. Anal. 29, 681–696 (1998)
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Rehim, M., Teng, Z. Mathematical analysis on nonautonomous droop model for phytoplankton growth in a chemostat: boundedness, permanence and extinction. J Math Chem 46, 459–483 (2009). https://doi.org/10.1007/s10910-008-9472-6
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DOI: https://doi.org/10.1007/s10910-008-9472-6