Global stability in a chemostat with multiple nutrients

Journal of Mathematical Biology

Volumn 52, Issue 4, 2006, Pages 419-438

  De Leenheer, Patrick ^a   Levin, Simon A ^b   Sontag, Eduardo D ^c   Klausmeier, Christopher A ^a  

^a UNIVERSITY OF FLORIDA   (United States)

^b PRINCETON UNIVERSITY   (United States)

^c Georgia Institute of Technology ^*

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHM; ARTICLE; BIOLOGICAL MODEL; BIOREACTOR;

ALGORITHMS; BIOREACTORS; MODELS, BIOLOGICAL;

EID: 33645076791     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-005-0344-4     Document Type: Article

References (34)

1. 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)
2. Caperon, J.: Population growth response of Isochrysis galbana to nitrate variation at limiting concentrations. Ecology 49, 866-872 (1968)
3. Clodong, S., Blasius, B.: Chaos in a periodically forced chemostat with algal mortality. Proc. R. Soc. of Lond. B 271, 1617-1624 (2004)
4. 12 and marine ecology. 4. The kinetics of uptake, growth and inhibition of Monochyrsis lutheri. J. Mar. Biol. Assoc. UK 48, 689-733 (1968)
5. Droop, M.R.: The nutrient status of algal cells in continuous culture. J. Mar. Biol. Assoc. UK 54, 825-855 (1974)
6. Droop, M.R.: Twenty-five years of algal growth kinetics, a personal view. Bot. Mar. 26, 99-112 (1983)
7. Huisman, J., Weissing, F.J.: Biodiversity of plankton by species oscillations and chaos. Nature 402, 407-410 (1999)
8. Hsu, S.B., Cheng, K.S., Hubbell, S.P.: Exploitative competition of microorganism for two complementary nutrients in continuous culture. SIAM J. Appl. Math. 41, 422-444 (1981)
9. Jang, S.R.-J., Baglama, J.: Qualitative behavior of a variable-yield simple food chain with an inhibiting nutrient. Math. Biosci. 164, 65-80 (2000)
10. Jiang, J.F.: On the global stability of cooperative systems. B. Lond. Math. Soc. 6, 455-458 (1994)
11. Kot, M., Sayler, G.S., Schultz, T.W.: Complex dynamics in a model microbial system. B. Math. Biol. 54, 619-648 (1992)
12. Klausmeier, C.A., Litchman, E., Levin, S.A.: Phytoplankton growth and stoichiometry under multiple nutrient limitation. Limnol. Oceanogr. 49, 1463-1470 (2004)
13. Klausmeier, C.A., Litchman, E., Daufresne, T., Levin, S.A.: Optimal N:P stoichiometry of phytoplankton. Nature 429, 171-174 (2004)
14. Lange, K., Oyarzun, F.J.: The attractiveness of the Droop equations. Mathe. Biosci. 111, 261-278 (1992)
15. Legović, T., Cruzado, A.: A model of phytoplankton growth on multiple nutrients based on the Michaelis-Menten-Monod uptake, Droop's growth and Liebig's law. Ecol. Model. 99, 19-31 (1997)
16. Leon, J.A., Tumpson, D.B.: Competition between two species for two complementary or substitutable resources. J. Theor. Biol. 50, 185-201 (1975)
17. Li, B.: Periodic coexistence in the chemostat with three species competing for three essential resources. Math. Biosci. 174, 27-40 (2001)
18. Li, B., Smith, H.L.: How many species can two essential resources support? SIAM J. Appl. Math. 62, 336-366 (2001)
19. Li, B., Smith, H.L.: Competition for essential resources: a brief review. Dynamical Systems and Their Applications in Biology. Edited by S. Ruan, G.S.K Wolkowicz, J. Wu. Fields Institute Communications 36, 213-227 (2003)
20. Li, B., Smith, H.L.: Periodic coexistence of four species competing for three essential resources. Math. Biosci. 184, 115-134 (2003)
21. Monod, J.: La technique de culture continue; theorie et applications. Ann. Inst. Pasteur 79, 390-410 (1950)
22. Oyarzun, F.J., Lange, K.: The attractiveness of the Droop equations II. Generic uptake and growth functions. Math. Biosci. 111, 261-278 (1992)
23. Pascual, M.: Periodic response to periodic forcing of the Droop equations for phytoplankton growth. J. Math. Biol. 32, 743-759 (1994)
24. Pascual, M., Caswell, H.: From the cell cycle to population cycles in phytoplankton-nutrient interactions. Ecology 78, 897-912 (1997)
25. Rhee, G.-Y.: Effects of N:P atomic ratios and nitrate limitation on algal growth, cell composition and nitrate uptake. Limnol. Oceanogr. 23, 10-25 (1978)
26. Smith, H.L., Waltman, P.: Competition for a single limiting resource in continuous culture: The varibale-yield model. SIAM J. Appl. Math. 54, 1113-1131 (1994)
27. Smith, H.L.: Monotone dynamical systems. American Mathematical Society. Providence, Rhode Island, 1995
28. Smith, H.L.: The periodically forced Droop model for phytoplankton growth in a chemostat. J. Math. Biol. 35, 545-556 (1997)
29. Smith, H.L., Waltman, P.: The Theory of the Chemostat. Cambridge University Press, 1995
30. Sontag, E.D.: Remarks on stabilization and factorization. Proc. IEEE Conf. Dec. Control, 1989, pp. 1376-1378
31. Sontag, E.D.: A remark on the converging-input converging-state property. IEEE Trans. Automatic Control 48, 313-314 (2003)
32. Tilman, D.: Resource Competition and Community Structure. Princeton University Press, 1982
33. Turpin, D.H.: Physiological mechanisms in phytoplankton resource competition. In: C.D. Sandgren, (ed.) "Growth and Reproductive Strategies of Freshwater Phytoplankton, Cambridge University Press, 1988, pp. 316-368
34. 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)