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Mathematical analysis of a chemostat system modeling the competition for light and inorganic carbon with internal storage

1 Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan
2 Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung, Keelung 204, Taiwan

Special Issues: Resource Explicit Population Models

This paper investigates a mathematical model of competition between two species for inorganic carbon and light in a well-mixed water column. The population growth of the species depends on the consumption of two substitutable forms of inorganic carbon, “CO2” (dissolved CO2 and carbonic acid) and “CARB” (bicarbonate and carbonate ions), which are stored internally. Besides, uptake rates also includes self-shading by the phytoplankton population, that is, an increase in population density will reduce light available for photosynthesis, and thereby reducing further carbon assimilation and population growth. We also incorporate the fact that carbon is lost by respiration, and the respiration rate is assumed to be proportional to the size of the transient carbon pool. Then we study the extinction and persistence of a single-species system. Finally, we show that coexistence of the two-species system is possible, depending on parameter values, and both persistence of one population.
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Keywords inorganic carbon; light; photosynthesis; internal storage; extinction and persistence; coexistence

Citation: Fu-Yuan Tsai, Feng-BinWang. Mathematical analysis of a chemostat system modeling the competition for light and inorganic carbon with internal storage. Mathematical Biosciences and Engineering, 2019, 16(1): 205-221. doi: 10.3934/mbe.2019011


  • 1. A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189–203.
  • 2. A. Cunningham and R. M. Nisbet, Transient and Oscillation in Continuous Culture, in Mathematics in Microbiology, M. J. Bazin, ed., Academic ress, New York, 1983.
  • 3. M. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis Lutheri, J. Mar. Biol. Assoc. UK, 48 (1968), 689–733.
  • 4. M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264–272.
  • 5. M. Droop, The nutrient status of algal cells in continuous culture, J. Mar. Biol. Assoc. UK, 54 (1974), 825–855.
  • 6. J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409–428.
  • 7. J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
  • 8. J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, The American Naturalist, 178 (2011), E 124–E 148.
  • 9. S.-B Hsu and C. J. Lin, Dynamics of two phytoplankton Species Competing for light and nutrient with internal storage, Discrete Cont. Dyn. S, 7 (2014), 1259–1285.
  • 10. S. B. Hsu, K. Y. Lam and F. B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol., 75 (2017), 1775–1825.
  • 11. J. Huisman, P. v. Oostveen and F. J.Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, The American Naturalist, 154 (1999), 46–67.
  • 12. S. B. Hsu, F. B.Wang, and X. Q. Zhao, A reaction-diffusion model of harmful algae and zooplankton in an ecosystem, J. Math. Anal. Appl., 451 (2017), 659–677.
  • 13. J. Jiang, On the global stability of cooperative systems, Bull London Math. Soc., 26 (1994), 455– 458.
  • 14. J. T. O. Kirk, Light and photosynthesis in aquatic ecosystems, 2nd edition, Cambridge University Press, Cambridge, 1994.
  • 15. P. D. Leenheer, S. A. Levin, E. D. Sontag and C. A. Klausmeier, Global stability in a chemostat with multiple nutrients, J. Math. Biol., 52 (2006), 419–438.
  • 16. F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol., 23 (1987), 137–150.
  • 17. H. Nie, S. B. Hsu and J. P. Grover, Algal Competition in a water column with excessive dioxide in the atmosphere, J. Math. Biol., 72 (2016), 1845–1892.
  • 18. H. L. Smith, Monotone Dynamical Systems:An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.
  • 19. H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755–763.
  • 20. D. B. V. deWaal, J. M. H. Verspagen, J. F. Finke, V. Vournazou, A. K. Immers,W. E. A. Kardinaal, L. Tonk, S. Becker, E. V. Donk, P. M. Visser and J. Huisman, Reversal in competitive dominance of a toxic versus non-toxic cyanobacterium in response to rising CO2, ISME J., 5 (2011), 1438–1450.
  • 21. X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.


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