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Critical conditions for phytoplankton blooms

Bulletin of Mathematical Biology volume 63, pages 1095–1124 (2001)Cite this article

Abstract

We motivate and analyse a reaction—advection—diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton—water—light-system characterized by the same dimensionless parameters will show the same dynamics.

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References

  • Abramowitz, M. and I. A. Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Nat. Bur. Standards Appl. Series 55, Washington, D.C.: U.S. Government Printing Office (paperback edition published by Dover, New York).

    MATH  Google Scholar 

  • Arrigo, K. R., D. H. Robinson, D. L. Worthen, R. B. Dunbar, G. R. DiTullio, M. VanWoert and M. P. Lizotte (1999). Phytoplankton community structure and the drawdown of nutrients and CO2 in the Southern Ocean. Science 283, 365–367.

    Article  Google Scholar 

  • Berman, T. and B. Shteinman (1998). Phytoplankton development and turbulent mixing in Lake Kinneret (1992–1996). J. Plankton Res. 20, 709–726.

    Google Scholar 

  • Britton, N. F. and U. Timm (1993). Effects of competition and shading in planktonic communities. J. Math. Biol. 31, 655–673.

    Article  MATH  Google Scholar 

  • Cloern, J. E. (1991). Tidal stirring and phytoplankton bloom dynamics in an estuary. J. Mar. Res. 49, 203–221.

    Article  Google Scholar 

  • Denny, M. W. (1993). Air and Water: The Biology and Physics of Life’s Media, Princeton: Princeton University Press.

    Google Scholar 

  • DiTullio, G. R., J. M. Grebmeier, K. R. Arrigo, M. P. Lizotte, D. H. Robinson, A. Leventer, J. P. Barry, M. L. VanWoert and R. B. Dunbar (2000). Rapid and early export of Phaeocystis antarctica blooms in the Ross Sea, Antarctica. Nature 404, 595–598.

    Article  Google Scholar 

  • Evers, E. G. (1991). A model for light-limited continous cultures: growth, shading, and maintenance. Biotechnol. Bioeng. 38, 254–259.

    Article  Google Scholar 

  • Falkowski, P. G., R. T. Barber and V. Smetacek (1998). Biogeochemical controls and feedbacks on ocean primary production. Science 281, 200–206.

    Article  Google Scholar 

  • Huisman, J. (1999). Population dynamics of light-limited phytoplankton: microcosm experiments. Ecology 80, 202–210.

    Article  Google Scholar 

  • Huisman, J., M. Arrayás, U. Ebert and B. Sommeijer (2001 or 2002). How do sinking phytoplankton species manage to persist? Am. Naturalist [to appear].

  • Huisman, J., R. R. Jonker, C. Zonneveld and F. J. Weissing (1999a). Competition for light between phytoplankton species: experimental tests of mechanistic theory. Ecology 80, 211–222.

    Article  Google Scholar 

  • Huisman, J., P. van Oostveen and F. J. Weissing (1999b). Critical depth and critical turbulence: two different mechanisms for the development of phytoplankton blooms. Limnology and Oceanography 44, 1781–1788.

    Article  Google Scholar 

  • Huisman, J., P. van Oostveen and F. J. Weissing (1999c). Species dynamics in phytoplankton blooms: incomplete mixing and competition for light. Am. Naturalist 154, 46–68.

    Article  Google Scholar 

  • Huisman, J. and F. J. Weissing (1994). Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model. Ecology 75, 507–520.

    Article  Google Scholar 

  • Ishii, H. and I. Takagi (1982). Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics. J. Math. Biol. 16, 1–24.

    MathSciNet  MATH  Google Scholar 

  • Jones, K. J. and R. J. Gowen (1990). Influence of stratification and irradiance regime on summer phytoplankton composition in coastal and shelf seas of the British Isles (UK). Estuarine Coastal and Shelf Science 30, 557–568.

    Google Scholar 

  • Kok, B. (1952). On the efficiency of Chlorella growth. Acta Botanica Neerlandica 1, 445–467.

    Google Scholar 

  • Landau, L. D. and E. M. Lifshitz (1966). Course of Theoretical Physics, Vol. 6: Fluid Dynamics, 3rd edn, Oxford: Pergamon.

    Google Scholar 

  • Monod, J. (1950). La technique de culture continue, theorie et applications. Annales de l’Institut Pasteur (Paris) 79, 390–410.

    Google Scholar 

  • Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models, Berlin: Springer.

    MATH  Google Scholar 

  • Platt, T., D. F. Bird and S. Sathyendranath (1991). Critical depth and marine primary production. Proceedings of the Royal Society of London B 246, 205–217.

    Google Scholar 

  • Platt, T., C. L. Gallegos and W. G. Harrison (1980). Photoinhibition of photosynthesis in natural assemblages of marine phytoplankton. J. Mar. Res. 38, 687–701.

    Google Scholar 

  • Reynolds, C. S. (1984). The Ecology of Freshwater Phytoplankton, Cambridge: Cambridge University Press.

    Google Scholar 

  • Reynolds, C. S., S. W. Wiseman, B. M. Godfrey and C. Butterwick (1983). Some effects of artificial mixing on the dynamics of phytoplankton populations in large limnetic enclosures. J. Plankton Res. 5, 203–234.

    Google Scholar 

  • Riley, G. A., H. Stommel and D. F. Bumpus (1949). Quantitative ecology of the plankton of the western North Atlantic. Bull. Bingham Oceanographic Collection 12, 1–169.

    Google Scholar 

  • Shigesada, N. and A. Okubo (1981). Analysis of the self-shading effect on algal vertical distribution in natural waters. J. Math. Biol. 12, 311–326.

    MathSciNet  Article  MATH  Google Scholar 

  • Sverdrup, H. U. (1953). On conditions for the vernal blooming of phytoplankton. Journal du Conseil Permanent International pour l’Exploration de la Mer 18, 287–295.

    Google Scholar 

  • Totaro, S. (1989). Mutual shading effect on algal distribution: a nonlinear problem. Nonlinear Anal. Theory Methods Appl. 13, 969–986.

    MATH  MathSciNet  Article  Google Scholar 

  • Visser, P. M., B. W. Ibelings, B. van der Veer, J. Koedood and L. R. Mur (1996a). Artificial mixing prevents nuisance blooms of the cyanobacterium Microcystis in Lake Nieuwe Meer, The Netherlands. Freshwater Biol. 36, 435–450.

    Article  Google Scholar 

  • Visser, P. M., L Massaut, J. Huisman and L. R. Mur (1996b). Sedimentation losses of Scenedesmus in relation to mixing depth. Archiv für Hydrobiologie 136, 289–308.

    Google Scholar 

  • Webb, W. L., M. Newton and D. Starr (1974). Carbon dioxide exchange of Alnus rubra: a mathematical model. Oecologia 17, 281–291.

    Article  Google Scholar 

  • Weissing, F. J. and J. Huisman (1994). Growth and competition in a light gradient. J. Theor. Biol. 168, 323–336.

    Article  Google Scholar 

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Authors and Affiliations

  1. CWI, P.O.Box 94079, 1090 GB, Amsterdam, The Netherlands

    Ute Ebert, Manuel Arrayás, Nico Temme & Ben Sommeijer

  2. Aquatic Microbiology, Institute for Biodiversity and Ecosystem Dynamics, Universiteit van Amsterdam, Nieuwe Achtergracht 127, 1018 WS, Amsterdam, The Netherlands

    Jef Huisman

  3. Institute Lorento, Universiteit Leiden, The Netherlands

    Manuel Arrayás

  4. Universidad Rey Juan Carlos, Madrid, Spain

    Manuel Arrayás

Authors
  1. Ute Ebert
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  2. Manuel Arrayás
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  3. Nico Temme
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  4. Ben Sommeijer
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  5. Jef Huisman
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Correspondence to Ute Ebert.

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Ebert, U., Arrayás, M., Temme, N. et al. Critical conditions for phytoplankton blooms. Bull. Math. Biol. 63, 1095–1124 (2001). https://doi.org/10.1006/bulm.2001.0261

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  • DOI: https://doi.org/10.1006/bulm.2001.0261

Keywords

  • Turbulent Diffusion
  • Phytoplankton Bloom
  • Phytoplankton Species
  • Phytoplankton Population
  • Critical Depth