2010, 7(4): 825-836. doi: 10.3934/mbe.2010.7.825

A stoichiometrically derived algal growth model and its global analysis

1. 

Department of Mathematical Sciences, Beijing Normal University, Beijing 100875

2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received  August 2010 Revised  August 2010 Published  October 2010

Organisms are composed of multiple chemical elements such as carbon, nitrogen, and phosphorus. The scarcity of any of these elements can severely restrict organismal and population growth. However, many trophic interaction models only consider carbon limitation via energy flow. In this paper, we construct an algal growth model with the explicit incorporation of light and nutrient availability to characterize both carbon and phosphorus limitations. We provide a global analysis of this model to illustrate how light and nutrient availability regulate algal dynamics.
Citation: Xiong Li, Hao Wang. A stoichiometrically derived algal growth model and its global analysis. Mathematical Biosciences & Engineering, 2010, 7 (4) : 825-836. doi: 10.3934/mbe.2010.7.825
References:
[1]

T. Andersen, "Pelagic Nutrient Cycles: Herbivores as Sourced and Sinks for Nutrients," Springer-Verlag, Berlin, 1997.

[2]

S. A. Berger, S. Diehl, T. J. Kunz, D. Albrecht, A. M. Oucible and S. Ritzer, Light supply, plankton biomass, and seston stoichiometry in a gradient of lake mixing depths, Limnol. Oceanogr., 51 (2006), 1898-1905. doi: doi:10.4319/lo.2006.51.4.1898.

[3]

S. Diehl, Phytoplankton, light, and nutrients in a gradient of mixing depths: Theory, Ecology, 83 (2002), 386-398. doi: doi:10.1890/0012-9658(2002)083[0386:PLANIA]2.0.CO;2.

[4]

S. Diehl, S. A. Berger and R. Wöhrl, Flexible algal nutrient stoichiometry mediates environmental influences on phytoplankton and its abiotic resources, Ecology, 86 (2005), 2931-2945. doi: doi:10.1890/04-1512.

[5]

M. R. 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. doi: doi:10.1017/S0025315400019238.

[6]

M. R. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.

[7]

J. P. Grover, Stoichiometry, herbivory and competition for nutrients: Simple models based on planktonic ecosystems, J. Theor. Biol., 214 (2002), 599-618. doi: doi:10.1006/jtbi.2001.2488.

[8]

D. O. Hessen and B. Bjerkeng, A model approach to planktonic stoichiometry and consumer-resource stability, Freshwater Biol., 38 (1997), 447-472. doi: doi:10.1046/j.1365-2427.1997.00224.x.

[9]

J. Huisman and F. J. Weissing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model, Ecology, 75 (1994), 507-520. doi: doi:10.2307/1939554.

[10]

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: A theoretical analysis, Am. Nat., 146 (1995), 536-564. doi: doi:10.1086/285814.

[11]

C. A. Klausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. doi: doi:10.4319/lo.2001.46.8.1998.

[12]

C. A. Klausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. doi: doi:10.4319/lo.2004.49.4_part_2.1463.

[13]

Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation, Mathematical Biosciences and Engineering, 1 (2004), 215-222.

[14]

L. D. J. Kuijper, B. W. Kooi, T. R. Anderson and S. A. L. M. Kooijman, Stoichiometry and food-chain dynamics, Theoretical Population Biology, 66 (2004), 323-339. doi: doi:10.1016/j.tpb.2004.06.011.

[15]

J. D. Logan, A. Joern and W. Wolesensky, Mathematical model of consumer homeostasis control in plant-herbivore dynamics, Mathematical and Computer Modelling, 40 (2004), 447-456. doi: doi:10.1016/j.mcm.2003.05.016.

[16]

I. Loladze, Y. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol., 62 (2000), 1137-1162. doi: doi:10.1006/bulm.2000.0201.

[17]

R. W. Sterner and J. J. Elser, "Ecological Stoichiometry - The Biology of Elements from Molecules to the Biosphere," Princeton University Press, 2002.

[18]

H. Wang, H. L. Smith, Y. Kuang and J. J. Elser, Dynamics of stoichiometric bacteria-algae interactions in the epilimnion, SIAM J. Appl. Math., 68 (2007), 503-522. doi: doi:10.1137/060665919.

[19]

H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model, Journal of Biological Dynamics, 2 (2008), 286-296. doi: doi:10.1080/17513750701769881.

[20]

H. Wang, K. Dunning, J. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, DCDS-B, 12 (2009), 481-493. doi: doi:10.3934/dcdsb.2009.12.481.

show all references

References:
[1]

T. Andersen, "Pelagic Nutrient Cycles: Herbivores as Sourced and Sinks for Nutrients," Springer-Verlag, Berlin, 1997.

[2]

S. A. Berger, S. Diehl, T. J. Kunz, D. Albrecht, A. M. Oucible and S. Ritzer, Light supply, plankton biomass, and seston stoichiometry in a gradient of lake mixing depths, Limnol. Oceanogr., 51 (2006), 1898-1905. doi: doi:10.4319/lo.2006.51.4.1898.

[3]

S. Diehl, Phytoplankton, light, and nutrients in a gradient of mixing depths: Theory, Ecology, 83 (2002), 386-398. doi: doi:10.1890/0012-9658(2002)083[0386:PLANIA]2.0.CO;2.

[4]

S. Diehl, S. A. Berger and R. Wöhrl, Flexible algal nutrient stoichiometry mediates environmental influences on phytoplankton and its abiotic resources, Ecology, 86 (2005), 2931-2945. doi: doi:10.1890/04-1512.

[5]

M. R. 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. doi: doi:10.1017/S0025315400019238.

[6]

M. R. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.

[7]

J. P. Grover, Stoichiometry, herbivory and competition for nutrients: Simple models based on planktonic ecosystems, J. Theor. Biol., 214 (2002), 599-618. doi: doi:10.1006/jtbi.2001.2488.

[8]

D. O. Hessen and B. Bjerkeng, A model approach to planktonic stoichiometry and consumer-resource stability, Freshwater Biol., 38 (1997), 447-472. doi: doi:10.1046/j.1365-2427.1997.00224.x.

[9]

J. Huisman and F. J. Weissing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model, Ecology, 75 (1994), 507-520. doi: doi:10.2307/1939554.

[10]

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: A theoretical analysis, Am. Nat., 146 (1995), 536-564. doi: doi:10.1086/285814.

[11]

C. A. Klausmeier and E. Litchman, Algal games: The vertical distribution of phytoplankton in poorly mixed water columns, Limnol. Oceanogr., 46 (2001), 1998-2007. doi: doi:10.4319/lo.2001.46.8.1998.

[12]

C. A. Klausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichiometry under multiple nutrient limitation, Limnol. Oceanogr., 49 (2004), 1463-1470. doi: doi:10.4319/lo.2004.49.4_part_2.1463.

[13]

Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation, Mathematical Biosciences and Engineering, 1 (2004), 215-222.

[14]

L. D. J. Kuijper, B. W. Kooi, T. R. Anderson and S. A. L. M. Kooijman, Stoichiometry and food-chain dynamics, Theoretical Population Biology, 66 (2004), 323-339. doi: doi:10.1016/j.tpb.2004.06.011.

[15]

J. D. Logan, A. Joern and W. Wolesensky, Mathematical model of consumer homeostasis control in plant-herbivore dynamics, Mathematical and Computer Modelling, 40 (2004), 447-456. doi: doi:10.1016/j.mcm.2003.05.016.

[16]

I. Loladze, Y. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol., 62 (2000), 1137-1162. doi: doi:10.1006/bulm.2000.0201.

[17]

R. W. Sterner and J. J. Elser, "Ecological Stoichiometry - The Biology of Elements from Molecules to the Biosphere," Princeton University Press, 2002.

[18]

H. Wang, H. L. Smith, Y. Kuang and J. J. Elser, Dynamics of stoichiometric bacteria-algae interactions in the epilimnion, SIAM J. Appl. Math., 68 (2007), 503-522. doi: doi:10.1137/060665919.

[19]

H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model, Journal of Biological Dynamics, 2 (2008), 286-296. doi: doi:10.1080/17513750701769881.

[20]

H. Wang, K. Dunning, J. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, DCDS-B, 12 (2009), 481-493. doi: doi:10.3934/dcdsb.2009.12.481.

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