2013, 10(3): 913-923. doi: 10.3934/mbe.2013.10.913

A flexible multivariable model for Phytoplankton growth

1. 

Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, United States, United States

2. 

School of Medicine, University of Alabama at Birmingham, Birmingham AL 35294, United States, United States

Received  May 2012 Revised  January 2013 Published  April 2013

We introduce a new multivariable model to be used to study the growth dynamics of phytoplankton as a function of both time and the concentration of nutrients. This model is applied to a set of experimental data which describes the rate of growth as a function of these two variables. The form of the model allows easy extension to additional variables. Thus, the model can be used to analyze experimental data regarding the effects of various factors on phytoplankton growth rate. Such a model will also be useful in analysis of the role of concentration of various nutrients or trace elements, temperature, and light intensity, or other important explanatory variables, or combinations of such variables, in analyzing phytoplankton growth dynamics.
Citation: Mohammad A. Tabatabai, Wayne M. Eby, Sejong Bae, Karan P. Singh. A flexible multivariable model for Phytoplankton growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 913-923. doi: 10.3934/mbe.2013.10.913
References:
[1]

M. E. Baird, S. M. Emsley and J. M. Meglade, Modelling the interacting effects of nutrient uptake, light capture and temperature on phytoplankton growth, J. Plankton Research 23, (2001), 840-2001.

[2]

B. Beardall, D. Allen, J. Bragg, Z. V. Finkel, K. J. Flynn, A. Quigg, T. A. V. Rees, A. Richardson and J. A. Raven, Allometry and stoichiometry of unicellular, colonial and multicellular phytoplankton, New Phytol., 181 (2009), 295-309.

[3]

Z. Bursac, M. Tabatabai and D. K. Williams, Non-linear hyperbolastic growth models and applications in cranofacial and stem cell growth, in "2005 Proceedings of the American Statistical Association Biometrics Section [CD-ROM]" (V. A. Alexandria), (2006), American Statistical Association, 190-197.

[4]

M. R. Droop, The nutrient status of algal cells in continuous culture, J. Mar. Biol. Assoc. UK, 54 (1974), 825-855.

[5]

P. Duarte, M. F. Macedo and L. C. da Fonseca, The relationship between phytoplankton diversity and community function in a coastal lagoon, Hydrobiologia, 555 (2006), 3-18.

[6]

W. Eby, M. Tabatabai and Z. Bursac, Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulfoxide, BMC Cancer, 10 (2010), pp. 509. doi: 10.1186/1471-2407-10-509.

[7]

G. T. Evans and M. A. Paranjape, Precision of estimates of phytoplankton growth and microzooplankton grazing when the functional response of grazers may be nonlinear, Mar. Ecol. Prog. Ser., 80 (1992), 285-290.

[8]

K. J. Flynn, A mechanistic model for describing dynamic multi-nutrient, light, temperature interactions in phytoplankton, J. Plankton Research, 23 (2001), 977-997.

[9]

K. Gao, Y. Wu, G. Li, H. Wu, V. E. Villafañe and E. W. Helbling, Solar UV radiation drives CO2 fixation in marine phytoplankton: A double-edged sword. Plant Physiol, 144 (2007), 54-59.

[10]

R. J. Greider, The relationship between steady state phytoplankton growth and photosynthesis, Limnol. Oceanogr., 35 (1990), 971-972.

[11]

T-Y. Ho, A. Quigg, Z. V. Finkel, A. J. Milligan, K. Wyman, P. G. Falkowski and F. M. M. Morel, The elemental composition of some marine phytoplankton, J. Phycol., 39 (2003), 1145-1159.

[12]

D. A. Kiefer and J. J. Cullen, Phytoplankton growth and light absorption as regulated by light, temperature, and nutrients, J. Plankton Research, 10 (1991), 163-172.

[13]

T. Kmeṫ, M. Straškraba and P. Mauersberger, A mechanistic model of the adaptation of phytoplankton photosynthesis, Bull. Math. Biol., 55 (1993), 259-275.

[14]

L. Mailleret, J-L. Gouzé and O. Bernard, Nonlinear control for algae growth models in the chemostat, Bioprocess Biosyst. Eng., 27 (2005), 319-327.

[15]

Z-P. Mei, Z. V. Finkel and A. J. Irwin, Light and nutrient availability affect the size-scaling of growth in phytoplankton, J. Theor. Biol., 259 (2009), 582-588.

[16]

J. Monod, La technique de culture continue: Théorie et applications, Annales de l'Inst. Pasteur, 79 (1950), 390-410.

[17]

R. K. Nagle and E. B. Saff, "Fundamentals of Differential Equations," Fourth Edition. Addison Wesley Publishing Company, 1996, Reading, MA.

[18]

C. Pahl-Wost and D. M. Imboden, DYPHORA-a dynamic model for the rate of photosynthesis of algae, J. Plankton Res., 12 (1990), 1207-1221.

[19]

D. L. Roelke, P. M. Eldridge and L. A. Cifuentes, A model of phytoplankton competition for limiting and nonlimiting nutrients: Implications for development of estuarine and near shore management schemes, Estuaries, 22 (1999), 92-104.

[20]

K. A. Safi and J. M. Gibbs, Importance of different size classes of phytoplankton in Beatrix Bay, Marlborough Sounds, New Zealand, and the potential implications for aquaculture of mussel, Prena canaliculus, New Zealand Journal of Marine and Freshwater Research, 37 (2003), 267-272.

[21]

E. Sakshang, K. Andresen and D. A. Kiefer, A steady state description of growth and light absorption in the marine planktonic diatom Skeletonema costatum, Limnol. Oceanogr., 34 (1989), 198-205.

[22]

W. V. Sobczak, J. E. Cloern, A. D. Jassby and A. B. Müller-Solger, Bioavailability of organic matter in a highly disturbed estuary: The role of detrital and algal resources, PNAS, 99 (2002), 8101-8105.

[23]

M. Tabatabai, Z. Bursac, W. Eby and K. Singh, Mathematical modeling of stem cell proliferation, Med. & Biol. Eng. & Comp., 49 (2011), 253-262. doi: 10.1007/s11517-010-0686-y.

[24]

M. Tabatabai, W. Eby and K. P. Singh, Hyperbolastic modeling of wound healing, Mathematical and Computer Modelling, 53 (2011), 755-768. doi: 10.1016/j.mcm.2010.10.013.

[25]

M. Tabatabai, D. K. Williams and Z. Bursac, Hyperbolastic growth models: Theory and application, Theoretical Biology and Medical Modeling, 2 (2005), 1-13.

[26]

M. Takahashi, J. Ishizaka, T. Ishimaru, L. P. Atkinson, T. N. Lee, Y. Yamaguchi, Y. Fujita and S. Ichimura, Temporal change in nutrient concentrations and phytoplankton biomass in short time scale local upwelling around the Izu Peninsula, Japan. J. Plankton Res., 8 (1986), 1039-1049.

[27]

S. C. Wofsy, A simple model to predict extinction coefficients and phytoplankton biomass in eutrophic waters, Limnology and Oceanography, 28 (1983), 1144-1155.

show all references

References:
[1]

M. E. Baird, S. M. Emsley and J. M. Meglade, Modelling the interacting effects of nutrient uptake, light capture and temperature on phytoplankton growth, J. Plankton Research 23, (2001), 840-2001.

[2]

B. Beardall, D. Allen, J. Bragg, Z. V. Finkel, K. J. Flynn, A. Quigg, T. A. V. Rees, A. Richardson and J. A. Raven, Allometry and stoichiometry of unicellular, colonial and multicellular phytoplankton, New Phytol., 181 (2009), 295-309.

[3]

Z. Bursac, M. Tabatabai and D. K. Williams, Non-linear hyperbolastic growth models and applications in cranofacial and stem cell growth, in "2005 Proceedings of the American Statistical Association Biometrics Section [CD-ROM]" (V. A. Alexandria), (2006), American Statistical Association, 190-197.

[4]

M. R. Droop, The nutrient status of algal cells in continuous culture, J. Mar. Biol. Assoc. UK, 54 (1974), 825-855.

[5]

P. Duarte, M. F. Macedo and L. C. da Fonseca, The relationship between phytoplankton diversity and community function in a coastal lagoon, Hydrobiologia, 555 (2006), 3-18.

[6]

W. Eby, M. Tabatabai and Z. Bursac, Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulfoxide, BMC Cancer, 10 (2010), pp. 509. doi: 10.1186/1471-2407-10-509.

[7]

G. T. Evans and M. A. Paranjape, Precision of estimates of phytoplankton growth and microzooplankton grazing when the functional response of grazers may be nonlinear, Mar. Ecol. Prog. Ser., 80 (1992), 285-290.

[8]

K. J. Flynn, A mechanistic model for describing dynamic multi-nutrient, light, temperature interactions in phytoplankton, J. Plankton Research, 23 (2001), 977-997.

[9]

K. Gao, Y. Wu, G. Li, H. Wu, V. E. Villafañe and E. W. Helbling, Solar UV radiation drives CO2 fixation in marine phytoplankton: A double-edged sword. Plant Physiol, 144 (2007), 54-59.

[10]

R. J. Greider, The relationship between steady state phytoplankton growth and photosynthesis, Limnol. Oceanogr., 35 (1990), 971-972.

[11]

T-Y. Ho, A. Quigg, Z. V. Finkel, A. J. Milligan, K. Wyman, P. G. Falkowski and F. M. M. Morel, The elemental composition of some marine phytoplankton, J. Phycol., 39 (2003), 1145-1159.

[12]

D. A. Kiefer and J. J. Cullen, Phytoplankton growth and light absorption as regulated by light, temperature, and nutrients, J. Plankton Research, 10 (1991), 163-172.

[13]

T. Kmeṫ, M. Straškraba and P. Mauersberger, A mechanistic model of the adaptation of phytoplankton photosynthesis, Bull. Math. Biol., 55 (1993), 259-275.

[14]

L. Mailleret, J-L. Gouzé and O. Bernard, Nonlinear control for algae growth models in the chemostat, Bioprocess Biosyst. Eng., 27 (2005), 319-327.

[15]

Z-P. Mei, Z. V. Finkel and A. J. Irwin, Light and nutrient availability affect the size-scaling of growth in phytoplankton, J. Theor. Biol., 259 (2009), 582-588.

[16]

J. Monod, La technique de culture continue: Théorie et applications, Annales de l'Inst. Pasteur, 79 (1950), 390-410.

[17]

R. K. Nagle and E. B. Saff, "Fundamentals of Differential Equations," Fourth Edition. Addison Wesley Publishing Company, 1996, Reading, MA.

[18]

C. Pahl-Wost and D. M. Imboden, DYPHORA-a dynamic model for the rate of photosynthesis of algae, J. Plankton Res., 12 (1990), 1207-1221.

[19]

D. L. Roelke, P. M. Eldridge and L. A. Cifuentes, A model of phytoplankton competition for limiting and nonlimiting nutrients: Implications for development of estuarine and near shore management schemes, Estuaries, 22 (1999), 92-104.

[20]

K. A. Safi and J. M. Gibbs, Importance of different size classes of phytoplankton in Beatrix Bay, Marlborough Sounds, New Zealand, and the potential implications for aquaculture of mussel, Prena canaliculus, New Zealand Journal of Marine and Freshwater Research, 37 (2003), 267-272.

[21]

E. Sakshang, K. Andresen and D. A. Kiefer, A steady state description of growth and light absorption in the marine planktonic diatom Skeletonema costatum, Limnol. Oceanogr., 34 (1989), 198-205.

[22]

W. V. Sobczak, J. E. Cloern, A. D. Jassby and A. B. Müller-Solger, Bioavailability of organic matter in a highly disturbed estuary: The role of detrital and algal resources, PNAS, 99 (2002), 8101-8105.

[23]

M. Tabatabai, Z. Bursac, W. Eby and K. Singh, Mathematical modeling of stem cell proliferation, Med. & Biol. Eng. & Comp., 49 (2011), 253-262. doi: 10.1007/s11517-010-0686-y.

[24]

M. Tabatabai, W. Eby and K. P. Singh, Hyperbolastic modeling of wound healing, Mathematical and Computer Modelling, 53 (2011), 755-768. doi: 10.1016/j.mcm.2010.10.013.

[25]

M. Tabatabai, D. K. Williams and Z. Bursac, Hyperbolastic growth models: Theory and application, Theoretical Biology and Medical Modeling, 2 (2005), 1-13.

[26]

M. Takahashi, J. Ishizaka, T. Ishimaru, L. P. Atkinson, T. N. Lee, Y. Yamaguchi, Y. Fujita and S. Ichimura, Temporal change in nutrient concentrations and phytoplankton biomass in short time scale local upwelling around the Izu Peninsula, Japan. J. Plankton Res., 8 (1986), 1039-1049.

[27]

S. C. Wofsy, A simple model to predict extinction coefficients and phytoplankton biomass in eutrophic waters, Limnology and Oceanography, 28 (1983), 1144-1155.

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