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, 23 (2001), 840. Google Scholar

[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. Google Scholar

[3]

Z. Bursac, M. Tabatabai and D. K. Williams, Non-linear hyperbolastic growth models and applications in cranofacial and stem cell growth,, in, (2006), 190. Google Scholar

[4]

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

[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. Google Scholar

[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). doi: 10.1186/1471-2407-10-509. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[17]

R. K. Nagle and E. B. Saff, "Fundamentals of Differential Equations,", Fourth Edition. Addison Wesley Publishing Company, (1996). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

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, 23 (2001), 840. Google Scholar

[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. Google Scholar

[3]

Z. Bursac, M. Tabatabai and D. K. Williams, Non-linear hyperbolastic growth models and applications in cranofacial and stem cell growth,, in, (2006), 190. Google Scholar

[4]

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

[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. Google Scholar

[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). doi: 10.1186/1471-2407-10-509. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[17]

R. K. Nagle and E. B. Saff, "Fundamentals of Differential Equations,", Fourth Edition. Addison Wesley Publishing Company, (1996). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[1]

Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587

[2]

Sze-Bi Hsu, Feng-Bin Wang. On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479

[3]

Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297

[4]

Sze-Bi Hsu, Chiu-Ju Lin. Dynamics of two phytoplankton species competing for light and nutrient with internal storage. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1259-1285. doi: 10.3934/dcdss.2014.7.1259

[5]

Zhipeng Qiu, Huaiping Zhu. Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2703-2728. doi: 10.3934/dcdsb.2016069

[6]

Ming Chen, Meng Fan, Xing Yuan, Huaiping Zhu. Effect of seasonal changing temperature on the growth of phytoplankton. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1091-1117. doi: 10.3934/mbe.2017057

[7]

Ebraheem O. Alzahrani, Yang Kuang. Nutrient limitations as an explanation of Gompertzian tumor growth. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 357-372. doi: 10.3934/dcdsb.2016.21.357

[8]

Linfeng Mei, Sze-Bi Hsu, Feng-Bin Wang. Growth of single phytoplankton species with internal storage in a water column. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 607-620. doi: 10.3934/dcdsb.2016.21.607

[9]

Danfeng Pang, Hua Nie, Jianhua Wu. Single phytoplankton species growth with light and crowding effect in a water column. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 41-74. doi: 10.3934/dcds.2019003

[10]

Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky, Yang Kuang. Tumor growth dynamics with nutrient limitation and cell proliferation time delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3771-3782. doi: 10.3934/dcdsb.2017189

[11]

Khalid Boushaba. A multi layer method applied to a model of phytoplankton. Networks & Heterogeneous Media, 2007, 2 (1) : 37-54. doi: 10.3934/nhm.2007.2.37

[12]

Hermann J. Eberl, Messoud A. Efendiev, Dariusz Wrzosek, Anna Zhigun. Analysis of a degenerate biofilm model with a nutrient taxis term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 99-119. doi: 10.3934/dcds.2014.34.99

[13]

Hayden Schaeffer, John Garnett, Luminita A. Vese. A texture model based on a concentration of measure. Inverse Problems & Imaging, 2013, 7 (3) : 927-946. doi: 10.3934/ipi.2013.7.927

[14]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417

[15]

Juan Li, Yongzhong Song, Hui Wan, Huaiping Zhu. Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge. Mathematical Biosciences & Engineering, 2017, 14 (2) : 529-557. doi: 10.3934/mbe.2017032

[16]

Zhiqi Lu. Global stability for a chemostat-type model with delayed nutrient recycling. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 663-670. doi: 10.3934/dcdsb.2004.4.663

[17]

Magali Tournus, Aurélie Edwards, Nicolas Seguin, Benoît Perthame. Analysis of a simplified model of the urine concentration mechanism. Networks & Heterogeneous Media, 2012, 7 (4) : 989-1018. doi: 10.3934/nhm.2012.7.989

[18]

Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221

[19]

Jun Yang. Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 965-994. doi: 10.3934/dcds.2011.30.965

[20]

Reiner Henseler, Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. A kinetic model for grain growth. Kinetic & Related Models, 2008, 1 (4) : 591-617. doi: 10.3934/krm.2008.1.591

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

[Back to Top]