A model for the interaction of phytoplankton aggregates and the environment: approximation and parameter estimation

References

• Banse K . Comment: does iron really limit phytoplankton production in the offshore subarctic Pacific? Limnol Oceanogr. 1990;35(3):772–775.
   Web of Science ®Google Scholar
• Gregg WW , Ginoux P , Schopf PS , et al . Phytoplankton and iron: validation of a global three-dimensional ocean biogeochemical model. Deep Sea Res II. 2003;50:3143–3169.
   Web of Science ®Google Scholar
• Jackson GA , Waite AM , Boyd PW . Role of algal aggregation in vertical carbon export during SOIREE and in other low biomass environments. Geophys Res Lett. 2005;32:1–4.
   Web of Science ®Google Scholar
• Martin JH , Gordon RM , Fitzwater SE . The case for iron. Limnol Oceanogr. 1991;36(8):1793–1802.
   Web of Science ®Google Scholar
• Miller CB , Frost BW , Booth BB , et al . Ecological processes in the subarctic Pacific: iron limitation cannot be the whole story. Oceanography. 1991;4(2):71–78.
   Google Scholar
• Mills MM , Ridame C , Davey M , et al . Iron and phosphorous co-limit nitrogen fixation in the eastern tropical North Atlantic. Nature. 2004;429:292–294.
   PubMed Web of Science ®Google Scholar
• Sunda W , Huntsman SA . Interrelated influence of iron, light and cell size on marine phytoplankton growth. Nature. 1997;390:389–392.
   Web of Science ®Google Scholar
• Timmermans KR , Gerringa LJA , de Baar HJW , et al . Growth rates of large and small Southern Ocean diatoms in relation to availability of iron in natural seawater. Limnol Oceanogr. 2001;46(2):260–266.
   Web of Science ®Google Scholar
• Laurenceau-Cornec EC , Trull TW , Davies DM , et al . The relative importance of phytoplankton aggregates and zooplankton fecal pellets to carbon export: insights from free-drifting sediment trap deployments in naturally iron-fertilised waters near the Kergeulen Plateau. Biogeosciences. 2015;12:1007–1027.
   Web of Science ®Google Scholar
• Chakraborty S , Lohrenz S . Phytoplankton community structure in the river-influenced continental margin of the northern Gulf of Mexico. Mar Ecol Prog Ser. 2015;521:31–47.
   Web of Science ®Google Scholar
• Penta B , Ko D , Gould R , et al. Using coupled models to study the effects of river discharge on biogeochemical cycling and hypoxia in the northern Gulf of Mexico. Proceedings of the Oceans ‘09 MTS/IEEE Conference; Biloxi. 2009.
   Google Scholar
• North RL , Guildford SJ , Smith REH , et al . Evidence for phosphorous, nitrogen, and iron colimitation of phytoplankton communities in Lake Erie. Limnol Oceanogr. 2007;52(1):315–328.
   Web of Science ®Google Scholar
• Takeda S , Tsuda A . An in situ iron-enrichment experiment in the western subarctic Pacific (SEEDS): introduction and summary. Prog Oceanogr. 2005;64:95–109.
   Web of Science ®Google Scholar
• Banse K . Rates of phytoplankton cell division in the field iron enrichment experiments. Limnol Oceanogr. 1991;36(8):1886–1898.
   Web of Science ®Google Scholar
• Bhattacharjee M , Siemann E . Low algal diversity systems are a promising method for biodiesel production in wastewater fed open reactors. Algae. 2015;30(1):67–79.
   Web of Science ®Google Scholar
• Hannon M , Gimpel J , Tran M , et al . Biofuels from algae: challenges and potential. Biofuels. 2010;1(5):763–784.
   PubMedGoogle Scholar
• Hinckley S , Coyle KO , Gibson G , et al . A biophysical NPZ model with iron for the Gulf of Alaska: reproducing the differences between an oceanic HNLC ecosystem and a classical northern temperate shelf ecosystem. Deep Sea Res II. 2009;56:2520–2536.
   Web of Science ®Google Scholar
• Huesemann MH , Van Wagenen J , Miller T , et al . A screening model to predict microalgae biomass growth in photobioreactors and raceway ponds. Biotechnol Bioeng. 2013;110:1583–1594.
   PubMed Web of Science ®Google Scholar
• Kunjapur AM , Eldridge RB . Photobioreactor design for commercial biofuel production from microalgae. Ind Eng Chem Res. 2010;49:3516–3526.
   Web of Science ®Google Scholar
• Milledge JJ , Heaven S . A review of the harvesting of micro-algae for biofuel production. Rev Environ Sci Biotechnol. 2012;12:165–178.
   Web of Science ®Google Scholar
• Quinn JC , Yates T , Douglas N , et al . Nannochloropsis production metrics in a scalable outdoor photobioreactor for commercial applications. Biores Tech. 2012;117:164–171.
   PubMed Web of Science ®Google Scholar
• Ackleh AS , Ferdinand R . A nonlinear phytoplankton aggregation model with light shading. SIAM J Appl Math. 1999;60:316–336.
   Web of Science ®Google Scholar
• Ackleh AS , Hallam TG , Muller-Landau HC . Estimation of sticking and contact efficiencies in aggregation of phytoplankton: The SIGMA reactor experiment. Deep Sea Res. 1993;60(1995):185–201.
   Google Scholar
• Ackleh AS , Hallam TG , Smith WO Jr . Influences of aggregation and grazing on phytoplankton dynamics and fluxes: an individual-based modeling approach. Nonlinear World. 1994;1:473–492.
   Google Scholar
• Ackleh AS , Fitzpatrick BG . Modeling aggregation and growth processes in an algal population model: analysis and computations. J Math Biol. 1997;35:480–502.
   Web of Science ®Google Scholar
• Jackson GA , Kiørboe T . Maximum phytoplankton concentrations in the sea. Limnol Oceanogr. 2008;53(1):395–399.
   Web of Science ®Google Scholar
• Jackson GA , Lochmann SE . Effect of coagulation on nutrient and light limitation of an algal bloom. Limnol Oceanogr. 1992;37(1):77–89.
   Web of Science ®Google Scholar
• Kiørboe T . Small-scale turbulence, marine snow formation, and planktivorous feeding. Sci Mar. 1997;61(Supl.1):141–158.
   Google Scholar
• Riebesell U . Particle aggregation during a diatom bloom. Mar Ecol Prog Ser. 1991;69:281–291.
   Web of Science ®Google Scholar
• Arino O , Rudnicki R . Phytoplankton dynamics. C R Biol. 2004;327:961–969.
   PubMed Web of Science ®Google Scholar
• Banasiak J , Noutchie S , Rudnicki R . Global solvability of a fragmentation-coagulation equation with growth and restricted coagulation. J Nonlinear Math Phys. 2009;16(1):13–26.
   Google Scholar
• Banasiak J , Lamb W . Coagulation, fragmentation and growth processes in a size structured population. Discret Contin Dyn Syst -- Ser B. 2009;11(3):563–585.
   Web of Science ®Google Scholar
• Noutchie SC . Analysis of the effects of fragmentation-coagulation in planktology. C R Biol. 2010;333:789–792.
   PubMed Web of Science ®Google Scholar
• Arrigo K . Marine microorganisms and global nutrient cycles. Nature. 2005;437:349–355.
   PubMed Web of Science ®Google Scholar
• Xue Z , He R , Fennel K , et al . Modeling ocean circulation and biogeochemical variability in the Gulf of Mexico. Biogeosciences. 2013;10:7219–7234.
   Web of Science ®Google Scholar
• Yu L , Fennel K , Laurent A , Murrell MC , Lehrter JC . Numerical analysis of the primary processes controlling oxygen dynamics on the Louisiana shelf. Biogeosciences. 2015;12:2063–2076.
   Web of Science ®Google Scholar
• Zhao Y , Quigg A . Nutrient limitation in Northern Gulf of Mexico (NGOM): phytoplankton communities and photosynthesis respond to nutrient pulse. PLoS One. 2014;2(9):e88732. DOI:10.1371/journal.pone.0088732.
   Google Scholar
• Bortz DM , Jackson TL , Taylor KA , et al . Klebsiella pneumonia flocculation dynamics. Bull Math Biol. 2008;70(3):745–768.
   PubMed Web of Science ®Google Scholar
• Brandt BW , Kooijman SALM . Two parameters account for the flocculated growth of microbes in biodegradation assays. Biotechnol Bioeng. 2000;70(6):677–684.
   PubMed Web of Science ®Google Scholar
• Fredrickson AG . Population balance equations for cell and microbial cultures revisited. AIChE J. 2003;49(4):1050–1059.
   Web of Science ®Google Scholar
• Smoller J . Shock waves and reaction-diffusion equations. 2nd ed. New York (NY): Springer-Verlag; 1994.
   Google Scholar
• Melzak ZA . The effect of coalescence in certain collision processes. Q Appl Math. 1953;11(2):231–234.
   Web of Science ®Google Scholar
• Melzak ZA . A scalar transport equation. Trans Am Math Soc. 1957;85(2):547–560.
   Google Scholar
• Smoluchowski PM . Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Physik Z. 1916;17:557–571 and 587--599.
   Google Scholar
• Farley K , Morel FM . Role of coagulation in the kinetics of sedimentation. Environ Sci Technol. 1986;20:187–195.
   PubMed Web of Science ®Google Scholar
• Sinko JW , Streifer W . A new model for age-size structure of a population. Ecology. 1967;48:910–918.
   Web of Science ®Google Scholar
• Filbet F , Laurençot P . Numerical simulation of the smoluchowski coagulation equation. SIAM J Sci Comput. 2004;25(6):2004–2028.
   Web of Science ®Google Scholar
• Ackleh AS , Ma B , Miller R . A general nonlinear model for the interaction of a size-structured population and its environment: well-posedness and approximation. Q Appl Math. 2016;74:671–704.
   Web of Science ®Google Scholar
• Alldredge AL , Gotschalk C , Passow U , et al . Mass aggregation of diatom blooms: insights from a mesocosm study. Deep Sea Res II. 1995;42(1):9–27.
   Web of Science ®Google Scholar
• Banks HT , Catenacci J , Hu S . Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems. J Inverse Ill-Posed Prob. 2016;24:413–433.
   Web of Science ®Google Scholar
• Cintron-Arias A , Banks HT , Capaldi A , et al . A sensitivity matrix based methodology for inverse problem formulation. J Inverse Ill-Posed Prob. 2009;17(6):545–564.
   Web of Science ®Google Scholar
• Martins JRRA , Strudza P , Alonso JJ . The complex-step derivative approximation. ACM T Math Soft. 2003;29(3):245–262.
   Web of Science ®Google Scholar
• Ackleh AS . Parameter estimation in a structured algal coagulation-fragmentation model. Nonlinear Anal. Theory Methods Appl. 1997;28(5):837–854.
   Web of Science ®Google Scholar
• Bache D . Floc rupture and turbulence: A framework for analysis. Chem Eng Sci. 2004;59(12):2521–2534.
   Web of Science ®Google Scholar
• Byrne E , Dzul S , Solomon M , et al . Postfragmentation density function for bacterial aggregates in laminar flow. Phys Rev E. 2011;83(4): 041911.
   Web of Science ®Google Scholar
• Doumic M , Tine LM . Estimating the division rate for the growth-fragmentation equation. J Math Biol. 2012;67(1):69–103.
   PubMed Web of Science ®Google Scholar
• Liu J , Zhou M , Zhou J , et al. Derivation and verification of the breakage rate coefficient during flocculation process based on shear strength. In: Kao JCM , Sung WP , and Chen R , editors. Materials, transportation and environmental engineering, Pts 1 and 2. Vols. 779--780, Stafa-Zurich: Trans Tech Publications Ltd; 2013. p. 1482–1489. WOS:000336106300288.
   Google Scholar
• Mirzaev I , Byrne EC , Bortz DM . An inverse problem for a class of conditional probability measure-dependent evolution equations. Inverse Prob. 2016;32(9):0950005.
   Web of Science ®Google Scholar
• Ziff RM , McGrady ED . The kinetics of cluster fragmentation and depolymerisation. J Phys A: Math Gen. 1985;18(15):3027–3037.
   Google Scholar
• Arasan S , Akbulut S , Hasiloglu AS . The relationship between the fractal dimension and shape properties of particles. KSCE J Civ Eng. 2011;15(7):1219–1225.
   Web of Science ®Google Scholar
• Droop MR . Comment: in defence of the cell quota model of micro-algal growth. J Plankton Res. 2003;25(1):103–107.
   Web of Science ®Google Scholar
• Buitenhuis ET , Geider RJ . A model of phytoplankton acclimation to iron-light colimitation. Limnol Oceanogr. 2010;55(2):714–724.
   Web of Science ®Google Scholar
• Pahlow M , Oschlies A . Optimal allocation backs Droop’s cell-quota model. Mar Ecol Prog Ser. 2013;473:1–5.
   Web of Science ®Google Scholar
• Brauer F , Noel J . The qualitative theory of ordinary differential equations. New York (NY): Dover; 1989.
   Google Scholar