June  2021, 26(6): 3143-3175. doi: 10.3934/dcdsb.2020223

A mathematical model to restore water quality in urban lakes using Phoslock

1. 

Department of Mathematics, University of Kalyani, Kalyani - 741235, India

2. 

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India

3. 

Department of Mathematics, Presidency University, Kolkata - 700073, India

4. 

Science and Mathematics Faculty, Arizona State University Mesa, AZ 85212, USA

* Corresponding author: akmisra@bhu.ac.in

Received  April 2020 Revised  May 2020 Published  June 2021 Early access  July 2020

Urban lakes are the life lines for the population residing in the city. Excessive amounts of phosphate entering water courses through household discharges is one of the main causes of deterioration of water quality in these lakes because of the way it drives algal productivity and undesirable changes in the balance of aquatic life. The ability to remove biologically available phosphorus in a lake is therefore a major step towards improving water quality. By removing phosphate from the water column using Phoslock essentially deprives algae and its proliferation. In view of this, we develop a mathematical model to investigate whether the application of Phoslock would significantly reduce the bio-availability of phosphate in the water column. We consider phosphorus, algae, detritus and Phoslock as dynamical variables. In the modeling process, the introduction rate of Phoslock is assumed to be proportional to the concentration of phosphorus in the lake. Further, we consider a discrete time delay which accounts for the time lag involved in the application of Phoslock. Moreover, we investigate behavior of the system by assuming the application rate of Phoslock as a periodic function of time. Our results evoke that Phoslock essentially reduces the concentration of phosphorus and density of algae, and plays crucial role in restoring the quality of water in urban lakes. We observe that for the gradual increase in the magnitude of the delay involved in application of Phoslock, the autonomous system develops limit cycle oscillations through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations.

Citation: Pankaj Kumar Tiwari, Rajesh Kumar Singh, Subhas Khajanchi, Yun Kang, Arvind Kumar Misra. A mathematical model to restore water quality in urban lakes using Phoslock. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3143-3175. doi: 10.3934/dcdsb.2020223
References:
[1]

An overview of Phoslock and use in aquatic environments, https://www.sepro.com/documents/Phoslock/TechInfo/Phoslock%20Technical%20Bulletin.pdf.

[2]

A. M. Beeton and W. T. Edmonsdon, The eutrophication problem, J. Fish. Res. Bd. Canada, 29 (1972), 673-682.  doi: 10.1139/f72-113.

[3]

J. M. Beman, K. R. Arrigo and P. A. Matson, Agricultural runoff fuels large phytoplankton blooms in vulnerable areas of the ocean, Nature, 434 (2005), 211.

[4]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, 4th edn., John Wiley & Sons, Inc. New York, 1989.

[5]

W. M. Bishop, T. McNabb, I. Cormican, B. E. Willis and S. Hyde, Operational evaluation of Phoslock phosphorus locking technology in Laguna Niguel Lake, California, Water Air Soil Pollut., 225 (2014), 2018. doi: 10.1007/s11270-014-2018-6.

[6]

D. F. Boesch, Harmful Algal Bloom in Coastal Waters: Options for Prevention, Control and Mitigation, 10, US Department of Commerce, National Oceanic and Atmospheric Administration, Coastal Ocean Office, 1997.

[7]

G. L. BowieW. B. MillsD. B. PorcellaC. L. CampbellJ. R. PagenkopfG. L. RuppK. M. JohnsonP. W. H. ChanS. A. GheriniC. E. Chamberlin and T. O. Barnwell, Rates, constants, and kinetics formulations in surface water quality modeling, EPA, 600 (1985), 3-85. 

[8]

S. R. Carpenter, Phosphorus control is critical to mitigating eutrophication, Proc. Nat. Acad. Sci. USA, 105 (2008), 11039-11040.  doi: 10.1073/pnas.0806112105.

[9]

S. ChakrabortyP. K. TiwariA. K. Misra and J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Biosci., 264 (2015), 94-100.  doi: 10.1016/j.mbs.2015.03.010.

[10]

S. ChakrabortyP. K. TiwariS. K. SasmalA. K. Misra and J. Chattopadhyay, Effects of fertilizers used in agricultural fields on algal blooms, Eur. Phys. J. Spec. Top., 226 (2017), 2119-2133.  doi: 10.1140/epjst/e2017-70031-7.

[11]

D. L. Correll, The role of phosphorus in the eutrophication of receiving waters: A review, J. Enviro. Qual., 27 (1998), 261-266.  doi: 10.2134/jeq1998.00472425002700020004x.

[12]

M. DokulilW. Chen and Q. Cai, Anthropogenic impacts to large lakes in china: The Taihu example, Aquat. Ecosyst. Health Manag., 3 (2000), 81-94. 

[13]

S. EgemoseK. ReitzelF. Ø. Andersen and M. R. Flindt, Chemical lake restoration products: Sediment stability and phosphorus dynamics, Environ. Sci. Technol., 44 (2010), 985-991.  doi: 10.1021/es903260y.

[14]

T. S. EpeK. Finsterle and S. Yasseri, Nine years of phosphorus management with lanthanum modified bentonite (Phoslock) in a eutrophic, shallow swimming lake in Germany, Lake Reserv. Manage., 33 (2017), 119-129.  doi: 10.1080/10402381.2016.1263693.

[15]

M. Fink, myAD: fast automatic differentiation code in Matlab, (2006) https://se.mathworks.com/matlabcentral/fileexchange/15235-automatic-differentiation-for-matlab.

[16]

M. FinkJ. J. Batzel and H. Tran, A respiratory system model: Parameter estimation and sensitivity analysis, Cardiovasc. Eng., 8 (2008), 120-134.  doi: 10.1007/s10558-007-9051-7.

[17]

P. J. S. Franks, Models of harmful algal blooms, Limnol. Oceanogr., 42 (1997), 1273-1282.  doi: 10.4319/lo.1997.42.5_part_2.1273.

[18]

K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its applications. 74, Kluwer Academic Pub. Dordrecht, 1992.

[19]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[20]

R. D. GulatiL. M. D. Pires and E. Van Donk, Lake restoration studies: Failures, bottlenecks and prospects of new ecotechnological measures, Limnologica, 38 (2008), 233-247.  doi: 10.1016/j.limno.2008.05.008.

[21]

J. K. Hale, Functional Differential Equations, Springer Berlin Heidelbergi, 1971.

[22]

J. K. Hale, L. Verduyn and M. Sjoerd, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[23]

A. D. Hasler, Eutrophication of lakes by domestic drainage, Ecology, 28 (1947), 383-395.  doi: 10.2307/1931228.

[24]

A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.

[25]

K. E. HavensT. L. EastS. J. HwangA. J. RoduskyB. Sharfstein and A. D. Steinman, Algal responses to experimental nutrient addition in the littooral community of a subtropical lake, Freshwater Biol, 42 (1999), 329-344. 

[26]

D.-W. HuangH.-L. WangJ.-F. Feng and Z.-W. Zhu, Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics, Appl. Math. Model., 32 (2008), 1318-1326.  doi: 10.1016/j.apm.2007.04.006.

[27]

A. HuppertB. Blasius and L. Stone, A model of phytoplankton blooms, Am. Nat., 159 (2002), 156-171.  doi: 10.1086/324789.

[28]

H. P. JarveC. Neal and P. J. A. Withers, Sewage-effluent phosphorus: A greater risk to river eutrophication than agricultural phosphorus?, Sci. Total Environ., 360 (2006), 246-253.  doi: 10.1016/j.scitotenv.2005.08.038.

[29]

R. A. Jones and G. F. Lee, Recent advances in assessing impact of phosphorus loads on eutrophication related water quality, Water Res., 16 (1982), 503-515.  doi: 10.1016/0043-1354(82)90069-0.

[30]

S. E. Jørgenson, A eutrophication model for a lake, Ecol. Model., 2 (1976), 147-165.  doi: 10.1016/0304-3800(76)90030-2.

[31]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York/Basel, 1989.

[32]

S. T. Larned, Nitrogen-versus phosphorus-limited growth and sources of nutrients for coral reef macroalgae, Marine Biol., 132 (1998), 409-421. 

[33]

J. W. G. Lund, Eutrophiocation, Proc. R. Soc. Lond. B, 180 (1972), 371-382. 

[34]

M. Lürling and Y. Tolman, Effects of lanthanum and lanthanum-modified clay on growth, survival and reproduction of Daphnia magna, Water Res., 44 (2010), 309-319.  doi: 10.1016/j.watres.2009.09.034.

[35]

H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology, CRC, 2008.

[36]

S. MeisB. M. SpearsS. C. Maberly and R. G. Perkins, Assessing the mode of action of Phoslock in the control of phosphorus release from the bed sediments in a shallow lake (Loch Flemington, UK), Water Res., 47 (2013), 4460-4473.  doi: 10.1016/j.watres.2013.05.017.

[37]

A. K. Misra, Modeling the depletion of dissolved oxygen in a lake due to submerged macrophytes, Nonlinear Anal. Model. Cont., 15 (2010), 185-198.  doi: 10.15388/NA.2010.15.2.14353.

[38]

A. K. Misra, Modeling the depletion of dissolved oxygen in a lake by taking Holling type-III interaction, Appl. Math. Comp., 217 (2011), 8367-8376. 

[39]

A. K. MisraR. K. SinghP. K. TiwariS. Khajanchi and Y. Kang, Dynamics of algae blooming: Effects of budget allocation and time delay, Nonlinear Dyn., 100 (2020), 1779-1807.  doi: 10.1007/s11071-020-05551-4.

[40]

A. K. Misra, P. K. Tiwari and P. Chandra, Modeling the control of algal bloom in a lake by applying some external efforts with time delay, Differ. Equ. Dyn. Syst., (2017). doi: 10.1007/s12591-017-0383-5.

[41]

T. Park, A matlab version of the lyapunov exponent estimation algorithm of Wolf et al. - physica16d, 1985, https://www.mathworks.com/matlabcentral/fileexchange/48084-lyapunov-exponent-estimation-from-a-time-series-documentation-added.

[42]

Phoslock: Patented phosphorus locking technology, https://www.sepro.com/aquatics/phoslock.

[43]

N. N. RabalaisR. J. DiazL. A. LevinR. E. TurnerD. Gilbert and J. Zhang, Dynamics and distribution of natural and human-caused coastal hypoxia, Biogeosciences Discussions, 6 (2009), 9359-9453. 

[44]

S. Rinaldi, R. Soncini-Sessa, H. Stehfest and H. Tamura, Modeling and Control of River Quality, McGraw-Hill Inc., U.K., 1979.

[45]

M. RobbB. GreenopZ. GossG. Douglas and J. Adeney, Application of Phoslock, an innovative phosphorus binding clay, to two Western Australian waterways: preliminary findings, Hydrobiologia, 494 (2003), 237-243. 

[46]

J. B. ShuklaA. K. Misra and P. Chandra, Modelling and analysis of the algal bloom in a lake caused by discharge of nutrients, Appl. Math. Comp., 196 (2008), 782-790.  doi: 10.1016/j.amc.2007.07.010.

[47]

Z. Teng and S. Chen, The positive periodic solutions of periodic kolmogorove type systems with delays, Acta Math. Appl. Sin., 22 (1999), 446-456. 

[48]

P. K. TiwariA. K. Misra and E. Venturino, The role of algae in agriculture: A mathematical study, J. Biol. Phys., 43 (2017), 297-314.  doi: 10.1007/s10867-017-9453-8.

[49]

P. K. TiwariS. RanaA. K. Misra and J. Chattopadhyay, Effect of cross-diffusion on the patterns of algal bloom in a lake: A nonlinear analysis, Nonlinear Stud., 21 (2014), 443-462. 

[50]

P. K. Tiwari, S. Samanta, J. D. Ferreira and A. K. Misra, A mathematical model for the effects of nitrogen and phosphorus on algal blooms, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950129, 30pp. doi: 10.1142/S0218127419501293.

[51]

F. van Oosterhout and M. Lürling, The effect of phosphorus binding clay (Phoslock) in mitigating cyanobacterial nuisance: a laboratory study on the effects on water quality variables and plankton, Hydrobiologia, 710 (2013), 265-277.  doi: 10.1007/s10750-012-1206-x.

[52]

M. L. G. Waajen, B. Engels and F. van Oosterhout, Effects of dredging and Lanthanum-modified clay on water quality variables in an enclosure study in a hypertrophic pond, Water, 9 (2017), 380. doi: 10.3390/w9060380.

[53]

H. WangA. Appan and J. S. Gulliver, Modelling of phosphorus dynamics in aquatic sediments: I-model development, Water Res., 37 (2003), 3928-3938. 

[54]

A. WolfJ. B. SwiftH. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.

[55]

X. YangL. Chen and J. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Comput. Math. Appl., 32 (1996), 109-116.  doi: 10.1016/0898-1221(96)00129-0.

[56]

M. ZamparasG. GavriilF. A. Coutelieris and I. Zacharias, A theoretical and experimental study on the P-absorption capacity of Phoslock, Appl. Surf. Sci., 335 (2015), 147-152. 

show all references

References:
[1]

An overview of Phoslock and use in aquatic environments, https://www.sepro.com/documents/Phoslock/TechInfo/Phoslock%20Technical%20Bulletin.pdf.

[2]

A. M. Beeton and W. T. Edmonsdon, The eutrophication problem, J. Fish. Res. Bd. Canada, 29 (1972), 673-682.  doi: 10.1139/f72-113.

[3]

J. M. Beman, K. R. Arrigo and P. A. Matson, Agricultural runoff fuels large phytoplankton blooms in vulnerable areas of the ocean, Nature, 434 (2005), 211.

[4]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, 4th edn., John Wiley & Sons, Inc. New York, 1989.

[5]

W. M. Bishop, T. McNabb, I. Cormican, B. E. Willis and S. Hyde, Operational evaluation of Phoslock phosphorus locking technology in Laguna Niguel Lake, California, Water Air Soil Pollut., 225 (2014), 2018. doi: 10.1007/s11270-014-2018-6.

[6]

D. F. Boesch, Harmful Algal Bloom in Coastal Waters: Options for Prevention, Control and Mitigation, 10, US Department of Commerce, National Oceanic and Atmospheric Administration, Coastal Ocean Office, 1997.

[7]

G. L. BowieW. B. MillsD. B. PorcellaC. L. CampbellJ. R. PagenkopfG. L. RuppK. M. JohnsonP. W. H. ChanS. A. GheriniC. E. Chamberlin and T. O. Barnwell, Rates, constants, and kinetics formulations in surface water quality modeling, EPA, 600 (1985), 3-85. 

[8]

S. R. Carpenter, Phosphorus control is critical to mitigating eutrophication, Proc. Nat. Acad. Sci. USA, 105 (2008), 11039-11040.  doi: 10.1073/pnas.0806112105.

[9]

S. ChakrabortyP. K. TiwariA. K. Misra and J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Biosci., 264 (2015), 94-100.  doi: 10.1016/j.mbs.2015.03.010.

[10]

S. ChakrabortyP. K. TiwariS. K. SasmalA. K. Misra and J. Chattopadhyay, Effects of fertilizers used in agricultural fields on algal blooms, Eur. Phys. J. Spec. Top., 226 (2017), 2119-2133.  doi: 10.1140/epjst/e2017-70031-7.

[11]

D. L. Correll, The role of phosphorus in the eutrophication of receiving waters: A review, J. Enviro. Qual., 27 (1998), 261-266.  doi: 10.2134/jeq1998.00472425002700020004x.

[12]

M. DokulilW. Chen and Q. Cai, Anthropogenic impacts to large lakes in china: The Taihu example, Aquat. Ecosyst. Health Manag., 3 (2000), 81-94. 

[13]

S. EgemoseK. ReitzelF. Ø. Andersen and M. R. Flindt, Chemical lake restoration products: Sediment stability and phosphorus dynamics, Environ. Sci. Technol., 44 (2010), 985-991.  doi: 10.1021/es903260y.

[14]

T. S. EpeK. Finsterle and S. Yasseri, Nine years of phosphorus management with lanthanum modified bentonite (Phoslock) in a eutrophic, shallow swimming lake in Germany, Lake Reserv. Manage., 33 (2017), 119-129.  doi: 10.1080/10402381.2016.1263693.

[15]

M. Fink, myAD: fast automatic differentiation code in Matlab, (2006) https://se.mathworks.com/matlabcentral/fileexchange/15235-automatic-differentiation-for-matlab.

[16]

M. FinkJ. J. Batzel and H. Tran, A respiratory system model: Parameter estimation and sensitivity analysis, Cardiovasc. Eng., 8 (2008), 120-134.  doi: 10.1007/s10558-007-9051-7.

[17]

P. J. S. Franks, Models of harmful algal blooms, Limnol. Oceanogr., 42 (1997), 1273-1282.  doi: 10.4319/lo.1997.42.5_part_2.1273.

[18]

K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its applications. 74, Kluwer Academic Pub. Dordrecht, 1992.

[19]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[20]

R. D. GulatiL. M. D. Pires and E. Van Donk, Lake restoration studies: Failures, bottlenecks and prospects of new ecotechnological measures, Limnologica, 38 (2008), 233-247.  doi: 10.1016/j.limno.2008.05.008.

[21]

J. K. Hale, Functional Differential Equations, Springer Berlin Heidelbergi, 1971.

[22]

J. K. Hale, L. Verduyn and M. Sjoerd, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[23]

A. D. Hasler, Eutrophication of lakes by domestic drainage, Ecology, 28 (1947), 383-395.  doi: 10.2307/1931228.

[24]

A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.

[25]

K. E. HavensT. L. EastS. J. HwangA. J. RoduskyB. Sharfstein and A. D. Steinman, Algal responses to experimental nutrient addition in the littooral community of a subtropical lake, Freshwater Biol, 42 (1999), 329-344. 

[26]

D.-W. HuangH.-L. WangJ.-F. Feng and Z.-W. Zhu, Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics, Appl. Math. Model., 32 (2008), 1318-1326.  doi: 10.1016/j.apm.2007.04.006.

[27]

A. HuppertB. Blasius and L. Stone, A model of phytoplankton blooms, Am. Nat., 159 (2002), 156-171.  doi: 10.1086/324789.

[28]

H. P. JarveC. Neal and P. J. A. Withers, Sewage-effluent phosphorus: A greater risk to river eutrophication than agricultural phosphorus?, Sci. Total Environ., 360 (2006), 246-253.  doi: 10.1016/j.scitotenv.2005.08.038.

[29]

R. A. Jones and G. F. Lee, Recent advances in assessing impact of phosphorus loads on eutrophication related water quality, Water Res., 16 (1982), 503-515.  doi: 10.1016/0043-1354(82)90069-0.

[30]

S. E. Jørgenson, A eutrophication model for a lake, Ecol. Model., 2 (1976), 147-165.  doi: 10.1016/0304-3800(76)90030-2.

[31]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York/Basel, 1989.

[32]

S. T. Larned, Nitrogen-versus phosphorus-limited growth and sources of nutrients for coral reef macroalgae, Marine Biol., 132 (1998), 409-421. 

[33]

J. W. G. Lund, Eutrophiocation, Proc. R. Soc. Lond. B, 180 (1972), 371-382. 

[34]

M. Lürling and Y. Tolman, Effects of lanthanum and lanthanum-modified clay on growth, survival and reproduction of Daphnia magna, Water Res., 44 (2010), 309-319.  doi: 10.1016/j.watres.2009.09.034.

[35]

H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology, CRC, 2008.

[36]

S. MeisB. M. SpearsS. C. Maberly and R. G. Perkins, Assessing the mode of action of Phoslock in the control of phosphorus release from the bed sediments in a shallow lake (Loch Flemington, UK), Water Res., 47 (2013), 4460-4473.  doi: 10.1016/j.watres.2013.05.017.

[37]

A. K. Misra, Modeling the depletion of dissolved oxygen in a lake due to submerged macrophytes, Nonlinear Anal. Model. Cont., 15 (2010), 185-198.  doi: 10.15388/NA.2010.15.2.14353.

[38]

A. K. Misra, Modeling the depletion of dissolved oxygen in a lake by taking Holling type-III interaction, Appl. Math. Comp., 217 (2011), 8367-8376. 

[39]

A. K. MisraR. K. SinghP. K. TiwariS. Khajanchi and Y. Kang, Dynamics of algae blooming: Effects of budget allocation and time delay, Nonlinear Dyn., 100 (2020), 1779-1807.  doi: 10.1007/s11071-020-05551-4.

[40]

A. K. Misra, P. K. Tiwari and P. Chandra, Modeling the control of algal bloom in a lake by applying some external efforts with time delay, Differ. Equ. Dyn. Syst., (2017). doi: 10.1007/s12591-017-0383-5.

[41]

T. Park, A matlab version of the lyapunov exponent estimation algorithm of Wolf et al. - physica16d, 1985, https://www.mathworks.com/matlabcentral/fileexchange/48084-lyapunov-exponent-estimation-from-a-time-series-documentation-added.

[42]

Phoslock: Patented phosphorus locking technology, https://www.sepro.com/aquatics/phoslock.

[43]

N. N. RabalaisR. J. DiazL. A. LevinR. E. TurnerD. Gilbert and J. Zhang, Dynamics and distribution of natural and human-caused coastal hypoxia, Biogeosciences Discussions, 6 (2009), 9359-9453. 

[44]

S. Rinaldi, R. Soncini-Sessa, H. Stehfest and H. Tamura, Modeling and Control of River Quality, McGraw-Hill Inc., U.K., 1979.

[45]

M. RobbB. GreenopZ. GossG. Douglas and J. Adeney, Application of Phoslock, an innovative phosphorus binding clay, to two Western Australian waterways: preliminary findings, Hydrobiologia, 494 (2003), 237-243. 

[46]

J. B. ShuklaA. K. Misra and P. Chandra, Modelling and analysis of the algal bloom in a lake caused by discharge of nutrients, Appl. Math. Comp., 196 (2008), 782-790.  doi: 10.1016/j.amc.2007.07.010.

[47]

Z. Teng and S. Chen, The positive periodic solutions of periodic kolmogorove type systems with delays, Acta Math. Appl. Sin., 22 (1999), 446-456. 

[48]

P. K. TiwariA. K. Misra and E. Venturino, The role of algae in agriculture: A mathematical study, J. Biol. Phys., 43 (2017), 297-314.  doi: 10.1007/s10867-017-9453-8.

[49]

P. K. TiwariS. RanaA. K. Misra and J. Chattopadhyay, Effect of cross-diffusion on the patterns of algal bloom in a lake: A nonlinear analysis, Nonlinear Stud., 21 (2014), 443-462. 

[50]

P. K. Tiwari, S. Samanta, J. D. Ferreira and A. K. Misra, A mathematical model for the effects of nitrogen and phosphorus on algal blooms, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950129, 30pp. doi: 10.1142/S0218127419501293.

[51]

F. van Oosterhout and M. Lürling, The effect of phosphorus binding clay (Phoslock) in mitigating cyanobacterial nuisance: a laboratory study on the effects on water quality variables and plankton, Hydrobiologia, 710 (2013), 265-277.  doi: 10.1007/s10750-012-1206-x.

[52]

M. L. G. Waajen, B. Engels and F. van Oosterhout, Effects of dredging and Lanthanum-modified clay on water quality variables in an enclosure study in a hypertrophic pond, Water, 9 (2017), 380. doi: 10.3390/w9060380.

[53]

H. WangA. Appan and J. S. Gulliver, Modelling of phosphorus dynamics in aquatic sediments: I-model development, Water Res., 37 (2003), 3928-3938. 

[54]

A. WolfJ. B. SwiftH. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.

[55]

X. YangL. Chen and J. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Comput. Math. Appl., 32 (1996), 109-116.  doi: 10.1016/0898-1221(96)00129-0.

[56]

M. ZamparasG. GavriilF. A. Coutelieris and I. Zacharias, A theoretical and experimental study on the P-absorption capacity of Phoslock, Appl. Surf. Sci., 335 (2015), 147-152. 

Figure 1.  Schematic diagram for the interactions among phosphorus, algae, detritus and Phoslock. Here, cyan color represents the impact of Phoslock on reduction of phosphorus concentration; blue color stands for the periodic introduction of Phoslock in the lake; red color represents the time delay involved in the application of Phoslock. The two dashed lines headed to each other indicate that Phoslock bind with phosphorus and becomes an inert component of the sediments
Figure 2.  Bifurcation diagram of the system (1) with respect to the uptake rate of phosphorus by algae, $ \beta_1 $. Rest of the parameters are at the same values as in Table Table 1 except $ \mu = 0.25 $. The algae-detritus-free equilibrium $ E_0 $ exchanges its stability with the interior equilibrium $ E^* $ when the parameter $ \beta_1 $ crosses its critical value from below
Figure 3.  The equilibrium values of phosphorus (first row) and algae (second row) as functions of $ \phi $ and $ q $ (first column), $ \mu $ and $ \beta_1 $ (second column), and $ \lambda $ and $ \pi_2 $ (third column). Parameters are at the same values as in Table Table 1 except $ \mu = 0.25 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825)
Figure 4.  Semi-relative sensitivities of the parameters using automatic differentiation. The observation window is [0,500] and the sensitivity of a parameter is identified by the maximum deviation of the state variable (along $ y $$ - $axis) and it also identifies the time intervals when the system is most sensitive to such changes. Parameters are at the same values as in Table Table 1 except $ \mu = 0.25 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825)
Figure 5.  Sensitivity quantification by calculating sensitivity coefficient through $ L^2 $ norm
Figure 6.  Bifurcation diagram of the system (20) with respect to $ \tau $. Here, the blue line represents the upper limit of the oscillation cycle and the red line represents the lower limit of the oscillation cycle. Parameters are at the same values as in Table Table 1, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825). The system exhibits Hopf-bifurcation through limit cycle oscillations for gradual increase in the delay parameter
Figure 7.  Stability region for the system (20) in (a) $ q $$ - $$ \tau $ and (b) $ \mu $$ - $$ \tau $ planes. Here, $ * $ represents the stable equilibrium for corresponding values of the parameters and $ * $ represents otherwise. Parameters are at the same values as in Table Table 1, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825)
Figure 8.  Simulation results of the nonautonomous system (29) at $ \tau = 0 $ day. Parameters are at the same values as in Table Table 1, and $ \mu_{11} = 0.03 $. For the periodic input of Phoslock, the system exhibits positive periodic solutions in the absence of time delay
Figure 9.  Simulation results of the nonautonomous system (29) at $ \tau = 10 $ days. Parameters are at the same values as in Table Table 1, and $ \mu_{11} = 0.03 $. For the periodic input of Phoslock, the system exhibits positive periodic solutions for lower values of time delay
Figure 10.  Global stability of positive periodic solution for the nonautonomous system (29) at $ \tau = 10 $ days. Parameters are at the same values as in Table Table 1, and $ \mu_{11} = 0.03 $. Figure shows that solution trajectories starting from three different initial points (1.2, 0.6, 0.2, 0.18), (2, 0.7, 0.3, 0.08) and (2.8 0.5 0.1 0.28) ultimately converge to a unique positive periodic solution
Figure 11.  Simulation results for the nonautonomous system (29) at $ \tau = 85 $ days. Parameters are at the same values as in Table Table 1 and $ \mu_{11} = 0.03 $. For the periodic input of Phoslock, the system exhibits chaotic dynamics for larger values of time delay
Figure 12.  Bifurcation diagram of the system (29) with respect to $ \tau $. Here, the blue line represents the upper limit of the oscillation cycle and the red line represents the lower limit of the oscillation cycle. Parameters are at the same values as in Table Table 1 and $ \mu_{11} = 0.03 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825). For the periodic input of Phoslock, the system enters into chaotic regime from positive periodic solutions for gradual increase in the delay parameter
Figure 13.  Poincaré map of the system (29) in $ A $$ - $$ D $$ - $$ C $ space at $ N = 0.8 $ $ \mu $g/L for $ \tau = 85 $ days. Parameters are at the same values as in Table Table 1 and $ \mu_{11} = 0.03 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825). The scattered distribution of the sampling points indicates the chaotic dynamics of the system
Figure 14.  Figure shows the maximum Lyapunov exponent of the nonautonomous delayed system (29) for $ \tau = 85 $ days. Parameters are at the same values as in Table Table 1 and $ \mu_{11} = 0.03 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825). In the figure, positive values of the maximum Lyapunov exponent confirms the occurrence of chaotic oscillation
Table 1.  Biological meanings of parameters in the system (1) and their values (hypothetical) used for numerical simulations
Names Descriptions Units Values
$q$ Input rate of phosphorus to the lake $\mu$g/L/day 0.1
$\alpha_0$ Per capita loss rate of phosphorus 1/day 0.002
$\beta_1$ Maximum uptake rate of phosphorus by algae 1/day 0.1
$\beta_{12}$ Half saturation constant for the uptake of phosphorus by algae $\mu$g/L/day 1
$\beta_{11}$ Proportionality constant 1
$\theta_1$ Algal growth due to phosphorus uptake 0.9
$\alpha_1$ Natural mortality and higher predation of algae 1/day 0.03
$\beta_{10}$ Algal mortality due to intraspecific competition L/$\mu$g/day 0.05
$\pi_1$ Algal conversion into detritus 0.3
$\delta$ Sinking rate of detritus to the bottom of the lake 1/day 0.05
$\pi_2$ Remineralization of detritus into nutrients 0.01
$\mu$ Introduction rate of Phoslock in the lake 1/day 0.085
$\mu_0$ Natural depletion rate of Phoslock 1/day 0.2
$\phi$ Reduction rate of Phoslock due to reaction with phosphorus L/$\mu$g/day 0.5
$\lambda$ Reduction of phosphorus due to reaction with Phoslock 0.5
Names Descriptions Units Values
$q$ Input rate of phosphorus to the lake $\mu$g/L/day 0.1
$\alpha_0$ Per capita loss rate of phosphorus 1/day 0.002
$\beta_1$ Maximum uptake rate of phosphorus by algae 1/day 0.1
$\beta_{12}$ Half saturation constant for the uptake of phosphorus by algae $\mu$g/L/day 1
$\beta_{11}$ Proportionality constant 1
$\theta_1$ Algal growth due to phosphorus uptake 0.9
$\alpha_1$ Natural mortality and higher predation of algae 1/day 0.03
$\beta_{10}$ Algal mortality due to intraspecific competition L/$\mu$g/day 0.05
$\pi_1$ Algal conversion into detritus 0.3
$\delta$ Sinking rate of detritus to the bottom of the lake 1/day 0.05
$\pi_2$ Remineralization of detritus into nutrients 0.01
$\mu$ Introduction rate of Phoslock in the lake 1/day 0.085
$\mu_0$ Natural depletion rate of Phoslock 1/day 0.2
$\phi$ Reduction rate of Phoslock due to reaction with phosphorus L/$\mu$g/day 0.5
$\lambda$ Reduction of phosphorus due to reaction with Phoslock 0.5
[1]

Hideo Ikeda, Masayasu Mimura, Tommaso Scotti. Shadow system approach to a plankton model generating harmful algal bloom. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 829-858. doi: 10.3934/dcds.2017034

[2]

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

[3]

Jiaxu Li, Yang Kuang, Bingtuan Li. Analysis of IVGTT glucose-insulin interaction models with time delay. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 103-124. doi: 10.3934/dcdsb.2001.1.103

[4]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633

[5]

Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial and Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175

[6]

Jaroslaw Smieja, Malgorzata Kardynska, Arkadiusz Jamroz. The meaning of sensitivity functions in signaling pathways analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2697-2707. doi: 10.3934/dcdsb.2014.19.2697

[7]

Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial and Management Optimization, 2020, 16 (4) : 2029-2044. doi: 10.3934/jimo.2019041

[8]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241

[9]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5227-5249. doi: 10.3934/dcdsb.2020341

[10]

Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022035

[11]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301

[12]

Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011

[13]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic and Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[14]

Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations and Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55

[15]

Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial and Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381

[16]

Hamed Azizollahi, Marion Darbas, Mohamadou M. Diallo, Abdellatif El Badia, Stephanie Lohrengel. EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity. Mathematical Biosciences & Engineering, 2018, 15 (4) : 905-932. doi: 10.3934/mbe.2018041

[17]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial and Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1

[18]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations and Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

[19]

Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521

[20]

Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks and Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (437)
  • HTML views (360)
  • Cited by (0)

[Back to Top]