American Institute of Mathematical Sciences

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

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

Pankaj Kumar Tiwari 1, , Rajesh Kumar Singh 2, , Subhas Khajanchi 3, , Yun Kang 4, and  Arvind Kumar Misra 2,,

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  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.

Keywords: Algal bloom, phosphorus, Phoslock, time delay, chaos, sensitivity analysis.
Mathematics Subject Classification: 91B76, 03C45, 34C23, 34C25, 34D08.
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 & Continuous Dynamical Systems - B, 2021, 26 (6) : 3143-3175. doi: 10.3934/dcdsb.2020223
References:
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An overview of Phoslock and use in aquatic environments, https://www.sepro.com/documents/Phoslock/TechInfo/Phoslock%20Technical%20Bulletin.pdf. Google Scholar

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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.  Google Scholar

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S. Chakraborty, P. K. Tiwari, S. K. Sasmal, A. 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.  Google Scholar

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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.  Google Scholar

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J. B. Shukla, A. 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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

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

[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.  Google Scholar

[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. Google Scholar

[7]

G. L. Bowie, W. B. Mills, D. B. Porcella, C. L. Campbell, J. R. Pagenkopf, G. L. Rupp, K. M. Johnson, P. W. H. Chan, S. A. Gherini, C. E. Chamberlin and T. O. Barnwell, Rates, constants, and kinetics formulations in surface water quality modeling, EPA, 600 (1985), 3-85.   Google Scholar

[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.  Google Scholar

[9]

S. Chakraborty, P. K. Tiwari, A. 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.  Google Scholar

[10]

S. Chakraborty, P. K. Tiwari, S. K. Sasmal, A. 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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

S. Egemose, K. Reitzel, F. 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.  Google Scholar

[14]

T. S. Epe, K. 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.  Google Scholar

[15]

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

[16]

M. Fink, J. 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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

R. D. Gulati, L. 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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

K. E. Havens, T. L. East, S. J. Hwang, A. J. Rodusky, B. 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.   Google Scholar

[26]

D.-W. Huang, H.-L. Wang, J.-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.  Google Scholar

[27]

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

[28]

H. P. Jarve, C. 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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[35]

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

[36]

S. Meis, B. M. Spears, S. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

A. K. Misra, R. K. Singh, P. K. Tiwari, S. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[42]

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

[43]

N. N. Rabalais, R. J. Diaz, L. A. Levin, R. E. Turner, D. Gilbert and J. Zhang, Dynamics and distribution of natural and human-caused coastal hypoxia, Biogeosciences Discussions, 6 (2009), 9359-9453.   Google Scholar

[44]

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

[45]

M. Robb, B. Greenop, Z. Goss, G. Douglas and J. Adeney, Application of Phoslock, an innovative phosphorus binding clay, to two Western Australian waterways: preliminary findings, Hydrobiologia, 494 (2003), 237-243.   Google Scholar

[46]

J. B. Shukla, A. 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.  Google Scholar

[47]

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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
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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 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
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Table 1 except $ \mu = 0.25 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825)">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)
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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 $$ - $axis) and it also identifies the time intervals when the system is most sensitive to such changes. Parameters are at the same values as in Table Table 1 except $ \mu = 0.25 $, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825)
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Figure 5.  Sensitivity quantification by calculating sensitivity coefficient through $ L^2 $ norm
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Table 1, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825). The system exhibits Hopf-bifurcation through limit cycle oscillations for gradual increase in the delay parameter">Figure 6.  Bifurcation diagram of the system (20) with respect to $ \tau $. Here, the blue line represents the upper limit of the oscillation cycle and the red line represents the lower limit of the oscillation cycle. Parameters are at the same values as in Table Table 1, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825). The system exhibits Hopf-bifurcation through limit cycle oscillations for gradual increase in the delay parameter
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Table 1, and the initial conditions are chosen as (0.3543, 0.8983, 0.6913, 0.1825)">Figure 7.  Stability region for the system (20) in (a) $ q $$ - $$ \tau $ and (b) $ \mu $$ - $$ \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)
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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 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
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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 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
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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 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
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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 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
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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 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
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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 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
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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">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
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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
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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
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