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Article Contents

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

• 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.

Mathematics Subject Classification: 91B76, 03C45, 34C23, 34C25, 34D08.

 Citation:

• 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

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