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

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