An optimal control model for the impact of phoslock on the mitigation of algal biomass in lakes

In this study, we investigate the effects of excessive inputs of bioavailable phosphorus into a lake from agricultural fields and households on algal bloom formation, and its potential management by using the lanthanum-modified clay Phoslock as a bioavailable phosphorus adsorbent. We also investigate the impact of time delay involved in the process of applying Phoslock after measuring the density of algal biomass in the lake. Moreover, the seasonal effects in the input of bioavailable phosphorus from the agricultural lands and the application rate of Phoslock have been investigated. Our simulation results show that the algal growth accelerates if the bioavailable phosphorus is excessively loaded through agricultural runoff and domestic discharges. However, algal biomass can be effectively controlled by employing Phoslock in a sufficiently large quantity. Further, we find that a delay in the application of Phoslock induces limit cycle oscillations. Furthermore, our findings show that the combined actions of delay and periodicity in the application of Phoslock bring forth dynamical complexity in the lake system by giving rise to higher periodic solutions and bursting patterns. Lastly, we investigate an optimal control problem to estimate the optimum dosage of Phoslock for the mitigation of algal biomass from the lake system.