Modeling the Control of Algal Bloom in a Lake by Applying Some External Efforts with Time Delay

One of the crucial issues in ecology is related to algal bloom in lakes. Vast amount of efforts has been channeled to combat this problem, but for successful implementation of any algae eradication effort, their impact on the dynamics of water body must be studied explicitly. In view of this, we propose and analyze a mathematical model to assess the effect of applying efforts for removal of algae and detritus from a lake on the occurrence of algal bloom. The system is governed by the interactions of four variables, namely, concentration of nutrients, density of algae, density of detritus and efforts applied to remove algae as well as detritus from the lake. The biologically feasible equilibria and their stability properties are analyzed and discussed. Considering the growth rate of applied efforts as time dependent, we also obtain the optimal control strategy to minimize both the algae and the associated costs. The optimality system is derived and solved numerically. Further, to make the system more realistic, two discrete time delays; one in conversion of detritus into nutrients and other in applying the control efforts are considered. We study various cases for the time delays and show that in general the system develops limit cycle oscillation through a Hopf-bifurcation for increasing time delays. However, we observe that in some cases, increase in time delays may diminish the oscillations. Our results indicate that high concentration of nutrient intensifies the occurrence of algal bloom and in that case applied efforts play a pivotal role in reducing the bloom.