Abstract
In this research work, a nonlinear mathematical model is proposed and analyzed to study the adverse effects of insects on agricultural productivity by controlling the insect population using insecticides. In the model formulation, it is assumed that agricultural crops grow logistically and the growth rate of insects wholly depends on agricultural crops with Holling type-II functional response. It is further assumed that insects uptake insecticides; thus, the amount of insecticides decreases at a rate proportional to its amount and the density of insect population, and the growth rate of insect population decrease in the same proportion. The feasibility of all non-negative equilibria and their stability properties are discussed. Stability analysis specifies that agricultural crop consumption rate destabilizes the system; however, the spraying rate of insecticides stabilizes the system. The conditions for the existence of pitchfork and Hopf-bifurcation are derived. Considering the spraying rate of insecticides as time-dependent, we have also discussed the optimal control strategy to minimize both insect density and the associated cost. The numerical simulation validates the analytical findings.
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