Mathematical Modeling of Pest Management with Optimal Control

Abstract

Pests are damaging organisms that affect both human activities and crops production. Agri cultural pests are the species that cause damage to crops or interfere with crops. One of the main goals of pest control is to maintain the density of the pest population at the equilibrium level below economic damages. This dissertation developed mathematical models for the control and management of crop pests with environmentally safe strategies to minimize crop damage from pests with optimal awareness levels. We proposed and analyzed mathematical models in crop pest management, considering crop biomass, pest and the effect of farming awareness and Holling type-II functional response. The desired goals are to identify trade offs between costs, impacts, and outcomes using biological control, chemical control, and farming awareness level. We proposed pest control models of a farming awareness-based in tegrated approach employing systems of nonlinear ordinary and delay differential equations to build the models. We further assumed that there may be some time delay in measuring the healthy pests in the crop field. Thus, we considered a mathematical model incorporat ing two-time delays. Qualitative analyses were applied to the equilibrium points to study the stability analysis. The existence and the stability criteria of the equilibria are studied. The local stability of equilibria was analyzed using the Routh-Hurwitz criterion. Moreover, we have used optimal control theory to provide the cost-effective outline of bio-pesticides and a global awareness campaign. Pontryagin’s minimum principle is employed to establish the necessary conditions for the existence and characterization of the optimal controls. A Pontryagin-type minimum principle for retarded optimal control problems applied for the cost-effectiveness of the delayed system. We use Runge-Kutta’s forward-backward sweep method to solve the optimal control systems. The forward and backward Euler is used to approximate the optimal control problem with delays. Time delays destabilize the system, but they have no effect when global awareness is high or when the consumption rate of the crop by the pest is low. If the density of awareness campaigns increases, the crop density increases, and as a result pest prevalence decline. Stability switches occur through Hopf bifurcation when bifurcation parameters cross critical values. The negative effect of time delays is seen minimized using the optimal control approach.