Optimal Control for Management in Gypsy Moth Models

The gypsy moth, Lymantria dispar (L.), is an invasive species and the most destructive forest defoliator in North America. Gypsy moth outbreaks are spatially synchronized over areas across hundreds of kilometers. Outbreaks can result in loss of timber and other forestry products. Greater losses tend to occur to the ecosystem services that forests provide, such as wildlife habitat, carbon sequestration, and nutrient cycling. The United States can be divided in three different areas: a generally infested area (populations established), an uninfested area (populations not established), and a transition zone between the two. There are different management programs matching these different areas: detection and eradication, the Slow-the-Spread program, and suppression of outbreaks in areas that are infested by the gypsy moth as a means to mitigate impacts. This dissertation focuses in optimal control techniques for models of areas where the population is established or in the invasion front. We develop an optimal control formulation for models of an established population of the invasive pest gypsy moth. The models include interaction with a pathogen and a generalist predator. The population of gypsy moth is assumed to be controlled with the pesticide Bt. The assumed objective functional minimizes cost due to gypsy moth and cost for suppressing the population of gypsy moth. Optimization techniques in our numerical results, suggest the timing and intensity of control. Our results are consistent over different parameter values and initial conditions. To model the population in the invasion front, we develop the theory of optimal control for a system of integrodifference equations. Integrodifference equations incorporate continuous space into a system of discrete time equations. We design an objective functional to minimize the cost generated by the defoliation caused by the gypsy moth and the cost of controlling the population. Existence and uniqueness results for the optimal control and corresponding states have been completed. We use a forward-backward sweep numerical method, and our numerical results suggest appropriate spatial and temporal location and intensity of optimal controls