Author

Mark Jackson

Graduation Semester and Year

2018

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Benito M Chen

Abstract

Plants are a food source for man and many species. They also are sources of medicines, fibers for clothes, and are essential for a healthy environment. But plants are subject to diseases many of which are caused by viruses. These viruses often kill the plant. As a result, billions of dollars are lost every year because of virus related crop loss. Most of the time, virus propagation is done by a vector, usually insects that bite infected plants, get themselves infected and then bite susceptible plants. To combat the vectors, and ultimately the viruses, pesticides are often used as a control. Unfortunately, chemicals in pesticides can have a harmful effect on their environment. An alternative method to control the insect population is to introduce a natural predator of the insect. These predators may be more expensive than insecticides, but they are more environmentally friendly. To understand the dynamics, a system of ordinary and delay differential equations modeling interactions between insects and plants is considered and analyzed. To analyze the system, the basic reproductive number is used along with numerical simulations to find bifurcations. Then, a predator is introduced to the model, and the dynamics are studied in a similar fashion. Because of the seasonality of insects, active in the warm months and almost dormant in the cooler ones, the model is then analyzed with periodic coefficients. To study this model, the basic reproductive number is used, but calculated in a couple of different ways: a time average approach and a linear operator one. Finally an optimal control problem is studied. In this problem, the goal is to minimize the cost of the insecticide, predator, and cost of an infected plant. To solve this problem, two approaches are taken: an indirect approach using Pontryagin maximum principle and a direct approach used in the BOCOP software package.

Keywords

Mathematical modeling, Delay differential equations, Optimal control

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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