Graduation Semester and Year

2022

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Hristo V Kojouharov

Abstract

Nonstandard finite difference (NSFD) methods have been widely used to numerically solve various problems in biology. In recent years, NSFD methods have been proposed that preserve essential properties of the solutions of general autonomous differential equations, such as positivity and elementary stability, among others. However, those methods are only of first-order accuracy. In the first part of this dissertation, we construct and analyze two second-order modified positive and elementary stable nonstandard (PESN) numerical methods for n-dimensional autonomous differential equations. The new PESN methods are generalized versions of the explicit Euler's method and second-order accurate, thereby improving the order of accuracy of the underlying numerical method. In the second part of this dissertation, we analyze several chemostat models with a constant input of one species. Chemostat models have been extensively used to represent microbial growth and competition in homogeneous environments. First, we consider a simple growth chemostat model for a donor bacteria and one limiting resource. Since competition is crucial in nature, we next propose a model when there is competition between a resident bacteria and the donor bacteria for a single limiting substrate in the presence of a lethal toxin. Resident bacteria can become donor bacteria by changing the genetics of the resident bacteria. This change can occur by a plasmid. Therefore, we propose a model when there is competition in the presence of a constant homogeneous plasmid. The proposed chemostat models are non-linear ordinary differential equations, and their exact solutions cannot be obtained analytically. Moreover, their solutions remain positive for all time. Therefore, using accurate and efficient numerical methods that also preserve the solutions' positivity property is essential when working with chemostat models. In the last part of the dissertation, the new second-order PESN (SOPESN) methods are used to approximate the solutions of the earlier presented chemostat models. In addition, the SOPESN methods are compared with several standard and nonstandard finite difference methods to numerically demonstrate their advantages when solving models in mathematical ecology.

Keywords

Nonstandard, Finite difference, Positivity, Elementary stable, Second-order, NSFD, PESN: Gut microbiome, Plasmids

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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