Ordinary Differential Equations and Applied Dynamics Commons^™

A Weak Fractional Calculus Theory And Numerical Methods For Fractional Differential Equations, Mitchell D. Sutton

Doctoral Dissertations

This dissertation is comprised of four integral parts. The first part comprises a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions.

The second part of this work presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the …

Data Driven Models Of Hemlock Woolly Adelgid Impacts And Biological Control, Hannah M. Thompson

Doctoral Dissertations

We present two models of the Adelges tsugae, the hemlock woolly adelgid, an invasive insect pest of Tsuga canadensis, eastern hemlock, in the eastern United States. An A. tsugae infestation often results in the death of T. canadensis within years, and has caused significant changes to hemlock forests. We construct two models composed of systems of ordinary differential equations with time dependent parameters to represent seasonality. The first model captures the coupled cycles in T. canadensis health and A. tsugae density. We use field data from Virginia to develop the model and to perform parameter estimation. The mechanisms …

Anthrax Models Involving Immunology, Epidemiology And Controls, Buddhi Raj Pantha

Doctoral Dissertations

This dissertation is divided in two parts. Chapters 2 and 3 consider the use of optimal control theory in an anthrax epidemiological model. Models consisting system of ordinary differential equations (ODEs) and partial differential differential equations (PDEs) are considered to describe the dynamics of infection spread. Two controls, vaccination and disposal of infected carcasses, are considered and their optimal management strategies are investigated. Chapter 4 consists modeling early host pathogen interaction in an inhalational anthrax infection which consists a system of ODEs that describes early dynamics of bacteria-phagocytic cell interaction associated to an inhalational anthrax infection.

First we consider a …

Dark-Bright Solitons And Vortices In Bose-Einstein Condensates, Dong Yan

Doctoral Dissertations

This dissertation focuses on the properties of nonlinear waves in Bose-Einstein condensates (BECs). The fundamental model here is the nonlinear Schrodinger equation, the so-called Gross-Pitaevskii (GP) equation, which is a mean-field description of BECs. The systematic analysis begins by considering the dark-bright (DB)-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component BECs. The interaction between two DB solitons in a homogeneous condensate and at the presence of the trap are both considered. Our analytical approximation relies in a Hamiltonian perturbation theory, which leads to an equation of motion of the centers of DB-soliton interacting pairs. Employing this equation, we demonstrate the …

Modeling The Genetic Consequences Of Mutualism On Communities, Carrie E. Eaton

Doctoral Dissertations

Three models of coevolutionary dynamics between mutualistically interacting species are developed. The first is a three loci, haploid model describing a general plant-pollinator system, such as Greya moth and its host plant. In this case, the system will always collapse to a single plant type and pollinator type. In a community with an mutant plant type, it is possible for a host-switch to occur, governed by the initial relative abundance plant type and the pollinator choosiness. In addition, genetic diversity can be maintained if the pollinator has no differential host preference, only adaptation to a host. Next, this model is …

Optimal Theory Applied In Integrodifference Equation Models And In A Cholera Differential Equation Model, Peng Zhong

Doctoral Dissertations

Integrodifference equations are discrete in time and continuous in space, and are used to model the spread of populations that are growing in discrete generations, or at discrete times, and dispersing spatially. We investigate optimal harvesting strategies, in order to maximize the profit and minimize the cost of harvesting. Theoretical results on the existence, uniqueness and characterization, as well as numerical results of optimized harvesting rates are obtained. The order of how the three events, growth, dispersal and harvesting, are arranged also affects the harvesting behavior.

Cholera remains a public health threat in many parts of the world and improved …

Optimal Control Of Species Augmentation Conservation Strategies, Erin Nicole Bodine

Doctoral Dissertations

Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. In this dissertation, species augmentation is analyzed in an optimal control setting to determine the optimal augmentation strategies given various constraints and settings. In each setting, we consider the effects on both the target/endangered population and a reserve population from which the individuals translocated in the augmentation are harvested. Four different optimal control formulations are explored. The first two optimal control formulations model the underlying population dynamics with a system of ordinary differential equations. Each of these …