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<p> The main objective of this dissertation is to study numerical methods and applications of structured population models. In Chapter 1, a quasilinear hierarchically size-structured population model is presented. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. The solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this chapter we developed a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method was provided. We also developed a high resolution second order scheme to compute the measure-valued solution of the model and performed a comparative study between the two schemes. In Chapter 2, a model that describes the dynamics of a frog population infected with chytridiomycosis disease is presented. Chytridiomycosis is an emerging disease caused by the fungal pathogen Batrachochydrium dendrobatidis (Bd) that poses a serious threat to frog populations worldwide. Several studies have shown that inoculation of bacterial species Janthinobacterium lividum (Jl) can mitigate the impact of the disease. However, there are many questions regarding this interaction that are unknown. Therefore, a mathematical model of a frog population infected with chytridiomycosis was developed to investigate how the inoculation of Jl could reduce the impact of Bd on frogs. The model also illustrates the important role of temperature in the diseases epidemiology. The model simulation results suggest possible control strategies for Jl to limit the impact of Bd in various scenarios. Finally, in Chapter 3, concluding remarks and discussion on our future work is provided.</p>

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<p> Geometric integrators yield high-fidelity numerical results by retaining conservation laws in the time advance. A particularly powerful class of geometric integrators is symplectic integrators, which are widely used in orbital mechanics and accelerator physics. An important application presently lacking symplectic integrators is the guiding center motion of magnetized particles represented by non-canonical coordinates. Because guiding center trajectories are foundational to many simulations of magnetically confined plasmas, geometric guiding center algorithms have high potential for impact. The motivation is compounded by the need to simulate long-pulse fusion devices, including ITER, and opportunities in high performance computing, including the use of petascale resources and beyond. </p><p> This dissertation uses a systematic procedure for constructing geometric integrators &mdash; known as variational integration &mdash; to deliver new algorithms for guiding center trajectories and other plasma-relevant dynamical systems. These variational integrators are non-trivial because the Lagrangians of interest are degenerate - the Euler-Lagrange equations are first-order differential equations and the Legendre transform is not invertible. The first contribution of this dissertation is that variational integrators for degenerate Lagrangian systems are typically <i>multistep methods.</i> Multistep methods admit parasitic mode instabilities that can ruin the numerical results. These instabilities motivate the second major contribution: degenerate variational integrators. By replicating the degeneracy of the continuous system, degenerate variational integrators avoid parasitic mode instabilities. The new methods are therefore robust geometric integrators for degenerate Lagrangian systems. </p><p> These developments in variational integration theory culminate in one-step degenerate variational integrators for non-canonical magnetic field line flow and guiding center dynamics. The guiding center integrator assumes coordinates such that one component of the magnetic field is zero; it is shown how to construct such coordinates for nested magnetic surface configurations. Additionally, collisional drag effects are incorporated in the variational guiding center algorithm for the first time, allowing simulation of energetic particle thermalization. Advantages relative to existing canonical-symplectic and non-geometric algorithms are numerically demonstrated. All algorithms have been implemented as part of a modern, parallel, ODE-solving library, suitable for use in high-performance simulations.</p>

This study is based on a 1994 survey of the nesting and territory preferences of the black-capped vireo, a small migratory songbird, within Fort Hood, Texas. By Federal Government Order 51 FR 44808-44812 of 1987 (Tazik et al., 1992), these birds were declared an endangered species based on a series of studies by J. D. Cornelius in 1985 and 1986 (Tazik et al., 1993). In this study, it has been shown that the vireo nesting and territory preferences are not uniform across Fort Hood installation grounds, rather the preference is towards the north to northwest. The vegetation in this region consists of oak woodlands, with grassy undergrowth; the elevation is at most 260 meters above sea level; and the geology yields loose and chalky limestone. In addition, the territory intensity, λij(s), for each point s(x,y) in the ijth grid cell on the military installation, has been estimated. For each such grid cell, the probability of a successful territory is given by Geographic Information Systems (GIS) raster color maps of territory distribution of the vireo have been produced. These maps were overlaid on the slope, aspect, soils, elevation, vegetation, and military activities coverages. With the density estimate of nesting success and the GIS maps, territory position preferences were identified, thus, helping in the Land Management Plan for the preservation of the black-capped vireo on the Fort Hood Military Installation.

This thesis concerns the implementation of the CORE GRAPHICS SYSTEM on the ZENITH Z-100 microcomputer, using the MS-DOS operating system and the MICROSOFT PASCAL programming language. The system gives the user the ability to create graphic displays via computer utilizing either a printer or a CRT. The basic features of the system, the problems encountered in its implementation, the general principles used, and an operations guide are documented within the paper. There are programming examples to illustrate the usage of the different system commands and there is documented source code of all system subroutines.

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With the increasing reliability of digital communication, the falling cost of hardware and increased computational power, the gathering and storage of data has become easier than at any other time in history. Commercial and public agencies are able to hold extensive records about all aspects of their operations. Witness the proliferation of point of sale (POS) transaction recording within retailing, digital storage of census data and computerized hospital records. Whilst the gathering of such data has uses in terms of answering specific queries and allowing visulisation of certain trends the volumes of data can hide significant patterns that would be impossible to locate manually. These patterns, once found, could provide an insight into customer behviour, demographic shifts and patient diagnosis hitherto unseen and unexpected. Remaining competitive in a modem business environment, or delivering services in a timely and cost effective manner for public services is a crucial part of modem economics. Analysis of the data held by an organisaton, by a system that "learns" can allow predictions to be made based on historical evidence. Users may guide the process but essentially the software is exploring the data unaided. The research described within this thesis develops current ideas regarding the exploration of large data volumes. Particular areas of research are the reduction of the search space within the dataset and the generation of rules which are deduced from the patterns within the data. These issues are discussed within an experimental framework which extracts information from binary data.