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<p> A model of two predators competing for the same prey also involving predation interaction between the two predators is considered. Coexistence in forms of equilibria and periodic orbits is obtained by using bifurcation and dynamical systems theory. Global dynamics is obtained by studying the survival functions and persistence is obtained by using a theorem of Freedman and Waltman. Finally, numerical results for a specific example demonstrate the above. A Hopf bifurcation at the interior equilibrium and its unstable periodic orbit are observed.</p> / Thesis / Master of Science (MSc)

Los problemas de dispersión de partículas, como son los de dos y tres cuerpos, tienen una relevancia crucial en física atómica, pues permiten describir diversos procesos de colisiones. Hoy en día, los casos de dos cuerpos pueden ser resueltos con el grado de precisión numérica que se desee. Los problemas de dispersión de tres partículas cargadas son notoriamente más difíciles pero aún así algo similar, aunque en menor medida, puede establecerse. El objetivo de este trabajo es contribuir a la comprensión de procesos Coulombianos de dispersión de tres cuerpos desde un punto de vista analítico. Esto no solo es de fundamental interés, sino que también es útil para dominar mejor los enfoques numéricos que se actualmente se desarrollan dentro de la comunidad de colisiones atómicas. Para lograr este objetivo, proponemos aproximar la solución del problema con desarrollos en series de funciones adecuadas y expresables analíticamente. Al hacer esto, desarrollamos una serie de herramientas matemáticas relacionadas con funciones Coulombianas, ecuaciones diferenciales de segundo orden homogéneas y no homogéneas, y funciones hipergeométricas en una y dos variables. En primer lugar, trabajamos con las funciones de onda Coulombianas radiales y revisamos sus principales propiedades. Así, extendemos los resultados conocidos para dar expresiones analíticas de los coeficientes asociados al desarrollo, en serie de funciones de tipo Laguerre, de las funciones Coulombianas irregulares. También establecemos una nueva conexión entre los coeficientes asociados al desarrollo de la función Coulombiana regular y los polinomios de Meixner-Pollaczek. Esta relación nos permite deducir propiedades de ortogonalidad y clausura para estos coeficientes al considerar la carga como variable. Luego, estudiamos las funciones hipergeométricas de dos variables. Para algunas de ellas, como las funciones de Appell o las confluentes de Horn, presentamos expresiones analíticas de sus derivadas respecto de sus parámetros. También estudiamos un conjunto particular de funciones Sturmianas Generalizadas de dos cuerpos construidas considerando como potencial generador el potencial de Hulthén. Contrariamente al caso habitual, en el que las funciones Sturmianas se construyen numéricamente, las funciones Sturmianas de Hulthén poseen forma analítica. Sus propiedades matem´aticas pueden ser analíticamente estudiadas proporcionando una herramienta única para comprender y analizar los problemas de dispersión y sus soluciones. Además, proponemos un nuevo conjunto de funciones a las que llamamos funciones Quasi-Sturmianas. Estas funciones se presentan como una alternativa para expandir la solución buscada en procesos de dispersi´on de dos y tres cuerpos. Se definen como soluciones de una ecuación diferencial de tipo-Schrödinger, no homogénea. Por construcción, incluyen un comportamiento asintótico adecuado para resolver problemas de dispersión. Presentamos diferentes expresiones analíticas y exploramos sus propiedades matemáticas, vinculando y justificando los desarrollos realizados previamente. Para finalizar, utilizamos las funciones estudiadas (Sturmianas de Hulthén y Quasi-Sturmianas) en la resolución de problemas particulares de dos y tres cuerpos. La eficacia de estas funciones se ilustra comparando los resultados obtenidos con datos provenientes de la aplicación de otras metodologías. / Two and three-body scattering problems are of crucial relevance in atomic physics as they allow to describe different atomic collision processes. Nowadays, the two-body cases can be solved with any degree of numerical accuracy. Scattering problem involving three charged particles are notoriously difficult but something similar –though to a lesser extentcan be stated. The aim of this work is to contribute to the understanding of three-body Coulomb scattering problems from an analytical point of view. This is not only of fundamental interest, it is also useful to better master numerical approaches that are being developed within the collision community. To achieve this aim we propose to approximate scattering solutions with expansions on sets of appropriate functions having closed form. In so doing, we develop a number of related mathematical tools involving Coulomb functions, homogeneous and non-homogeneous second order differential equations, and hypergeometric functions in one and two variables. First we deal with the two-body radial Coulomb wave functions, and review their main properties. We extend known results to give in closed form the Laguerre expansions coefficients of the irregular solutions, and establish a new connection between the coefficients corresponding to the regular solution and Meixner-Pollaczek polynomials. This relation allows us to obtain an orthogonality and closure relation for these coefficients considering the charge as a variable. Then we explore two-variable hypergeometric functions. For some of them, such as Appell and confluent Horn functions, we find closed form for the derivatives with respect to their parameters. We also study a particular set of two-body Generalized Sturmian functions constructed with a Hulth´en generating potential. Contrary to the usual case in which Sturmian functions are numerically constructed, the Hulth´en Sturmian functions can be given in closed form. Their mathematical properties can thus be analytically studied providing a unique tool to investigate scattering problems. Next, we introduce a novel set of functions that we name Quasi-Sturmian functions. They constitute an alternative set of functions, given in closed form, to expand the sought after solution of two- and three-body scattering processes. Quasi-Sturmian functions are solutions of a non-homogeneous second order Schr¨odinger-like differential equation and have, by construction, the appropriate asymptotic behavior. We present different analytic expressions and explore their mathematical properties, linking and justifying the developed mathematical tools described above. Finally we use the studied Hulth´en Sturmian and Quasi-Sturmian functions to solve some particular two- and three-body scattering problems. The efficiency of these sets of functions is illustrated by comparing our results with those obtained by other methods

Increased efficiency and durability of gas turbine components is driven by demands for reduced fuel consumption and increased reliability in aircraft and power generation applications. The complex flow near the endwall of an axial gas turbine has been identified as a significant contributing factor to aerodynamic loss and increased part temperatures. Three-dimensional (non-axisymmetric) contouring of the endwall surface has been shown to reduce aerodynamic losses, but the effect of the contouring on endwall heat transfer is not well understood. This research focused on understanding the general flow physics of contouring and the sensitivity of the contouring to perturbations arising from leakage features present in an engine. Two scaled low-speed cascades were designed for spatially-resolved measurements of endwall heat transfer and film cooling. One cascade was intended for flat and contoured endwall studies without considering typical engine leakage features. The other cascade modeled the gaps present between a stator and rotor and between adjacent blades on a wheel, in addition to the non-axisymmetric endwall contouring. Comparisons between a flat and contoured endwall showed that the contour increased endwall heat transfer and increased turbulence in the forward portion of the passage due to displacement of the horseshoe vortex. However, the contour decreased heat transfer further into the passage, particularly in regions of high heat transfer, due to delayed development of the passage vortex and reduced boundary layer skew. Realistic leakage features such as the stator-rotor rim seal had a significant effect on the endwall heat transfer, although leakage flow from the rim seal only affected the horseshoe vortex. The contours studied were not effective at reducing the impact of secondary flows on endwall heat transfer and loss when realistic leakage features were also considered. The most significant factor in loss generation and high levels of endwall heat transfer was the presence of a platform gap between adjacent airfoils. / Ph. D.

Electrical properties of Si- and Mg-doped InAs epitaxial layers grown by MOCVD were studied by performing magneto-transport measurements at different temperatures, from 300 K down to 1.2 K. The longitudinal magnetoresistance and Hall effect indicate a three-band system existing in n-type (p-type) InAs, which consists of the surface accumulation (inversion) layer, the bulk electron (hole) layer, and the nucleation layer. Therefore, a classical parabolic background in magnetoresistance due to multi-carrier occurs at low fields. With the magnetic field being further applied, a linear magnetoresistance caused by inhomogeneities is revealed. At liquid helium temperature, the Shubnikov-de Haas magneto-oscillations are also observed. These transport characterizations provide a means of analyzing the band structure at the InAs surface. In a set of n-type InAs epilayers with Si doped at different levels, the bulk electron density increases as the doping level increases. The increased ionized impurities lead to lower electron mobilities due to more Coulomb scatterings. For all the n-type InAs films, except the two active layers (surface and the bulk), the nucleation layer contributes to the film conductivity as well with an electron density of ~ 5 x 10¹⁷ cm⁻³ and a mobility of ~ 2000 cm²}/Vs. In a cooldown process, the electron density of each layer slightly and monotonically decreases whereas the mobility experiences a maximum from the competition between phonon scatterings and Coulomb scatterings. The phonon scattering overwhelms the Coulomb scatting at high temperatures, but declines as temperature decreases, thus the mobility increases. Around 100 K, the temperature-independent ionized impurity scattering becomes comparable with and starts exceeding the phonon scattering, as temperature further lowered, the screening effect of the Coulomb scattering is weakened because of the decreased carrier densities. As a result, the mobility starts dropping. The maximum mobility corresponds to a minimum resistance, which explains the non-trivial temperature dependence of the resistance in the cooldown history. For the p-type InAs film, the doping with Mg in the course of MOCVD growth allows us to obtain a large hole density and a low mobility at 300 K. At low temperatures, holes are frozen out, and a strong negative magnetoresistance with a dip at 0 field are observed, which is the antilocalization signal from accumulation electrons. This is a strong technique to probe the surface quantum states and derive the phase coherence length and the spin flip length of surface electrons. / Master of Science

Mars is one of the most intriguing planets in the solar system. It is the fourth terrestrial planet and is differentiated into a core, mantle and crust. The crust of Mars is divided into the Southern highlands and the Northern lowlands. The largest volcano in the solar system, Olympus Mons is found on the crustal dichotomy boundary. The presence of isolated volcanism on the surface suggests the importance of internal activity on the planet. In addition to volcanism in the past, there has been evidence of present day volcanic activity. Convective upwelling, including decompression melting, has remained an important contributing factor in melting history of the planet. In this thesis, I investigate the production of melt in the mantle for a Newtonian rheology, and compare it with the melt needed to create Tharsis. In addition to the melt production, I analyze the 3D structure of the mantle for a stagnant lithosphere. I vary different parameters in the Martian mantle to understand the production of low or high degree structures early on to explain the crustal dichotomy. This isothermal structure in the mantle contributes to the geoid and topography on the planet. I also analyze how much of the internal density contributes to the surface topography and areoid of Mars. In contrast to Mars, Ceres is a dwarf planet in the Asteroid belt. Ceres is an icy body and it is unclear if it is differentiated into a core, mantle and crust yet. However, studies show that it is most likely a differentiated body and the mantle consists of ice and silicate. The presence of brucite and serpentine on the surface suggests the presence of internal activity. Being a massive body and also believed to have existed since the beginning of the solar system, studying Ceres will shed light on the conditions of the early solar system. Ceres has been of great interest in the scientific community and its importance has motivated NASA to launch a mission, Dawn, to study the planet. Dawn will collect data from the dwarf planet when it arrives in 2015. In my modeling studies, I implement a similar technique on Ceres, as followed on Mars, and focus on the mantle convection process and the geoid and topography. The silicate-ice mixture in the mantle gives rise to a non-Newtonian rheology that depends on the grain size of the ice particle. The geoid and topography observed for different differentiated scenarios in my modeling can be compared with the data from the Dawn mission when it arrives at Ceres in 2015. / Ph. D.