This dissertation presents several applications of the multi-scaling (multi-timing) technique to the analysis of both single and two species population models where the defining parameters vary slowly with time. Although exact solutions in such cases would be preferred, they are almost always impossible to obtain when slow variation is involved. Numerical solutions can be obtained in these cases, however they are often time consuming and offer limited insight into what is causing the behaviour we see in the solution. Here an approximation method is chosen as it gives an explicit analytic approximate expression for the solutions of such population models. The multi-scaling method was chosen because the defining parameters are varying slowly compared to the response of the system. This technique is well-established in the physical and engineering sciences literature; however, it has rarely been applied in the area of population modelling. All single species differential equation population models incorporate parameters which define the model - for example, the growth rate r and the carrying capacity k, for the Logistic model. For constant parameter values an exact solution may be found, giving the population as a function of time. However, for arbitrary time-varying parameters, exact solutions are rarely possible, and numerical solution techniques must be employed. Here we will demonstrate that for a Logistic model where the growth rate and carrying capacity both vary slowly with time, an analysis with multiple time scales leads to approximate closed form solutions that are explicit. These solutions prove to be valid for a range of parameter values and compare favourably with numerically generated ones.

Radioimmunotherapy consists in the use of beta radioactive labeled monoclonal antibodies as selective carriers of radiation to tumors. Internal spatially distributed sources created at the disease sites would deliver high radiation doses to tumors while the normal tissues would not be exposed to the intense radiation as in conventional forms of cancer treatments. A rapid and accurate estimation of the spatial dose distribution from nonuniform sources is essential for the optimization of this form of cancer therapy. The method used for such calculations is based on the knowledge of dose distributions around a unit source, quantities referred to as dose kernels. Thus far, the Monte Carlo technique is the most accurate way of the dose kernel determinations. However, for routine dosimetry simpler and less time consuming methods of adequate accuracy may appear more preferable. The "scaling factor" method is used to determine the depth dose distribution in a medium based on data about the dose distribution in an arbitrary reference medium (e.g. air, water). The transformation of the dose distribution curves from the reference medium to the desired new medium is done using a constant, known as scaling factor or relative dose attenuation, and a closely related renormalization factor imposed by the energy conservation. This work investigates the accuracy of the scaling factor method using a statistical approach (generalized chi-squared test), focusing on a particular case of potential practical interest, the scaling factor water to bone. The work also investigates a procedure for extending the applicability of the scaling factor method to dosimetry in dissimilar media, as a first step, a planar interface. / Thesis / Master of Science (MS)

In this thesis, We study the driven Morse system in a strong time-periodic field. We are interested in the quasi-bound states, which live in the driven system with limit life-times, with an increasing field strength in a low frequency region. We found those states by using Floquet theory, and the exterior complex scaling method (ECCS), which widely use in the resonance system. Choosing the Morse potential with supports 3 bound states, we found that as we increase the time-periodic external field, the number of the quasi-bound states decrease to 2. The distributions of the quasi-bound states which represented by the Husimi distribution were also studied, and compared with the Poincaré surface of section plots of the system. / text

Une des grandes problématiques en écologie est d’identifier les liens qui existent entre ce qui se passe au niveau de la physiologie et du comportement des individus et les propriétés émergentes qui apparaissent au niveau de la population et des écosystèmes dans leur globalité.Dans cette thèse, nous avons abordé cette problématique à travers la modélisation du phénomène de prédation, en nous intéressant plus particulièrement à la représentation mathématique de la réponse fonctionnelle. Cette fonction représente la quantité de proies consommées par prédateur et par unité de temps. Elle synthétiseau niveau de la population un ensemble de processus survenant à différentes échelles d’organisation. La modélisation du phénomène de prédation rencontre diverses limitations liées à la complexité de ce processus biologique, et il existe donc une forte incertitude sur la nature de la réponse fonctionnelle à utiliser.A travers l’étude d’un modèle prédateur-proie en chemostat d’une part, et l’utilisation de méthodes de changement d’échelle sur un modèle prédateur-proie en patchs d’autre part, nous avons cherché à déterminer les sources de variations dans la représentation de cette réponse.Tout d’abord, nous avons mis en évidence l’influence de la variabilité des données sur la paramétrisation de la réponse fonctionnelle ainsi que sur la robustesse des sorties du modèle. Une étude de sensibilité a également permis de montrer la forte sensibilité structurelle du modèle face à cette formulation, qui peut-être plus importante que face à des changements de paramètres.De plus, il apparait que la représentation mathématique de la réponse fonctionnelle dépend fortement de l’échelle d’observation considérée. En effet, la nature de la réponse peut être modifiée lorsque l’on passe de l’échelle d’une population à celle de la communauté. / One of the major issues in ecology is to identify the links between what happens in terms of physiology and behavior of individuals and the emergent properties that appear at the population and ecosystems level. In this thesis, we addressed this problem through modeling of the phenomenon of predation, especially by focusing on the mathematical functional response representation. This function represents the amount of prey consumed by predator per unit time. It synthesizes at the population level a set processes occurring at different scales of organization. Modeling of the phenomenon of predation encounters various limitations related to the complexity of this biological process, and there is, therefore, considerable uncertainty aboutthe nature of the functional response to use.Through the study of a predator-prey model in chemostat on the one hand, and use of scaling methods in a patches predator-prey model on the other hand, we seek to determine sources of variations in therepresentation of that response.First, we demonstrated the influence of data variability on the parameterization of the functional response as well as the robustness of the model outputs. A sensitivity study has also demonstrated the high structural sensitivity of the model to the formulation of this response, which may be more important than to parameterchanges.In addition, it appears that the mathematical representation of the functional response depends strongly on the scale of observation considered. The nature of the response can, indeed, be modified when changing the scale from the population to the community level.

Active water management uses real-time information to continually respond and adjust to water management needs and situations. To support active water management, the Texas Commission on Environmental Quality (TCEQ) needs tools to access and understand data and to apply that understanding to operational decisions. The work described herein addresses two objectives in providing decision-support for the TCEQ: (1) methods for including environmental pulse flow regulations in water rights documents, and (2) improved ease of access to information needed for TCEQ watermaster operations, particularly in times of drought. A Pulse Scaling Method for calculating the trigger flow rate, volume, and duration of flow pulses, using known characteristics at a reference location A, that are appropriate at a target location B (with unknown characteristics) was developed from three key relationships found in the written environmental flow regulations for fifteen locations in the Trinity, San Jacinto, Sabine, and Neches basins. Applying the method and analyzing the results shows that the predictions are statistically consistent with original regulations. A Common Operating Picture is a layered web-map allowing simultaneous access to one or more spatially-related datasets that TCEQ watermaster staff need to consider in decision-making. By its very nature as a dynamic map with associated time series, the Common Operating Picture presents data as information in a way that can support water resource management and decision-making. The project is currently in the pilot stage, with a number of data sources included and an interface available, but with additional work planned and further testing needed before larger-scale implementation. / text