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This thesis investigates the analysis of the extended model of animal growth proposed by Oliviera et al (personal communication, July 2009). This mechanistic model of animal growth based on a detailed representation of energy dynamics focussing on the interaction between four compartment of body composition; nutrient level, fat content, visceral protein and non-visceral protein. The model is mathematically analysed and the behaviour of the model for different feeding level is examined. The animal growth model exhibits thresholds typical of nonlinear systems and multiple stable steady states which have distinct basins of stability which depend on the value of the large number of physiologically-determined parameters. These have not been previously explored theoretically and these are done in this thesis. The model demonstrates richer behaviour where path-following techniques are used to explore the distribution in parameter space of the varying phenomenology.

This thesis describes the Operational Harvest Scheduling (OHS) problem and develops an algorithm that solves instances of the problem. The solution to an OHS problem is an Operational Harvest Schedule (OHS). An OHS: ² assigns forest harvesting crews to locations within a forest in the short-term (4-8 weeks); ² instructs crews to harvest specific log-types and allocates these log-types to customers; ² maximises profitability while meeting customer demand. The OHS problem is modelled as a Mixed Integer Linear Program (MILP). The formulation given in this thesis differs significantly from previous literature, especially with regard to the construction of the problem variables. With this novel formulation, the problem can be solved using techniques developed in previous work on aircraft crew scheduling optimisation (Ryan 1992). These techniques include constraint branching and column generation. The concept of relaxed integer solutions is introduced. A traditional integer solution to the OHS problem will require harvesting crews to move between harvesting locations at the end of a week. However, a relaxed integer solution allows crews to move at any time during a week. This concept allows my OHS model to more effectively model the practical problem. The OHS model is formulated for New Zealand and Australian commercial forestry operations,though the model could be applied to other intensively managed production forests. Three case studies are developed for two companies. These case studies show improvements in profitability over manual solution methods and a significant improvement in the ability to meet demand restrictions. The optimised solutions increased profit (revenue less harvesting and transportation costs) by between 3-7%, while decreasing the total value of excess or shortfall logs by between 15-86%.

The analysis of presence/absence data of plant species by regression analysis is the subject of this thesis. A nonparametric approach is emphasized, and methods which take into account correlations between species are also considered. In particular, generalized additive models (GAMs) are used, and these are applied to species’ responses to greenhouse scenarios and to examine multispecies interactions. Parametric models are used to estimate optimal conditions for the presence of species and to test several niche theory hypotheses. An extension of GAMs called vector GAMs is proposed, and they provide a means for proposing nonparametric versions of the following models: multivariate regression, the proportional and nonproportional odds model, the multiple logistic regression model, and bivariate binary regression models such as bivariate probit model and the bivariate logistic model. Some theoretical properties of vector GAMs are deduced from those pertaining to ordinary GAMs, and its relationship with the generalized estimating equations (GEE) approach elucidated. / Whole document restricted, but available by request, use the feedback form to request access.

Monte Carlo simulation methods are becoming increasingly important for solving difficult optimization problems. Monte Carlo methods are often used when it is infeasible to determine an exact result via a deterministic algorithm, such as with NP or #P problems. Several recent Monte Carlo techniques employ the idea of importance sampling; examples include the Cross-Entropy method and sequential importance sampling. The Cross-Entropy method is a relatively new Monte Carlo technique that has been successfully applied to a wide range of optimization and estimation problems since introduced by R. Y. Rubinstein in 1997. However, as the problem size increases, the Cross-Entropy method, like many heuristics, can take an exponentially increasing amount of time before it returns a solution. For large problems this can lead to an impractical amount of running time. A main aim of this thesis is to develop the Cross-Entropy method for large-scale parallel computing, allowing the running time of a Cross-Entropy program to be significantly reduced by the use of additional computing resources. The effectiveness of the parallel approach is demonstrated via a number of numerical studies. A second aim is to apply the Cross-Entropy method and sequential importance sampling to biological problems, in particular the multiple change-point problem for DNA sequences. The multiple change-point problem in a general setting is the problem of identifying, given a particular sequence of numbers/characters, a point along that sequence where some property of interest changes abruptly. An example in a biological setting, is identifying points in a DNA sequence where there is a significant change in the proportion of the nucleotides G and C with respect to the nucleotides A and T. We show that both sequential importance sampling and the Cross-Entropy approach yield significant improvements in time and/or accuracy over existing techniques.

This thesis reports on a study of underground fluid flow and boiling processes which take place in hydrothermal eruptions. A conceptual model is presented for the eruptive process and a laboratory scale physical model confirming the effectiveness of this process is described. A mathematical formulation of the underground flow problem is given for two fluid flow regimes: two-phase homogeneous mixture (HM) flow and separable two-phase (SP) flow. Solutions to the system of equations obtained are solved under the simplifying assumptions of two-dimensional steady isothermal flow and transient non-isothermal horizontal flow. The main contribution of the study on steady isothermal flows is a description of how the ground flow may recover following a hydrothermal eruption. A numerical technique developed for plotting the streamlines in this case (and verified against analytic results) may also have applications in solving the steady non-isothermal flow problem. The main contribution of the study on the transient horizontal flow problem is a comparison of the differing predictions of HM and SP flow. The rate at which a boiling front progresses through a porous medium and the degree of boiling which occurs is described for each fluid flow regime. A set of horizontal physical experiments and numerical simulations have also been carried out for comparison with the mathematical model. Qualitative results for these three models agree. Suggestions given for improvements to the design of the physical experiment provide a basis for future study into the type of flow which occurs in hydrothermal eruptions

This thesis describes the Operational Harvest Scheduling (OHS) problem and develops an algorithm that solves instances of the problem. The solution to an OHS problem is an Operational Harvest Schedule (OHS). An OHS: ² assigns forest harvesting crews to locations within a forest in the short-term (4-8 weeks); ² instructs crews to harvest specific log-types and allocates these log-types to customers; ² maximises profitability while meeting customer demand. The OHS problem is modelled as a Mixed Integer Linear Program (MILP). The formulation given in this thesis differs significantly from previous literature, especially with regard to the construction of the problem variables. With this novel formulation, the problem can be solved using techniques developed in previous work on aircraft crew scheduling optimisation (Ryan 1992). These techniques include constraint branching and column generation. The concept of relaxed integer solutions is introduced. A traditional integer solution to the OHS problem will require harvesting crews to move between harvesting locations at the end of a week. However, a relaxed integer solution allows crews to move at any time during a week. This concept allows my OHS model to more effectively model the practical problem. The OHS model is formulated for New Zealand and Australian commercial forestry operations,though the model could be applied to other intensively managed production forests. Three case studies are developed for two companies. These case studies show improvements in profitability over manual solution methods and a significant improvement in the ability to meet demand restrictions. The optimised solutions increased profit (revenue less harvesting and transportation costs) by between 3-7%, while decreasing the total value of excess or shortfall logs by between 15-86%.

The analysis of presence/absence data of plant species by regression analysis is the subject of this thesis. A nonparametric approach is emphasized, and methods which take into account correlations between species are also considered. In particular, generalized additive models (GAMs) are used, and these are applied to species’ responses to greenhouse scenarios and to examine multispecies interactions. Parametric models are used to estimate optimal conditions for the presence of species and to test several niche theory hypotheses. An extension of GAMs called vector GAMs is proposed, and they provide a means for proposing nonparametric versions of the following models: multivariate regression, the proportional and nonproportional odds model, the multiple logistic regression model, and bivariate binary regression models such as bivariate probit model and the bivariate logistic model. Some theoretical properties of vector GAMs are deduced from those pertaining to ordinary GAMs, and its relationship with the generalized estimating equations (GEE) approach elucidated. / Whole document restricted, but available by request, use the feedback form to request access.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007. / Data analysis frequently involves the extraction (i.e. recognition) of parts that are important at the expense of parts that are deemed unimportant. Many mathematical perspectives exist for performing these separations, however no single technique is a panacea as the de nition of signal and noise depends on the purpose of the analysis. For data that can be considered a sampling of a smooth function with added 'well-behaved' noise, linear techniques tend to work well. When large impulses or discontinuities are present, a non-linear approach becomes necessary. The LULU operators, composed using the simplest rank selectors, are non-linear operators that are comparable to the well-known median smoothers, but are computationally e cient and allow a conceptually simple description of behaviour. De ned using compositions of di erent order LULU operators, the discrete pulse transform (dpt) allows the interpretation of sequences in terms of pulses of di erent scales: thereby creating a multi-resolution analysis. These techniques are very di erent from those of standard linear analysis, which renders intuitions regarding their behaviour somewhat undependable. The LULU perspective and analysis tools are investigated with a strong emphasis on practical applications. The LULU smoothers are known to separate signal and noise ef- ciently: they are idempotent and co-idempotent. Sequences are smoothed by mapping them into smoothness classes; which is achieved by the removal, in a consistent manner, of block-pulses. Furthermore, these operators preserve local trend (i.e. they are fully trend preserving). Di erences in interpretation with respect to Fourier and Wavelet decompositions are also discussed. The dpt is de ned, its implications are investigated, and a linear time algorithm is discussed. The dpt is found to allow a multi-resolution measure of roughness. Practical sequence processing through the reconstruction of modi ed pulses is possible; in some cases still maintaining a consistent multi-resolution interpretation. Extensions to two-dimensions is discussed, and a technique for the estimation of standard deviation of a random distribution is presented. These tools have been found to be e ective in the analysis and processing of sequences and images. The LULU tools are an useful alternative to standard analysis methods. The operators are found to be robust in the presence of impulsive and more 'well-behaved' noise. They allow the fast design and deployment of specialized detection and processing algorithms, and are possibly very useful in creating automated data analysis solutions.

Thesis (MSc (Mathematical Sciences. Computer Science))--University of Stellenbosch, 2007. / It is widely recognized that Traffic engineering (TE) mechanisms have to be added to the IP transport functionalities to provide QoS guarantees while ensuring efficient use of network resources. Traffic engineering is a network management technique which routes traffic to where bandwidth is available in the network to achieve QoS agreements between current and future demands and the available network resources. Multi-path routing has been proven to be a more efficient TE mechanism than Shortest Path First (SPF) routing in terms of proffit maximization and resource usage optimization. However the identiffication of set of paths over which traffic is forwarded from source to the destination and the distribution of traffic among these paths are two issues that have been widely addressed by the IP community but remain an open issue for the emerging generation IP networks. Building upon different frameworks, this thesis revisits the issue of multi-path routing to present and evaluate the performance of different traffic splitting mechanisms to achieve QoS routing in Multi-Protocol Label Switching (MPLS) and Wireless Sensor Networks (WSNs). Three main contributions are identified in this thesis. First, we extend an optimization model that used the M/M/1 queueing model on a simple network consisting of a single source-destination pair by using the M/M/s queueing model on a general network consisting of several source-destination pairs. The model solves a multi-path routing problem by defining a Hamiltonian as a function of delay incurred and subjecting this Hamiltonian to Pontryagin's cost minimization to achieve efficient diffusion of traffic over the available parallel paths. Second, we revisit the problem of cost-based optimization in a multi-path setting by using a Game theoretical framework to propose and evaluate the performance of competitive and cooperative multi-path routing schemes and the impact of the routing metric (cost) on the difference between these two schemes. Finally, building upon a previously proposed optimization benchmark, we propose an Energy constrained QoS routing scheme for Wireless Sensor Networks and show through simulation that our scheme outperforms the benchmark scheme.