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Mathematical modeling in the sustainable use of natural resources
(University of the Western Cape, 2015) Mthombeni, Lestinah
The sustainable use of natural resources is of utmost importance for every community. In particular, it is important for every given generation to plan in such a way that proper provision is made for future generations. The scientific understanding of resources use and appreciation for its life-supporting capacity is therefore essential. Mathematical modeling has proved useful to inform the planning and management of strategies for sustainable use of natural resources. Some specific topics in resource management has been studied intensively through many decades. In particular, mining, fisheries, forestry and water resources are among these. Instead of presenting a study of the latter topics, this dissertation presents a variety of cases of mathematical modeling in resource management. The aim is to improve the general understanding of the relevant problems. We expand on existing literature, papers of other authors, and add to such studies by focusing on specific items in the work, illuminating it with further explanations and graphs, or by modifying the models through the introduction of stochastic perturbations. In particular this dissertation makes contributions by giving more explanation, on the so-called environmental Fisher information or EFI for brevity (Section 2.4 and Chapter 6), and by introducing stochasticity into a pest control model (Chapter 4) and into a savanna vegetation model (Chapter 5). In Chapter 3 we present a model from the literature pertaining to the problem of shifting cultivation, i.e, the use of forest land when used for subsistence level agricultural purposes, until the land is so degraded that the occupants abandon it and move on to a new stand. The model used to study the shifting period is similar to the forest rotation problem. A model, already in the literature, for biological control of a pest is studied in Chapter 4. Onto the deterministic model we impose a stochastic perturii bation, so that we obtain a stochastic differential equation model. We prove stochastic stability of the disease-free state, when the basic reproduction number of the pest is below unity. We have performed simulations of solutions of the stochastic system. In Chapter 5 we review an existing ordinary differential equation model for the competition between trees and grass in savanna environment. The competition between them is for soil water, fed by annual rainfall. On the other hand, trees and grass are perturbed by fire, and some other environmental forcings such as herbivores. For this ODE model, we introduce stochastic perturbations. The stochastic perturbations are in the form of three mutually independent Brownian motions. Simulations to illustrate the effect of the stochasticity are shown. We present a three-tiered predator-prey model and consider its stability in terms of Fisher information. This appears as Chapter 6. The Fisher information is defined on the basis of the so-called sustainable measures hypotheses. The model is already in the literature and in the dissertation we present several computations to show the influence of carrying capacity of prey and of mortality rate on EFI. Another problem that we consider, in Chapter 7, is that of lake eutrophication caused by excessive phosphorus inflow. The computation illustrates the management of the runoff nutrients into or out of the lake. Necessary and the sufficient conditions for an optimal utility management are obtained using standard optimal control theory. The results of this dissertation demonstrate the modeling techniques in the sustainable use of natural resources. Sustainability is the quest for equal opportunities over all generations. The manner in which this sustainability is quantified in models is being debated and improved all the time. The discourse on sustainability is especially important in view of a growing world population, and with forcings such as climate change. The most important original contribution in this dissertation is the stochastic analysis on the pest control model and the savanna model.
Mathematical model of performance measurement of defined contribution pension funds
(University of the Western Cape, 2015) Kelekele, Liloo Didier Joel
The industry of pension funds has become one of the drivers of today’s economic activity by its important volume of contribution in the financial market and by creating wealth. The increasing importance that pension funds have acquired in today’s economy and financial market, raises special attention from investors, financial actors and pundits in the sector. Regarding this economic weight of pension funds, a thorough analysis of the performance of different pension funds plans in order to optimise benefits need to be undertaken. The research explores criteria and invariants that make it possible to compare the performance of different pension fund products. Pension fund companies currently do measure their performances with those of others. Likewise, the individual investing in a pension plan compares different products available in the market. There exist different ways of measuring the performance of a pension fund according to their different schemes. Generally, there exist two main pension funds plans. The defined benefit (DB) pension funds plan which is mostly preferred by pension members due to his ability to hold the risk to the pension fund manager. The defined contributions (DC) pension fund plan on the other hand, is more popularly preferred by the pension fund managers due to its ability to transfer the risk to the pension fund members. One of the reasons that motivate pension fund members’ choices of entering into a certain programme is that their expectations of maintaining their living lifestyle after retirement are met by the pension fund strategies. This dissertation investigates the various properties and characteristics of the defined contribution pension fund plan with a minimum guarantee and benchmark in order to mitigate the risk that pension fund members are subject to. For the pension fund manager the aim is to find the optimal asset allocation strategy which optimises its retribution which is in fact a part of the surplus (the difference between the pension fund value and the guarantee) (2004) [19] and to analyse the effect of sharing between the contributor and the pension fund. From the pension fund members’ perspective it is to define a optimal guarantee as a solution to the contributor’s optimisation programme. In particular, we consider a case of a pension fund company which invests in a bond, stocks and a money market account. The uncertainty in the financial market is driven by Brownian motions. Numerical simulations were performed to compare the different models.
Entrepreneurial learning, Opportunity Recognition and Development – Evaluating the Impact of a Training Programme at tsiba Education, Cape Town, South Africa
(University of the Western Cape, 2017) Samsodien, Adeeb
The youth of South Africa is currently faced with high levels of unemployment and poverty. This raises concerns about the future of the South African people and its economy. Consequently, much is needed to develop the youth to allow for a prosperous future. A decrease in poverty and unemployment was found to be linked to an increase in education. Entrepreneurial education can have a significant impact on entrepreneurial success. This research paper aims to assess the influence of entrepreneurial education and training on students' entrepreneurial development. To ensure a comprehensive assessment of the development of students, I utilised a mixed methodology to assess students both quantitatively and qualitatively. The study assessed a sample of students from South Africa and the United States of America who jointly partook in a two-week entrepreneurial education and training programme at TSiBA Education, Cape Town, South Africa. Entrepreneurial education and training had a positive effect on the students' entrepreneurial development, entrepreneurial orientation and intentions to start a business. While entrepreneurial orientation aids the exploitation of business opportunities, before opportunities can be exploited they must be recognised. Bringing into question the link between students' entrepreneurial orientation and their abilities to recognise and develop opportunities into successful ventures. To explore this I interviewed two participants and found the high entrepreneurially orientated participant to exhibit greater opportunity recognition and development abilities, compared to the low entrepreneurially orientated participant. I concluded by finding that although differences do exist between high and low entrepreneurially orientated students, entrepreneurial education and training could positively influence students' entrepreneurial orientation and assist in the development of entrepreneurs. The results suggest that TSiBA Education should attempt to increase the duration of the entrepreneurial programme to maximize the influence of the entrepreneurial education and training.
Character tables of some selected groups of extension type using Fischer-Clifford matrices
(University of the Western Cape, 2015) Monaledi, R.L.
The aim of this dissertation is to calculate character tables of group extensions. There are several well developed methods for calculating the character tables of some selected group extensions. The method we study in this dissertation, is a standard application of Clifford theory, made efficient by the use of Fischer-Clifford matrices, as introduced by Fischer. We consider only extensions Ḡ of the normal subgroup N by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in Ḡ , if N is abelian. This is indeed the case if Ḡ is a split extension, by a well known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of Fischer-Clifford matrices is discussed in detail. Some of the properties of these Fischer-Clifford matrices which make their calculation much easier, are also given. We restrict ourselves to split extension groups Ḡ = N:G in which N is always an elementary abelian 2-group. In this thesis we are concerned with the construction of the character tables (by means of the technique of Fischer-Clifford matrices) of certain extension groups which are associated with the orthogonal group O+10(2), the automorphism groups U₆(2):2, U₆(2):3 of the unitary group U₆(2) and the smallest Fischer sporadic simple group Fi₂₂. These groups are of the type type 2⁸:(U₄(2):2), (2⁹ : L₃(4)):2, (2⁹:L₃(4)):3 and 2⁶:(2⁵:S₆).
Formal Mentorship and Entrepreneurial Learning The Case of a Support Programme in the Western Cape Clothing Sector
(University of the Western Cape, 2017) Davies, Clint
How entrepreneurs learn to cope and survive in the South African clothing sector, with its high levels of macro-environment turbulence, may engender particular lessons for entrepreneurial learning and related outcomes such as business innovation. Although SME support measures worldwide offer mentorship to assist firm survival and growth, little is known about how entrepreneurs learn under the guidance of a mentor. Formal mentorship is employed with increased frequency as a training intervention suited to entrepreneurs. Entrepreneurial learning is linked to experiential learning in the personal development of the entrepreneur and development of the business venture. Formal mentorship as a medium to enhance entrepreneurial learning is the focus of this study. Past research does not adequately address entrepreneurial learning in the context of prolonged turbulent competitive environments, and the role of formal mentorship as a significant contributor to entrepreneurial learning. This qualitative case study is set within the clothing industry of the Western Cape, which is affected by high levels of competitive turbulence. Entrepreneurs and their mentor’s accounts are collected through unstructured and semi-structured personal interviews and analysed using thematic analysis. The mentors are contracted to an organisation that provides business development support to SMEs within the clothing sector. Key participants within this organisation, and their sponsor, are interviewed to study strategic influences on formal mentorship. This constitutes the case and a purposive-snowball sampling strategy was employed.