Doctoral Degrees (Mathematics and Applied Mathematics)

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Browsing Doctoral Degrees (Mathematics and Applied Mathematics) by Author "Schoombie, S. W."

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    Adaptive dynamics for an age-structured population model with a Shepherd recruitment function
    (University of the Free State, 2013-06-07) Ellis, Michelle Heidi; Schoombie, S. W.
    English: In this study the evolution of the genetic composition of certain species will be replaced by the evolution of the traits that represent these genetic compositions. Depending on the nature of the trait of interest, a scalar valued parameter called the strategy parameter will be assigned to this trait making the simulation of strategy evolution possible. The trait of interest, and therefore the strategy associated, will be the ability of a population to keep its densities within the carrying capacity of the environment they find themselves in. The Shepherd function, on account of its wide use in population simulations as well as composing of exactly such a density parameter, will be the density curbing mechanism of choice in the age-structured population model designed here. An algorithm will be designed to simulate strategy evolution towards an evolutionary stable strategy or ESS that will ensure not only an optimal fit for this environment but also render the population immune against future invasion by other members of the population practising slight variations of this strategy. There are two ways to come by such an optimal strategy without directly involving genetics. The first is game theory, allowing strategists to compete for this position, and the second is with the use of adaptive dynamics, converting winning and loosing instead into tangible mathematics. Combining these two classics will show that the quest is an exercise in strategy optimization, not only from the point of view of an already established population but also from the point of view of an initially small one. It will be interesting!
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    Analysis of numerical approximation algorithms for nonlinear differential equations using a discrete multiple scales technique
    (University of the Free State, 2002-12) Maré, Eben; Schoombie, S. W.
    English: Perturbation techniques for the solution of differential equations form an essential ingredient of the tools of mathematics as applied to physics, engineering, finance and other areas of applied mathematics. A natural extension would be to seek perturbation-type solutions for discrete approximations of differential equations. The main objective of the research project is to develop a perturbation technique for approximation. The spurious behavior, predicted theoretically, is shown to be present experimentally, independent of temporal discretization. We also detail some comparisons of central difference solutions of different orders of approximation to the KdV equation. The results show a clear benefit of higher order central differences relative to lower order methods. The benefit of the central difference methodology would also extend to more general regions over which we would solve partial differential equations. We also show that the method of multiple scales can provide an adequate explanation for spurious behavior in a difference scheme for the Van der Pal equation. the solution of discrete equations. We discuss the well-known method of multiple scales and show its use for the solution of the Korteweg-de Vries (KdV), Regularized Long Wave (RLW) and Van der Pol equations. In particular, for the KdV and RLW equations the analysis shows that the envelopes of modulated waves are governed by the nonlinear Schrödinger equation. We present a variation of the multiple scales technique which presents an ideal framework from which we devise a discrete multiple scales analysis methodology. We discuss a discrete multiple scales methodology derived by Schoombie [111], as applied to the Zabusky-Kruskal approximation of the KdV equation. This discrete multiple scales analysis methodology is generalized and applied to the solution of a generalized finite difference approximation of the KdV equation. We show the consistency of the method with the continuous analysis as the discretization parameters tend to zero. The discrete multiple scales technique is a powerful tool for the examination of modulational properties of the KdV equation. In the case of certain modes of the carrier wave, the discrete multiple scales analysis breaks down, indicating that the numerical solution deviates in behavior from that of the KdV equation. Several numerical experiments are performed to examine the spurious behavior for different orders of approximation. The spurious behavior, predicted theoretically, is shown to be present experimentally, independent of temporal discretization. We also detail some comparisons of central difference solutions of different orders of approximation to the KdV equation. The results show a clear benefit of higher order central differences relative to lower order methods. The benefit of the central difference methodology would also extend to more general regions over which we would solve partial differential equations. We also show that the method of multiple scales can provide an adequate explanation for spurious behavior in a difference scheme for the Van der Pal equation.