Year

2004

Degree Name

Doctor of Philosophy (PhD)

Department

School of Mathematics and Applied Statistics - Faculty of Infomatics

Abstract

In this thesis, we reinforce the validity of using reaction-diffusion equations with cubic source terms to describe the change in frequency of alleles in a gene pool. In a population with two possible alleles at the locus in question, the Fitzhugh-Nagumo equation is shown to be appropriate when there is no dominance, whereas the Huxley equation is appropriate when one of the alleles is completely dominant. The difference between the Huxley equation (with cubic source term) and the Fisher-Kolmogorov equation (with quadratic source term) is explained numerically and analytically. Using the method of nonclassical symmetry analysis, we construct some practical analytic solutions to the Fitzhugh-Nagumo and Huxley equations. The solutions satisfy specific boundary conditions and are different from previously derived travelling wave solutions. We derive a system of reaction-diffusion equations describing the case of three possible alleles at the locus in question. By introducing a nonlinear transformation, we are able to construct an exact travelling wave solution. We also extend the model to include the case of spatially dependent reproductive success rates. We use classical and nonclassical symmetry methods to discover what forms of explicit spatial variability will enable us to find exact solutions to our equations. A number of solutions are constructed for various forms of spatial variability. Finally, we demonstrate the benefits of systematic symmetry analysis by studying two related systems of reaction-diffusion equations.

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