A GEOMETRIC APPROACH TO MODELLING INTERACTING POPULATIONS
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The purpose of this thesis is to present a different approach to the formulation of differential equation mathematical models for interacting populations. It does this through considering a geometric method. The solution to the differential equations are forced, through theuformulation, to lie on the surface of well known three-dimensional shapes. It is this that allows detailed analysis of how, and why, the solution of the equations behave as they do. The thesis firstly reviews some of the skills and techniques used in the formulation and alysis of differential equation models to give a background for some of the analysis used the geometric approach to modelling. The geometrical approach is then presented using two three-dimensional surfaces, the ellipsoid and the torus. Also examined is an extension of the basic shape to higher dimensions. Using the three-dimensional shape as a reference, a four-dimensional representation is formulated. This increase in the number of variables in the model allows more situations to be modelled. The thesis concludes by discussing the use of the type of model produced by the geometric approach by addressing some of the advantages and disadvantages of this approach. It ends with some extensions to this study of geometric models for future work.
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