Numerical studies of reaction-diffusion in physiological processes

Abstract

The aim of this study is to conduct a numerical investigation into mathematical models representing two physiological processes, both being examples of reaction-diffusion processes. The first of these processes comes from the field of population genetics when two types of individuals are allowed to mate at random. The governing equation of the relevant model is Fisher's equation [1] The second process comes from the field of neurobiology and concerns the conduction of an impulse in the nervous system, the principal model being the Hodgkin-Huxley system of differential equations [45] with simplifications due to FitzHugh [28] and Nagumo, Arimoto and Yoshizawa [64]. Chapter 1 is an introductory chapter, introducing the reader to reactiondiffusion equations, mathematical modelling in general and giving an exposition as to the purpose of the present study. In Chapter 2 the relevant two physiological processes are discussed and a description is given of the construction of the mathematical models which represent these processes respectively. Assumptions are motivated and refinements to the original models are discussed. In Chapter 3 a general system of reaction-diffusion equations is given, of which the governing equations of the various models are particular examples. A literature survey is then presented which covers both the pure analytical results and numerical results available on the various models.