An analytical and numerical study of Galton-Watson branching processes relevant to population dynamics.

摘要

Galton-Watson branching processes of relevance to human population dynamics are the subject of this thesis. We begin with an historical survey of the invention of the invention of this model in the middle of the 19th century, for the purpose of modelling the extinction of unusual surnames in France and Britain. We then review the principal developments and refinements of this model, and their applications to a wide variety of problems in biology and physics.; Next, we discuss in detail the case where the probability generating function for a Galton-Watson branching process is a geometric series, which can be summed in closed form to yield a fractional linear generating function that can be iterated indefinitely in closed form.; We then describe the matrix method of Keyfitz and Tyree, and use it to determine how large a matrix must be chosen to model accurately a Galton-Watson branching process for a very large number of generations, of the order of hundreds or even thousands.; Finally, we show that any attempt to explain the recent evidence for the existence thousands of generations ago of a 'mitochondrial Eve' and a 'Y-chromosomal Adam' in terms of a the standard Galton-Watson branching process, or indeed any statistical model that assumes equality of probabilities of passing one's genes to one's descendents in later generations, is unlikely to be successful. We explain that such models take no account of the advantages that the descendents of the most successful individuals in earlier generations enjoy over their contemporaries, which must play a key role in human evolution.