Bienyeme‐Galton‐Watson Branching Process

Abstract

The study of the growth and development of many species of animals, plants and other organisms over time may be approached in the following manner. At some point in time a set of individuals called ancestors or the zeroth generation is identified. These produce offspring and the collection of offspring of all the ancestors constitutes the first generation. The offspring of these first generation individuals constitute the second generation and so on. If one specifies the rules by which the offspring production takes place then one could study the behavior of the long time evolution of such a process, called a branching process. Questions of interest are the long term survival of such a process or its extinction, the growth rate of such a population, fluctuation of population sizes, effects of control mechanicisms etc. In this article several simple mathematical models and some results for these will be discussed. Many of the assumptions made, especially the one about independence of lines of descent, are somewhat idealistic and unrealistic. Nevertheless, the models do prove useful in answering some general questions, since many results about long term behavior deviate somewhat from the basic assumptions.