Coalescence in branching processes with age dependent structure in population

Abstract

Branching process and their variants are a widely used mathematical model in the study of population dynamics, in which all individuals in a given generation produces some random number of individuals for the next generation. In the recent past, branching process has also found applications in areas like operations research, marketing, finance, genetics etc. A problem that has caught attention in the context of coalescence in branching process is as follows: Assume that the branching process is started by one individual in 0th generation and the population size of the tree obtained by branching process in generation n is greater than 1. Next, pick two individuals from n th generation at random and trace their lines of descent back till they meet. Call that random generation by Xn . The objective is to study the properties of Xn . While this problem has been studied by many authors for simple and multitype discrete time branching processes, not much attention has been given for the realistic extension when one individual is allowed to survive for more than one generation and can also give birth more than once. We study this problem for some deterministic and random cases. Explicit expressions about some mathematical properties of Xn have been derived for broad classes of deterministic trees. For random trees, we provide explicit expression for some special cases. We also derive properties of Xn as n goes to infinity. Additionally, simulation analysis has also been performed and some interesting insights are discussed.