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Title: Coalescence in branching processes with age dependent structure in population Authors:  Sumit Kumar Yadav - Indian Institute of Technology Roorkee (India) [presenting]
Prof Arnab K Laha - Indian Institute of Mangement Ahmedabad (India)
Abstract: Branching processes and their variants are widely used mathematical models in studying population dynamics. In the recent past, branching processes have 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 processes is as follows: Assume that one individual starts the branching process in 0-th generation and the population size of the tree obtained by the 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 $X(n)$. The objective is to study the properties of $X(n)$. While this problem has been studied by many authors for simple and multitype discrete time branching processes, not much attention has been given to 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 $X(n)$ have been derived for broad classes of deterministic trees. For random trees, we provide an explicit expression for some special cases. We also derive properties of $X(n)$ as $n$ goes to infinity. A simulation analysis has also been performed, and some interesting insights are discussed.