Extinction in branching processes with countably many types

Multitype branching processes describe the evolution of populations in which individuals give birth independently according to a probability distribution that depends on their type. In this thesis, we consider the extinction of branching processes with countably infinitely many types. We begin by developing a class of iterative methods to compute the global extinction probability vector q. In particular, we construct a sequence of truncated and augmented branching processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some su cient conditions, the corresponding sequence of extinction probability vectors converges to the infinite vector q. Besides giving rise to a family of algorithmic methods, our approach leads to new su cient conditions for q “ 1 and q † 1. When then turn our attention to a specific class of branching processes which we refer to as lower Hessenberg branching processes. These are branching processes with typeset X “ t0, 1, 2, . u, in which individuals of type imay give birth to o↵spring of type j § i ` 1 only. For this class of processes, we study the set of fixed points of the progeny generating vector. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vector q̃. We derive a computationally e cient partial extinction criterion. In the case where q̃ “ 1, we derive a global extinction criterion, and in the case where q̃ † 1, we develop necessary and su cient conditions for q “ q̃ . Finally, we consider a more general class of structured branching processes which we refer to as block lower Hessenberg branching processes. For these processes, we derive partial and global extinction criteria, and we study the probability of extinction qpAq in any set of types A. In particular, we develop conditions for qpAq to be di↵erent from the global and partial extinction probability vectors, we present an iterative method to compute the iii vectors qpAq, and we investigate their location in the set of fixed points of the progeny generating vector. Declaration This is to certify that: i. the thesis comprises only my original work towards the PhD except where indicated in the Preface, ii. due acknowledgment has been made in the text to all other material used, iii. the thesis is fewer than 100 000 words in length, exclusive of tables, maps, bibliographies and appendices. Signed Peter Timothy Braunsteins Preface This thesis contains original research in Chapters 3, 4, and 5. Chapter 3 is based on the following paper, which is currently in revision. • [16]: Braunsteins, P., Decrouez, G. , and Hautphenne, S. (2018). A pathwise approach to the extinction of branching processes with countably many types. In revision. Chapter 4 is based on the following paper, which is currently in revision. • [15]: Braunsteins, P., and Hautphenne, S. (2018). Extinction in lower Hessenberg branching processes with countably many types. In revision. Chapter 5 is based on the following working paper. • [14]: Braunsteins, P., and Hautphenne, S. (2018). Extinction in block lower Hessenberg branching processes. Working paper. All co–authorship has taken place in accordance with the Graduate Research Training Policy of the University of Melbourne. Acknowledgments I would like to thank my supervisors, Professor Peter Taylor and Dr. Sophie Hautphenne. Thank you Peter for your invaluable advice; thank you for encouraging me to pursue the pathwise arguments that form the nucleus of this thesis; and thank you for supporting me between masters and doctorate degrees when I was unsure which pathway to take. Thank you Sophie for your enthusiastic guidance; thank you for pushing me to present my research; and most of all, thank you for your unwavering support; I thoroughly enjoyed working with you. I am grateful to Geo↵rey Decrouez for his help on Chapter 3, and for his detailed feedback on Chapters 1 and 2. I am also grateful to Nicholas Read for his level headed advice. My research was supported by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers. ix