First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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Library of Congress Control Number: 2017956887

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ISBN 978-1-78630-253-3

Foreword

Since the 1940s, when Kolmogorov introduced the acclaimed term “branching”, the theory of branching processes has been developing in a “supercritical” fashion, spreading in various directions. Nowadays, branching processes have a prominent role in stochastic modeling with their important applications. From nuclear reactions and cosmic rays to cell proliferation and digital information, branching models are used in explaining very interesting real-world stochastic phenomena.

In the beginning, classical branching processes with independent individual evolutions were investigated. Traditionally, they were interpreted as evolutionary models for isolated populations that develop using only their own resources. For the most part, the mathematical theory explored the method of probability-generating functions in obtaining intricate results. The fundamental investigations were summarized in several monographs, appropriately included in the references of the present book.

However, in many real phenomena, the evolutions of the individuals (particles) are not independent. Various approaches to modeling dependent evolutions exist in the literature. The subject matter of this monograph focuses on one of them in detail – random control. In this first book on “controlled branching processes” (CB processes), the authors study the most well-developed class of CB processes having single-type individuals and discrete-state space. The CB processes can be interpreted as models of population dynamics taking into account outside (environmental) factors. For instance, a random part of one generation could be eliminated and would not take part in further evolution (emigration) and/or new individuals could join the population and participate in the reproduction process (immigration). It is interesting to point out that, as a result of the random control (or regulation) mechanism, new effects arise in the population dynamics which were not observed in the classical theory.

The book does not require any previous knowledge of branching processes. From the first chapter, the reader will be introduced to the basic ideas and results of the classical theory. Undergraduate level stochastic education will be enough to follow the exposition. The book can be used in specialized courses on branching processes or as a supplementary text for a general course on applied stochastic processes. The monograph will be also very useful for researchers who would like to apply some form of control (regulation) in modeling concrete population dynamics. The specialists in branching processes will also be surprised to find some interesting ideas and open problems. For instance, Chapter 5 discusses some recent statistical investigations on CB processes. It seems that one of the main directions for further investigations is the development of the theory of time-continuous CB processes.

In conclusion, let me commend the authors for the easy-to-follow mathematical exposition accompanied with appropriate discussions and illustrations.

Nikolay M. YANEV
Sofia, Bulgaria

Preface

“All theory is gray, my friend. But forever green is the tree of life”.

– Johann Wolfgang von Goethe, Faust: First Part

The intent of this book is to provide stimulating discussion and insights into the available results for discrete-time branching processes with random control functions. These stochastic models of population dynamics have evolved from the classical branching processes as models which assume a state-dependent reproduction of the population. A fundamental assumption in traditional branching processes is that each individual’s reproduction or survival is independent of the chance of others. As a result, branching processes are good models of the evolution of small populations, in which resource limitations, for example, do not play an important role. Naturally, we are also interested in processes in which generation size increases or decreases depending on the available resources or interactions with other populations.

Since the 1960s, a number of models allowing different forms of population size regulations have been introduced and studied. In 1974, Sevastyanov and Zubkov [SEV 74] proposed a class of branching processes in which the number of reproductive individuals in one generation decreases or increases depending on the size of the previous generation through a set of control functions. The individual reproduction law (offspring distribution) is not affected by the control and remains independent of the population size. These processes are known as controlled branching processes (CB processes). In 1975, Yanev [YAN 75] (no relation to the third author of this book) essentially extended the class of CB processes by considering random control functions. CB processes constitute a very large class of stochastic processes, which take into account different conditions for immigration and emigration. These have been successfully used as modeling tools in a wide range of applications outside of mathematics. Within mathematics, CB processes are a fascinating research field of their own with thought-provoking unanswered questions. Over the years, a number of particular subclasses of CB processes have been introduced and investigated in detail. At the same time, fruitful connections were established to other types of branching processes, including two-sex processes and population-size-dependent processes. We consider the general properties of CB processes rather than their applicability to any real-world system. In particular, we turn our attention to (i) the probability of extinction, (ii) criticality, (iii) limit theorems, and (iv) statistical inferences.

CB processes are discrete-time and discrete-state stochastic population models. The two qualifiers, discrete and stochastic, simultaneously provide richness and technical challenges in terms of the measurements that can be made. Population development is modeled in two phases: the reproductive phase, when the individuals’ produce offspring, and the control phase, when the number of potential progenitors is determined (see the diagram above).

This book is divided into three distinct parts. The first part, consisting of two chapters, discusses particular sub-classes of CB processes of varying generality. Chapter 1 is devoted to classical models, including Bienaymé–Galton–Watson processes, processes that allow for unrestricted immigration, processes with immigration at zero only, and processes with time-dependent immigration. Chapter 2 presents in detail one class of processes with migration (immigration and emigration). An extension, connecting processes with migration and alternating regenerative processes, is also discussed. The second part includes Chapters 3 and 4, in which CB processes are treated in their generality. Chapter 3 addresses the fundamental problems of extinction and classification into subcritical, critical, and supercritical subclasses. Chapter 4 presents a variety of limit theorems depending on the criticality of the processes. The third part, consisting of Chapter 5, addresses statistical estimation procedures for certain parameters of CB processes. Each chapter ends with some background and bibliographical notes. For easy reference, some classical and auxiliary results needed in the proofs are given in the appendices.

The authors express their appreciation to N.M. Yanev (who is the third author’s academic advisor) for his continual support and mentorship. The first two authors would like to express their gratitude to the members of the research group Branching Processes and their Applications at the University of Extremadura (Badajoz, Spain), especially to M. Molina, R. Martínez and C. Minuesa for their contributions to the development of the theory of CB processes. The third author thanks his colleagues and teachers I. Rahimov and J. Stoyanov for their support and helpful discussions.

Part of the research included in this book was supported by the Spanish Ministry of Economy and Competitiveness (Ministerio de Economía y Competitividad) through the Grant MTM2015-70522-P and by the NFSR 190 at the MES of Bulgaria, Grant No DFNI-I02/17. The research of the first two authors has also been partially supported by the Junta de Extremadura, Grant IB16099, and the Fondo Europeo de Desarrollo Regional (FEDER). Working on this book, the third author was on leave from the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences.

We extend our sincere thanks to E. Yarovaya, the editor of the series of which this book is a part. We also express gratitude to the ISTE editorial office for all the services they provided.

Miguel GONZÁLEZ
Inés M. DEL PUERTO
George P. YANEV
October 2017