Controlled Branching Processes

By Miguel González Velasco, Inés María Del Puerto García and George Petrov Yanev

The purpose of this book is to provide a comprehensive discussion of the available results for discrete time branching processes with random control functions. The independence of individuals’ reproduction is a fundamental assumption in the classical branching processes. Alternatively, the controlled branching processes (CBPs) allow the number of reproductive individuals in one generation to decrease or increase depending on the size of the previous generation.

Generating a wide range of behaviors, the CBPs have been successfully used as modeling tools in diverse areas of applications.

• Language: English
• Publisher: Wiley
• Release date: Dec 27, 2017
• ISBN: 9781119484561

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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

1

Classical Branching Models

"Common sense is not so common".

– Voltaire

In this chapter, some classical theorems for the ordinary Bienaymé–Galton–Watson process and its modifications allowing immigration are discussed. These are well-studied models for which a comprehensive mathematical theory exists. It is not our goal to present the most general models and results available, but rather to discuss selected processes with and without immigration, relevant to the material included in the following chapters. Questions of interest for each model include (1) extinction criteria and criticality classification, (2) asymptotic behavior of the survival probability, (3) mean population growth rate, (4) limit theorems for the population size and (5) effects on the population of possible immigration.

1.1. Bienaymé–Galton–Watson process

In this section, the simple branching process with discrete time and one type of individuals is described and its basic results are summarized. This classical model is called Galton–Watson (GW) process before Bienaymé’s priority was recognized; now it is often called Bienaymé–Galton–Watson process. In the beginning, the fundamental equation for the probability generating functions (p.g.f.s) associated with the process is derived. Next, the first two moments as well as the coefficient of variation are calculated. The processes are classified according to the value of their offspring mean and the extinction probability in each class is studied. Limit theorems for the population size are discussed with either detailed or outlined proofs.

Let (Ω, A, P) be a probability space on which an array of non-negative integer-valued random variables {Xn,i : n = 0, 1, … ; i = 1, 2, …} is given, where {Xn,i} are independent and identically distributed (i.i.d.) with a common probability mass function (p.m.f.) {pk}k≥0. To avoid trivialities, we assume that p0 > 0, p0 + p1 < 1 and therefore pj ≠1 for any j.

DEFINITION 1.1.– The (Bienaymé–)Galton–Watson branching process is a discrete time homogeneous Markov chain {Zn}n≥0 defined inductively by Z0 = 1 and for n = 0, 1, …

[1.1]

By convention, equality between random variables stands for equality in distribution. Alternatively, we will also use the notation .

The most common interpretation of this formal definition is in terms of the evolution of a population. The state space consists of a number of individuals, where an individual might be an animal or a plant, but also a cell or an elementary particle – the defining property is that it gives birth, splits into, or somehow generates new individuals (see [HAC 05, p. 6]). In the beginning, the population consists of one individual (the ancestor), Z0 = 1, with unit lifetime. In the end of its life, the ancestor produces a random number X0,1 offspring (direct descendants) with p.m.f. {pk}k≥0. Every direct descendant (if there is any) also has a unit lifetime and in its end produces (independently of the other descendants) a random number of offspring according to the same offspring distribution {pk}k≥0. Proceeding in this way and interpreting n as the generation index and Xn,i as the number of direct offspring of the ith individual from the nth generation, we can say that Zn+1 represents the size of the next (n + 1)st generation, i.e. the GW process counts the generation sizes.

Note that the above-mentioned terms birth and offspring are used in a general sense. For example, in an epidemic outbreak, the number of infected individuals represents the states of the branching process and a birth means a new infection and offspring means the number of new infections produced. Generally, in a GW process, the birth of offspring implies death of the parent (non-overlapping generations). However, if the parent gives birth to j offspring and survives, then the number of new offspring, counting the surviving parent, is j + 1 with corresponding probability pj+1. This latter assumption can be thought of as a death of the parent and replacement by an identical substitute. Such an assumption is reasonable in an epidemic outbreak, where the infected parent continues to spread infection until death or recovery (see [ALL 15, p. 2]).

An equivalent to [1.1] representation is (see [KIM 15, p. 38]) Z0 = 1 and

[1.2]

where the random variables are i.i.d. copies of Zn. The relation [1.2] reflects the fact that the (n + 1)st generation is the sum of the nth generations of subprocesses initiated by the first-generation offspring of the ancestor.

Alternatively, the GW process {Zn}n≥0 is defined as a Markov chain on the non-negative integers with stationary transition probabilities given by

where δij is the Kronecker delta and

is the jth term of the ith fold convolution of the sequence {pk}k≥0.

Yet another characterization of GW process is through the so-called branching (additive) property. Let N+ be the set of positive integers. If Zn(i) denotes a GW process initiated by i ∈ N+ ancestors, the family of processes {Zn(i), i ∈ N+}n≥0 is such that

where and are two independent copies of Zn(·). It can be shown that this property uniquely characterizes GW processes. It is inherited by more general classes of branching processes (e.g. continuous state processes, e.g. [LAM 67]).

1.1.1. Moments and probability of extinction

The offspring p.g.f. and the generation size p.g.f. Fn(s) :=E [sZn], 0 ≤ s ≤ 1, are the main tools in the study of the process. Recall that, in order to exclude trivial situations, we assumed, p0 = f(0) > 0 and p0 + p1 < 1. This last condition implies that f(s), which is increasing on [0, 1], is a strictly convex function. The following fundamental equations hold.

THEOREM 1.1.– For any n = 0, 1, …

[1.3]

where fn(s) is the nth functional iterate of the offspring p.g.f. f(s).

Proof. The initial condition reflects the assumption Z0 = 1. By the definition [1.1] of the process and the independence of the offspring random variables {Xn,i : n = 0, 1, … ; i = 1, 2, …}, using the law of total expectation, we have

Similarly, using [1.2] and first-step analysis we obtain

Since F1(s) = f(s), iterating we have Fn(s) = fn(s).

The offspring mean and variance are denoted by m and σ², respectively. Throughout this chapter, we assume that m < ∞, non-explosive case (sample paths do not approach infinity for finite time). Unless stated otherwise, for the offspring variance we assume σ² < ∞. Next, we will calculate the mean and variance of Zn in terms of m and σ².

THEOREM 1.2.– The mean and variance of Zn are E[Zn] = mn and

[1.4]

Proof. Differentiating each side of [1.3] and evaluating the derivatives at s = 1, we have

Thus, in general, E[Zn] = mn. Differentiating each side of [1.3] twice and evaluating at s = 1, we have

Now, it is not difficult to calculate

[1.5]

Since V ar[Zn] = E[Zn(Zn − 1)] + E[Zn] − (E[Zn])², referring to [1.5] and the formula for E[Zn], we obtain [1.4].

For the coefficient of variation of the process, CV [Zn] say, we obtain

[1.6]

where the limit is as n → ∞.

The asymptotic behavior of E[Zn] and V ar[Zn] varies depending on the offspring mean m. If m < 1, then both mean and variance of the population size decrease to zero, and the process should eventually die out. If m = 1, then the mean remains one, whereas the variance experiences a linear growth. In this case, a more delicate analysis of the asymptotic behavior of the process is required. Finally, if m > 1, then both mean and variance increase exponentially, such that the coefficient of variation stabilizes and thus the population experiences a steady expansion. Depending on its asymptotic behavior, the GW processes form three distinct classes as follows.

DEFINITION 1.2.– A GW process with offspring mean m is said to be subcritical if m