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Asymptotic Analysis of Some Stochastic Models from

Population Dynamics and Population Genetics

by

Todd Parsons

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c! 2012 by Todd Parsons

Abstract

Asymptotic Analysis of Some Stochastic Models from Population Dynamics and

Population Genetics

Todd Parsons

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2012

Near the beginning of the last century, R. A. Fisher and Sewall Wright devised an el-

egant, mathematically tractable model of gene reproduction and replacement that laid

the foundation for contemporary population genetics. The Wright-Fisher model and its

extensions have given biologists powerful tools of statistical inference that enabled the

quantification of genetic drift and selection. Given the utility of these tools, we often for-

get that their model - for reasons of mathematical tractability - makes assumptions that

are violated in many real-world populations. In particular, the classical models assume

fixed population sizes, held constant by (unspecified) sampling mechanisms.

Here, we consider an alternative framework that merges Moran’s continuous time

Markov chain model of allele frequencies in haploid populations of fixed size with the

density dependent models of ecological competition of Lotka, Volterra, Gause, and Kol-

mogorov. This allows for haploid populations of stochastically varying – but bounded –

size. Populations are kept finite by resource limitation. We show the existence of limits

that naturally generalize the weak and strong selection regimes of classical population

genetics, which allow the calculation of fixation times and probabilities, as well as the

long-term stationary allele frequency distribution.

ii

Acknowledgements

While any e!ort is inevitably shaped by every interaction, there are a few people to

whom I am eternally grateful for the significant contributions they made to the writing

of this dissertation and my overall intellectual development. First among them is Peter

Abrams. To me he will always epitomize what a senior scientist should be. I will never

forget that he gave me a second chance after I left a PhD in pure mathematics. I hope

that I will follow his example providing essential guidance while encouraging students

to find their own paths. I am equally indebted to Joe Repka, who has been my link to

the mathematics department and has been consistently available for advice, comment,

reading and critique of my writing, and without whom even submitting a thesis would be

an impossibility. David Earn was a fantastic external examiner, and I’m deeply grateful

for his careful reading of this thesis, and the many thoughtful suggestions he made for

improving its content and readability.

Most of the questions posed and addressed here emerged from joint work with Christo-

pher Quince, with whom I hope to have many more years of fruitful collaboration. Much

of the rest emerged from conversations with Joshua Plotkin, in whose Mathematical Bi-

ology Group I was employed during a portion of the time this thesis was written. Both

provided support both moral and financial, introduced me to a broad community of pop-

ulation geneticists and a host of fascinating problems, and constantly reminded me that

a mathematical formalism is not an end in itself, but a means to obtain insight about

the world around us.

My understanding and appreciation of mathematical population genetics has been

deepened by many conversations with Warren Ewens, who provided invaluable mentor-

ship, for which I am forever grateful.

My thanks also to Jeremy Quastel and Balig Virant for allowing me to participate in

the Probability Seminar, for answering my innumerable naive questions, and for clarifying

many misconceptions on my part.

iii

Peter Ralph read and identified errors in earlier versions of Theorems 4.1 and 4.2.

Amaury Lambert provided many insightful comments on early drafts of Chapters 3

and 5.

Jesse Taylor pointed out that global weak convergence could be proven (as opposed

to just convergence of finite dimensional distributions), which led to Corollary 4.3. Rick

Durrett for showed me a draft of [39], which inspired considerable improvements to

Theorems 4.2 and 4.2. Frank Ball introduced me to the notion of coupling and pointed

me to [174].

Section 4.3.3 reproduces text and figures from [151] which was written in collaboration

with Christopher Quince and Joshua B. Plotkin. All analytical results (and the errors

therein) are my own contribution; I am grateful to Chris and Josh for helping me distill

concrete biological insights out of an abstract formalism.

This project would have never been conceived, and I would likely never have met

Peter Abrams, if I hadn’t first met Jonathan Dusho! and Lee Worden, who introduced

me first to the field of mathematical biology and then to their supervisor, Simon Levin,

who encouraged me to read and explore the field, and who directed me to Peter.

Finally, my sincerest thanks to Ida Bulat, who exhibited Buddha-like patience with my

many, many requests, and to my dear friends Michael Buckley, Jeremy Draghi, Hayssam

Hulays, Sergey Kryazhimskiy, Eden Legree, Jebney Lewis, Melanie Redman, Martin Reis,

Phillip Smith, Alexander Stewart, and Katie Wood, who have all provided invaluable

support, both moral and material, at important turns in the thinking, writing, and

rewriting process.

iv

Contents

1 Introduction 1

1.1 Density Dependence and the Lotka-Volterra-Gause Model . . . . . . . . . 3

1.1.1 Kolmogorov’s Population Equations and Competitive Dynamical

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Classical Population Genetics . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Wright-Fisher Model . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Neutrality, Strong vs. Weak Selection, and Kimura’s Di!usion . . 11

1.2.3 The Neutral Theory of Molecular Evolution and the Neutralist/Selectionist

Debate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.4 Moran’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.5 Cannings’ Models and Neutrality . . . . . . . . . . . . . . . . . . 18

1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Birth-Death-Mutation Processes 25

2.1 Density Dependent Population Processes . . . . . . . . . . . . . . . . . . 26

2.1.1 Stochastic Di!erential Equation Representation . . . . . . . . . . 28

2.1.2 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Density-dependent Birth-Death-Mutation Processes . . . . . . . . . . . . 36

2.2.1 Individual Fecundity . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Strong and Weak Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 43

v

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Invasion Dynamics and Strong Selection 49

3.1 Continuous Time Markov Branching Process in a Varying Environment . 51

3.1.1 Moments of Z(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.2 The Probability Generating Function and Extinction Probabilities 54

3.1.3 Criticality for the Markov Branching Process in Varying Environ-

ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.4 Supercritical Processes . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.5 Subcritical Processes . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.6 Critical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1.7 Time Asymptotics for Processes with a Malthusian Parameter . . 66

3.2 Application to Density-Dependent Birth-Death-Mutation Processes . . . 72

3.2.1 A Coupling Result . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2.2 Invasion and Fixation in the Supercritical Case . . . . . . . . . . 79

3.2.3 Fixation Time Asymptotics When the Resident Dynamics have an

Attracting Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . 89

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Weak Selection 99

4.1 Convergence to a Di!usion on C . . . . . . . . . . . . . . . . . . . . . . . 101

4.1.1 Di!usion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2 Computing the Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.2 Explicit Expressions in Co-dimension One . . . . . . . . . . . . . 131

4.2.3 Energy-based Approaches . . . . . . . . . . . . . . . . . . . . . . 136

4.3 Weakly Selected Density Dependent Birth-Death-Mutation Processes . . 139

vi

4.3.1 The Relative Frequency Process and Projection to the Standard

Simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.3.2 Exchangeable Processes, Neutrality and Kimura’s Di!usion . . . . 145

4.3.3 The Generalized Moran Model . . . . . . . . . . . . . . . . . . . . 152

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5 Evolution in the SIR Epidemic Model 161

5.1 Birth-Death-Mutation Processes with Immigration and Transmutation . 162

5.2 The SIR Model with Host Demography . . . . . . . . . . . . . . . . . . . 163

5.2.1 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . 164

5.2.2 Strong Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.2.3 Weak Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6 Conclusion 178

A Probability Glossary 183

A.1 Random Variables and Stochastic Processes . . . . . . . . . . . . . . . . 184

A.2 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

A.3 Convergence of Random Variables . . . . . . . . . . . . . . . . . . . . . . 187

A.4 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

A.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

A.6 Quadratic Variation and Covariation . . . . . . . . . . . . . . . . . . . . 195

A.7 Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

A.8 Counting Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Bibliography 200

vii

List of Tables

1.1 Ecological interactions and the corresponding conditions on interaction

coe"cients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.1 A comparison of the neutral Wright-Fisher model and the quasi-neutral

Generalized Moran model. . . . . . . . . . . . . . . . . . . . . . . . . . 155

viii

List of Figures

2.1 Generalized Moran model under weak selection . . . . . . . . . . . . . . 45

3.1 Fixation probabilities for the Generalized Moran model under strong se-

lection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.1 Fixation probabilities and times for the Generalized Moran model under

weak selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.2 Stationary distribution for the Generalized Moran model . . . . . . . . . 158

5.1 Compartmental model of a two-strain SIR epidemic with demography . . 165

ix

Chapter 1

Introduction: Two Streams of

Mathematical Biology

I have deeply regretted that I did not proceed far enough at least to under-

stand something of the great leading principles of mathematics; for men thus

endowed seem to have an extra sense.

– Charles Darwin

As we approach the centenary of modern mathematical biology, a retrospective look

shows that there have been two streams of research that have dominated most of its

development, both of which trace back to the very first e!orts in the area. The first,

originating in the work of Lotka [130] and Volterra [182, 183] and later Gause [65], is

informed by ecology, and concerns itself with the dynamics of interacting populations of

predators, prey, competitors, and mutualists, subject to resource limitation. Its princi-

pal mathematical tools have been ordinary, and later, partial di!erential equations and

di!erence equations. Populations are typically modelled deterministically and treated as

continuous quantities, rather than discrete individuals.

The second stream begins with the foundational work in population genetics by Hal-

dane [71], Fisher [59], and Wright [186]. Concerned with the survival and increase in

1

Chapter 1. Introduction 2

frequency of rare mutations, their approach was stochastic, assuming discrete individ-

uals and using Markov chains, including the Bienayme–Galton–Watson process [184].

Whereas Lotka and Volterra and their successors were concerned with the mechanisms

by which populations are regulated, the early population geneticists were content – for

reasons of analytical tractability – to allow populations to either grow to infinity, or

assume that they were held fixed in size by some unspecified sampling mechanism.

Beginning with Fisher [59], there has long been interest in understanding how incorpo-

rating more realistic population dynamics might complicate population genetics, usually

by assuming that the total population size either varies deterministically (e.g., [52, 106,

147]), or alternates between a discrete set of values according to an a priori Markov chain

(e.g., [111, 94, 32, 93, 92, 85]).

More recently, it has been acknowledged that such approaches are insu"cient, and

there is a need to more fully integrate the two steams of mathematical biology: to

incorporate notions from ecology such as density dependence into population genetics,

and to develop a stochastic, individual based approach to population dynamics [143,

48, 123, 90]. Some steps towards this path have already been taken, from the first,

deterministic approach to density dependent population genetics [165, 28], to recent

work incorporating density dependence into the Bienayme–Galton–Watson process [121,

122]. In what follows, we present a complementary approach to incorporating density

dependence into population genetics, taking Moran’s continuous time Markov formulation

of population genetics [142] and Kurtz’s density-dependent population processes [116,

117, 118, 119] as a point of departure.

The balance of this chapter is a survey of the relevant concepts and models in mathe-

matical biology that will be assumed in subsequent chapters, concluding with a summary

of the results herein. Some of these results, along with a discussion of their biological

significance, have appeared previously [148, 149, 150, 151]. In our previous publications,

they were derived via heuristic singular perturbation methods. While the questions con-


sidered here were inspired by those previous studies, the proofs presented are novel and

rigorous, and the models considered here substantially generalize the ones in those publi-

cations. Where relevant, we show how previous results may be obtained as special cases

of those presented here.

Throughout, we assume familiarity with the language of measure-theoretic probabil-

ity, stochastic processes and stochastic integration. We refer the uninitiated to Appendix

A for a brief summary of the relevant definitions and results and some useful references.

1.1 Density Dependence and the Lotka-Volterra-Gause

Model

The notion of density dependence was first introduced by Verhulst [181], as a correction

to the unrealistic exponential growth considered in Malthus’ An essay on the principle

of population [134]. His logistic equation, familiar to any freshman student of calculus,

captures the ecological notion of resource limitation in two simple parameters: if X(t) is

the number of individuals in of a population at time t, then

X = rX

!

1"X

N

"

, (1.1)

where r is the intrinsic growth rate of the population, the rate of growth in the absence

of competition, and N is the carrying capacity, the stable equilibrium the population will

asymptotically approach, reflecting the saturation of resources by the population. The

logistic equation is the foundation of mathematical ecology, and continues to be used

extensively in the present. It is not uncommon to see the logistic equation in rescaled

form: setting Y (t) = X(t)N

, we obtain an expression independent of N ,

Y = rY (1" Y ) . (1.2)


Indeed, that the dynamics only depend on the rescaled variable may be taken as the

definition of density-dependence.

Remark 1.1. In the theoretical ecology literature, the carrying capacity is frequently

denoted by K, but to avoid confusion with the standard notation of population genetics,

where K is often used to denote the number of species, we use N . Given that we shall

consider limits as N #$, the choice of letters seems mathematically appropriate.

Verhulst’s logistic equation would eventually be generalized for the study of commu-

nities of organisms, consisting of predators and prey, competitors and mutualists, in the

form of the Lotka-Volterra-Gause [130, 182, 183, 65] equations,

Yi =

#

bi +K$

j=1

aijYj

%

Yi, (1.3)

describing the dynamics of a population of K interacting species. Species i has intrinsic

growth rate bi and has interaction coe!cient aij with species j, leading to a net per

capita individual growth rate

fi(Y) = bi +K$

j=1

aijYj (1.4)

for all individuals of type i. Various ecological relationships can be captured by appro-

priate choices for the interaction coe"cients, summarized in Table 1.1:

Interaction ConditionPredation/Parasitism aij < 0 and aji > 0 or aij > 0 and aji < 0Competition aij < 0 and aji < 0Mutualism aij > 0 and aji > 0

Table 1.1: Ecological interactions and the corresponding conditions on interaction coe!cients.

With their simplicity and versatility in describing community dynamics, the Lotka-

Volterra-Gause equations have been employed frequently and investigated extensively

(see e.g., [82] and references therein), although many natural questions regarding the


qualitative properties of the solutions remain unsolved e.g.,necessary and su"cient con-

ditions for the survival of species i remain unavailable for K > 3. They have been

shown to give good fits in a few empirical studies of competition in microbial populations

(see e.g., [65, 180, 152]), and as a result have become a standard model in theoretical

population ecology.

For our purposes, the Lotka-Volterra-Gause equations are the prototypical model of

species competing for limited resources, and we shall return to them throughout the

following chapters. Nonetheless, they have distinct limitations, both biologically and

mathematically: they are a drastically oversimplified model of interspecies interactions,

whilst the set of equations of the form (1.3) is not structurally stable, so that even arbi-

trarily small nonlinear perturbations in individual growth rates can result in qualitatively

di!erent solutions [139].

1.1.1 Kolmogorov’s Population Equations and Competitive Dy-

namical Systems

Inspired by the work of Volterra, Kolmogorov [112] considered generalized equations for

continuous population dynamics,

Yi = fi(Y)Yi. (1.5)

Following [169], we will refer to such equations as being of Kolmogorov type. As with

the Lotka-Volterra-Gause equations, fi(Y) may be interpreted as the per capita growth

rate of individuals of type i.

Following [81, 172, 78, 79, 80], we have the following definitions:

Definition 1.1. 1.5 A dynamical system of Kolmogorov type is

(i) dissipative if there is a compact set K that uniformly attracts each compact set of

initial values,


(ii) irreducible if the matrix (!jfi) is irreducible for all x %!

RK+ ,

(iii) competitive if !fi!xj

& 0 for all i '= j, and totally competitive if !fi!xj

< 0 for all i '= j.

Each of these has a reasonable biological interpretation: the first requires that the

population remain finite, the second that the population cannot be separated into dis-

tinct, non-interacting subpopulations (which could then be studied independently), and

the third that individual growth rates decline in the presence of competitors. In what

follows, we will limit our attention to Kolmogorov systems with all of these properties,

as well as imposing the additional requirement that the dynamics have a source at the

origin, so that populations will grow when started from small initial densities.

The family of dynamical systems of Kolmogorov type, unlike those satisfying the

Lotka-Volterra-Gause equations, has the advantage of being topologically closed. Unfor-

tunately, the search for meaningful results at this level of generality is somewhat Quixotic,

as indicated by the following construction, due to Smale [172]:

Example 1.1. Let

#0 =

&

x % RK :

K$

i=1

xi = 0

'

(1.6)

and

#1 =

&

x % RK+ :

K$

i=1

xi = 1

'

, (1.7)

and let h0 : #1 # #0 be any C" map. Let h : RK+ # #0 be a C" extension of h0

and let " : R # R be a C" function which is identically 1 in a neighbourhood of 1 and

vanishes o! of(12 ,

32

)

. Set

fi(x) = 1"K$

j=1

xj + #1

xi

"(K$

j=1

xj)

#K*

j=1

xj

%

h(x). (1.8)

Then, for # > 0 su"ciently small, the corresponding Kolmogorov population system is

competitive. Moreover, #1 is an attractor for all initial conditions in!

RK+ , and, on #1,


the dynamics are, up to a constant, those of Y = h0(Y) i.e., any dynamics are possible

within a competitive dynamical system.

Nonetheless, some general results can be obtained. Indeed, in Smale’s construction,

the use of the simplex is in some sense generic: Hirsch [78, 79, 80] showed there is an

invariant set C, the carrying simplex, that is homeomorphic to the (K " 1)-dimensional

simplex #1, and is a global attractor of every point excluding the origin. This carrying

simplex contains all of the asymptotic dynamics of the system.

In what follows, we will consider a class of stochastic processes that have competitive

Kolmogorov systems as a functional Law of Large Numbers, specializing as needed to

more specific models (in particular the Lotka-Volterra-Gause model) to obtain more

precise results.

1.2 Classical Population Genetics

We now turn to the second stream, giving a brief introduction to the classical theory of

population genetics. The literature is far too extensive to provide an exhaustive survey,

so the discussion is limited to the handful of essential results necessary to provide the

context for subsequent chapters. In particular, the discussion – and all results herein – are

limited to panmictic populations of haploid individuals i.e., any individual can potentially

interact with all other individuals, and reproduction is clonal, so that, absent mutation,

o!spring are identical to their parent. Somewhat egregiously, we omit the coalescent

[108, 109] and subsequent developments in genealogical processes. See [53, 38, 49] for

surveys of population genetics intended for a mathematical audience that address the

coalescent, diploid populations, etc.


1.2.1 Wright-Fisher Model

The Wright-Fisher model [59, 186] is the foundational model in mathematical population

genetics. While it seems surprising from our current perspective, at the beginning of the

20th century, Darwin’s theory of evolution seemed in peril, with its principal challenger

being Mendel’s theory of inheritance [163, 53]. Like many so-called “gradualists”, Darwin

believed that an individuals traits were a blend of those of its parents. As his “saltation-

ist” opponents pointed out, over time, random mating would lead to entirely uniform

populations incapable of further adaptation. They further argued that the small changes

proposed by Darwin would not under su"cient selection to change the composition of

the population, and in any case could only produce di!erence in quantity, but not kind.

After the immediate selective force ceased, they argued, reversion to the mean would

return the population to its original state [163]

In contrast, the saltationists, informed by Mendel’s breeding experiments, argued that

traits were inherited discretely through pairs of alleles, one from one each parent. They

argued that natural selection, to the extent it acted at all, was primarily to emphasize pre-

existing variants, or upon novel mutations that conveyed large benefits. Such a model

maintained variation, but also encountered di"culty: without mutations that create

“hopeful monsters” with su"ciently large fitness advantage, eventually all alleles would

reach their Hardy-Weinberg equilibrium, and again, adaptation would cease. Wright and

Fisher sought to merge of Darwinian evolution with Mendel’s discrete heredity in what we

now refer to, following Huxley [84], as the Modern Synthesis : rare mutations give rise to

new alleles encoding new traits which may or may not confer a selective advantage, which

could in turn be small or large. Adaptation proceeds by the fixation of advantageous

novel alleles, as new new traits displace existing ones. For this to occur, it is necessary

that newly emerged advantageous alleles avoid stochastic extinction.

To demonstrate that this was possible, Wright and Fisher independently introduced

their eponymous model. In its simplest form, the neutral model , we assume a panmictic


population consisting of N haploid individuals and discrete generations. That is, we

assume that individuals reproduce clonally, and barring mutation, the o!spring is ge-

netically identical to the parent. Each individual has an allelotype (or type for brevity).

When Wright and Fisher were devising their theory of heredity, DNA was yet to be dis-

covered, but Mendel’s work gave a remarkably prescient picture of di!ering traits arising

from the presence of di!erent alleles at specific heritable loci. For simplicity, they as-

sumed that genes at specific loci were inherited more or less independently, and focused

on a single locus at which one of several distinct alleles might be present.

To model reproduction, they assumed that the n + 1st is formed by choosing N

individuals to produce an o!spring, uniformly at random with replacement from the nth

generation. Thus, in a population with K types, writing X(N)i (n) for the number of

individuals of type i in the nth generation, the Wright-Fisher model is a Markov chain

with multinomial transition probabilities:

P

+

X(N)1 (n+ 1) = j1, . . . , X

(N)K (n + 1) = jK

,,,X

(N)1 (n) = i1, . . . , X

(N)K = iK

-

=N !

j1! · · · jK !

!i1N

"j1

· · ·!iKN

"jK

. (1.9)

More generally, the model may be extended to include mutation by assuming that

with probability µij, the o!spring of a parent of type i is of type j, and selection, by

assuming that individuals of type i are chosen with probability proportional to 1+si, the

relative fitness of that type. The parameter si is referred to as the selection coe!cient. In

the absence of mutation, the Wright-Fisher model has K absorbing states, corresponding

to X(N)i (n) = N and X

(N)j (n) = 0 for all t, for some i; when the chain is in the ith

absorbing state, we say that type i has fixed.

There are myriad generalizations of the Wright-Fisher model, allowing for e.g.,diploid

organisms, subdivided populations, multiple loci, etc. which have led to the development

of a rich literature in population genetics, but the fundamental questions remain the


same:

(i) In the absence of mutation, what is the probability that type i eventually fixes?

What is the expected absorption time, the first time the population is monomorphic

(i.e., composed entirely of individuals of one type), and what is the expected time

to fixation of type i, that is, the the absorption time conditioned on the fixation of

type i?

(ii) If mutation between types occurs, what is the stationary distribution of types?

For Wright and Fisher, the goal in answering the first question was to show that a

single individual with even a vanishingly small fitness advantage – in the Wright-Fisher

model, an unspecified trait that increased the individual’s probability of contributing to

subsequent generations – had a non-zero probability of eventually fixing, and that the

rate of fixation of advantageous alleles was su"ciently rapid that evolutionary processes

could explain how observed diversity had arisen from simple organisms within the lifetime

of the Earth.

A vast literature on fixation probabilities and times has since arisen, attempting

to understand how complicated life-histories and ecological interactions will a!ect the

process of adaptation. The question of which traits might arise remains a largely empirical

one. Fixation probabilities, however, provide a quantitative measure of which traits we

expect to survive, should they appear, and thereby a means to understand the trajectory

of adaptation. An explicit expression for the fixation probability of a novel allele, based

upon the traits it encodes, is an essential component of many theoretical approaches

to predicting long-term evolution, such as the currently popular canonical equation of

Adaptive Dynamics (see e.g., [31, 26, 25, 27]).

Answering the second question, regarding the stationary distribution, gives an un-

derstanding of how and when mutation and selection can maintain biological diversity

and explain observed diversity, while simultaneously providing a means by which the


strength of selection and the population rate of mutation may be inferred from samples

from a population assumed to be at equilibrium [102, 166, 76, 22, 30]. In particular, with

models incorporating additional biological detail, inferred parameters can be compared

with a priori values obtained either from other summary statistics or in vitro to compare

models and reject hypotheses (e.g., [175, 64, 54].)

1.2.2 Neutrality, Strong vs. Weak Selection, and Kimura’s Dif-

fusion

Unfortunately, even as simple a model as the Wright-Fisher model is analytically almost

intractable, and much of the subsequent development of mathematical population ge-

netics would have to wait for the introduction of approximating di!usions by Feller [55]

and Kimura [98]. In order to obtain a di!usion approximation, we will assume that the

mutation rate is O(

1N

)

i.e.,we will assume that there exist constants $ij , independent of

N , so that

µij =$ijN

. (1.10)

This assumption is not unreasonable for a su"ciently small mutational target in a microbe

with DNA chromosomes, or a higher eukaryote, where estimates of per base pair mutation

rates vary on the order of 10#7–10#11, e.g., 5.4 ( 10#10 for Escherichia coli [35]. Given

typical gene lengths of order 102–103 [83], this assumption is valid for N of order 104–109.

By contrast, population sizes were of the order of 5( 108 in Lenski’s E. coli Long-term

Experimental Evolution Project [126].

The (Nearly) Neutral Theory

Let X(N)(n) = (X(N)1 (n), . . . , X(N)

K (n)) (in general, we will write X for the vector with

components Xi). If we assume that the selection coe"cients, si, are O(

1N

)

, then, the


vector of frequencies

P(N)(t) =1

NX(N)()Nt*), (1.11)

converges in in distribution as N # $ to P(t), a di!usion process with infinitesimal

generator1

(Gf)(p) =K$

i=1

#

"

#K$

j=1

$ij

%

pk +K$

j=1

$jipj

%

(!if)(p)

+1

2

K$

i=1

K$

j=1

pi (%ij " pj) (!i!jf)(p). (1.12)

This di!usion process is commonly referred to as the Wright-Fisher di"usion or as

Kimura’s di"usion. It was introduced in this form in [98], whilst convergence in dis-

tribution was proved in [51]. Using this di!usion process, it becomes possible to provide

explicit, if asymptotic, answers to Wright and Fisher’s questions, at least in some cases

(see e.g., [53]):

(i) Suppose that µij = 0 for all i, j. Then, the states P(t) = ei are absorbing for the

di!usion process, and the absorption time

Tabs = inf {t > 0 : P(t) = ei for some i} (1.13)

is a stopping time. Now, under the assumption of neutrality, P(t) is a martingale,

and we can use the optional stopping theorem2 to conclude that

pi = E [Pi(Tabs)|P(0) = p] = P {P(Tabs) = ei|P(0) = p} . (1.14)

1see A.4 for a discussion of infinitesimal generators and their uses.2see A.5 for a discussion of martingales, stopping times, and the optional stopping theorem


Thus, if hi(p) is the fixation probability for type i conditioned on P(0) = p, then

hi(p) = pi. (1.15)

Moreover, the Dirichlet problems3 G&(p) = "1 and G(hi&)(p) = "hi(p) can be

solved to yield [104, 103]

E [Tabs|P(0) = p] =K$

i=1

"(1" pi) ln (1" pi) (1.16)

and

E [Tabs|Pi(Tabs) = 1, |P(0) = p] = "(1 " pi)

piln (1" pi). (1.17)

(ii) The Dirichlet problem, G$&(p) = 0, for the adjoint

(G"f)(p) = "K$

i=1

!i

.#

"

#K$

j=1

$ij

%

pk +K$

j=1

$jipj

%

f(p)

/

+1

2

K$

i=1

K$

j=1

!i!j [pi (%ij " pj) f(p)] . (1.18)

can sometimes be solved to yield the stationary distribution,

P

+

limt%"

P(t) = p-

. (1.19)

In particular, when K = 2, the stationary distribution for the Wright-Fisher di!u-

sion with mutation is the beta distribution [186]:

P

+

limt%"

P(t) = (p, 1" p)-

=$($12 + $21)

$($12)$($21)p#12#1(1" p)#21#1. (1.20)

3see A.4 for a discussion of Dirichlet problems for the generator and their application to exit proba-bilities and exit times.


If the mutation rates depend only on the parent type, so $ij = $i for all j, G$&(p) =

0, may be solved for arbitrary K to yield a Dirichlet distribution [185],

$(0K

i=1 $i)1K

i=1 $($i)

K*

i=1

p#i#1i . (1.21)

Weak Selection

If selection is weak,

si ='i

N, (1.22)

then, proceeding as above, the limiting di!usion P (t) has infinitesimal generator

(Gf)(p) = "K$

i=1

#

"

#K$

j=1

$ij

%

pk +K$

j=1

$jipj + pi

#

'i "K$

j=1

'jpj

%%

(!if)(p)

+1

2

K$

i=1

K$

j=1

pi (%ij " pj) (!i!jf)(p). (1.23)

Here, however, fewer exact results are available, but we still have the following two results

when K = 2,

(i) Suppose µ12 = µ21 = 0. Then [99] (see also [186, 60]),,

P {P1(Tabs) = 1|P1(0) = p} =1" e#2($1#$2)p

1" e#2($1#$2)

= p+ ('1 " '2)p(1" p) +O(('1 " '2)2).

(1.24)

(ii) When mutation rates are non-zero, we have [186],

P

+

limt%"

P(t) = (p, 1" p)-

+ p#12#1(1" p)#21#1e($1#$2)p. (1.25)

Remark 1.2. The expressions given above are superficially di!erent from those which

commonly appear in population genetics, where the convention is to assume that '1 = '


and '2 = 0.

Strong Selection

In the strong selection regime, si = O(1), and P(N)i (t) no longer converges weakly to a

di!usion, but it can still be coupled4 over bounded time intervals to a di!usion process

P(N)i (t) with generator

G(N)f(p) = "K$

i=1

#

"

#K$

j=1

$ij

%

pk +K$

j=1

$jipj + pi

#

si "K$

j=1

sjpj

%%

(!if)(p)

+1

2N

K$

i=1

K$

j=1

pi (%ij " pj) (!i!jf)(p). (1.26)

such that for each fixed T > 0 there is a constant CT > 0 such that [118]

limN%"

Pp

2

supt&T

333P(N)(t)" P(N)(t)

333 >

CT lnN

N

4

= 0. (1.27)

While this does not necessarily justify the use of P (N)i (t) as an approximation to P

(N)i (t)

for all t & Tabs, P(N)i (t) can nonetheless be used to obtain a very accurate approximation

to the fixation probability when mutation rates are set to 0, which is, using the notation

above [99] (see also [186, 60]),

P {P1(Tabs) = 1|P1(0) = p} =1" e#2N(s1#s2)p

1" e#2N(s1#s2). (1.28)

In particular, assuming s1" s2 > 0, if p = 1N, as N #$ this approaches 1" e#2(s1#s2) =

2(s1 " s2) + O(

|s1 " s2|2)

. The latter estimate was previously obtained by Haldane

[71], who approximated the number of individuals of the novel type by a supercritical

Bienayme–Galton–Watson process, in which each in each generation, each individual

produces o!spring with numbers distributed according to a Poisson random variable

4see A.2 for a discussion of coupling


with rate 1 + (s1 " s2). In what follows, we will make use of such branching process

approximations, which will be rigorously justified using a coupling argument.

1.2.3 The Neutral Theory of Molecular Evolution and the Neu-

tralist/Selectionist Debate

Estimating substitution rates from hemoglobin, cytochrome c, and triosephosphate de-

hydrogenase, Kimura found them to be higher than would be plausible under selection,

prompting him to speculate that genetic diversity was overwhelmingly the result of sam-

pling e!ects among selectively neutral genes [100]. This hypothesis would be further de-

veloped (e.g., [107, 105]), and lead to the “neutralist/selectionist” debate as to whether

natural selection or neutral genetic drift is the primary driver of evolution. This debate

continues, as evidence has arisen for widespread selection on amino acids [138]. Further,

the neutral theory substantially overestimates the observed diversity (see [114] and ref-

erences therein), and evidence is lacking for the strong correlation between population

sizes and genetic diversity, as would be expected in a neutral population [131]. While

some hypotheses exist to explain the observed level of genetic diversity (e.g., the nearly

neutral theory [146] or genetic hitchhiking [69]), it remains an interesting an important

question to determine mechanisms that give rise to the observed, intermediate level of

diversity.

1.2.4 Moran’s Model

In [142], Moran introduced an alternative formulation for mathematical population ge-

netics, which, although largely neglected by biologists, is in many ways preferable to the

Wright-Fisher model for the purposes of mathematical analysis. Moran’s formulation is

a continuous time Markov chain, in which individuals of type i reproduce and replace


individuals of type j with intensity

##

1"$

j '=i

µij

%

"iX(N)i (t) +

$

j '=i

µji"jX(N)j (t)

%#

X(N)j (t)

N

%

, (1.29)

i.e., individuals of type i produce o!spring at rate "i, and these o!spring are of type j '= i

with probability µij. Each o!spring replaces an individual chosen uniformly at random

from the population, insuring the population size remains fixed at N .

In addition to being a more biologically appropriate model for e.g.,microorganisms,

which tend to have overlapping generations and reproduce by binary fission, the Moran

model has the advantage of being analytically tractable without recourse to the di!usion

approximation. For example, when K = 2 and µ12 = µ21 = 0, adopting the notation

from the previous section, we have [142]

P

+

X(N)1 (Tabs) = N

,,,X

(N)1 (0) = n

-

=1"5

%2

%1

6n

1"5

%2

%1

6N. (1.30)

N.B. For the (discrete) Moran model, we take the time to first absorption (i.e.,fixation

or extinction of type 1) to be

Tabs = inf+

t , 0 : X(N)1 (t) % {0, N}

-

. (1.31)

In the weak selection regime, the Moran model and the Wright-Fisher model are

asymptotically equivalent: if we let "i = 1+si and Ne =N2 , and assume that $ij = 2Neµij

and 'i = 2Nesi, then1NX(N)(Net) converges in distribution to P (t), the same di!usion

as for the Wright-Fisher model, with generator (1.12). Ne is referred to as the e"ective

population size for the Moran model. For appropriate choices of Ne, a wide array of

population genetic models converge to Kimura’s di!usion [141, 170].

We shall use Moran’s model as a point of departure for a modelling framework that in-


tegrates Kolmogorov’s approach to population dynamics with population genetics, mod-

elling allele frequencies via continuous time Markov processes. Unlike Moran, we decouple

birth and death events, but allow their rates to depend on the number and type of all

individuals composing the population.

1.2.5 Cannings’ Models and Neutrality

We conclude our brief survey of mathematical population genetics with Cannings’ models

[24]. There have been a number of di!erent notions of neutrality that have arisen in the

population genetic literature, usually in the context of di!usion limits, not all of which

are interchangeable, which has led to occasional confusion [162]. As we will have need of

a suitable null model in later chapters, we wish to fix one notion of neutrality. Perhaps

the most robust of these are the exchangeable models proposed by Cannings. Recall that

a random vector (v1, . . . , vN) is exchangeable if (v&(1), . . . , v&(N)) has the same distribution

as (v1, . . . , vN) for all permutations ( of {1, . . . , N} i.e., relabelling the components leaves

the underlying random vector unchanged. Suppose we have a haploid population of

N panmictic individuals with discrete generations. Label the individuals in the nth

generation by 1, . . . , N , and write vi(n) for the number of o!spring of the ith individual

in the n+ 1st generation; by assumption, vi(n) % N0 := {0, 1, 2, . . .} and

N$

i=1

vi(n) = N. (1.32)

for all n , 0. This describes a Cannings’ Model if (v1(n), . . . , vN(n)) is an exchangeable

random vector for all n, and, moreover, {(v1(n), . . . , vN(n)) : n % N0} are identically

and independently distributed. For example, the Wright-Fisher model is the Cannings’

model for which (v1(n), . . . , vN(n)) is multinomial distributed with N trials, each with

probability of success 1N.

All individuals in a Cannings’ model are e!ectively interchangeable, and if individuals


are assigned types, those types are merely labels, and have no e!ect on the change in

allele frequency. Cannings’ models thus provide a suitable mathematical definition for a

neutral population.

Suppose that we have a Cannings’ model with K types, and, as before, let X(N)i (n)

be the number of individuals of type i in the nth generation and let

X(N)(n) = (X(N)i (n), . . . , X(N)

i (n)). (1.33)

Let cN be the probability that two individuals independently sampled uniformly from

generation n+ 1 have the same parent in generation n, so

cN =E [v1(n)(v1(n)" 1)]

N " 1. (1.34)

Recently, Mohle [141] showed that, provided

E [v1(n)(v1(n)" 1)v2(n)(v2(n)" 1)]

N2cN# 0 (1.35)

as N #$, then X(N)57

tcN

86

converges to a di!usion satisfying Kimura’s generator

(Gf)(p) =1

2

K$

i=1

K$

j=1

pi (%ij " pj) (!i!jf)(p). (1.36)

In Chapter 4, we will follow Cannings’ example, and use exchangeable processes as a

point of departure for analyzing processes in the weak selection regime.

1.3 Outline of Thesis

Our goal is the development of a mathematical formalism that is su"ciently simple so

as to be analytically tractable, and amenable to the exact computation of fixation prob-


abilities, fixation times, and stationary allele frequencies, while remaining rich enough

to encompass diverse ecological dynamics. This thesis is an e!ort in this direction. In

particular, we will develop a class of models which we call density-dependent birth-death-

mutation processes. such that

• selection is at the level of individuals that reproduce and die, rather than abstracted

genes,

• rather than imposing fixed, deterministically varying, or exogenously determined

population sizes or compositions, we will insist that the total population number

varies with the population dynamics, in the same stochastic events that cause allele

numbers to vary,

• populations will be regulated by density dependent competition for limited re-

sources rather than being “sampled” by unspecified external mechanisms.

• rather than assuming a priori selection coe"cients (constant or time-varying), we

allow the selection coe"cients to emerge mechanistically from the population dy-

namics, and seek to interpret them in terms of life-history strategies.

The lack of fixed population sizes is a particularly important feature of the pop-

ulation models considered in this thesis. At the risk of stating the obvious, natural

populations are not fixed, but are subject to internally generated demographic stochas-

ticity and exogenous environmental stochasticity. There has been long and sustained

interest in determining the e!ects of variable population size on the probability of allele

fixation (e.g., [59, 111, 52, 94, 101, 106, 147, 154, 160, 23, 121, 122, 153, 45, 124], among

many others). Here, we provide a general framework for investigating the e!ects of de-

mographic stochasticity which allows us to consider diverse community dynamics, and

thus provides a first step towards the generalized theory of coevolution that incorporates

complex, multi-trophic interactions that some have called community genetics [6, 7, 143].


We take philosophical guidance from Nisbet and Gurney [144], and focus on obtaining

a robust approach to demographic stochasticity, into which environmental stochasticity

can later be incorporated. While it is beyond the scope of this thesis, we propose that the

problem of intermediate diversity that drives some of the neutralist/selectionist debate

might in part be explained by ecological mechanisms of coexistence, that arise when mul-

tiple species interact with nonlinear birth and death rates (e.g., [9, 139, 10]) and varying

population sizes. This is a question to which we intend to return; the results here are a

foundation for that e!ort.

Density-dependent birth-death-mutation processes are a hybrid of multi-type birth

and death processes (so that rates are proportional to the number of individuals; we do

not, however, assume binary reproduction) and the density-dependent population pro-

cesses considered by Thomas Kurtz ([116, 117, 118, 119], see also [50]), modified to allow

weak mutation. In those papers, Kurtz derived a functional law of large numbers and

central limit theorem valid for times less than a fixed T that is independent of N . Un-

fortunately, Kurtz’s results on density dependent population processes are not directly

useful in finding the probability of the extinction/fixation events that drive adaptive evo-

lution. In what follows, we identify and prove convergence for the limits and appropriate

to population genetics.

As in classical population genetics, we will derive our results as a parameter, N ,

tends to infinity. Unlike the classical theory, N is not a number of individuals at which

the population is fixed, but rather an analogue of the carrying capacity in the logistic

equation,

X = rX

!

1"X

N

"

, (1.37)

which measures the strength of interaction between individuals and is proportional to the

equilibrium population size, where net per-capita growth rates drop to zero. This system

size, N , will be interpreted slightly di!erently in di!erent models. In a Gause-Lotka-

Volterra model, it is an abstract measure of the number of individuals an environment


can sustain. In an explicit resource model e.g., of a chemostat, it might measure the

volume of the reaction vessel. In the epidemic models contained herein, it is proportional

to the number of uninfected hosts. It is important to realize that we do not require that

the total number of individuals of all types sum toN . Instead, we will consider individuals

to be common or rare when their numbers are O(N), or asymptotically smaller than N ,

respectively, and allow those numbers to vary stochastically.

In Chapter 2, we formally define density-dependent birth-death-mutation processes

and prove a functional law of large numbers. This tells us that in the limit of large

N , common types (i.e., those with initial numbers proportional to N) behave essentially

deterministically – if there are su"ciently many individuals of a given type, the fluctua-

tions are e!ectively “averaged out”. More precisely, there exists a dynamical system of

Kolmogorov type,

Y = F(Y) (1.38)

such that for times t < C lnN , for a constant C depending on the vector field F, the

number of individuals of type i is NYi(t) +O(N r) for some 12 < r < 1.

We are thus left with considering two situations: as in classical population genetics,

we will be concerned with the ability of rare types (i.e., those with initial numbers - N

to become common. Further, we will be interested in the long-term dynamics, when our

functional law of large numbers no longer applies. We note that for N large, Y will come

arbitrarily close to its asymptotic attractor, which will be integral to understanding the

stochastic dynamics. Unfortunately, disparate topological classes will have to be analyzed

by qualitatively di!erent means, and Smale’s construction shows that this analysis can

never be exhaustive. Instead, we focus on two cases which are most naturally analogous

to the strong and weak selection regimes of population genetics, and which encompass

many results of classical population genetics, which we may obtain as simple corollaries.

More importantly, our approach leads to many novel results.


Chapter 3 discusses fixation times and fixation probabilities in the strong selection

regime. Here, this is equivalent to Y = F(Y) having a unique, globally stable axial

fixed point, which without loss of generality we take to be x",1 = x",11 e1. Under this

assumption, without mutation, if the number of individuals of the advantageous type 1

reaches )N , i.e., if type 1 becomes common, then it will eventually fix with probability

approaching 1 as N #$. (more precisely, the probability that type 1 fails to fix decays

exponentially in N). We thus consider the problem of fixation of a small number of novel

individuals (usually, but not necessarily, assumed to be one) invading a community of

resident species which are assumed much more numerous. We compute the probability of

hitting )N , which we show is a function of the average birth rate and death rate of type

1, into which all other types only enter via the functional Law of Large Numbers. We also

derive an expression the expected time to fixation for type 1. Analogously to Haldane’s

branching process approximation [71], these results are obtained using a coupling to a

time-inhomogeneous Markov branching processes; the asymptotic probability of hitting

)N is then the probability that the coupled branching process grows indefinitely.

In Chapter 4 we consider a generalization of the weak-selection regime, which here

is the degenerate case when the set of interior fixed points of Y = F(Y) is a is globally

asymptotically stable embedded submanifold of RK+ (recall, K is the number of distinct

types). We show that suitably modified and rescaled, the density of types converges in

distribution (i.e., the corresponding probability measures on Skorokhod space DRK+[0,$)5

converge in the weak topology) to a di!usion on that manifold. We show that the di!usion

limit we obtain reduces to Kimura’s di!usion when the initial birth-death-mutation pro-

cess is exchangeable, and consider a number of departures from exchangeability in which

a di!usion limit is still available. These encompass, generalize, and provide rigorous justi-

fication for a number of previously considered population genetic models (e.g.,Gillespie’s

models of fecundity variance polymorphism [66, 67, 68, 61]) while providing an explicit

5see A.3 for the definition of Skorokhod space D.


population genetic formulation of more ecological notions such as r vs. K selection

[132, 155, 156] and resource ratio theory [133, 178, 179]. We then consider what we call

the Generalized Moran model, which we use to consider a fecundity-mortality tradeo!s

as a simple example of a life-history strategy. When the number of species, K, is equal

to two, we use the di!usion approximation to the Generalized Moran model to obtain

analytical expressions for the fixation probabilities and expected time to fixation, and if

mutation is allowed, the quasi-stationary distribution of allele frequencies. Using these,

we find selection favouring reduced fecundity, and interestingly, that the stationary allele

frequency distribution is indistinguishable from that for the Wright-Fisher model, so that

inference techniques based upon the allele frequency distribution may not be adequate

to distinguish between di!erent hypotheses.

Chapter 5 demonstrates the versatility of the methods developed in Chapter 3 and

Chapter 4, by applying them to an example of epidemiological interest, the evolution

of virulence in the SIR (Susceptible-Infective-Recovered) model with demography. We

consider a two-strain epidemic, where the pathogens convey cross-immunity and thus

compete for hosts. We consider the fixation probability for a novel strain invading a

population of hosts in which a “wild-type” strain has reached endemic equilibrium, and

show this can be simply expressed as the ratio of the two strains’ basic reproductive

numbers. The basic reproductive number, the expected number of new infections caused

by a single infected individual entering a population of susceptibles, has long been used

in mathematical epidemiology to characterizes the deterministic evolutionary dynamics

of pathogens [127, 2, 3, 20, 136, 110, 137]). Finally, we look at the weakly selected two-

strain epidemic, which provides an interesting co-dimension two application of the results

of Chapter 4, and also suggests that stochasticity in viral evolution may give new life to

the currently out-of-favour notion of selection for reduced virulence.

Finally, we conclude with a discussion of future work.

Chapter 2

Density Dependent

Birth-Death-Mutation Processes

In this chapter, we introduce the primary object of this study, density dependent birth-

death mutation processes. These are a class of continuous time Markov jump processes

generalizing the models considered by Wright, Fisher and Moran, such that their func-

tional Law of Large Numbers gives rise to a competitive Kolmogorov dynamical system.

We begin by recalling a broader class of processes, the density dependent population

processes that were introduced by T. G. Kurtz ([116, 117, 118, 119], see also [50]) to de-

scribe families of jump Markov processes arising in chemistry, population dynamics and

population genetics. We introduce a slight generalization of Kurtz’s density dependent

population processes that allows us to consider O(

1N

)

e!ects like mutation. We then

derive a simple extension of the functional Law of Large Numbers for these processes,

Proposition 2.1, which shows that, provided the initial number of individuals of type i is

O(N), then the rescaled process N#1X(N)i (t) is approximately deterministic. This was

originally proved by Kurtz [116] for times t % [0, T ] for a fixed T > 0, independent of N .

Because we wish to consider extinction and fixation, we require a result valid for times

that scale with N , and an estimate of the error in our approximation, hence Proposition

25

Chapter 2. Birth-Death-Mutation Processes 26

2.1, which uses a di!erent method of proof from [116].

This result shapes our overall strategy: from it, we know that once the population

exceeds )N for any ) > 0 independent of N , it can be approximated by a deterministic

process, for a time span su"ciently long that both the original process and the functional

Law of Large Numbers approximation are in a o(1)-neighbourhood of the attractor of the

deterministic dynamics. Thus, we need only answer two questions: what are the dynamics

when the number of individuals is o(N), and, what happens when the stochastic process

is started near the asymptotic attractor. Chapter 3 addresses the former question.

Unfortunately, Smale’s construction (Example 1.1) shows that a complete answer

to the latter question is impossible, as essentially arbitrary asymptotic dynamics are

possible. We thus proceed by identifying two asymptotic regimes, the strong and weak

selection regimes that generalize the eponymous regimes from classical population genet-

ics. In the strong selection regime, there is a single type that is guaranteed to fix in the

deterministic limit; in this case, fixation is equivalent to the selected type reaching )N

before going extinct, and thus is completely understood by the results in Chapter 3. The

weak selection regime corresponds to the existence of a manifold of fixed points at which

deterministic coexistence is possible, and is the subject of Chapter 4.

Finally, we conclude the chapter by considering su"cient conditions for being in these

two regimes in the Lotka-Volterra-Gause equations.

2.1 Density Dependent Population Processes

Consider a population consisting of a finite number of discrete individuals. Each indi-

vidual has a type, which is indexed by an integer in the set {1, . . . , K}. All individuals

of a given type are identical.


Denote the number of individuals of type i % {1, . . . , K} at time t by X(N)i (t) and let

X(N)(t) =5

X(N)1 (t), . . . , X(N)

K (t)6

. (2.1)

Definition 2.1. Let {*(N)n (x)}n(ZK be a collection of non-negative functions defined on

a subset E . RK . Set

E(N) := E / {N#1k : k % ZK}, (2.2)

and assume that x % E(N) and *(N)n (x) > 0 imply x + N#1n % E(N). The density

dependent family corresponding to the *(N)n (x) is a sequence {X(N)} of jump Markov

processes such that X(N) has state space E(N) and intensities

q(N)x,x! = N*

(N)N(x!#x)(x), x,x) % E(N). (2.3)

Remark 2.1. As we discussed in the Introduction, the index N , sometimes referred to

as the “system size”, plays the role of the fixed population size N in the Wright-Fisher

and Moran models, and of the carrying capacity N in Verhulst’s logistic equation (1.1).

It should not, however, be thought of as a fixed population size – we allow the total

number of individuals to vary stochastically, but, under our assumption of density de-

pendence, after a transient phase, the total number of individuals will fluctuate about

an “asymptotic equilibrium” value that is proportional to N .

Remark 2.2. The class of density dependent population processes includes examples

which would not be considered density dependent in the ecologist’s sense. For exam-

ple, the simple birth-death process with rates

q(N)X,X+1 = "X, q

(N)X,X#1 = %X (2.4)


corresponds to the density dependent family

*(N)1 (x) = "x, *

(N)#1 (x) = %x. (2.5)

However, if " > %, then this process will grow without bound with probability '%[96]. To

exclude this, and similar examples, we introduce a narrower class of density dependent

birth-death-mutation processes in the next section. Nonetheless, whenever possible, we

will prove results at the level of generality of density dependent population processes,

specializing to the latter class as necessary to obtain more refined results.

2.1.1 Stochastic Di!erential Equation Representation

Henceforth, we adopt the following assumptions

Assumption 2.1. For all compact sets K 0 RK+ ,

$

n(ZK

1n12 supx(K

*(N)n (x) <$ and

$

n(ZK

1n122 supx(K

*(N)n (x) <$. (2.6)

Assumption 2.2. There exist functions {*n(x)}n(ZK such that for all compact sets K 0

RK+ ,

limN%"

$

n(ZK

1n12 supx(K

,,*(N)

n (x)" *n(x),, = 0 (2.7)

and

limN%"

N$

n(ZK

1n122 supx(K

,,*(N)

n (x)" *n(x),, <$. (2.8)

Remark 2.3. Assumption 2.1 guarantees that the jumps in X(N)(t) have finite mean and

variance for all N . For our purposes, this will mean that the total number of o!spring

per individual, per reproductive event, has finite mean and variance – an eminently

plausible assumption for natural populations (although, see e.g., [42] for a discussion of

the biological applicability of models in which variance in individual reproductive output


tends to infinity with population size).

Remark 2.4. Assumption 2.2, together with the functional form of the transition rates

*(N)(x), Equation 2.3, combine to give us a notion of density dependence: like the logistic

equation, (1.1), the number of individuals of each type only appear in the rates rescaled

by the system size N – we could have equally well expressed the rates in terms of the

rescaled process Y(N)(t) = N#1X(N)(t). Assumption 2.2 requires that those transition

rates approach a finite limit as N #$; in particular, we require that the rates converge

in such a way that the mean and variance of the jump sizes also converge to finite limits,

which is a minimal requirement for a biologically meaningful limit. As we saw previously,

this notion of density-dependence is weaker than that of the logistic equation, as it does

not guarantee that populations approach a finite equilibrium density. We will discuss the

appropriate additional assumptions to guarantee finiteness below.

We then have the following representation of the process X(N): let H(N)n (t) be a

counting process1 with intensity *(N)n (X(N)(t)) , 0, and compensated process

H(N)n (t) = H(N)

n (t)"9 t

0

*(N)n (X(N)(s")) ds (2.9)

is a square-integrable local martingale with quadratic variation process2

[H(N)n ](t) = H(N)

n (t). (2.10)

We assume that the processes H(N)n (t) are orthogonal, so

[H(N)n , H(N)

m ](t) = 0 for n '= m. (2.11)

1see A.8 for a discusssion of counting processes.2see A.6 for the definition of the quadratic variation and covariation processes


We can then write

X(N)(t) = X(N)(0) +$

n(ZK

H(N)n (t), (2.12)

It will be convenient to decompose the semimartingale X(N)(t) into finite-variation

and local martingale components3,

X(N)(t) = X(N)(0) +

9 t

0+

NF(N)5

X(N)(s)N

6

ds+M(N)(t) (2.13)

where

F(N)(x) =$

n(ZK

n*(N)n (x), (2.14)

and

M(N)(t) =$

n(ZK

nH(N)n (t). (2.15)

Under Assumption 2.1, M(N)(t) is a square-integrable local martingale with tensor

quadratic variation4 [140]

!M(N)"(t) =$

n(ZK

nn*H(N)n (t) (2.16)

and Meyer process

22M(N)33(t) =9 t

0+

$

n(ZK

nn**(N)n (X(N)(s)) ds. (2.17)

2.1.2 Law of Large Numbers

Let

F(x) =$

n(ZK

n*n(x). (2.18)

3see A.7 for a discussion of semimartingales and the Doob-Meyer decomposition4see A.7.


As a consequence of Assumptions 2.1 and 2.2, F(N) # F uniformly on compact sets. We

will assume that F is C3 and thus locally Lipschitz.

We will henceforth let +t denote the flow of

Y(t) = F(Y(t)), (2.19)

and set

Y(N)(t) := 1NX(N)(t). (2.20)

Notation 2.1. Here, and in subsequent chapters, we will be interested in initial condi-

tions X(N)i (0) such that

limN%"

1

NX

(N)i (0) = x (2.21)

for some fixed x % RK+ . We thus adopt the notation

Px {·} = P

2

·,,,,lim

N%"

1

NX

(N)i (0) = x

4

, (2.22)

and

Ex [·] = E

:

·,,,,lim

N%"

1

NX

(N)i (0) = x

;

. (2.23)

Let x %!

RK+ . By assumption, +tx is eventually in a compact set, so we may choose a

compact set Kx 0!

RK+ such that the forward orbit of x, ,+

x := {+tx : t , 0} is contained

in the interior of Kx,!Kx. We may thus fix ) > 0 such that

Kx,((t) := {x) : 1x) " +tx1 < )} (2.24)

and

Kx,( :=<

t+0

Kx,((t) (2.25)

are contained in Kx. Since F is locally Lipschitz, there exists a constant Lx > 0 such


that

1F(x1)" F(x2)1 < Lx 1x1 " x21 (2.26)

for all x1,x2 % Kx.

In what follows, we will make extensive use of Kx and the tubular )-neighbourhood

of the deterministic trajectory ,+x , Kx,(. Our arguments in this and subsequent chapters

will hinge upon the fact that, provided that N is su"ciently large, the rescaled process

N#1X(N)(t) will remain inside Kx,( with probability approaching 1 as N #$.

To be precise, we have the following functional Law of Large Numbers for Y(N)(t),

which states that for times of order O(lnN), N#1X(N)(t) may be approximated arbitrar-

ily well by the deterministic flow +tx.

Proposition 2.1. Fix 0 < r < s < 1 and assume that E=33Y(N)(0)" x

33>

= O(N#1).

Then,

(i) for N su!ciently large,

Px

?

@

Asup

t& (1"s)

4L2x

lnN

33Y(N)(t)" +tx

332> N#r

B

C

D< N#r, (2.27)

(ii) and, for all fixed T > 0,

limN%"

supt&T

33Y(N)(t)" +tx

33 = 0 Px " a.s. (2.28)

Remark 2.5. In [119], (ii) is proved under the assumptionsY(N)(0)# x, *(N)n (x) 4 *n(x),

and

supx(K

$

n(ZK

1n12 *(N)n (x) <$. (2.29)


Unfortunately, that proof relies on the compactness of [0, T ], whereas we shall require

estimates when t is O(lnN) throughout the next two chapters.

Proof. As above, we may write Y(N)(t) as

Y(N)(t) = Y(N)(0) +

9 t

0+

F(N)(Y(N)(s)) ds+1

NM(N)(t). (2.30)

Define a stopping time

T (N)( := inf{t , 0 :

33Y(N)(t)" +tx

33 > )}, (2.31)

so T(N)( is the first time that the process exits Kx,(. We will consider the stopped process

Y(N)(t 5 T(N)( ) (recall, x 5 y := max{x, y}):

Y(N)(t 5 T (N)( )" +

t,T (N)!

x = Y(N)(0)" x+

9 t,T (N)!

0+

F(Y(N)(s))" F(+sx) ds

+

9 t,T (N)!

0+

F(N)(Y(N)(s))" F(Y(N)(s)) ds+1

NM(N)(t 5 T (N)

( ). (2.32)

Applying the Cauchy–Bunyakovsky–Schwarz inequality gives

333Y(N)(t 5 T (N)

( )" +t,T (N)

!x333

2

& 4

#

33Y(N)(0)" x

332+

33333

9 t,T (N)!

0+


33333

2

+

33333

9 t,T (N)!

0+

F(N)(Y(N)(s))" F(Y(N)(s)) ds

33333

2

+1

N2

33M(N)(t 5 T (N)

( )332

%

. (2.33)

We consider each of the terms individually:

1. By assumption, we have E

E33Y(N)(0)" x

332F

< CN

for some constant C.


2. Applying Jensen’s inequality gives

supt&T

33333

9 t,T (N)!

0+


33333

2

& L2xT

9 T,T (N)!

0+

33Y(N)(s))" +sx

332ds

(2.34)

and

supt&T

33333

9 t,T (N)!

0+

F(N)(Y(N)(s))" F(Y(N)(s)) ds

33333

2

& T

9 T,T (N)!

0+

33F(N)(Y(N)(s))" F(Y(N)(s))

332ds, (2.35)

whilst by assumption, we have33F(N)(x)" F(x)

33 = O

(1N

)

uniformly on Kx.

3. Lastly, Doob’s inequality gives

E

:

supt&T

33M(N)(t 5 T (N)

( )332;

& 4E=

2M(N)3(T 5 T (N)( )>

& N

9 T,T (N)!

0+

$

n(ZK

1n12 E=

*(N)n (Y(N)(t))

>

ds, (2.36)

and again, by assumption,0

n(ZK 1n12 EE

*(N)n (Y(N)(t))

F

is uniformly bounded on

Kx.

Combining these, we have

E

:

supt&T

333Y(N)(t 5 T (N)

( )" +t,T (N)

!x333

2;

&4C +Bx,(T

N+ 4L2

xT

9 T,T (N)!

0

E

E33Y(N)(s)" +sx

332F

ds

(2.37)


for some constant Bx,(. Applying Gronwall’s inequality, we have

E

:

supt&T

333Y(N)(t 5 T (N)

( )" +t,T (N)

!x333

2;

&4C +Bx,(T

Ne4L

2xT , (2.38)

from which, taking T = (1#s)4L2

x

lnN , we obtain

supt& (1"s)

4L2x

lnN

Ex

333Y(N)(t 5 T (N)

( )" +t,T (N)

!x333

2&

4C + (1" s)Bx,(

4L2x

(lnN)N#s < N#r

(2.39)

for N su"ciently large.

To complete the proof of (i), we observe

Px

G

T (N)( < t

H

= Px

2333Y(N)(t 5 T (N)

( )" +t,T (N)

!x333

2, )24

(2.40)

&Ex

:333Y(N)(t 5 T

(N)( )" +

t,T (N)!

x333

2;

)2(2.41)

<4C + (1" s)Bx,(

4L2x)

2(lnN)N#s < N#r (2.42)

for N su"ciently large.

To prove (ii), write (%,F ,P) for the underlying probability space5, and suppose that

supt&T

33Y(N)(-, t)" +tx

33 '# 0 for some - % %. Then, there exists % > 0 and Nk # $

such that

supt&T

33Y(Nk)(-, t)" +tx

33 > % for all k. (2.43)

Without loss of generality, we may assume that for all k, N#rk < % and T & (1#s)

4L2x

lnNk.

Thus

- %"I

k=1

{- :33Y(Nk)(-, t)" +tx

33 > N#r

k }. (2.44)

By (i), this set has measure 0.

5see Appendix A for a discussion of the probability triple (",F ,P).


2.2 Density-dependent Birth-Death-Mutation Pro-

cesses

Here we define to the principal object of interest, a subset of the density dependent birth

death processes that we will call density dependent birth-death-mutation processes. As

in the previous section, X(N)i (t), i = 1, . . . , K is the number of individuals of type i. The

number of individuals of each type changes when there is a

• birth/mutation event in which some individual of type i produces a collection con-

sisting of |n| := n1+ · · ·+nK o!spring, of which nj % N0 are of type j, which occur

at a per capita rate

"(N)i,n

5X(N)(t)

N

6

(2.45)

or

• the death of an individual of type i, which occurs at per capita rate

%(N)i

5X(N)(t)

N

6

(2.46)

As before, the Markov chain X(N)(t) may be represented as

X(N)(t) = X(N)(0) +K$

i=1

$

n(NK0

nB(N)i,n (t)"

K$

i=1

eiD(N)i (t) (2.47)

where B(N)i,n (t) and D

(N)i (t) are counting processes with intensities "

(N)i,n

5X(N)(t)

N

6

X(N)i (t)

and %(N)i

5X(N)(t)

N

6

X(N)i (t), respectively. As before, we write B

(N)i,n and D

(N)i for the cor-

responding compensated processes (see Equation 2.9).

A density dependent birth-death-mutation process corresponds to a density dependent

family

*(N)n (x) =

K$

i=1

"(N)i,n (x)xi, *

(N)#ei = %(N)(x)xi, (2.48)


for which we continue to assume Assumption 2.1. We replace Assumption 2.2 by the

stronger

Assumption 2.3. As in classical population genetics, we will assume that the rate of

mutation is O(

1N

)

. In our context, this is equivalent to the existence a family of functions

"i,n(x) =

?

JJ@

JJA

"i,n(x) if n = nei

0 otherwise

(2.49)

such that for all compact sets K 0 RK+ ,

limN%"

$

n(ZK

1n12 supx(K

,,,"

(N)i,n (x)" "i,n(x)

,,, = 0 (2.50)

and

limN%"

N$

n(ZK

1n122 supx(K

,,,"

(N)i,n (x)" "i,n(x)

,,, <$. (2.51)

Thus, the rates at which an individual of type i produces one or more o!spring of some

type j '= i, "(N)i,n , n '= nei, are always bounded above by some constant multiple of 1

Non

any compact set.

We also assume that there exists %i(x) such that

limN%"

supx(K

,,,%

(N)i (x)" %i(x)

,,, = 0 and lim

N%"N sup

x(K

,,,%

(N)i (x)" %i(x)

,,, <$. (2.52)

Remark 2.6. While Assumption 2.3 is made for technical reasons, the conditions imposed

are eminently plausible from a biological perspective, as they simply require that the

mean and variance in the number of o!spring produced in any single reproductive event

are both finite. Both are readily satisfied by assuming some fixed maximum on the total

number of o!spring per reproduction event that is independent of N .


Under Assumptions 2.1 and 2.3, the functional Law of Large Numbers, Proposition

2.1 holds. Setting

"i(x) :="$

i=1

n"in(x) (2.53)

and

fi(x) := "i(x)" %i(x), (2.54)

so fi is the average per capita net reproductive rate, we see that the limiting process

Y(t) satisfies

Yi = fi(Y)Yi, (2.55)

i.e., the corresponding dynamical system, Y = F(Y) is of Kolmogorov type.

Definition 2.2. We will say that the birth-death-mutation process is density-dependent if

this dynamical system corresponding to F is dissipative, and competitive if the dynamical

system is competitive in Kolmogorov’s sense (Definition 1.5).

In the next two chapters, we will repeatedly revisit the following example of a density-

dependent birth-death-mutation process, which we call the Generalized Moran Model

[148, 149, 150, 151]. The Generalized Moran model is a hybrid of the Lotka-Volterra-

Gause and Moran models. It was given a heuristic analysis in those papers, and is the

inspiration and motivation for the more general and rigorous results derived here. It

provides a concrete instance of a density-dependent birth-death-mutation process which

is simultaneously rich enough to provide some biological insights, simple enough to be

analytically tractable (at least when K = 2) and open to intuitive interpretation.

Example 2.1 (Generalized Moran Model). Consider the density-dependent birth-death-


mutation process with intensities

"(N)i,n (x) =

?

JJJJJJ@

JJJJJJA

5

1"0K

j=1#ijN

6

"i if n = ei

#ijN"i if n = ej

0 otherwise

(2.56)

corresponding to reproduction by binary fission, and death rate

%(N)i (x) = %i

#

1 +K$

j=1

xj

%

. (2.57)

for "i, %i > 0.

The corresponding functional Law of Large Numbers is of Lotka-Volterra-Gause type

and competitive:

fi(x) = ("i " %i)"K$

j=1

%ixj . (2.58)

Remark 2.7. Although our primary interpretation of the Generalized Moran model will

be interspecies competition in an environment with a carrying capacity proportional to

N , an analogous model has been used to analyze the dynamics of an epidemic passing

through a population of fixed size N , in which recovery does not entail immunity, the

SIS (Susceptible-Infective-Susceptible) model (see e.g., [5, 4] and references therein). We

will not discuss it extensively, but our results for the Generalized Moran model have an

alternative interpretation regarding evolution in multi-strain epidemics. However, see

Chapter 5 for analysis and discussion of the closely related stochastic SIR (Susceptible-

Infective-Recovered) model.

2.2.1 Individual Fecundity

We conclude this section with a few simple results regarding individual fecundity, which

will be useful in subsequent chapters to compare our results to classical population genetic


models. In particular, we will be interested in individual lifetime expected reproduction,

or fitness, and the variance in total individual fecundity.

Let L(N)

i,X(N)(0)be the life span of an individual of type i, born at time t = 0 into a

population in state X(N)(0), so

P

+

L(N)

i,X(N)(0), t-

= EX(N)(0)

E

e#! t0 '

(N)i (X(N)(s#)) ds

F

(2.59)

and let

O(N)i,X (t) =

$

n(NK0

1n11 Pn

#9 t,L(N)

i,X

0

"(N)i,n (X(N)(s")) ds

%

(2.60)

be the total reproductive output of that individual up to time t.

We begin with a simple lemma:

Lemma 2.1. Let (S, d) be a Polish space and let C(S) be the space of continuous

real-valued functions on S, endowed with the compact uniform topology. Suppose that

F (N) D"# F in C(S) and that X(N) D"# X in DS[0,$). Then F (N) 6X(N) D"# F 6X in

DR[0,$).

Proof. Define & : C(S)( DS[0,$) "# DR[0,$) by &(f, x) = f 6 x. By the continuous

mapping theorem [17], it su"ces to show that & is continuous at all pairs (f, x). To that

end, suppose that fn # f uniformly on compact sets, and that xn # x in the Skorokhod

topology. Thus, there exist *n # id such that ess supt+0 |ln*)n(t)| <$ and

limn%"

supt&T

d(xn(t), (x 6 *n)(t)) = 0 (2.61)

for all fixed T > 0. Moreover, {xn} . DS[0,$) is compact, so there exist a compact set


$T . S such that xn(t), (x 6 *n)(t) % $T for all t & T 5 supn *n(T ). Then

supt&T

|(fn 6 xm)(t)" (f 6 x)(*n(t))|

& supt&T

|fn(xn(t))" f(xn(t))|+ supt&T

|f(xn(t))" f((x 6 *n)(t))| . (2.62)

Now, fn # f uniformly on $T , so the former quantity vanishes as n # $, whilst f , as

the limit of a sequence of functions converging uniformly on $T , is uniformly continuous

on $T , and thus the latter term above vanishes as n#$ also. Thus fn 6 xn # f 6 x in

DR[0,$), and the result follows.

In particular, "(N)i,n (x) and %

(N)i (x) converge in the compact uniform topology, so

"(N)i,n (Y(N))

D"#

?

JJ@

JJA

"i,n(Y) if n = nei,

0 otherwise

(2.63)

and %(N)i (Y(N))

D"# %i(Y) as N #$. Moreover, F : DR[0,$)# DR[0,$) given by

F (x)(t) =

9 t

0

xn(s") ds (2.64)

is continuous, so9 ·

0

%(N)i (Y(N)(s")) ds D"#

9 ·

0

%i(Y(s")) ds (2.65)

etc. We thus have

Proposition 2.2. Suppose 1NX(N)(0)# x. Then

(i) L(N)

i,X(N)(0)

D"# Li,x, where

P {Li,x , t} = e#! t0 'i()sx) ds. (2.66)


(ii) O(N)

i,X(N)(0)(t)

D"# Oi,x(t) :=0"

n=1 nPnei

5K t,Li,x

0 "i,nei(+sx) ds6

, and

(iii) O(N)

i,X(N)(0)(L(N)

i,X(N)(0))

D"# Oi,x(Li,x).

Proof. The first follows immediately from the Portmanteau theorem, since, by the defi-

nition of weak convergence,

EX(N)(0)

E

e#! t0 '

(N)i (Y(N)(s#)) ds

F

# Ex

E

e#! t0 'i(Y(s)) ds

F

. (2.67)

The latter two follow from standard results on random time change [50].

Computing the expectation and variance of Oi,x(Li,x) using the Law of Total Expec-

tation and the Law of Total Variance, e.g.,

E

."$

n=0

nPn

!9 Li,x

0

"i,n(+sx) ds

"/

= E

."$

n=0

nPn

!9 Li,x

0

"i,n(+sx) ds

"/

(2.68)

= E

.

E

."$

n=0

nPn

!9 Li,x

0

"i,n(+sx) ds

",,,,,Li,x

//

(2.69)

= E

:9 Li,x

0

"i(+sx) ds

;

(2.70)

=

9 "

0

e#! t0 'i()sx) ds"i(+tx) dt, (2.71)

we thus have the following asymptotic results on expected lifetime reproductive fitness

and individual fecundity variance:

Corollary 2.1. (i) limN%"

E

E

O(N)

i,X(N)(0)(L(N)

i,X(N)(0))F

=

9 "

0

e#! t0 'i()sx) ds"i(+tx) dt, and

limN%"

Var5

O(N)

i,X(N)(0)(L(N)

i,X(N)(0))6

=

9 "

0

e#! t0 'i()sx) ds"i(+tx) dt

+2

9 "

0

e#! t0 'i()sx) ds"i(+tx)

!9 t

0

"i(+sx) ds

"

dt"!9 "

0

e#! t0 'i()sx) ds"i(+tx) dt

"2

.

(2.72)


(ii) In particular, if x = x" is a rest point for F, then

limN%"

E

E

O(N)

i,X(N)(0)(L(N)

i,X(N)(0))F

="i(x")

%i(x")= 1 (2.73)

and

limN%"

Var5

O(N)

i,X(N)(0)(L(N)

i,X(N)(0))6

="i(x")

%i(x")+

!"i(x")

%i(x")

"2

="i(x")

"i(x")+ 1 (2.74)

2.3 Strong and Weak Selection

In light of Smale’s construction (Example 1.1), if there are three or more species, es-

sentially any dynamics can arise as the Law of Large Numbers for a density-dependent

population process, so attempting to obtain a complete understanding of such processes

is a fool’s errand. Nonetheless, it is possible to identify special cases, generalizing the

weak and strong selection regimes of classical population genetics, in which something

can be said.

Definition 2.3. A competitive density-dependent birth-death-mutation process is in the

strong selection regime if the dynamical system arising as the Law of Large Numbers for

the process has a unique globally attracting axial fixed point. It is in the weak selection

regime if the --limit set for the Law of Large Numbers is an attracting embedded co-

dimension one submanifold, C of RK+ .

Example 2.2. For the Generalized Moran model, Example 2.1, type 1 is strongly selected

if f %1

'1>

%j

'jfor all j > 1, in which case x",1 =

5%1

'1" 16

e1 is the globally asymptotically

stable axial fixed point. Weak selection corresponds to case when %i

'i= %j

'jfor all i, j, in

which case every point on

C =

&

x % RK+ :

K$

i=1

xi = 1

'

(2.75)


is a fixed point for the Law of Large Numbers (2.58), corresponding to coexistence of

all types. Figure 2.1 shows deterministic flows and sample paths for the weakly selected

Generalized Moran model.

Conditions for global stability in the strong and weak selection regimes for the Gen-

eralized Moran model follow from Theorem 2.1 and Proposition 2.3 below, respectively.

The strong selection corresponds to the case in which one type (without loss of gen-

erality, type 1) will fix under the deterministic dynamics given by the Law of Large

numbers, whereas the weak selection case corresponds to deterministic coexistence. In

the former case, the problem of determining fixation probabilities and times is thus re-

duced to determining the probability that, starting from small numbers (i.e.,O(1)), the

number of individuals of type 1 will reach )N for some ) > 0 prior to extinction; once the

number of type 1 individuals is O(N), the dynamics are essentially deterministic, and

fixation is guaranteed. In the weak selection regime, assuming that the population does

not vanish at small numbers due to demographic stochasticity, there is an approximately

deterministic transient phase, at the end of which all species will be present (determinis-

tic coexistence). After this, there is a much longer period, during which stochasticity in

births and deaths causes the densities of each type to vary. On this longer time-scale, the

dynamics are asymptotically given by a di!usion on the set of deterministic equilibria.

Given a general process with corresponding Kolmogorov equations Yi = fi(Y)Yi, there

is no general criterion on the fi that guarantee that one is in the strong or weak selection

regimes. Even when the Law of Large Numbers is of Lotka-Volterra-Gause type, finding

necessary and su"cient conditions for the existence of a unique globally attracting fixed

point remains an open problem [187, 188]. For the strong selection case, however, we

have the following su"cient (but not necessary and su"cient, see [187]) conditions:

Theorem 2.1 (Zeeman [188]). Consider a community of K mutually competing species


Figure 2.1: Sample paths and flows of the Law of Large Numbers for the Generalized Moranmodel, Example 2.1, under weak selection. Flow lines were generated using (2.58) for parametervalues !1 = 2, "1 = 1, !2 = 1, and "2 = 0.5. The dashed line gives the C = {x1 + x2 = 0.5},and the dotted line, x1 + x2 = 1. Two simulations of the birth-death-mutation process arealso shown in red for the same parameter values and N = 1000. The simulations had initial

conditions X(N)1 (0) = 10 and x2 = 50. One simulation shows type 1 fixing, and the other type

2. Both trajectories have the same qualitative behaviour, approximately following the flowsof the Law of Large Numbers prior to reaching a small neighbourhood of C, and subsequentlyevolving as a di#usion confined to a neighbourhood about C (reproduced from [149], originallyprepared by Christopher Quince)

modelled by the autonomous Lotka-Volterra-Gause system

Yi =

#

bi "K$

j=1

aijYj

%

Yi. (2.76)

If, up to permutation of the indices, the system satisfies

bjajj

<biaij

for all i < j (2.77)


andbjajj

>biaij

for all i > j (2.78)

then the axial fixed point5

b1a11

, 0, . . . , 06

is globally attracting on!

RK+ .

A global stability result is easily obtained in the weak selection regime:

Proposition 2.3. If bjajj

= biaij

for all i, j & K, then

C =

&

x % RK+ :

K$

j=1

ajjbj

xj = 1

'

(2.79)

is asymptotically stable in RK+ " {0}.

Proof. Clearly, every point of C is a fixed point. Let

V (x) =1

2

#

1"K$

j=1

ajjbj

xj

%2

. (2.80)

Then

V (x) =K$

i=1

!V

!xi

#

bi "K$

j=1

aijxj

%

xi = "

#

1"K$

j=1

ajjbj

xj

%2# K$

i=1

aiixi

%

< 0 (2.81)

on RK+ " ({0} 7 C). The result follows by Lyapunov’s stability theorem.

2.4 Summary

The intent of this chapter to define the birth-death-processes that are our object of study

and to identify the strategy we will use in analyzing them. We begin (Section 2.1) with

a notion of density dependent population processes that we adapt from Kurtz [116, 117,

118, 119], for which we can prove a functional Law of Large Numbers, Proposition 2.1.


While the result is only a slight generalization of the Kurtz’s result [116], it is crucial to

our e!orts in three regards:

(i) By associating a dynamical system to our stochastic processes, we are able to trans-

late crucial notions from the deterministic theory to the stochastic context, in par-

ticular, dissipativity, which for us is an appropriately general formalization of the

physical constraints that prevent populations from growing indefinitely, and the

very general definition of competition given by Kolmogorov.

(ii) The domain of applicability of our deterministic approximation (initial numbers

of order O(N) and times of order O(lnN)) indicate which limiting regimes must

be analyzed to understand the dynamics of the underlying birth-death-mutation

process. This indicates our overall strategy: we know that once the population

reaches )N for any ) > 0 independent of N , it can e!ectively be approximated by a

deterministic process, for a time span long enough that the approximation only fails

once the process is in a o(1) neighbourhood of the attractor of the deterministic

dynamics. We thus need to determine the probability the population reaches )N

prior to “fizzling out”, and to analyze the stochastic dynamics in the vicinity of the

attractor if the deterministic dynamics.

(iii) The bifurcation structure of the deterministic limit indicates the di!erent qualitative

dynamics possible in the stochastic system. In particular, we identify two regimes

of asymptotic behaviour that generalize the notions of weak and strong selection in

classical population genetics (Section 2.3). For our purposes, the former corresponds

to deterministic fixation of one type, whereas the latter corresponds to deterministic

coexistence.

The next two chapters will implement this strategy. Chapter 2 uses branching process

approximations to the address the problem of escaping extinction at small numbers and


applies these results in the strong selection regime. Chapter 3 considers the weak selection

regime using di!usion approximations, analogously to classical population genetics.

Chapter 3

Invasion Dynamics and Strong

Selection

In this chapter, we consider the case when a small initial number of individuals of (without

loss of generality) type 1 attempt to invade a resident population of types 2, . . . , K. For

present purposes, type 1 is said to successfully invade, if starting from O(1) individuals

(usually taken to be a single individual, as our primary interest is the fixation of a novel

allele), the number of individuals of type i is eventually greater than )N for some ) > 0, at

which point the dynamics of that type are e!ectively given by the Law of Large Numbers

of the previous chapter. i.e., throughout this chapter,

limN%"

1

NX

(N)i (0) =

?

JJ@

JJA

0 if i = 1

xi if i = 2, . . . , K.

(3.1)

As we saw previously in Proposition 2.1, under these assumptions, the rescaled process

Y(N)(t) =1

NX(N)(t) (3.2)

converges almost surely to the deterministic flow +tx, x = (0, x2, . . . , xK), for the vector

49

Chapter 3. Invasion Dynamics and Strong Selection 50

field F,

Fi(x) = fi(x)xi =(

"i(x)" %i(x))

xi.

In particular, writing +tx = (+1t x, . . . ,+

Kt x), we have +1

tx 4 0 for all t > 0.

In this chapter, we will limit our focus to examples without mutation, so

"(N)i,n (x) 4 0 (3.3)

for all n '= nei. Under this assumption, when X(N)1 (0)- N , we can approximate X(N)

i (t)

by a time-inhomogeneous Markov branching process Z1(t) (i.e., a Markov branching pro-

cess in a varying environment) with transition rates

"1,n(t) = "i,n(0,+2tx, . . . ,+

Kt x) (3.4)

and

%1(t) = %i(0,+2tx, . . . ,+

Kt x), (3.5)

in a sense that we will make precise via a coupling argument in Section 3.2.1. When

mutation is allowed, coupling with a branching process with immigration is still possible

(see [34, 33]), but will not be discussed here.

Armed with these results, we will obtain explicit expressions for the probability that

a single individual with an advantageous mutation can invade – that is, exceed )N

individuals prior to extinction – a resident population that is already abundant (i.e., the

number of individuals of all types j > 1 is O(N)). The resident species enter into our

expressions only through the deterministic approximation obtained as the functional Law

of Large Numbers, Proposition 2.1, which allows us to use insights previously obtained

from deterministic studies in theoretical population ecology.

As we saw in the previous chapter, once the number of individuals of type 1 exceed

)N , their subsequent dynamics are essentially deterministic. In particular, if type 1 is


strongly selected, then a successful invasion implies fixation with probability approaching

1 as N # $. We will use this to obtain expressions for the the fixation probability

and the expected fixation time in the strong selection regime. We find that when the

resident population is at a stable fixed point, we recover results analogous to Haldane’s

2s approximation [71]. Unlike classical results, however, we do not require the resident

“wild-type” population to be at equilibrium. In particular, our expressions provide an

excellent tool for investigating the ultimate fate of mutants arising in a population that

has undergone a recent bottleneck, in which case we find that the life-history strategy of

an invading mutant influences its probability of fixation.

First, however, we develop some results regarding branching processes in a varying

environment that will be useful in the sequel. These are simple adaptations of results for

the Bienayme–Galton–Watson in varying environments in [87, 29, 36]. We then prove the

coupling lemma, which allows us to apply the results obtained for branching processes

to find the probability that a novel mutant is able to invade an established “wild-type”

population in a birth-death-mutation process, and thus the fixation probability of a

strongly selected mutant. We then specialize our results to the case when the established

population is approaching a globally stable fixed point in the hyperplane x1 = 0, in which

case we obtain an asymptotic for the time to fixation of a strongly selected mutant.

3.1 Continuous Time Markov Branching Process in

a Varying Environment

In this section, we will assume Z(t) is a non-autonomous birth-and-death process, with

time varying transition intensities

qZ,Z+n(t) = "n(t) and qZ,Z#1(t) = %(t). (3.6)


so the generator of the process Z(t) is

(Gtf)(z) ="$

n=1

"n(t)z (f(z + n)" f(z)) + %(t)z (f(z " 1)" f(z)) (3.7)

for functions f : N0 # R.

We begin by recalling the following su"cient condition for non-explosion, which will

be assumed throughout:

Condition 3.1 (Non-explosion Condition, [74]). Assume that

"(t) ="$

n=1

n"n(t), (3.8)

converges uniformly in t on compact intervals. Then

P{Z(t) <$} = 1 (3.9)

for all t, and there is a unique process Z(t) with generator Gt.

3.1.1 Moments of Z(t)

Taking f(z) = zk, we have

(Gtf)(z,x) ="$

n=1

"n(t)z(

(z + n)k " zk)

+ %(t)z(

(z " 1)k " zk)

(3.10)

=k$

j=1

!k

j " 1

"#

"$

n=1

nk#j+1"n(t) + ("1)k#j+1%(t)

%

zj . (3.11)

Thus, provided that the series

"(m)(t) ="$

n=1

nm"n(t), (3.12)


converges uniformly for m = 1, . . . , k, we have

d

dtE=

Z(t)k,,Z(s) = z

>

=k$

j=1

!

k

j " 1

"(

"(k#j+1)(t) + ("1)k#j+1%(t))

E=

Z(t)j,,Z(.) = z

>

= k(

"(t)" %(t))

Ez

=

Z(t)k>

+k#1$

j=1

!

k

j " 1

"(

"(k#j+1)(t) + ("1)k#j+1%(t))

E=

Z(t)j,,Z(.) = z

>

(3.13)

whence

E=

Z(t)k,,Z(s) = z

>

= ek!

t

s!(u)!"(u) du

.

zk

+

9 t

s

e!k!

!

s!(u)!"(u) du

k!1$

j=1

!k

j " 1

"5

!(k!j+1)(#) + ("1)k!j+1"(#)6

Ez

=

Z(#)j,,Z(#) = z

>

d#

/

. (3.14)

In particular,

E [Z(t)|Z(s) = z] = e! ts%(u)#'(u) duz,

E=

Z(t)2,,Z(s) = z

>

= e2! ts %(u)#'(u) du

:

z2 +

9 t

s

e#2! "s %(u)#'(u) du

5

"(.) + %(.)6

d.

;

,

and

Var (Z(t)|Z(s) = z) =

9 t

s

e2! t" %(u)#'(u) du

5

"(.) + %(.)6

d.

where

"(t) ="$

n=0

n2"n(t). (3.15)

In what follows, we adopt the notation

m(s, t) := E [Z(t)|Z(s) = 1] = e! ts %(u)#'(u) du (3.16)


and m(t) := m(0, t).

3.1.2 The Probability Generating Function and Extinction Prob-

abilities

Let

Fz(/, s, t) := E=

/Z(t),,Z(s) = z

>

. (3.17)

Then, Fz = F z1 , and assuming Condition 3.1, F1(/, s, t) is the unique solution to the

backward equation!F1

!t= "GtF1 (3.18)

with initial condition F1(t, t", t) = /1 [74].

We begin with the appropriate generalization of the key lemma for the Bienayme–

Galton–Watson process from [77]:

Lemma 3.1. Let F (/, t) := F1(/, 0, t). Then,

9 t

s

"(.)m(., t) d. &m(s, t)

1" F1(/, s, t)"

1

1" /&9 t

s

m(., t)

#

"(.) + "(.)

2

%

d., (3.19)

with equality if and only if "(t) = "(t),which occurs if and only if "n(t) 4 0 for n > 1.

1We will use the notational shorthand

f(t") = lims"t!

f(s)

throughtout.


Proof. From 3.18, we have

$F1

$#= "

.#$

n=1

!n(#)(

F1(%, #, t)n+1 " F1(%, #, t)

)

+ "(#) (1" F1(%, #, t))

/

= " (1" F1(%, #, t))

#

"(#) "#$

n=1

!n(#)n$

m=1

F1(%, #, t)m

%

= " (1" F1(%, #, t))

#

"(#) " !(#) "#$

n=1

!n(#)n$

m=1

(F1(%, #, t)m " 1)

%

= " (1" F1(%, #, t))

(

#

"(#) " !(#) + (1" F1(%, #, t))

##$

n=1

!n(#)n$

m=1

m!1$

k=0

F1(%, #, t)k

%%

= " (1" F1(%, #, t))

(

#

"(#) " !(#) + (1" F1(%, #, t))

##$

n=1

!n(#)n!1$

k=0

(n" k)F1(%, #, t)k

%%

.

(3.20)

Thus,!

!.

m(., t)

1" F1(/, ., t)=

(%(.)" "(.)m(., t)

1" F1(/, ., t)+

m(., t)!F1!*

1" F1(/, ., t)

= "m(., t)"$

n=1

"n(.)n#1$

k=0

(n" k)F1(/, ., t)k,

(3.21)

and integrating, recalling m(t, t) = 1 and F1(/, t, t) = /, we have

1

1" /"

m(s, t)

1" F1(/, s, t)= "9 t

s

m(., t)"$

n=1

"n(.)n#1$

k=0

(n" k)F1(/, ., t)k d.. (3.22)

Since 0 & F1(/, ., t) & 1, we have

n &n#1$

k=0

(n" k)F1(/, ., t)k &

n#1$

k=0

(n" k) =n(n+ 1)

2. (3.23)

and thus

9 t

0

m(., t)"$

n=1

n"n(.) d. &m(0, t)

1" F1(/, 0, t)"

1

1" /

&9 t

0

m(., t)"$

n=1

n(n+ 1)

2"n(.) d.. (3.24)


The result follows.

We then have

Proposition 3.1 (Extinction Probabilities). Let

q(s, t) := P {Z(t) = 0|Z(s) = 1} . (3.25)

Then,

K t

se#

! "s %(u)#'(u) du%(.) d.

1 +K t

se#

! "s %(u)#'(u) du%(.) d.

& q(s, t)

&

K t

se#

! "s

%(u)#'(u) du5

%(.) + %(*)#%(*)2

6

d.,

1 +K t

se#

! "s

%(u)#'(u) du5

%(.) + %(*)#%(*)2

6

d.. (3.26)

Proof. This follows immediately from the lemma, as q(s, t) = F (0, s, t).

As remarked previously, when "1(t) = "(t) and "n(t) 4 0 for n > 1, we have "(t) =

"(t) andm(t)

1" F (/, t)"

1

1" /=

9 t

0

e! ts%(u)#'(u) du"(s) ds, (3.27)

from which the exact form of the probability generating function may be obtained:

Proposition 3.2. Suppose "1(t) = "(t) and "n(t) 4 0 for n > 1. Then, Z(t) has

probability generating function

F (/, t) = 1"m(t)

11#+

+K t

0 e#

! ts %(u)#'(u) du"(s) ds

. (3.28)

In particular,

q(t) =

K t

0 e#

! s0 '(u)#%(u) du%(s) ds

1 +K t

0 e#

! s0 '(u)#%(u) du%(s) ds

, (3.29)

and, setting

0t :=e! t0 e

! ts #(u)"$(u)du'(s) ds%(s) ds

1 +K t

0 e! ts '(u)#%(u) du%(s) ds

, (3.30)


and Pn(t) := P {Z(t) = n}, we have

P0(t) = q(t) and Pn(t) = (1" P0(t))(1" 0t)0n#1t . (3.31)

Moreover, taking t#$, we have

Corollary 3.1. The probability of ultimate extinction, q($) is

K"0 e

! s0 '(u)#%(u) du%(s) ds

1 +K"0 e

! s0 '(u)#%(u) du%(s) ds

, (3.32)

which is equal to 1 if and only if the integral I :=K"0 e

! s0 '(u)#%(u) du%(s) ds diverges.

Remark 3.1. Proposition 3.2 and Corollary 3.1 originally appeared in [97], with a di!erent

proof based upon the forward equation.

3.1.3 Criticality for the Markov Branching Process in Varying

Environments

Unlike the time-inhomogeneous case, there doesn’t appear to be an equally tidy number

like the Malthusian parameter to characterize of the asymptotic behaviour of processes

in varying environments. Proposition 3.1 gives us a necessary condition for the process

started from 1 individual at time s to persist indefinitely, namely that

9 "

s

e#! "s

%(u)#'(u) du"(.) d. <$ (3.33)

and a su"cient condition, as well, that

9 "

s

e#! "s %(u)#'(u) du

#

"(.) + "(.)

2

%

d. <$, (3.34)

but unfortunately not a necessary and su"cient condition (though, see Corollary 3.1).

(3.33) holds whenever % is integrable, a case we will exclude, as otherwise, we could have


individuals that never perish. We will henceforth assume that

9 "

s

%(.) d. (3.35)

diverges for all s, and say that a process is supercritical if (3.33) holds, in which case

m(s, t) # $ as t # $ for all fixed s. The latter is unfortunately not a su"cient

condition for supercriticality, as the following simple example indicates:

Example 3.1. Let "1(t) = 1 + 1(t+2) ln (t+2) , "n(t) 4 0 for n > 1, and %(t) = 1. Then, the

corresponding birth and death process in a varying environment satisfies

(i) m(s, t) = ln (t+ 2)" ln (s+ 2)#$ as t#$ for all fixed s, whereas

(ii)K t

s'(*)

m(s,*) d. =K t

sd*

ln (t+2)#ln (s+2) , which diverges,

so the process goes extinct with probability one.

We subdivide those processes for whichK"s

e#! "s

%(u)#'(u) du"(.) d. diverges into crit-

ical and subcritical as

0 < inft+0

m(t) & supt+0

m(t) <$ (3.36)

and m(s, t) # 0 as t " s # $ respectively. In the next three section, we will develop

analogues, where available to the limit theorems of Harris, Sevastyanov, Yaglom and

Kolmogorov under appropriate second moment conditions.

3.1.4 Supercritical Processes

We next recall the standard result for the long-term behaviour of a branching process.

Let

W (t) :=Z(t)

m(t). (3.37)


W (t) is a martingale with E [W (t)] = 1. By Doob’s martingale convergence theorem,

W := limt%"

W (t) (3.38)

exists pointwise almost surely, and, provided

E=

W (t)2,,Z(s) = 1

>

= 1 +

9 t

s

e#2! "0 %(u)#'(u) du

5

"(.) + %(.)6

d., (3.39)

is bounded for all s, t, W (t)#W in L2(P) as well. In particular, if supt E [W (t)2] <$,

we have E [W ] = 1. Presumably, the latter holds under weaker moment conditions than

those imposed above, but we will not consider that question here, as a second moment

condition is eminently plausible in the biological problems of interest here.

Proposition 3.3. Assume that

infs+0

9 "

s

e#! "s

"#n=1 %n(u)+'(u) du%(.) d. > p > 0 (3.40)

i.e., the probability of an individual dying without o"spring is bounded below. Then,

q($) := limt%"

q(t) = P {W = 0} . (3.41)

We begin with a pair of simple lemmas

Lemma 3.2. Assume (3.40). Then,

P

+

limt%"

Z(t) = 0-

+ P

+

limt%"

Z(t) = +$-

= 1 (3.42)

Proof. This is a simple consequence of Theorem 2 in [89], we need only show that for


any M > 0, there exists % > 0 such that

P

+

limt%"

Z(t) = 0,,,Z(s) < M

-

> % (3.43)

for all s < t. This follows immediately from our assumption, as

P

+

limt%"

Z(t) = 0,,,Z(s) < M

-

> pZ(s) > p-M. > 0. (3.44)

Lemma 3.3. Assume (3.39). Then

P {W = 0|Z(s) = 1} &

K"s

e#2! "s %(u)#'(u) du

5

"(.) + %(.)6

d.

1 +K"s

e#2! "s %(u)#'(u) du

5

"(.) + %(.)6

d.(3.45)

Proof. This is a simple consequence of the Cauchy–Bunyakovsky–Schwarz inequality:

1 = E [W |Z(s) = 1]2 = E [ W>0W |Z(s) = 1]2

& E=

2W>0

,,Z(s) = 1

>

E=

W 2,,Z(s) = 1

>

= P {W > 0|Z(s) = 1}E=

W 2,,Z(s) = 1

>

.

(3.46)

The result follows upon rearranging.

Proof of Proposition 3.3. We begin by observing that

{Z(s) = 0} . {Z(t) = 0} . {W = 0} (3.47)

for all s & t, so q(t) = P{Z(t) = 0} is increasing and bounded above i.e., q($) exists

and q($) & P{W = 0}.


For the other inequality, let wt = P {W = 0|Z(t) = 1} and let

w := sups+0

K"s

e#2! "s %(u)#'(u) du

5

"(.) + %(.)6

d.

1 +K"s

e#2! "s %(u)#'(u) du

5

"(.) + %(.)6

d.< 1. (3.48)

Using Lemma 3.3, we have

P {W = 0} = E [P {W = 0|Z(t)}] = E

E

wZ(t)t

F

& E=

wZ(t)>

. (3.49)

Applying Lebesgue’s dominated convergence theorem, in light of Lemma 3.2, gives

limt%"

E=

wZ(t)>

= E=

{limt$# Z(t)=0}>

= q($). (3.50)

We end this section by observing that when "1(t) = "(t) and "n(t) 4 0 for n > 1, we

can explicitly characterize the process W in the supercritical case:

Proposition 3.4. Suppose I <$. Then W has characteristic function

E=

e##W>

=I

1 + I+

I

(1 + I)21

$ + 11+I

(3.51)

i.e.,P {W & x} = I1+I

+ 11+I

5

1" e#x

1+I

6

and P {W & x|W > 0} = 1" e#x

1+I .

Proof. Substitute / = e#%

m(t) into (3.27) and take the limit as t#$.

3.1.5 Subcritical Processes

In the subcritical case, we have the following weaker analogue of the Yaglom-Kolmogorov

theorem, which is an immediate consequence of Lemma 3.1:



lim supt%"

9 t

0

#

"(.) + "(.)

2

%

m(., t) d. <$. (3.52)

Then

0 < lim inft%"

1" q(t)

m(t)& lim sup

t%"

1" q(t)

m(t)< 1. (3.53)

In [87, 88], a stronger result is claimed for a Galton Watson process in varying en-

vironments, Zn, under similar second moment conditions on the o!spring distribution,

namely that if mn = E [Zn], then

limn%"

P{Zn > 0}mn

(3.54)

exists and is constant. The equivalent assertion does not hold in our case, as the following

example shows.

Example 3.2. Let "1(t) = 1 + 12 sin t+

12 cos t, "n(t) 4 0 for n > 1, and %(t) = 1 + "(t).

Then, the corresponding birth and death process in a varying environment satisfies

(i) "(t) = "(t) = "1(t),

(ii) m(., t) = e#(t#*) # 0 as t" . #$,

(iii)K t

0 e#

! "0 '(u)#%(u) du%(.) d. =

(

2(1" e#t) + 12 sin t

)

et #$, so q(t)# 1,

(iv)K t

0

5%(*)+%(*)

2

6

m(., t) d. =K t

0 "1(.)m(., t) d. = 1" e#t + 12 sin t, so

lim supt%"

9 t

0

#

"(.) + "(.)

2

%

m(., t) d. =3

2, (3.55)

and

(v) m(t)1#q(t) = 1 +

K t

0 "1(.)m(., t) d. .


However, we then have

3

2= lim inf

t%"

m(t)

1" q(t)< lim sup

t%"

m(t)

1" q(t)=

5

2, (3.56)

in contradiction to the continuous time analogue of the claim in [87, 88].

3.1.6 Critical Processes

For the sake of completeness, we conclude this discussion with an analogue of Yaglom’s

theorem in the critical case. Unlike the previous results, we shall not have need of this

in the sequel.


0 < inft+0

m(t) & supt+0

m(t) <$ (3.57)

inft+0

"(t) > 0, (3.58)

and

"$

n=1

n2"n(t) converges uniformly in t. (3.59)

(3.60)

Then

(i) P {Z(t) > 0} 8 1! t0

#

$(")+$(")2

$

m(*,t) d*

2 as t#$.

(ii) P

+Z(t)

m(t)I(t) > u,,,Z(t) > 0

-

# e#u as t#$.

2we use the notation f(t) 8 g(t) if

limt"#

f(t)

g(t)= 1.


Proof. For simplicity of notation, let

I(t) =

9 t

0

#

"(.) + "(.)

2

%

m(., t) d.. (3.61)

We begin by observing that assumptions (3.57) and (3.58) imply that

9 t

s

"(.)m(., t) d. (3.62)

and thus9 t

s

#

"(.) + "(.)

2

%

m(., t) d. (3.63)

diverge as |t" s|#$. Recalling, Lemma 3.1,

0 & 1" F1(/, s, t) &1

K t

s"(.)m(., t) d.

(3.64)

so that F1(/, s, t)# 1 uniformly in / as |t" s|#$.

Similarly to the proof of Lemma 3.1, we have

m(t)

1" F (/, t)"

1

1" /= I(t) +

9 t

0

m(., t)"$

n=1

"n(.)n#1$

k=0

(n" k)(

1" F1(/, ., t)k)

d.. (3.65)

Now,

"n(.)n#1$

k=0

(n" k)(

1" F1(/, ., t)k)

& "n(.)n(n + 1)

2, (3.66)

so, by assumption (3.59), the former sum converges uniformly in (/, ., t). In particular,

lim+/1

"$

n=1

"n(.)n#1$

k=0

(n" k)(

1" F1(/, ., t)k)

= 0, (3.67)


uniformly in / and . as |t" . |#$. Fix ) > 0 and choose T such that

0 &1

m(.)

"$

n=1

"n(.)n#1$

k=0

(n" k)(

1" F1(/, ., t)k)

< ) (3.68)

for all t" . > T . Then,

K t

0 m(., t)0"

n=1 "n(.)0n#1

k=0(n" k)(

1" F1(/, ., t)k)

d.

I(t)

=

K t#T

01

m(*)

0"n=1 "n(.)

0n#1k=0(n" k)

(

1" F1(/, ., t)k)

d.

I(t)

+

K t

t#T1

m(*)

0"n=1 "n(.)

0n#1k=0(n" k)

(

1" F1(/, ., t)k)

d.

I(t)

<t" T

t

2 sup*+0m(.)

inf*+0("(.) + "(.)))+ T

sup"%0(%(*)+%(*))

2 inf"%0 m(*)

I(t)(3.69)

i.e.,! t0 m(*,t)

"#n=1 %n(*)

"n"1k=0 (n#k)(1#F1(+,*,t)k) d*I(t) can be made arbitrarily small as t # $,

uniformly in /.

In particular, we have

1I(t)

1" F (0, t)= 1 +

K t

0 m(., t)0"

n=1 "n(.)0n#1

k=0(n" k)(

1" F1(0, ., t)k)

d.

I(t)# 1 (3.70)

as t#$, proving (i)

Next, rearrange (3.65) as

m(t)

1" F (/, t)= 1 +

1

m(t)I(t)(1" /)

+

K t

0 m(., t)0"

n=1 "n(.)0n#1

k=0(n" k)(

1" F1(/, ., t)k)

d.

I(t). (3.71)


Now, recall from the previous theorem that

E=

/Z(t),,Z(t) > 0

>

= 1"1" F (/, t)

1" F (0, t), (3.72)

so that

E

E

e%Z(t)

m(t)I(t)

,,,Z(t) > 0

F

= 1"1" F

5

e%

m(t)I(t) , t6

1" F (0, t)

= 1"1 + 1

m(t)I(t) +1

m(t)

! t0 m(*,t)

"#n=1 %n(*)

"n"1k=0 (n#k)(1#F1(0,*,t)k) d*I(t)

1 + 1

m(t)I(t)(1#e%

m(t)I(t) )+ 1

m(t)

! t0 m(*,t)

"#n=1 %n(*)

"n"1k=0 (n#k)

%

1#F1(e%

m(t)I(t) ,*,t)k&

d*

I(t)

,

(3.73)

which converges to 1" 11+ 1

%

= 11+#

as t#$. (ii) follows.

Remark 3.2. We briefly note that the integral appearing in the asymptotic for the extinc-

tion probability in the critical case, i, is the same as that which appears in our expression

for the extinction probability in general (Proposition 3.1). This suggests (but does not

prove) that the upper bound in Proposition 3.1 is tight.

3.1.7 Time Asymptotics for Processes with a Malthusian Pa-

rameter

Throughout this section, we assume that "(t) and %(t) converge to finite, non-zero limits

bar"($) and %($) as t # $, and moreover that "(t) " "($) and %(t) " %($) are

integrable. Set

' := "($)" %($). (3.74)


In a slight abuse of terminology, we refer to ' as theMalthusian parameter for the process

Z(t). We then have

& = limt%"

e#$tm(t) = e! t0 (%(u)#%("))#('(u)#'(")) du (3.75)

exists and is bounded. In particular, we have

limt%"

E1

=

e#$tZ(t)>

= & (3.76)

and, if M > 0,

limt%"

e#$tZ(t) = &W (3.77)

in L2.

This latter limit gives us the following asymptotic first hitting time result in the super-

critical case, a simple adaptation of the result in [88] for the general (time-homogeneous)

branching process:

Proposition 3.7. Suppose ' > 0 and let

.M = inf{t : Z(t) ,M}. (3.78)

Then, as M #$,

.M 91

'(lnM " lnW " ln&) , (3.79)

where we understand the right hand side to be +$ if W = 0.

Proof. By definition

Z(.M " %) < M < Z(.M) (3.80)

for all % > 0. Thus

e#$'e#$(*M#')Z(.M " %) < e#$*MM & e#$*MZ(.M). (3.81)


Now, Z(t) is non-explosive, so .M # $ as M # $. Thus, taking M # $ and % # 0,

both the left and right sides converge to &W , and

limM%"

e#$*MM = &W. (3.82)

The result follows by taking logarithms.

With some additional conditions on the process W , which are adequate for the follow-

ing examples, but somewhat unsatisfying (we would prefer to assert, rather than assume

them, but at present, we are unable to prove that these conditions hold in general), we

can give the following asymptotic for the expected first hitting time of M > 0:

Corollary 3.2. Suppose that W has a density w(x) that is continuous on (0,$) and

that there exists , < 1 such that

lim infx%0

x,w(x) (3.83)

exists. Then, as M #$,

E [.M |W > 0] 91

'lnM. (3.84)

Remark 3.3. Proposition 3.4 shows that assumption (3.83) holds in the case when "n(x) 4

0 for all n > 1, which is adequate for the examples we will consider in this chapter and

in Chapter 5. The result for the time homogeneous case suggests that (3.83) holds in

general. In that case, W has a density, w(x), which is continuous on (0,$). Moroever,

the Laplace-Stieltjes transform of W can be completely characterized [73, 168]: if '($) =

E=

e##W>

,

a ="$

n=1

"n + % (3.85)

is the cumulative rate of all births and deaths, and f is the generating function for the


individual o!spring distribution, i.e.,

f(/) ="$

n=1

"n

a/n +

%

a, (3.86)

then

'#1(x) = (1" x)e#

! 1x

#

f !(1)"1f(&)"& + 1

1"&

$

d+. (3.87)

In particular, if , = f !(q)#1f !(1)#1 , then , < 1, and a simple calculation using l’Hopital’s rule

shows

limx%q

(x" q)1''#1(x) (3.88)

exists and is finite. Now, lim#%" '($) = q, so there exists a constant A such that

'($) 8 q + A$, as $ # $. Applying the Hardy-Littlewood Tauberian theorem (see

e.g., [57]), we then have

limx%0

x,w(x) (3.89)

exists, as in assumption (3.83). We have yet to verify or refute the analogous result for

the time-inhomogeneous case.

Proof. It only remains to show that E [lnW |W > 0] is bounded.

The upper bound is trivial: we begin by observing that for all x > 0, ln x & x" 1, so

that

E [lnW |W > 0] & E [W |W > 0]" 1 =q

1" q, (3.90)

since E [W |W > 0]P {W > 0} = E [W ] = 1, and E [W |W > 0] = 11#q

.

For the lower bound, let lim infx%0 x,w(x) = C and fix ) > 0. Choose % > 0 so that


x,w(x) , C " ) for all x % [0, %). Then,

E [lnW |W > 0] =1

1" q

9 "

0

w(x) lnx dx ,1

1" q

9 1

0

w(x) lnx dx

=1

1" q

9 '

0

w(x) ln x dx+1

1" q

9 1

'

w(x) ln x dx

, (C " ))1

1" q

9 '

0

x#, lnx+ C1

1" q

9 1

'

lnx dx, (3.91)

for some constant C, as w(x) is continuous, and thus bounded below on [%, 1]. Now,

9 1

0

x#, ln x dx = "1

(, " 1)2. (3.92)

for all , < 1, so E [lnW |W > 0] is bounded below as well. The result follows.

An asymptotic for the time to extinction, starting from large numbers, can also be

obtained in the subcritical case, similarly to the result for the Harris-Bellman process in

[91]:

Proposition 3.8. Suppose ' < 0 and let

.0 = inf{t : Z(t) = 0}. (3.93)

Then, as M #$,

EM [.0] 91

|'|lnM. (3.94)

Proof. From Proposition 3.5, there are constants C# and C+ such that

0 < C# = lim inft%"

e#$t(1" q(t)) & lim supt%"

e#$t(1" q(t)) = C+ (3.95)

Fix ) > 0, and choose T > 0 such that

C+ + ) > e#$t(1" q(t)) > C# " ) > 0 (3.96)


for all t > T and

(C+ + ))e$T < 2. (3.97)

Then,

EM [.0] =

9 "

0

PM {.0 > t} dt (3.98)

=

9 "

0

PM {Z(t) > 0} dt (3.99)

=

9 T

0

PM {Z(t) > 0} dt+

9 "

T

PM {Z(t) > 0} dt. (3.100)

Now, for any fixed T , we may apply the dominated convergence theorem to conclude

that

limM%"

9 T

0

PM {Z(t) > 0} dt = T, (3.101)

leaving us with the task of computing the tail,

9 "

T

PM {Z(t) > 0} dt =

9 "

T

1" q(t)M dt (3.102)

By assumption,

9 "

T

1"(1"(C++))e$t)M dt &9 "

T

1"q(t)M dt &9 "

T

1"(1"(C#"))e$t)M dt. (3.103)

We are thus left with the task of computing

9 "

T

1" (1" Ce$t)M dt =

9 "

T

M#1$

i=0

Ce$t(1" Ce$t)i (3.104)


for C = C± ± ). After a change of variable, x = Ce$t, this equals

"1

'

9 Ce(T

0

M#1$

i=0

(1" x)i dx =1

|'|

#M#1$

i=0

1

i+ 1"

M#1$

i=0

(1" Ce$T )i+1

i+ 1

%

(3.105)

=1

|'|

5

lnM + , + ln Ce$T +O(

1M

)6

(3.106)

=1

|'|

5

lnM + , + ln C +O(

1M

)6

" T, (3.107)

where , is Euler’s constant [8]. By assumption,,,1" Ce$T

,,, is within the radius of con-

vergence of the logarithm. The result follows.

In Section ??, we will use these estimates to obtain an asymptotic value (in N) for

the fixation time of an advantageous allele for a strongly selected density dependent

birth-death-mutation process X(N)(t).

3.2 Application to Density-Dependent Birth-Death-

Mutation Processes

We now return to the primary task of studying the process density dependent birth-

death-mutation process X(N)1 (t). We take our inspiration from [174], and use a coupling

argument, constructing a probability space upon which X(N)1 (t) is defined, along with

branching processes Z(()1,x,+ and Z

(()1,x,# that bound X

(N)1 (t) above and below, respectively.

Equipped with these bounds, we can then use the results of the preceding section to

obtain estimates of the invasion probability of a novel mutant and the probability of

fixation and expected time to fixation of a strongly advantageous mutant.

Remark 3.4. For clarity of exposition, we have chosen to present the case when only

X(N)1 (0) is of order O(1). All the results of this and subsequent sections continue to hold


when X(N)1 (0), . . . , X(N)

d (0) are all O(1). In that case, we would have

limN%"

1

NX(N)(0) = x (3.108)

for x = (0, . . . , 0, xd+1, . . . , xK , and +itx 4 0 for i = 1, . . . , d. We could then construct

independent time-inhomogeneous branching processes Z(()i,x,±(t), i = 1, . . . , d, such that

Z(()i,x,#(t) & X

(N)i (t) & Z

(()i,x,#(t) (3.109)

for i = 1, . . . , d.

3.2.1 A Coupling Result

Recall that in Section 2.1.2 we introduced a compact set Kx 0!

RK+ such that the forward

orbit of x, ,+x := {+tx : t , 0}, is contained in

!Kx.

Let

1(N)n (Kx) = sup

1,x!(Kx

,,,"

(N)1,n (x))" "1,n(x

)),,, (3.110)

and

1(N)(Kx) = sup1,x!(Kx

,,,%

(N)1 (x))" %1(x

)),,, . (3.111)

Then, by Assumption 2.3,

limN%"

$

n(ZK+

1n12 1(N)n (Kx) = 0 (3.112)

and

limN%"

N$

n(ZK+

1n12 1(N)n (Kx) <$. (3.113)

etc.


Fix ) > 0 su"ciently small so that

Kx,((t) = {x) % RK+ : 1x) " +tx1 < )} . Kx (3.114)

for all t, and choose N( such that

$

n(ZK+

1n12 1(N)n (Kx) < ) (3.115)

for all N , N(.

Let

"(()1,n,x,+(t) := sup

1,x!(Kx,!(t)"1,n(x

)) + 1(N!)n (Kx), (3.116)

"(()1,n,x,#(t) := inf

1,x!(Kx,!(t)"1,n(x

))" 1(N!)n (Kx), (3.117)

%(()1,x,+(t) := sup

1,x!(Kx,!(t)%1(x

)) + 1(N!)(Kx), (3.118)

and

%(()1,x,#(t) := inf

1,x!(Kx,!(t)%1(x

))" 1(N!)(Kx). (3.119)

(3.120)

We will also write

"1,n,x(t) := "(0)1,x,n,±(t) = "1,n(+tx) (3.121)

and

%1,x(t) := %(0)1,x,±(t) = %1(+tx). (3.122)

Finally, let

T (N)( = inf{t , 0 :

33Y(N)(t)" +tx

33 > )}. (3.123)


By definition, provided N , N( and t < T(N)( , we have

"(()1,n,x,#(t) < "

(N)1,ne1(Y

(N)(t)) < "(()1,n,x,+(t)

%(()1,x,#(t) < %

(N)1 (Y(N)(t)) < %

(()1,x,+(t).

Remark 3.5. Given (3.113), all of the inequalities above continue to hold if ) is replaced

by )N , where {)N}"N=1 is a monotonically decreasing sequence, provided {N)N}"N=1 is

increasing, and N)N #$ as N #$. We shall use this extensively in the next section.

We then have

Proposition 3.9. There exist nonautonomous Markov processes Z(()1,x,+ and Z

(()1,x,#, and

counting processes B(()1,n,x,#(t), B

(()1,n,x,+(t), D

(()1,x,#(t), and D

(()1,x,+(t) with intensities "(()

1,n,x,#(t)Z(()1,x,+(t),

"(()1,n,x,+(t)Z

(()1,x,+(t), %

(()1,x,#(t)Z

(()1,x,+(t), and %

(()1,x,+(t)Z

(()1,x,+(t), respectively such that

Z(()1,x,#(t) = X1(0) +

"$

n=0

nB(()1,n,x,#(t)"D1,x,+(t),

Z(()1,x,+(t) = X1(0) +

"$

n=0

nB(()1,n,x,+(t)"D1,x,#(t),

and, if

T(,N = sup{t : X(N)(t) < )N}, (3.124)

then

B(()1,n,x,#(t) & B

(N)1,ne1(t) & D

(()n,x,+(t),

D(()1,x,#(t) & D

(N)1 (t) & D

(()1,x,+(t),

and

Z(()1,x,#(t) & X

(N)1 (t) & Z

(()1,x,+(t),


in distribution for all

t < T(,N 5 T (N)( (3.125)

and N su!ciently large.

Proof. We construct the coupled processes inductively up to an arbitrary time T ; the

existence of the process for all time then follows from Kolmogorov’s extension theorem

(see e.g., [37]). To begin, let

B(()1,n,x,#(0) = B

(N)1,ne1(0) = B

(()1,n,x,+(0) = 0,

D(()1,x,#(0) = D

(N)1 (0) = D

(()1,x,+(0) = 0,

and

Z(()1,x,#(0) = Z

(()1,x,+(0) = X

(N)1 (0).

and let Uj and Vj , j = 1, 2, . . . be be i.i.d. random variables, uniformly distributed on

[0, 1]. Now, suppose the processes have been constructed up to a jump time .j . Let .j+1

be a random variable with distribution

P{.j+1 > t} = e#! t0

#

"#n=1 %

(!)1,n,x,+(s)+'

(!)1,x,+(s)

$

Z(!)1,x,+(s,*j) ds (3.126)

For .j & t < .j+1, let

B(()1,n,x,#(t) = B

(()1,n,x,#(.j),

B(N)1,ne1(t) = B

(N)1,ne1(.j),

B(()1,n,x,+(t) = B

(()1,n,x,+(.j),


etc. Now, if

0 & Uj+1 &%(()1,x,+(.j+1)Z

(()1,x,+(.j+1")

50"

n=1 "(()1,n,x,+(.j+1) + %

(()1,x,+(.j+1)

6

Z(()1,x,+(.j+1")

(3.127)

then let

D(()1,x,+(.j+1) = D

(()1,x,+(t)(.j) + 1 and

Z(()1,x,#(.j+1) = Z

(()1,x,#(.j)" 1

if Vj+1 &%(()1,x,+(.j+1)Z

(()1,x,#(.j+1")

%(()1,x,+(.j+1)Z

(()1,x,+(.j+1")

D(N)1 (.j+1) = D

(N)1 (.j) + 1 and

X(N)1 (.j+1) = X

(N)1 (.j)" 1

if Vj+1 &%(N)1 (Y(N)(.j+1"))X(N)

1 (.j+1")%(()1,x,+(.j+1)Z

(()1,x,+(.j+1")

and

D(()1,x,#(t)(.j+1) = D

(()1,x,#(t)(.j) + 1 and

Z(()1,x,+(.j+1) = Z

(()1,x,+(.j)" 1

if Vj+1 &%(()1,x,#(.j+1)Z

(()1,x,+(.j+1")

%(()1,x,+(.j+1)Z

(()1,x,+(.j+1")

.

Similarly, if

50m#1

n=1 "(()1,n,x,+(.j+1) + %

(()1,x,+(.j+1)

6

Z(()1,x,+(.j+1")

50"

n=1 "(()1,n,x,+(.j+1) + %

(()1,x,+(.j+1)

6

Z(()1,x,+(.j+1")

& Uj+1 &

50m

n=1 "(()1,n,x,+(.j+1) + %

(()1,x,+(.j+1)

6

Z(()1,x,+(.j+1")

50"

n=1 "(()1,n,x,+(.j+1) + %

(()1,x,+(.j+1)

6

Z(()1,x,+(.j+1")

(3.128)

then, let B(()m,+(.j+1) = B

(()m,+(.j) + 1, Z(()

1,x,+(.j+1) = Z(()1,x,+(.j) +m and

B(N)1,m,xe1

(.j+1) = B(N)1,m,xe1

(.j) + 1 and

X(N)1 (.j+1) = X

(N)1 (.j) +m

if Vj+1 &"(N)1,me1

(Y(N)(.j+1"))X(N)1 (.j+1")

"(()1,m,x,+(.j+1)Z

(()1,x,+(.j+1")

and

B(()m,#(.j+1) = B

(()m,#(.j) + 1 and

Z(()1,x,+(.j+1) = Z

(()1,x,+(.j) +m

if Vj+1 &"(()1,m,x,#(.j+1)Z

(()1,x,#(.j+1")

"(()1,m,x,+(.j+1)Z

(()1,x,+(.j+1")

.


Proceeding in this manner, we obtain the desired Markov chains.

Remark 3.6. Given an arbitrary sequence (finite or countably infinite) of values 0 & )0 <

)1 < )2 · · · , we may proceed as in the proof of Lemma 3.9 to construct a family of coupled

processes B((j)1,n,x,±(t), D

((j)1,x,±(t) and Z

((j)1,x,±(t) such that

· · · & Z((1)1,x,#(t) & Z

((0)1,x,#(t) & Z

((0)1,x,+(t) & Z

((1)1,x,+(t) & · · · (3.129)

etc. In particular, in what follows, we shall routinely assume that we have coupled

processes

Z(()1,x,#(t) & Z1,x(t) := Z

(0)1,x,#(t) = Z

(0)1,x,+(t) & Z

(()1,x,#(t). (3.130)

We shall need the mean and extinction probabilities of the processes Z(()1,x,#(t), Z1,x(t),

and Z(()1,x,#(t), which we denote as follows:

Definition 3.1. For fixed ) > 0, we will write

m(()1,x,±(t) := E

E

Z(()1,x,±(t)

,,,Z

(()1,x,±(0) = 1

F

m1,x(t) := E

E

Z1,x(t),,,Z

(()1,x(0) = 1

F

q(()1,x,±(t) := P

+

Z(()1,x,±(t) = 0

,,,Z

(()1,x,±(0) = 1

-

and

q1,x(t) := P

+

Z1,x(t) = 0,,,Z

(()1,x(0) = 1

-

We write q(()1,x,±($) = limt%" q1,x,±(t) and q1,x($) = limt%" q1,x(t) for the probability of

eventual extinction of the processes Z(()1,x,± and Z1,x respectively.


3.2.2 Invasion and Fixation in the Supercritical Case

Recall that the process Z1,x(t) has mean e!#0 %1,x(s)#'1,x(s) ds, where

"1,x(t) ="$

n=1

n"1,n,x(t). (3.131)

We now assume that

9 "

0

"1,x(s)" %1,x(s) ds = +$, (3.132)

so that m1,x(t)#$ as t#$. Let

21,x :=

,,,,supt(R+

"i(t)" %i(t)

,,,,. (3.133)

21,x is the maximum of "(&tx)" %(&tx) over the compact trajectory {&tx : t % R+}, and

is thus finite, and positive by our assumption of supercriticality.

Recall that L1,x > 0 is a constant such that

1F(y1)" F(y2)1 < L1,x 1y1 " y21 (3.134)

for all y1,y2 % Kx. Fix 0 < r < s < 1 as in Proposition 2.1 and choose %,0 > 0 such

that 0 < 0 < r2 and 0 5 -1,x(1#s)

4L21,x

< # < 1" 0. Let

tN =1" 0" #

21,xlnN (3.135)

and

)N = N#.. (3.136)

We then have

(i) tN #$,


(ii) m1,x(tN)#$,

(iii) tN & (1#s)4L2

1,xlnN ,

(iv) )N # 0,

(v) Nr2)N #$, and thus, N)N #$, and

(vi) )N tN # 0,

as N #$.

Let

W1,x = limt%"

Z1,x(t)

m1,x(t). (3.137)

As we observed in Section 3.1.4, this limit holds almost surely and in L1(P), and E[W1,x] =

1. We then have

Lemma 3.4. As N #$,

W(N)1,x,± :=

Z((N )1,x,+(tN )

m((N )1,x,±(tN )

# W1,x,± (3.138)

in L1(P) as N #$, and E [W1,x,±] = 1, where

W1,x,#D= W1,x,+

D= W1,x. (3.139)

Proof. The existence of the limits and the topology of convergence follow from Doob’s

martingale convergence theorem, as in 3.1.4. To characterize the limit, let

F(()1,x,±(/, t) := E

E

/Z(!)1,x,±(t)

F

(3.140)

and

F1,x(/, t) := E=

/Z1,x(t)>

(3.141)


be the probability generating functions of Z(()1,x,± and Z1,x respectively. Then,

lim(%0

F(()1,x,±(/, t) = F1,x(/, t) (3.142)

uniformly in t. In particular, the W(N)1,x,± have characteristic functions

3(N)($) = F((N )1,x,±

#

e

i%

m(!N )1,x,±(tN ) , tN

%

, (3.143)

and

limN%"

3(N)($) = 3($), (3.144)

where 3($) = limt%" F1,x

5

ei%

m1,x(t) , t6

is the characteristic function for W1,x. The result

follows by Levy’s convergence theorem [37].

Lemma 3.5. As N #$, P

2

X(N)1 (tN )(NN

> 1

4

# 0.

Proof. We begin by observing that, provided tN < T(N)(N ,

X(N)1 (tN)

)NN&

Z((N )1,x,+(tN )

)NN=

m((N )1,x,+(tN)

)NN

Z((N )1,x,+(tN )

m((N )1,x,+(tN)

=m

((N )1,x,+(tN)

)NNW

(N)1,x,+. (3.145)

We consider the two terms in the product independently. First,

m((N )1,x,+(tN)

)NN= e

! tN0 %

(!N )1,x,+(u)#'

(!N )1,x,"(u) du#lnN#ln (N (3.146)

Now, "1,ne1(x) and %1(x) are C1 and thus Lipschitz on the compact set Kx,(N (t), and, by

definition, 1+tx" x)1 < )N for all x) % Kx,(N (t), so there exists a constant C such that

"((N )1,x,+(t)" %

((N )1,x,#(t) & "1,x(t)" %1,x(t) + C)N (3.147)


whence

m((N )1,x,+(tN )

)NN& etN supt%0|%1,x(t)#'1,x(t)|+C(N tN#lnN#ln (N (3.148)

= e-1,xtN+C(N tN#lnN+. lnN (3.149)

= e#/ lnN+C(N tN 8 N#/ (3.150)

as N #$.

On the other hand, by the previous lemma, W (N)1,x,± #W1,x. Thus, for any 0 < #) < #,

P

&

Z((N )1,x,+(tN)

m((N )1,x,+(tN )

> N/!

'

&E

E

W(N)1,x,+

F

N/!# 0 (3.151)

as N #$, so the productm

(!N )1,x,+(tN )

(NNW

(N)1,x,+ is o(1) provided tN < T

(N)(N .

Lastly, recalling the proof of Proposition 2.1, we have

P1,x

G

T (N)(N

< tNH

<N#r

)2N(3.152)

which tends to 0 as N #$. The result follows.

We then have

Proposition 3.10. As N #$

X(N)i (tN)

m1,x(tN)#W1,x (3.153)

almost surely and in L2(P).

Proof. This follows immediately from the coupling, the preceding lemma and its proof;

we have P

2

X(N)i (tN )(NN

> 1

4

# 0, so for N su"ciently large, we have

Z((N )1,x,#(tN ) & X(N)(tN ) & Z

((N )1,x,+(tN) (3.154)


with probability tending to 1 as N #$. Thus,

m((N )1,x,#(tN)

m1,x(tN )

Z((N )1,x,#(tN )

m((N )1,x,#(tN)

&X(N)(tN)

m1,x(tN)&

m((N )1,x,+(tN)

m1,x(tN )

Z((N )1,x,+(tN )

m((N )1,x,+(tN)

. (3.155)

with probability tending to 1 as N #$ as well.

From above, we haveZ

((N )1,x,+(tN )

m((N )1,x,+(tN)

#W1,x,+D= W1,x, (3.156)

and in a similar manner,Z

((N )1,x,#(tN )

m((N )1,x,#(tN)

#W1,x,#D= W1,x. (3.157)

Proceeding as in the previous lemma, it is readily shown that

m1,x(tN)

m((N )1,x,#(tN )

# 1 andm1,x(tN )

m((N )1,x,+(tN )

# 1. (3.158)

The result follows.

Now, fix ) > 0 independent of N . Let

.((,N)1 = inf{t , 0 : X(N)

1 (t) , )N}. (3.159)

Then we have

Corollary 3.3. As N #$,

P1,x{. ((,N)1 = +$} = P1,x{W1,x = 0} = q1,x($) (3.160)

Proof. Proceeding as in Lemma 3.9, choosing ) > 0 su"ciently small, we may construct

i.i.d. supercritical branching processes Z((,j)1,x,#(t) and Z

((,j)1,x,+(t), j = 1, . . . , X(N)(tN ), such

thatX(N)(tN )$

j=1

Z((,j)1,x,#(t) & X(N)(tN + t) &

X(N)(tN )$

j=1

Z((,j)1,x,+(t) (3.161)


for all t such that tN + t < .((,N)1 . Then, if

q(())tN

x,±(t) := P

+

Z((,1)1,x,±(t) = 0

,,,Z

((,1)1,x,±(0) = 1

-

, (3.162)

we have that

q(())tN

x,#(t)X(N)(tN ) = P

?

@

A

X(N)(tN )$

j=1

Z((,j)1,x,#(t) = 0

B

C

D& PG

X(N)(tN + t) = 0H

& P

?

@

A

X(N)(tN )$

j=1

Z((,j)1,x,#(t) = 0

B

C

D= q

(())tN

x,#(t)X(N)(tN ). (3.163)

From Proposition 3.10, we have

X(N)1 (tN )

m1,x(tN )#W1,x. (3.164)

Thus, for any - '% {W1,x = 0}, we can pick % > 0 such that % < W1,x(-), and there exists

N' such that

(W1,x(-)" %)m1,x(tN) < X(N)(tN) < (W1,x(-) + %)m1,x(tN), (3.165)

for all N > N'. Since m1,x(tN)#$, X(N)(tN )#$ as well. In particular,

q(())tN

x,#(t)X(N)(tN ) # 0 (3.166)

and

q(())tN

x,#(t)X(N)(tN ) # 0, (3.167)

so, for t < .((,N)1 , we have P

G

X(N)(tN + t) = 0H

# 0.

Moreover, from Lemma 3.2, we know that either there exists T such that Z(()1,x,#(t) = 0

for all t , T , or Z(()1,x,#(t) # $. If the latter holds, we must have .

((,N)1 < $, as


Z(()1,x,#(t) & X(N)(t) for t < .

((,N)1 .

Lastly, once X(N)(t) , )N , Proposition 2.1 tells us that the dynamics are e!ectively

deterministic, giving us

Corollary 3.4. If x",i = riiei is the unique stable attractor of Y(t) = F(Y(t)), i.e., type i

can exclude other types, then type i fixes with probability approaching 1"q1,x as N #$.

Remark 3.7 (Fecundity Variance Polymorphism). From Lemma 3.1 and Corollary 3.4 we

have that

1

1 +K"0 e#

! s0 %1()ux)#'1()ux) du%1(+sx) ds

, 1" q1,x

,1

1 +K"0 e#

! s0 %1()ux)#'1()ux) du

5

%1(+sx) +%1()sx)#%1()sx)

2

6

ds. (3.168)

In particular, the upper bound is equal to the fixation probability for type 1 in a density-

dependent birth-death-mutation process with reproduction rates

limN%"

"(N)1,n (x) =

?

JJ@

JJA

"1(x) if n = e1,

0 otherwise

, (3.169)

i.e.,with probability approaching 1 as N # $, all reproductions produce exactly one

o!spring of the same type as the parent. Moreover, unless this condition holds, the lower

bound (and q1,x) is strictly smaller than the upper bound. In particular, among all types

with the same mean growth rate, those that achieve that growth rate by minimizing

the variance in the number and frequency of o!spring per reproductive event have the

highest probability of invasion. Thus, when the invading types are able to exclude their

competitors, then the trend in evolution is towards minimizing fecundity variance. This

phenomenon has been observed previously via heuristic analyses of the di!usion limit of

the Wright-Fisher process in the weak selection regime [66, 171, 67, 68, 61]. Corollary


3.4 generalizes these results to the a wide class of processes in the strong selection case.

Recall that if , is a curve in RK and x : [0, t] # RK is any parameterization of that

curve, then9

,

f(x) dx =

9 t

0

f(x(s)) 1x(s)1 ds. (3.170)

Since ddt+tx = F(+tx), we then have the following

Corollary 3.5. Let ,+x = {+tx : t , 0} be the forward orbit of x. Then,

K

,+xe#

! !1,x

$(")"#(")&F(")& d! '(")

0F(")0 , d!

1 +K

,+xe#

! !1,x

$(")"#(")&F(")& d! '(")

0F(")0 , d!& q1,x &

K

,+xe#

! !1,x

$(")"#(")&F(")& d! '(")+ $(!)"$(!)

20F(")0 d!,

1 +K

,+xe#

! !1,x

$(")"#(")&F(")& d! '(")+ $(!)"$(!)

20F(")0 d!

.

(3.171)

In particular, in the case of binary reproduction, "1,n(x) 4 0 for all n '= e1, so "1(x) =

"1(x) = "1,e1(x), we have

q1,x =

K

,+xe#

! !1,x

$(")"#(")&F(")& d! '(")

0F(")0 , d!

1 +K

,+xe#

! !1,x

$(")"#(")&F(")& d! '(")

0F(")0 , d!. (3.172)

Example 3.3. In general, the curve ,+x above will be quite complicated, but consider

for a moment the Generalized Moran model, Example 2.1, without mutation, for K = 2.

In that case,

"(N)n (x) =

?

JJ@

JJA

"i if n = ei,

0 otherwise

(3.173)

and

%(N)i (x) = %i(1 + x1 + x2), (3.174)

so that Fi(x) = xi ("i " %i(1 + x1 + x2)).

Assuming X1(0) individuals invade a resident population of type 2 with initial density


x2, then x = (0, x2), and, as t#$,

+tx# x",2 (3.175)

where x",21 = 0 and x",2

2 = %2

'2" 1. Here, ,+

x = {! : /1 = 0, x2 & /2 & x"2}, so F1 4 0 on

,+x . The extinction probability is then

L

M

K x)2x2

e! &x2

$1"#1(1+*)|*($2"#2(1+*))|

d0 '1(1++)+(%2#'2(1++)) d/

1 +K x)2x2

e! &x2

$1"#1(1+*)|*($2"#2(1+*))|

d0 '1(1++)+(%2#'2(1++)) d/

N

O

X1(0)

. (3.176)

Let

'(x2) :=

9 x)2

x2

e! &x2

$1"#1(1+*)|*($2"#2(1+*))|

d0 %1(1 + /)

/("2 " %2(1 + /))d/ (3.177)

=

9 x)2

x2

,,,,

"2 " %2(1 + x2)

"2 " %2(1 + /)

,,,,

$1#2"#1$2$2($2"#2)

,,,,

/

x2

,,,,

$1"#1$2"#2 %1(1 + /)

/("2 " %2(1 + /))d/. (3.178)

After some rearranging, the extinction probability is

!

1 +1

'(x2)

"#X1(0)

, (3.179)

which, as Figure 3.1 shows, provides an excellent match to numerically determined fixa-

tion probabilities, across a range of initial densities, for N as small as 1000.

In contrast, the heuristic argument of [148] gave a fixation probability of e#X1(0)!(x2) , which

arises from Feller’s di!usion approximation to the branching process [55, 56]. The two are

approximately equal for '(x2) large, but the heuristic result consistently underestimates

the fixation probability.

We obtain a particularly intuitive expression for the fixation probability when a single

individual of type 1 invades a resident population at equilibrium, i.e., limN%"

1

NX

(N)2 (0) = x"

2.


Figure 3.1: Fixation probability, &(p) of an advantageous type 1 mutant as a function of itsinitial frequency p. The solid lines give the asymptotic approximations (Equation (3.179)), andthe symbols the exact numerical calculations. The symbols always lie on the correspondingcurve. This is shown for three di#erent initial population densities, n := x1 + x2 = 0.2, 0.4, 0.6.The parameter values are !1 = 1.0, "1 = 0.5,!2 = 1.0, "2 = 0.6, andN = 1000 (reproduced from[148], originally prepared by Christopher Quince).

In this case,

limx2%x)2

'(x2) =1

1"$1#1$2#2

, (3.180)

so, if we set

1 + s =%1

'1%2

'2

, (3.181)

the fixation probability is1

1" s= s+O(s2), (3.182)

echoing Haldane’s result [71]: %i

'iis the expected lifetime reproductive success of type

i absent competition, so ' is the relative fitness of type 1 to type 2, less one, which

corresponds to the classical selection coe"cient. Numerical integration of '(x2) shows

that in contrast to the classical result from population genetics, the intrinsic birth and

death rates contribute independently to the fixation probability, so that two types may

have the same carrying capacity, but di!ering fixation probabilities, contrary to the

arguments made in [165]. In particular, comparing fixation probabilities for two invaders


which have the same fitness relative to the resident population (i.e., same ratio 1 + s),

but di!erent birth and death rates, one finds that the invader with the higher birth rate

(and correspondingly elevated death rate) is more likely to fix in a growing population

(x2 < x",22 ) and less likely to fix in shrinking populations (x2 > x",2

2 ) than the mutant

with lower birth and death rates, as was conjectured by Fisher [59]. We thus see r vs. K

selection in this example of a fecundity-mortality tradeo!. We refer the interested reader

to [148] for further discussion of the biological implications of this result.

3.2.3 Fixation Time Asymptotics When the Resident Dynamics

have an Attracting Fixed Point

??

For a subset I . {1, . . . , K}, we will write RI for the space R|I|, which we identify

with the coordinate hyperplane {x % RK : xi = 0 for all i '% I}, and FI for the vector

field (Fi)i(I on RI , which can be identified with the vector field generated by F on the

hyperplane xi = 0, i '% I. For each I . {1, . . . , K}, RI+ is an invariant subspace for F,

and thus for x % RI+, +t is also a flow map for FI .

In this section, we will be interested in the case when I = I1 := {2, . . . , K} and the

dynamical system corresponding to FI has a fixed point x" that is globally attracting on!

RI1+ . Under this assumption, functions of the form f(+tx) converge exponentially fast,

and are integrable:

Lemma 3.6. Assume that x" is a locally stable fixed point for some vector field G and

x is in the basin of attraction of x", and that f : Rd # R is Lipschitz. Then,

limt%"

9 t

0

f(+ux)" f(x") du (3.183)

exists and is finite. Moreover, regarded as a function of x, it is continuous at x".


Proof. By assumption, x is hyperbolic, so applying the Hartman-Grobman theorem,

there exists a positively invariant neighbourhood U of x" and a di!eomorphism h : U #

Rd such that h(x") = 0 and

h(+tx) = etDG(x))h(x) (3.184)

for all t such that t % U . Moreover, +tx# x", so there exists T > 0 such that t % U for

all t > T . It su"ces to show that the tail

9 t

T

f(+ux)" f(x") du (3.185)

is finite for all t. Now, f is Lipschitz and h is a di!eomorphism, so f 6h#1 is also Lipschitz,

and

,,,,

9 t

T

f(+ux)" f(x") du

,,,,=

,,,,

9 t

T

(f 6 h#1)(h(+ux))" (f 6 h#1)(h(x")) du

,,,,

(3.186)

=

,,,,

9 t

T

(f 6 h#1)(etDG(x))h(x))" (f 6 h#1)(0) du

,,,,

(3.187)

&9 t

T

L33etDG(x))h(x)

33 du (3.188)

& L

9 t

T

Met10 1h(x)1 du (3.189)

&LM

-01h(x)1

(

et10 " eT10)

, (3.190)

where the growth bound for DG,

-0 := limt10

1

tln33etDG(x))

33 = inf{:(*) : * % ((DG(x"))}, (3.191)

is negative, since DG is stable (see e.g., [44], Theorem 3.6). Hence

limt%"

9 t

T

f(+ux)" f(x") du (3.192)


is bounded.

Lastly, if x % U , then we may take T = 0, and, from the previous calculation, we

have,,,,

9 "

0

f(+ux)" f(x") du

,,,,&

LM

|-0|1h(x)1 , (3.193)

which, by the continuity of h, tends to 0 as x# x", proving continuity at x".

In particular, we have "i(+tx) " "i(x") and %i(+tx) " %i(x") are integrable for all i,

and, if we constrict branching processes Zi,x as in Section 3.2.1, then Zi,x is Malthusian

with parameter

fi(x") = "i(x

")" %i(x"). (3.194)

Now, by assumption, the limiting dynamical system is dissipative, so the forward orbit

,+x = {+tx : t , 0} is compact. Thus, "i, ", and %i are bounded, and provided %i > 0

on ,+x , there exists %# such that %i(x)) > %# for all x) % ,+

x . Under these assumptions,

Proposition 3.1 implies qi,x($) < 1 if and only if fi(x") > 0.

Notice that the sign of the values fi(x") also determine the local stability of x". For

x % RId and i % {1, . . . , d},

(!jFi)(x) = xj(!jfi)(x) + %ijfi(x) = %ijfi(x), (3.195)

so that

(DF)(x") =

L

PPPPPPPPPPPPPPM

f1(x")

. . . 0

fd(x")

;(

DFId)

(x")

N

QQQQQQQQQQQQQQO

. (3.196)

By assumption,(

DFId)

(x") is a stable matrix, so (DF)(x") is stable if and only if


fi(x") < 0 for all i % {1, . . . , d}, i.e., if and only if the corresponding branching processes

Zi,x are subcritical.

In particular, if K = 2, and type 1 is strongly selected, then x",1 is globally attracting

on R{1}7!R2

+, whereas x",2 is a saddle point, and we can use Corollary 3.2 and Proposition

3.8 to obtain an asymptotic for the expected time to fixation (i.e., the expected time to

absorption, conditioned on type 1 fixing):

Proposition 3.11. Let . (N) := inf{t > 0 : X(N)1 (t) = 0 5 X

(N)2 (t) = 0}. Then, as

N #$,

E

:

. (N)

,,,,X

(N)2 (. (N)) = 0, X(N)

1 (0) = 1, limN%"

1

NX

(N)2 (0) = x2

;

9!

1

f1(x"2)

+1

f2(x"1)

"

lnN.

(3.197)

Proof. We consider three phases:

(i) X(N)1 (t) < )N ,

(ii) X(N)1 (t) , )N and X

(N)2 (t) , )N , and

(iii) X(N)2 (t) < )N .

In phases (i) and (iii), we will approximate X(N)1 and X

(N)2 by super- and subcritical

branching processes respectively. By assumption, we have f1(x"2) > 0, so for ) > 0

su"ciently small and N su"ciently large,

f(()1,+ := "

(()1,x,+($)" %

(()1,x,#($) > 0 (3.198)

and

f(()1,# := "

(()1,x,#($)" %

(()1,x,+($) > 0, (3.199)

where the rates "(()1,x,± and %

(()1,x,# are defined as in Equation 3.116. Thus, the branching

processes Z(()1,x,± are both supercritical. Similarly, f2(x"

1) < 0, so, for su"ciently small


) > 0 and large N , we have

f(()2,+ := "

(()2,x,+($)" %

(()2,x,#($) > 0 and f (()

2,# := "(()2,x,#($)" %

(()2,x,+($) > 0 (3.200)

where the rates "(()2,x,± and %

(()2,x,# are defined analogously. Hence, the corresponding

branching processes Z(()2,x,± are subcritical. In what follows, we assume that a fixed ) > 0

has been chosen so that all these inequalities hold, and as usual that a neighbourhood of

radius ) of the forward orbit ,+x is contained in Kx.

We now consider phase (i). Recall that

.((,N)1 := inf{t : X(N)

1 (t) , )N} (3.201)

and define

.((,N)1,± := inf{t : Z(()

1,x,±(t) , )N} (3.202)

Now, for t < .((,N)1 ,

Z(()1,x,#(t) & X

(N)1 (t) & Z

(()1,x,+(t), (3.203)

so we must have .((,N)1,+ & .

((,N)1 & .

((,N)1,# and

E

E

.((,N)1,+

,,,Z

(()1,x,+(.

((,N)1,+ ) , )N

F

< E

E

.((,N)1

,,,X

(N)1 (. ((,N)

1 ) , )NF

< E

E

.((,N)1,#

,,,Z

(()1,x,#(.

((,N)1,# ) , )N

F

(3.204)

Then, by Corollary 3.2,

E

E

.((,N)1,±

,,,Z

(()1,x,±(.

((,N)1 ) > )N

F

91

f(()1,±

ln )N. (3.205)


We next consider region (ii). Let

x(() := limN%"

1

NX(N)(. ((,N)

1 ), (3.206)

so x(()1 , ). We begin by considering the time required for the deterministic limiting

process to reach a neighbourhood of the stable attractor x"1. Let

.(()1%2 := inf{t , 0 : +2

tx(() = )}, (3.207)

where +tx = (+1tx,+

2t x), and fix some arbitrary point x) in the trajectory {+tx(() : 0 <

t < .(()1%2}. Then, .

(()1%2 = .

(()1,x!%1 + .

(()1,x!%2, where

.(()1,x!%2 := inf{t , 0 : +2

tx) < )} and .

(()1,x!%1 := inf{t , 0 : +1

#tx) < )}. (3.208)

We start by estimating .(()1,x!%2. Now,

d

dt+2t x

) = +2tx

)f2(+tx)), (3.209)

so

ln+2

*(!)1,x!$2

x) " ln x)2 =

9 *(!)1,x!$2

0

f2(+sx)) dt. (3.210)

Further, as t#$, +tx) # x",1, so applying Lemma 3.6, we see that f2(+tx))" f2(x",1)

is integrable. In particular,

9 *(!)

1,x!$2

0

f2(+sx)) dt =

9 *(!)

1,x!$2

0

f2(+sx))" f2(x

",1) dt+ f2(x",1). (()

1,x!%2, (3.211)

and9 *

(!)

1,x!$2

0

f2(+sx))" f2(x

",1) dt <

9 "

0

f2(+sx))" f2(x

",1) dt <$. (3.212)


Moreover, we have +2

*(!)1,x!$2

= ), so that

.(()1,x!%2 "

1

f2(x",1)ln ) (3.213)

is bounded.

To determine .(()1,x!%1, we first observe +1

#t is the flow of the vector field "F, which

has a stable attractor at x",2. Proceeding as above, we can conclude that

.(()1,x!%1 +

1

f1(x",2)ln ) (3.214)

is bounded. Thus,

.(()1%2 = "

!1

|f2(x",1)|+

1

f1(x",2)

"

ln )+O(1). (3.215)

We complete the analysis of phase (ii) by observing that for fixed ), we may apply

Proposition 2.1, (ii),

limN%"

supt&*

(!)1$2

3333

1

NX(N)(t)" +tx

3333= 0 P1,x " a.s. (3.216)

to conclude that

limN%"

1

NX

(N)2 (. (()

1%2) & ). (3.217)

We are thus in phase (iii), and it only remains to determine the time to extinction of

type 2. As with phase (i) we may bound X(N)2 above and below by subcritical branching

processes Z(()2,x,±, and proceed similarly, observing that

E

E

.((,N)1,+

,,,Z

(()2,x,+(.

((,N)1,+ ) , )N

F

< E

E

.((,N)1

,,,X

(N)2 (. ((,N)

1 ) , )NF

< E

E

.((,N)1,#

,,,Z

(()2,x,#(.

((,N)1,# ) , )N

F

, (3.218)


and, recalling Proposition 3.8, that

E

E

.((,N)1

,,,Z

(()2,x,±(.

((,N)1 ) > )N

F

91,,,f

(()2,±

,,,

ln )N. (3.219)

Combining all the above, we see that for N su"ciently large,

.(()1%2 +

1,,,f

(()1,+

,,,

ln )N +1,,,f

(()2,+

,,,

ln )N & E(1,Nx2)

E

. (N),,,X

(N)2 (. (N)) = 0

F

& .(()1%2 +

1,,,f

(()1,#

,,,

ln )N +1,,,f

(()2,#

,,,

ln )N (3.220)

By definition of f (()i,±, we have that

,,,f1(x",2)" f

(()1,±

,,, = O()), and

,,,f2(x",1)" f

(()2,±

,,, = O())

so that

#

1

f(()1,±

"1

f1(x",2)

%

ln ) # 0, and

#

1

f(()2,±

"1

f2(x",1)

%

ln )# 0 (3.221)

as )# 0. The result follows.

Example 3.4. Applying this result to the Generalized Moran model, Example 2.1, for

which x",i =5

%i

'i" 16

ei, we get the following simple and appealing form for the fixation

time,!

1

%1+

1

%2

"1

%1

'1" %2

'2

lnN.

Note that 1'i

is the expected lifetime of an individual of type i in the absence of compe-

tition, whereas

11'1+ 1

'2

is the harmonic mean of the individual intrinsic death rates (absent competition). In

particular, we can interpret the leading term in the fixation time, 1'1+ 1

'2, as an e!ective

expected individual life span in a mixed population. Thus, measured in units of “e!ec-


tive life spans”, the fixation time depends only on the di!erence in per-capita expected

lifetime reproductive success %1

'1" %2

'2, and not on individual life histories. This is an in-

teresting contrast to the fixation probabilities (Corollary 3.5), which may be understood

via Equation 3.180. There, we saw that if the mutant invades when the wild-type is

already at equilibrium, then the fixation probability was determined by di!erence in per-

capita expected lifetime reproductive success. Now, fixation probabilities are determined

in the very short term, where numbers are small and the resident may still be far from

equilibrium. Here, we have conditioned on non-extinction of the invading type, so we

know it will survive. It will also, crucially, spend considerable time growing from small

to intermediate numbers, and during this period the resident will have long since arrived

at equilibrium. Similarly, the balance of the fixation time will be spent with the final few

wild-type individuals fizzling out while type 1 is at it’s equilibrium density – the proof of

Proposition 3.11 shows that the time of traverse of phase space from )N individuals of

type 1 to )N individuals of type 2 is negligible in comparison to the required to get from

1 to )N individuals of type 1 or )N to no individuals of type 2. Thus, the fixation time

is determined by the dynamics near the unstable attractor x",2 and the stable attractor

x",1.

3.3 Summary

In this chapter, we sought to understand the dynamics of a birth-death-mutation pro-

cess when a single mutant of type 1 invades an established wild-type population. The

crucial tool was Proposition 3.9, which showed that when the number of individuals of

type 1 is less than )N , their numbers can be stochastically bounded above and below

by time-inhomogeneous Markov branching processes. We could thus adapt previously

obtained results on the survival probability of a branching process (Corollary 3.1) to

obtain the probability that type 1 successfully invades, that is, the probability that the


number of individuals of type 1 hits )N prior to extinction. In the last chapter, we saw

that should this happen, the subsequent dynamics are essentially deterministic. We thus

have a relatively complete picture of the dynamics in the strong selection case, where

invasion implies fixation. In particular, we can obtain explicit expressions for the fixation

probabilities (Corollary 3.5) and fixation times (Proposition 3.11) for quite general mod-

els. When the mutant invades a wild-type population at its demographic equilibrium,

we recover Haldane’s classical result, that the fixation probability is approximately the

relative fitness of the invading type. In departure from the classical case, we see that life

history strategy matters in determining fixation probabilities when the wild-type is away

from equilibrium. We find that r vs. K selection occurs, and that among types with the

same expected lifetime reproductive success, those that achieve it through higher birth

rates (and correspondingly elevated death rates) do better when the resident population

has dropped below equilibrium, whereas longer-living types that forego allocation to re-

production in favour of self-preservation are more likely to successfully invade a resident

population near carrying capacity. This has interesting implications for the study of

bottlenecks and the way they shape evolution.

Chapter 4

Weak Selection

In this chapter, we consider the case of weak selection, when the dynamical system

arising as the functional Law of Large Numbers (Proposition 2.1) for a density dependent

birth-death-mutation process X(N)(t) has an attracting embedded submanifold of fixed

points, C. As we observed previously, each point of C is an equilibrium point for the

deterministic dynamics at which two or more types stably coexist. In this chapter, we

will see that, unlike the approximating deterministic process, the stochastic birth-death-

mutation process does not remain at rest, but continues to traverse C until all but one

of the types disappear.

To be precise, we assume that the initial number of individuals of all types is propor-

tional to N , and show that the time-rescaled process

Z(N)(t) :=1

NX(N)(Nt) (4.1)

converges in distribution to a di!usion process Z(t) on C. In fact, we will show that this

is true for a broader class of density dependent processes, of which birth-death-mutation

processes are only a subset.

We then show how the infinitesimal generator of the limiting di!usion may be com-

puted from the transition rates of the original discrete process, and give an explicit

99

Chapter 4. Weak Selection 100

expression for the generic generator when C has co-dimension one.

Both the proof of existence and characterization of the limiting process critically

involve a projection map, "(x). As before, we take +t to be the flow map for the

deterministic dynamical system that arises in the functional Law of Large Numbers.

Given a point x % RK+ ,

"(x) = limt%"

+tx,

so "(x) is the stationary point on C at which the deterministic process comes to rest when

started from an initial point x. We will show that, given a family of density dependent

processes X(N)(t) such that

limN%"

1

NX(N)(t) = x,

the limiting process Z(t) jumps instantaneously from its initial condition Z(0) = x to

"(x), after which it di!uses across C. Knowing the first and second partial derivatives

of " will be crucial in characterizing Z, so we dedicate some e!ort to devising several

means of computing them from the vector field F that defines +t.

We then specialize these results to the case of a weakly selected density dependent

birth-death-mutation processes in co-dimension one. We show that when the process

X(N)(t) is exchangeable, we recover Kimura’s di!usion after an appropriate time-change.

We then consider the consequences of relaxing the various assumptions that constitute

exchangeability, seeing how di!erent individual traits give rise to novel drift terms in the

limiting di!usion, and how they can be interpreted as selection coe"cients. Finally, we

end the chapter by giving explicit expressions for the fixation probabilities, absorption

times, and fixation times for the two-type Generalized Moran model without mutation,

compute the quasi-stationary distribution when mutation between types is possible, and

discuss the potential implications of more biologically realistic life-history strategies for

current approaches to inferring natural selection.


4.1 Convergence to a Di!usion on C

Throughout this section, we assume that X(N) is the family of density dependent pop-

ulation processes corresponding to the family of rate functions {*(N)n (x)}, subject to

Assumptions 2.1 and 2.2. Thus,

F(N)(x) =$

n(ZK

n*(N)n (x)# F(x) =

$

n(ZK

n*n(x) (4.2)

and

#(N)(x) := N$

n(ZK

n(

*(N)n (x)" *n(x)

)

# #(x) (4.3)

uniformly on compact sets.

Throughout this section, we assume that F % C2(RK+ ,R

K) and that the --limit set,

C, of the corresponding dynamical system is a embedded submanifold (possibly with

corners) of RK+ consisting of fixed points of F. By assumption, the dynamical system

corresponding to the vector field F is dissipative, so C is compact.

We also set

M(N)(t) :=1

N

$

n(ZK

nH(N)n (Nt), (4.4)

where H(N)n (t) is a counting process with intensity *

(N)n (X(N)(t)) and H

(N)n is the corre-

sponding centered process. Thus,

Z(N)(t) = Z(N)(0) +

9 t

0+

NF(N)(Z(N)(s")) ds+M(N)(t), (4.5)

while M(N)(t) is a square integrable local martingale with tensor quadratic variation1

!M(N)"(t) =1

N2

$

n(ZK

nn*H(N)n (Nt). (4.6)

1see A.7 for a definition of the tensor quadratic variation and Meyer process


and corresponding Meyer process

22M(N)33(t) =1

N2

9 t

0+

$

n(ZK

nn**(N)n (Z(N)(s")) ds. (4.7)

As always, we write +t for the flow map of the dynamical system

+tx = F(+tx). (4.8)

Set

"(x) := limt%"

+tx, (4.9)

so 4 is a projection onto the --limit set for F, taking the initial state x to the corre-

sponding equilibrium. Note that +t – and thus " – is also C2 [81].

We now turn to the main result of this chapter, which is to characterize the limit of

"(

Z(N)(t))

as N #$. The proof proceeds in several steps:

(i) We begin with a local convergence result, showing that if the time-rescaled process

starts on C (to be precise, Z(N)(0)D"# Z(0) % C), then the process will remain on

C. We proceed by constructing a Lyapunov function, V , for +t in a neighbourhood

of C, and then show that V (Z(N)(t))D"# 0.

(ii) From the previous step, we conclude that "(Z(N)(t))D"# "(Z(t)), and show that

the latter is a di!usion process on C.

(iii) Finally, we observe that when Z(0) '% C, the process Z(t) fails to be Feller, but that

the process

Z(N)(t) := Z(N)(t)" +NtZ(N)(0) + "(Z(N)(0)), (4.10)

which gives the fluctuations about the deterministic trajectory, converges weakly

to a di!usion process on C.

Throughout, we assume that the first and second derivatives of "(x) are known. In the


next section, we will turn to the task of computing these derivatives, given the vector

field F(x).

Definition 4.1. Let U 0 RK+ be a neigbourhood of C. A function V : U # R+ is a

strong Lyapunov function for F if

(i) V (x) , 0 and V (x) = 0 if and only if x % C, and

(ii) There exists a constant * < 0 such that

V (x) := DV (x) · F(x) & *V (x) (4.11)

for all x % U .

If we assume that C is an embedded submanifold, then constructing a strong Lyapunov

function near C is a straightforward generalization of the construction of a Lyapunov

function near a hyperbolic fixed point (see e.g., [70]).

Proposition 4.1. Assume that C is an embedded submanifold of RK. Then, there is a

strong Lyapunov function for F.

Proof. The proof proceeds in two steps. First, we consider a fixed x" % C and construct

a strong Lyapunov function Vx) in a neigbourhood Ux) of x". We then show that if C

is compact, then finitely many such functions may be pieced together to form a global

Lyapunov function.

Let "x) be a slice chart defined in a neigbourhood Ux) of x", so "x)(x") = 0 and

"d+1x) (x) = · · · = "d+n

x) (x) = 0 for all x % C / Ux) . (4.12)

Let

G(x) := (D"x))("#1x) (x))F("

#1x) (x)). (4.13)

Then,


(i) G(0) = 0,

(ii) (DG)(0) is similar to (DF)(x"), with change of basis matrix (D"x))(x"), and

(iii) Ker(DG)(0) = {x % RK : xd+1 = · · ·xn = 0}.

Moreover, without loss of generality, "x) may be chosen so that (DG)(0) is in real Jordan

canonical form, i.e.,

(DG)(0) =

R

SSSSSSST

J0

J1

. . .

Jp

U

VVVVVVVW

, (4.14)

where J0 is a d( d matrix of zeroes and J1, . . . , Jp are non-zero di ( di Jordan blocks.

We introduce a further change of coordinates. Suppose Ji corresponds to a real

eigenvalue *i,

Ji =

R

SSSSSSST

*i 1

*i. . .

. . . 1

*i

U

VVVVVVVW

. (4.15)

Then set

Qi :=

R

SSSSSSST

)#1

)#2

. . .

)#di

U

VVVVVVVW

. (4.16)

If Ji corresponds to a complex conjugate pair of eigenvalues, *i ± i-i, i.e.,

Ji =

R

SSSSSSST

(i I222

(i. . .

. . . I222

(i

U

VVVVVVVW

(4.17)


for (i =

R

ST

*i "-i

-i *i

U

VW, then let

Qi :=

R

SSSSSSST

)#1I222

)#2I222

. . .

)#di2 I222

U

VVVVVVVW

. (4.18)

Finally, let

Q :=

R

SSSSSSST

Id2d

Q1

. . .

Qp

U

VVVVVVVW

(4.19)

and let G(x) := QG(Q#1x). Then (DG)(0) is a block-diagonal matrix with blocks

QiJiQ#1i =

R

SSSSSSST

*i )

*i. . .

. . . )

*i

U

VVVVVVVW

, (4.20)

or

QiJiQ#1i =

R

SSSSSSST

(i )I222

(i. . .

. . . )I222

(i

U

VVVVVVVW

(4.21)

if the corresponding eigenvalues are real or complex, respectively. Let mi = d+0#1

j=1 dj.


If Ji corresponds to a real eigenvalue, then

5

(DG)(0)x6

mi+j=

?

JJ@

JJA

*ixmi+j + )xmi+j+1 if j < di

*ixmi+di otherwise,

(4.22)

whereas if Ji corresponds to a complex eigenvalue,

5

(DG)(0)x6

mi+j=

?

JJJJJJJJJJ@

JJJJJJJJJJA

*ixmi+j " -ixmi+j+1 + )xmi+j+2 if j = 2k " 1 < di " 1

-ixmi+j#1 + *ixmi+j + )xmi+j+2 if j = 2k < di

*ixmi+di#1 " -ixmi+di if j = di " 1

-ixmi+di#1 + *ixmi+di if j = di.

(4.23)

Now, let

Vx)(x) =n$

i=d+1

x2i . (4.24)

We claim that Vx)(x) is a Lyapunov function for G(x). Firstly,

DVx)(x) · G(x) =n$

j=d+1

xjGj (4.25)

can be considered as the sum of terms corresponding to the blocks QiJiQ#1i . If if Ji


corresponds to a real eigenvalue,

di$

j=1

xmi+jGmi+j(x) = *i

di$

j=1

x2mi+j + )

di#1$

j=1

xmi+jxmi+j+1 +O(

|x|3)

=5

*i ")

2

6(

x2mi+1 + x2

mi+di

)

+ (*i " ))di$

j=1

x2mi+j +

)

2

di#1$

j=2

(xmi+j + xmi+j+1)2 +O

(

|x|3)

&5

*i ")

2

6(

x2mi+1 + x2

mi+di

)

+ (*i " ))di$

j=1

x2mi+j + )

di#1$

j=2

(

x2mi+j + x2

mi+j+1

)

+O(

|x|3)

=5

*i +)

2

6(

x2mi+1 + x2

mi+di

)

+ (*i + ))di$

j=1

x2mi+j +O

(

|x|3)

(4.26)

Proceeding similarly, the same inequality holds for terms arising from a block correspond-

ing to a complex eigenvalue. In particular, if we take max{*1, . . . ,*r} < *x) < 0 and

choose ) > 0 such that (*i + )) < *x) , then

n$

j=d+1

xjGj(x) < *x)

di$

j=1

x2mi+j +O

(

|x|3)

. (4.27)

Lastly, since *x)Vx)(x) < 0 for all x '% C, we see that for x in a su"ciently small

neighbourhood V of 0, DVx)(x) · G(x) & *x)Vx)(x).

Now, let Ux) = Ux) /"#1x) (Q

#1V) and Vx)(x) = Vx)(Q"x)(x)). Clearly, Vx)(x) = 0 if

and only if x % Ux) / C. Direct calculation shows that

DVx)(x) · F(x) & *x)Vx)(x) for all x % Ux) . (4.28)

To complete the proof, take a family {(Ux) , Vx))}x)(C of strong Lyapunov functions

along C, and for each x" % C let +x)(x) be bump function vanishing o! Ux) and equal to

1 on a neighbourhood Wx) 0 Ux) of x". Since C is compact, {Wx)}x)(C admits a finite


subcover {Wx)1, . . . ,Wx)s}. Then

V (x) = limy%x

10s

j=11

)x)j(y)Vx)

j(y)

(4.29)

is a strong Lyapunov function on the neighbourhood U =Xs

j=1Wx)jof C for * =

max{*x)1, . . . ,*x)s}.

Remark 4.1. A su"cient condition for C to be an embedded manifold of co-dimension d,

which will be adequate for the examples contained herein, is the existence of a submersion

f : RK # RN#d such that C = f#1(0). To see this, consider x" % C. Reordering the

coordinates, if necessary, we can assume that !(f1,...,fN"d)!(xd+1,...,xN )(x

") is invertible, so

"x)(x) = (x1 " x"1, . . . , xd " x"

d, f(x)) (4.30)

has rank N at x", and thus has a di!erentiable inverse in a neighbourhood Ux) , yielding

the desired slice chart.

Remark 4.2. If f : RK # RK is not a submersion, its zero set need not be an embedded

submanifold, e.g., let f : R2 # R2 be given by

fi(x) = ci(x21 " x3

2), (4.31)

for fixed constants c1, c2. Then f#1(0) has a cusp at 0, and is thus not an embedded

submanifold.

Now, recall from Section 2.1.2 that, for x %!

RK+ , Kx is assumed to be a compact set

containing a neighbourhood of the forward orbit ,+x , and Lx > 0 is a constant such that

1F(x1)" F(x2)1 < Lx 1x1 " x21 (4.32)

for all x1,x2 % Kx.


Corollary 4.1. Let V : U # R+ be a strong Lyapunov function for F with corresponding

constant *, and fix r, r) and s such that 0 < r) < |2|(1#s)4L2

x,!and r) < r < s < 1. Then

P

+,,,V5

Y(N)5

(1#s)4L2

x,!lnN66,,, > N#r!

-

< N#r (4.33)

for N su!ciently large.

Proof. From the previous proposition, we have

d

dtV (+tx) & *V (+tx) (4.34)

for all x % U .

Choose % > 0 su"ciently small that C' := V #1([0, %]) . U . By definition, +tx is

eventually in C' for all x %!

RK+ . Thus, there exists T',x such that +tx % U for all t , T',x.

Then, using Gronwall’s inequality, we have

V (+tx) & V (+T#,xx)e2(t#T#,x) for t > T',x (4.35)

In particular, for N su"ciently large, (1#s)4L2

x,!lnN > T',x, and

V

!

+ (1"s)

4L2x,!

lnNx

"

& V (+T#,xx)e#2T#,xN

+(1"s)

4L2x,! . (4.36)

Now, V is locally Lipschitz on C', so there exists L' such that |V (x)" V (y)| & L' 1x" y1

for all x,y % C'. Thus,

,,,V5

Y(N)5

(1#s)4L2

x,!lnN66,,, &,,,,V5

Y(N)5

(1#s)4L2

x,!lnN66

" V

!

+ (1"s)

4L2x,!

lnNx

",,,,+

,,,,V

!

+ (1"s)

4L2x,!

lnNx

",,,,

& L'

3333Y(N)

5(1#s)4L2

x,!lnN6

" + (1"s)

4L2x,!

lnNx

3333+ V(

+T#,xx)

e#2T#,xN+(1"s)

4L2x,! . (4.37)


The result then follows from Proposition 2.1, as

P

23333Y(N)

!(1" s)

4L2x,(

lnN

"

" + (1"s)

4L2x,!

lnNx

3333> N#r

4

< N#r. (4.38)

From the Corollary, we see that as as N # $ the process will come arbitrarily

close to the manifold of fixed points . We next show that in the limit, the process is

then e!ectively confined to the --limit set; in the next section, we will characterize the

process there.

Theorem 4.1. Let V : U # R+ be a strong Lyapunov function for F with constant *,

and assume Z(N)(0)D"# Z(0) % C. Then,

V (Z(N)(t))D"# 0 (4.39)

in DR[0,$).

Remark 4.3. It follows immediately [17] that V (Z(N))P"# 0 as N #$.

Proof. Choose % > 0 su"ciently small that C' := V #1([0, %]) . U , and define a family of

stopping times

T(N)' = inf{t > 0 : V (Z(N)(t)) > %}. (4.40)

We will show that the stopped processes V (Z(N)(t 5 T(N)' )) converge in L2 for t % [0, T ]

for any T > 0. Weak convergence to 0 in DR[0, T ] is immediate, whilst weak convergence

in DR[0,$) follows from Theorem 3) in [128]. Thus (see e.g., [17])

V (Z(N)(t 5 T(N)' ))

P"# 0 (4.41)


as N #$ also. Given this, the theorem follows immediately, as

P

+

t > T(N)'

-

= P

+

V (Z(N)(t 5 T(N)' )) > %

-

# 0 as N #$. (4.42)

Applying Ito’s formula [161] to the pure jump process e#N2tV (Z(N)(t 5 T(N)' )) and

rearranging yields

V (Z(N)(t 5 T(N)' )) = eN2tV (Z(N)(0))

+

9 t,T (N)#

0+

NeN2(t#s)(

DV (Z(N)(s")) · F(N)(Z(N)(s"))" *V (Z(N)(s")))

ds

+

9 t,T (N)#

0+

eN2(t#s)DV (Z(N)(s")) · dM(N)(s) + )(N)(t)

= eN2tV (Z(N)(0))+

9 t,T (N)#

0+

NeN2(t#s)(

DV (Z(N)(s")) · F(Z(N)(s"))" *V (Z(N)(s")))

ds

+

9 t,T (N)#

0+

NeN2(t#s)DV (Z(N)(s")) ·(

F(N)(Z(N)(s"))" F(Z(N)(s")))

ds

+

9 t,T (N)#

0+

eN2(t#s)DV (Z(N)(s")) · dM(N)(s) + )(N)(t 5 T(N)' ) (4.43)

where

)(N)(t) =$

0<s&t

G

V (Z(N)(s))" V (Z(N)(s"))"DV (Z(N)(s")) ·#Z(N)(s)H

. (4.44)

Now, V is a strong Lyapunov function and Z(N)(t) % U for t & T(N)' , so DV (Z(N)(s")) ·

F(Z(N)(s"))" *V (Z(N)(s")) & 0 and

0 & V (Z(N)(t 5 T(N)' )) & V (Z(N)(0))

+

9 t,T (N)#

0+


F(N)(Z(N)(s"))" F(Z(N)(s")))

ds

+

9 t,T (N)#

0+

eN2(t#s)DV (Z(N)(s")) · dM(N)(s) + )(N)(t 5 T(N)' ). (4.45)


In particular, applying the Cauchy–Bunyakovsky–Schwarz inequality, we have

0 & V (Z(N)(t 5 T(N)' ))2 & 4

.

V (Z(N)(0))2

+

#9 t,T (N)

#

0+

NeN2((t,T (N))#s)DV (Z(N)(s")) ·(

F(N)(Z(N)(s"))" F(Z(N)(s")))

ds

%2

Y Z[ \

➀

+

#9 t,T (N)

#

0+

eN2((t,T (N))#s)DV (Z(N)(s")) · dM(N)(s)

%2

Y Z[ \

➁

+5

)(N)(t 5 T(N)' )62/

. (4.46)

Applying Jensen’s inequality, we have

➀ & (t 5 T(N)' )

9 t,T (N)#

0+


F(N)(Z(N)(s"))" F(Z(N)(s")))2

ds.

(4.47)

Note that for t & T(N)' , Z(N)(t) % C', a compact set. By Assumption 2.1 and 2.2,

F(N)(x)"F(x) = O(

1N

)

uniformly on on compacts, so ➀ is bounded above by a constant

multiple of

(t 5 T(N)' )

9 t,T (N)

0+

e2N2(s#(t,T (N)) ds =1

2N*

5

eN2(t,T (N)# ) " 1

6

, (4.48)

i.e., supt&T

➀ &C1,'T

2N |*|for some constant C1,'.

Using Doob’s Inequality, we have

E

:

supt&T

➁

;

& E

R

T

#9 T,T (N)

#

0+

eN2((T,T (N)# )#s)DV (Z(N)(s")) · dM(N)(s)

%2U

W

= E

.9 T,T (N)

#

0+

e2N2((T,T (N)# )#s)

K$

i=1

K$

j=1

DiV (Z(N)(s"))DjV (Z(N)(s"))d2M (N)i ,M

(N)j 3(s)

/

(4.49)


Moreover,

2M (N)i ,M

(N)j 3(t) =

9 t

0+

$

n

ninj*(N)n (Z(N)(s")) ds (4.50)

and by Assumption 2.1,

supx(K

$

n(ZK

1n122 *(N)n (x) (4.51)

is uniformly bounded on C', so for t & T(N)' , there exists a constant C' such that

2M (N)i ,M

(N)j 3(t) < C't. and a constant C2,' such that

E

:

supt&T

➁

;

&C2,'

2N |*|. (4.52)

Lastly, Taylor’s theorem gives

V (Z(N)(s))" V (Z(N)(s"))"DV (Z(N)(s")) ·#Z(N)(s)

=K$

i=1

K$

j=1

!i!jV ($)#Z(N)i (s)#Z

(N)j (s) (4.53)

for some $ such that33Z(N)(s")" $

33 &

33#Z(N)(s)

33 & 1

N, so, as before, there is a

constant C such that V (Z(N)(s))" V (Z(N)(s"))"DV (Z(N)(s")) ·#Z(N)(s) is bounded

above by

C

K$

i=1

K$

j=1

,,,#Z

(N)i (s)

,,,

,,,#Z

(N)j (s)

,,, & CK

K$

i=1

,,,#Z

(N)i (s)

,,,

2

. (4.54)

Thus,

,,)(N)(t)

,,2 & C2K2

#

$

0<s&t

K$

i=1

eN2(s#t),,,#Z

(N)i (s)

,,,

2%2

= C2K2

#K$

i=1

9 t

0+

eN2(s#t) d[M (N)i ](s)

%2

(4.55)


Recalling (4.4). (4.6) and (4.7), we have

[M (N)i ](t) = 2M (N)

i 3(t) +M(N)i (t) (4.56)

so that

,,)(N)(t)

,,2 & 2C2K3

K$

i=1

.!9 t

0+

eN2(s#t) dM(N)i (s)

"2

+

!9 t

0+

eN2(s#t) d2M (N)i 3(s)

"2/

(4.57)

and, proceeding as as above, there exists a constant CT,' such that

E

R

T supt&T,T (N)

#

,,)(N)(t)

,,2

U

W &CT,'

2N |*|. (4.58)

Combining all the above, we see that for any T > 0,

E

R

T supt&T,T (N)

#

,,V (Z(N)(t))

,,2

U

W = O(

1N

)

. (4.59)

The result follows.

Corollary 4.2. If Z(N)(0)D"# Z(0) % C, then Z(N) " "(Z(N))

D"# 0 for all t.

4.1.1 Di!usion Process

In this section, we characterize the limiting process on C. We begin by introducing some

notation that will simplify subsequent expressions. For f : RK # R, let

(D2f)(x)2X,Y3 = Y*(D2f)(x)X for X,Y % RK , (4.60)


and, if f : RK # RK , let

(D2f)(x)2X,Y3 =

R

SSSST

(D2f1)(x)2X,Y3...

(D2fK)(x)2X,Y3

U

VVVVW

, (4.61)

so that f has Taylor expansion

f(y) = f(x) + (Df)(x)(y " x) + (D2f)(x)2y " x,y " x3+O(

1y " x13)

. (4.62)

Further, following [43], given semimartingales X(t) and Y(t), taking values in a subset

of Rd, we adopt the notation

9 t

0+

(D2f)(X(s"))2dY(s), dY(s)3 :=9 t

0+

d$

i=1

d$

j=1

(!i!jf)(X(s)) d[Yi, Yj](s) (4.63)

and write9 t

0+

(D2f)(X(s"))2dY(s), dY(s)3 (4.64)

for the vector-valued process with components

9 t

0+

d$

i=1

d$

j=1

!i!jfk(X(s")) d[Yi, Yj](s). (4.65)

With this notation, Ito’s formula becomes

f(Y(t)) = f(Y(0)) +

9 t

0

(Df)(Y(s")) dY(s) +1

2

9 t

0

(D2f)(X(s"))2dY(s), dY(s)3

+$

s&t

f(Y(s))" f(Y(s"))" (Df)(Y(s"))#Y(s)"1

2(D2f)(Y(s"))2#Y(s),#Y(s)3.

(4.66)

Theorem 4.2. Let {Wn : n % ZK} be independent Brownian motions. Assume that


Z(N)(0)D"# Z(0) % C and that that the solution to

Z(t) = Z(0) +

9 t

0+

(D")(Z(s"))#(Z(s")) +$

n(ZK

1

2(D2")(Z(s))2n,n3*n(Z(s")) ds

+ (D")(Z(s"))

#

$

n(ZK

n

9 t

0+

]

*n(Z(s")) dWn(s)

%

(4.67)

exists and is unique. Then Z(N)(t))D"# Z(t).

Proof. We begin by recalling that

Z(N)(t) = Z(N)(0) +

9 t

0

NF(N)(Z(N)(s")) ds+1

N

$

n(ZK

nHn(Nt). (4.68)

We start with the martingale terms. For each n % ZK , let

W (N)n =

1

N

9 t

0+

dH(N)n (Ns)

]

*(N)(Z(N))(s"))(4.69)

so that

1

NHn(Nt) =

9 t

0+

^

*(N)n (Z(N)(s")) dW (N)

n (s), (4.70)

whilst

[W (N)n ]t =

1

N2

9 t

0+

d[H(N)n ]Ns

*(N)(Z(N))(s"))(4.71)

=1

N2

9 t

0+

dH(N)n (Ns)

*(N)(Z(N))(s")), (4.72)

which, decomposing H(N)n into a local martingale and a finite variation components

(Equation 2.9), can be written

=

9 t

0+

ds+1

N

9 t

0+

dH(N)n (Ns)

*(N)(Z(N))(s")). (4.73)


Now,

E

R

T

#

1

N

9 t

0+

dH(N)n (Ns)

*(N)(Z(N))(s"))

%2U

W =1

N2E

.9 t

0+

d[H(N)n ]Ns

(*(N)(Z(N))(s")))2

/

(4.74)

=1

NE

:9 t

0+

ds

*(N)(Z(N))(s"))

;

, (4.75)

so [W (N)n ]t # t as N # $. Further, since Hn, Hm are orthogonal for n '= m, we have

[Wn,Wm]t = 0. Since each W(N)n (t) is a martingale with jumps tending to 0 as N #$,

it follows from the Martingale central limit theorem that5

W(N)n

6

n

D"# (Wn)n, where

the Wn are independent Brownian motions.

We then have, using Ito’s formula,

Z(N)(t) = "(Z(N)(t)) +(

Z(N)(t)" "(Z(N)(t)))

= "(Z(N)(0))+

9 t

0

(D")(Z(N)(s")) dZ(N)(s)+1

2

9 t

0

(D2")(Z(N)(s"))2dZ(N)(s), dZ(N)(s)3

+ #(N)(t) +(

Z(N)(t)" "(Z(N)(t)))

= "(Z(N)(0)) +

9 t

0

N(D")(Z(N)(s"))F(N)(Z(N)(s")) ds

+

9 t

0+

(D")(Z(N)(s"))

#

$

n(ZK

^

*(N)n (Z(N)(s")) dW (N)

n (s)

%

+1

2

9 t

0

(D2")(Z(N)(s"))2dZ(N)(s), dZ(N)(s)3+ #(N)(t) +(

Z(N)(t)" "(Z(N)(t)))

= Z(N)(0) +(

"(Z(N)(0))" Z(N)(0))

+

9 t

0

(D")(Z(N)(s"))(

NF(Z(N)(s")) + #(N)(Z(N)(s")))

ds

+

9 t

0+

(D")(Z(N)(s"))

#

$

n(ZK

^

*(N)n (Z(N)(s")) dW (N)

n (s)

%

+1

2

9 t

0

(D2")(Z(N)(s"))2dZ(N)(s), dZ(N)(s)3+ #(N)(t) +(

Z(N)(t)" "(Z(N)(t)))

(4.76)


where

#(N)(t) :=$

s&t

"(Z(N)(s))" "(Z(N)(s"))" (D")(Z(N)(s"))#Z(N)(s)

"1

2(D2")(Z(N)(s"))2#Z(N)(s),#Z(N)(s)3 (4.77)

Thus,

Z(N)(t) = Z(N)(0) +

9 t

0+

(D")(Z(N)(s"))#(Z(N)(s"))

+$

n(ZK

1

2(D2")(Z(N)(s))2n,n3*n(Z

(N)(s")) ds

+ (D")(Z(N)(s"))

#

$

n(ZK

n

9 t

0+

^

*n(Z(N)(s")) dW (N)n (s)

%

+ )(N)(t) (4.78)

where

)(N)(t) :=(

"(Z(N)(0))" Z(N)(0))

Y Z[ \

➀

+

9 t

0

N(D")(Z(N)(s"))F(Z(N)(s")) dsY Z[ \

➁

+

9 t

0

(D")(Z(N)(s"))(

5(N)(Z(N)(s"))" #(Z(N)(s")))

ds

Y Z[ \

➂

+

9 t

0+

(D")(Z(N)(s"))

#

$

n(ZK

^

*(N)n (Z(N)(s"))"

^

*n(Z(N)(s"))

%

dW (N)n (s)

Y Z[ \

➃

+1

2

9 t

0

(D2")(Z(N)(s"))2dZ(N)(s), dZ(N)(s)3 "$

n(ZK

1

2(D2")(Z(N)(s))2n,n3*n(Z

(N)(s")) ds

Y Z[ \

➄

+ #(N)(t)Y Z[ \

➅

+(

Z(N)(t)" "(Z(N)(t)))

Y Z[ \

➆

(4.79)


Fix B > 0 and let

T(N)B := inf{t > 0 : 1Z(N)(t)1 > B}. (4.80)

Then, using Theorem 5.4 in [120] (we use the theorem as it is stated in [115]), provided

)(N)(t5T (N)B )

D"# 0, Z(N)(t5T (N)B ) converges weakly to a local solution of (4.67) stopped

at

TC := inf{t > 0 : 1Z(t)1 > C}, (4.81)

for any C < B. Moreover, if (4.67) has a unique global solution Z, then Z(N) D"# Z.

Thus, our our result follows upon showing that )(N)(t 5 T(N)B )

D"# 0. We will show that

the terms individually converge weakly to 0, and thus in probability also, from which we

can conclude the sum converges weakly to 0 as well.

We begin by observing that ➀D"# 0 by assumption.

Next, observe that by definition, "(+tx) = "(x), whence

0 =d

dt"(+tx) = (D")(+tx)F(+tx) (4.82)

for all t. In particular, (D")(x) · F(x) = 0 and thus, ➁ is identically zero.

For t < T(N)B , Z(N)(t) is compactly contained, so ➂ converges uniformly to 0 by

Assumptions 2.1 and 2.2.

Using Doob’s inequality, we have

E

.

supt&T

9 t,T (N)B

0+

(D")(Z(N)(s"))

#

$

n(ZK

^

*(N)n (Z(N)(s"))"

^

*n(Z(N)(s"))

%

dW (N)n (s)

/

& E

R

T

#9 T,T (N)

B

0+

(D")(Z(N)(s"))

#

$

n(ZK

^

*(N)n (Z(N)(s"))"

^

*n(Z(N)(s"))

%

dW (N)n (s)

%2U

W

& E

R

T

9 T

0+

#

(D")(Z(N)(s"))

#

$

n(ZK

^

*(N)n (Z(N)(s"))"

^

*n(Z(N)(s"))

%%2

d[W (N)n ]s

U

W .

(4.83)


We observed above that [W (N)n ]t

P"# t, whilst the integrand converges uniformly to 0 on

compact sets, so ➃ converges to 0.

Next, we observe that

9 t

0

(D24k)(Z(N)(s"))2dZ(N)(s), dZ(N)(s)3

=1

N2

9 t

0+

$

n(ZK

K$

i=1

K$

j=1

(!i!j4k)(Z(N)(s"))ninj d

E

H(N)n

F

Ns

=1

N2

9 t

0+

$

n(ZK

(D24k)(Z(N)(s"))2n,n3 d

E

H(N)n

F

Ns

(4.84)

whilst

1

N2

E

H(N)n

F

Nt=

9 t

0

*(N)n (Z(N)(s")) ds+

1

N2H(N)

n (Nt). (4.85)

Thus

➄ =1

N2

9 t

0+

$

n(ZK

1

2(D2")(Z(N)(s"))2n,n3 dH(N)

n (Ns)

+

9 t

0+

$

n(ZK

1

2(D2")(Z(N)(s"))2n,n3

(

*(N)n (Z(N)(s"))" *n(Z

(N)(s")))

ds. (4.86)

Now, D2" is bounded on compact sets, so

1

N2supt&T

,,,,,

9 t,T (N)B

0+

$

n(ZK

1

2(D2")(Z(N)(s"))2n,n3 dH(N)

n (Ns)

,,,,,

& sup{z:0z02&B}

(D2")(z)$

n(ZK

1n122 supt&T

,,,H(N)

n (N(t 5 T(N)B ))

,,,

&1

Nsup

{z:0z02&B}(D2")(z)

($

n(ZK

1n122 supt&T

#,,,,

1

NH(N)

n (N(t 5 T(N)B ))

,,,,+

,,,,,

9 t,T (N)B

0

*(N)n (Z(N)(s")) ds

,,,,,

%

. (4.87)


We are unable to use Doob’s inequality here without making stronger assumptions on

the summability of the *(N)n . Instead, we adapt the approach of Theorem 11.2.1 in [50],

and use the functional Law of Large Numbers for the counting process:

limN%"

supt&T

,,,,,

1

NH(N)

n (N(t 5 T(N)B ))"

9 t,T (N)B

0

*(N)n (Z(N)(s")) ds

,,,,,= 0. (4.88)

In particular,

$

n(ZK

1n122 limN%"

supt&T

#,,,,

1

NH(N)

n (N(t 5 T(N)B ))

,,,,+

,,,,,

9 t,T (N)B

0

*(N)n (Z(N)(s")) ds

,,,,,

%

=$

n(ZK

1n122 29 T,T (N)

B

0

*(N)n (Z(N)(s")) ds, (4.89)

which is, by assumption, convergent, so we can interchange sum and limit to conclude

that

limN%"

1

N2supt&T

,,,,,

9 t,T (N)B

0+

$

n(ZK

1

2(D2")(Z(N)(s"))2n,n3 dH(N)

n (Ns)

,,,,,= 0. (4.90)

The second integral,

9 t,T (N)B

0+

$

n(ZK

1

2(D2")(Z(N)(s"))2n,n3

(

*(N)n (Z(N)(s"))" *n(Z

(N)(s")))

ds, (4.91)

vanishes since(

*(N)n (z)" *n(z)

)

# 0 (4.92)

uniformly on compact sets. Thus ➄D"# 0.


Lastly, from Taylor’s Theorem, we have

4k(x+#x)" 4k(x)"K$

i=1

(!i4k)(x)#xi(s)

=1

2

$

1&i,j&K

#xi#xj

9 1

0

(1" t)2(!i!j4k)(x + t#x) dt (4.93)

so that

#(N)k (t) =

$

0<s&t

21

2

$

1&i,j&K

#Z(N)i (s)#Z

(N)j (s)

(9 1

0

(1" t)2(!i!j4k)(Z(N)(s") + t#Z(N)(s))" (!i!j4k)(Z

(N)(s")) dt4

(4.94)

and

,,,#

(N)k (t)

,,, &$

0<s&t

21

2

$

1&i,j&K

#Z(N)i (s)#Z

(N)j (s)

sup{h:0h0& 1

N}

,,(!i!j4k)(Z

(N)(s") + h)" (!i!j4k)(Z(N)(s"))

,,

4

. (4.95)

By assumption, (!i!j4k)(x) is continuously di!erentiable, and thus locally Lipschitz con-

tinuous, so,,,#

(N)k (t)

,,,-

1

N[Z(N)

k ](t) =1

N3

$

n(ZK

n2k[H

(N)n ]Nt, (4.96)

so,,,#

(N)k (t)

,,, weakly converges to 0 as N #$, by arguments similar to those above.

Finally, ➆D"# 0 as a consequence of Corollary 4.2, completing the proof.

Remark 4.4. Equivalently, if we let

b(x) := (D")(x)#(x) +$

n(ZK

1

2(D2")(x)2n,n3*n(x) (4.97)

a(x) :=$

n(ZK

nn**n(x) (4.98)


and

a(x) := (D")(x)a(D")*(x) (4.99)

then, provided Z(N)(0)D"# Z(0) % C, Z(t) has generator

(G&)(x) =K$

i=1

bi(x)!i&(p) +1

2

K$

i=1

K$

j=1

aij(x)!i!j&(x), (4.100)

for & % C2(C).

Remark 4.5. We refer the interested reader to [46, 47] for a discussion of existence and

uniqueness of solutions of the Cauchy problem for the infinitesimal generator, (4.100).

Remark 4.6. Note that the assumption that Z(N)(0)D"# Z(0) % C is necessary to obtain

weak convergence in the results above; Propositions 2.1 and Theorem 4.1 tell us that in

general,

limN%"

Z(N)(0+) = "(Z(0)) '= Z(0). (4.101)

Thus, the limiting process has a jump discontinuity at t = 0 fails to satisfy the Feller

property. We may still obtain a global convergence result, by considering a variant of the

original process.

Corollary 4.3. Let

Z(N)(t) := Z(N)(t)" +NtZ(N)(0) + "(Z(N)(0)). (4.102)

Then Z(N) D"# Z, a di"usion process satisfying (4.67) above.

Remark 4.7. We introduce Z(N) to avoid technical di"culties arising from the disconti-

nuity above. Intuitively Z(N) is obtained by removing the “deterministic part”, +Ntx,

from Z(N), and then shifting the resulting process so that it starts at "(Z(N)(0)).


Proof. Let )N = 1#s4L2

x

lnNN

, where 0 < r) < r < s < 1 are as in Proposition 2.1 and

Corollary 4.1. From those results, we have

P{ supt&(N

,,Z(N)(t)" +NtZ

(N)(0),, > N#r} & N#r, (4.103)

while

P{V (Z(N)()N) > N#r!} < N#r. (4.104)

In particular Z(N)()N)D"# "(Z(0)) % C. Now, consider the process

Z(N)(t) = Z(N)(t+ )N). (4.105)

By the above, Z(N)(0)D"# Z(0) % C, while Z(N) satisfies (4.5). Thus, we may apply

Theorem 4.2 to conclude that Z(N) D"# Z, where the limiting process satisfies (4.67).

Moreover,

Z(N)(t + )N)" Z(N)(t) = +Nt+N(NZ(N)(0)" "(Z(N)(0)). (4.106)

Since N)N # $ as N # $, the right hand side converges to 0 uniformly in t, so we

must have Z(N)(·+ )N)" Z(N) D"# 0.

Lastly, define ,(N) : R+ # R+ by

,(N)(t) :=

?

JJ@

JJA

0 0 & if t & )N

t" )N otherwise.

(4.107)

Then, ,(N)(t)# t uniformly in N , while

Z(N)(,(N)(t)) =

?

JJ@

JJA

Z(N)(0) 0 & if t & )N

Z(N)(t) otherwise.

(4.108)

Now, the map (f, g)# f 6 g : DC[0,$)( DR+ [0,$)# DC[0,$) is continuous at points


where the pair f, g are continuous. Since Z is continuous, we may apply the continuous

mapping theorem to conclude that Z(N) 6 ,(N) D"# Z as N #$; on the other hand,

supt

,,,Z(N)(t)" Z(N)(,(N)(t))

,,, = sup

t&(N

,,,Z(N)(t)" Z(N)()N)

,,, (4.109)

which converges uniformly to 0. Thus Z(N) " Z(N) D"# 0, and the result follows.

We will henceforth focus on the process Z.

4.2 Computing the Derivatives

In practice, to use Theorem 4.2, it will be necessary to explicitly compute the partial

derivatives of "(x). In this section, we consider several approaches. The first, which

we present in below, uses Duhamel’s principle to characterize the derivatives of " in

terms of F. While generally applicable in principle, it proves unwieldy in practice, so we

conclude this section with a second approach that uses conservation laws to characterize

the derivatives, which proves useful when the functional Law of Large Numbers for the

process of interest is of Lotka-Volterra-Gause type.

4.2.1 Variational Formulation

We begin by observing that

+tx =d

dtF(+tx) = (DF)(+tx)+tx, (4.110)

so that

(D")(x) = limt%"

e#! t0 (DF)()sx) ds. (4.111)


In particular, if x" % C, then +tx" = x" for all t and

(D")(x") = limt%"

e#t(DF)(x)). (4.112)

Let Px) be an invertible matrix such that (DF)(x") is in real Jordan canonical form

i.e., (DF)(x") = Px)AP#1x) , for

A =

R

SSSSSSST

J0

J1

. . .

Jp

U

VVVVVVVW

. (4.113)

where, as in the proof of Proposition 4.1, J0 = 0d2d and Ji is a di ( di Jordan block

corresponding to the ith eigenvalue. Then, e#t(DF)()s(x))) = Px)etAP#1

x) , while

etA = etJ0 < · · · etJp , (4.114)

where < is the direct sum of linear operators.

Now, etJ0 = Id2d, while if Ji corresponds to a real eigenvalue, then

etJi =

R

SSSSSSST

et2i tet2i · · · tdidi!et2i

et2i. . .

...

. . . tet2i

et2i

U

VVVVVVVW

. (4.115)

so that limt%" etJi = 0di2di, and the same holds if Ji corresponds to a complex eigenvalue,

since we have assumed that x" is locally asymptotically stable, and thus (DF)(x") is a

stable matrix. Then,

limt%"

etA = Id2d < 0K#d2K#d, (4.116)


i.e., (D")(x") is a projection onto Tx)C and may be computed explicitly:

Lemma 4.1. Let Px) be a change of basis matrix such that P#1x) (DF)(+s(x"))Px) is in

real Jordan canonical form. Then (D")(x") = Px)(Id2d < 0K#d2K#d)P#1x) .

Equivalently, writing (DF)(x")† for the matrix with

(DF)(x")†|Im((DF)(x))) =(

(DF)(x")|Im((DF)(x)))

)#1(4.117)

and

(DF)(x")†|Ker((DF)(x))) = 0, (4.118)

we have

(D")(x") = I" (DF)(x")†(DF)(x"). (4.119)

In a similar manner, we can find the second derivatives. As before,

d

dt(Dij+t)(x) = (D2F)(+tx)2(Di+t)(x), (Dj+t)(x)3+ (DF)(+tx)(Dij+t)(x). (4.120)

Taking t#$ gives

0 = (D2F)("(x))2(Di")("(x)), (Dj")("(x))3+ (DF)("(x))(Dij")("(x)), (4.121)

from which we conclude that (D2F)("(x))2(Di")("(x)), (Dj")("(x))3 is in Im((DF)("(x))),

and

(Dij")("(x))" (DF)("(x))†(D2F)("(x))2(Di")("(x)), (Dj")("(x))3

% Ker((DF)("(x))). (4.122)

To determine the component in Ker((DF)("(x))), we apply Duhamel’s principle to


(4.120), to obtain

(Dij+t)(x) =

9 t

0

e! ts (DF)()u(x)) du(D2F)(+sx)2(Di+s)(x), (Dj+s)(x)3 ds. (4.123)

As before, we wish to find the derivatives at points x" % C, for which

(Dij+t)(x") =

9 t

0

e(DF)(x))(t#s)(D2F)(x")2e(DF)(x))sei, e(DF)(x))sej3 ds. (4.124)

Now, (D")(x") is a projection onto Ker((DF)("(x))), so the quantity we seek is

(D")(x")

9 t

0

e(DF)(x))(t#s)(D2F)(x")2e(DF)(x))sei, e(DF)(x))sej3 ds

=

9 t

0

(D")(x")(D2F)(x")2e(DF)(x))sei, e(DF)(x))sej3 ds (4.125)

since e(DF)(x))(t#s) acts as the identity on Ker((DF)("(x))). This in turn is equal to

9 t

0

(D")(x")5

(D2F)(x")2e(DF)(x))sei, e(DF)(x))sej3

" (D2F)(x")2(Di")(x"), (Dj")(x

")36

ds (4.126)

which, using the bilinearity of the Hessian, is equal to

=

9 t

0

(D")(x")5

(D2F)(x")2(

e(DF)(x))s " (D")(x"))

ei,(

e(DF)(x))s " (D")(x"))

ej3

+ (D2F)(x")2(

e(DF)(x))s " (D")(x"))

ei, (Dj")(x")3

+ (D2F)(x")2(Di")(x"),(

e(DF)(x))s " (D")(x"))

ej36

ds. (4.127)

We now consider the three terms in the integral separately. For the latter two, we may


use the bilinearity of the Hessian to interchange the form and the integral. Then,

9 t

0

e(DF)(x))s " (D")(x") ds = Px)

!9 t

0

eAs " Id2d < 0K#d2K#d ds

"

P#1x)

= Px)

!9 t

0

0d2d < eJ1s < · · ·< eJps ds

"

P#1x)

= Px)(

0d2d < J#11

(

eJ1t " Id12d1

)

< · · ·< J#1p

(

eJpt " Idp2dp

))

P#1x) (4.128)

which, taking t#$, yields

"Px)(

0d2d < J#11 < · · ·< J#1

p

)

P#1x) = "(DF)(x")†. (4.129)

Thus in the limit, the second and third terms yield

"(D2F)(x")2(DF)(x")†ei, (Dj")(x")3 " (D2F)(x")2(Di")(x

"), (DF)(x")†ej3. (4.130)

For the remaining term, we are left with computing

9 t

0

(D2F)(x")2(

e(DF)(x))s " (D")(x"))

ei,(

e(DF)(x))s " (D")(x"))

ej3 ds. (4.131)

For conciseness, let A = J1 < · · ·< Jp, so that (DF)(x") = Px)

5

0d2d < A6

P#1x) and

e(DF)(x))s " (D")(x") = Px)

5

0d2d < eAs6

P#1x) . Then, the kth component of the Hessian

above is the ijth entry of

(P*x))

#1

!9 t

0

5

0d2d < eA's6

P*x)(D

2Fk)(x")Px)

5

0d2d < eA's6

ds

"

P#1x)

= (P*x))

#1

!!

0d2d <9 t

0

eA'sBke

As ds

""

P#1x) (4.132)


where

(0d2d <Bk) = (0d2d < IK#d2K#d)P*x)(D

2Fk)(x")Px)(0d2d < IK#d2K#d). (4.133)

i.e.,

(P*x))

#1(0d2d < Bk)P#1x) = (I" (D")(x"))*(D2Fk)(x

")#1(I" (D")(x")). (4.134)

Now, A is a negative definite matrix, so

Xk = limt%"

9 t

0

eA'sBke

As ds (4.135)

exists and is the unique solution to

A*Xk +XkA = "Bk, (4.136)

which may be obtained by solving the linear system

(A= Id2d + Id2d = A)vec(Xk) = "vec(Bk), (4.137)

where= denotes the Kronecker product, and vec(X) is the vector in Rd2 with vec(X)(i#1)d+j =

Xij [16].

Writing Hij for the vector with

H ijk = e*i (P

*x))

#1 (Xk < 0d2d)P#1x) ej, (4.138)


we then have,

limt%"

9 t

0

(D2F)(x")2(

e(DF)(x))s " (D")(x"))

ei,(

e(DF)(x))s " (D")(x"))

ej3 ds = Hij.

(4.139)

Summarizing the above, we have

Lemma 4.2. Using the notation above, for x" % C, we have

(Dij")(x") = "(DF)(x")†(D2F)(x")2(Di")(x

"), (Dj")(x")3

+ (D")(x")(

Hij " (D2F)(x")2(DF)(x")†ei, (Dj")(x")3

" (D2F)(x")2(Di")(x"), (DF)(x")†ej3

)

. (4.140)

4.2.2 Explicit Expressions in Co-dimension One

Here, we consider the case where C = {x : f(x) = 0} for a C2 function f : RK+ # R

for which (Df)(x) '= 0 for all x % RK . Fix x" % C, and, without loss of generality,

assume that (!Kf)(x") '= 0. Then, recalling the proof of Proposition 4.1 and the remarks

following,

"x)(x) = (x1 " x"1, . . . , xK#1 " x"

K#1, f(x)) (4.141)

is a slice chart for C in a neighbourhood Ux) of x". Then, "x)(Ux) / C) . {x : xK = 0},

and if

G(x) = (D"x))("#1x) (x))F("

#1x) (x)), (4.142)

is the vector field topologically conjugate to F under "x) , then G(x1, . . . , xK#1, 0) = 0,

i.e.,G(x) = xKH(x) for some H, and writing

r(x) = (D"x))(x)#1H("x)(x)) (4.143)


we see, that, provided f is at least C2, r is at least C1, and

F(x) = f(x)r(x) (4.144)

near x". Moreover, since x" is arbitrary and f(x) is globally defined, r : RK+ # RK is

globally defined (r(x) = 1f(x)F(x)) and is C1.

Thus,

(DF)(x) = f(x)(Dr)(x) + (Df)(x)= r(x), (4.145)

where u= v is the rank-one linear operator defined by

(u= v)X = (u ·X)v (4.146)

for X % RK . If x" % C, (DF)(x") = (Df)(x") = r(x"), and (DF)(x") has two distinct

eigenvalues, 0 and *(x"), where

*(x) := (Df)(x) · r(x). (4.147)

The former has corresponding eigenspace (Df)(x")3 = Tx)C, whilst the latter corre-

sponds to the single eigenvector r(x"). Thus, the projection onto Im((DF)(x")) is

1

*(x")(DF)(x") =

1

*(x")(Df)(x")= r(x") (4.148)

and

(DF)(x")† =1

*2(x")(Df)(x")= r(x"), (4.149)

(which is simply multiplication by 12(x)) when restricted to Im((DF)(x"))), so that

(D")(x") = I"1

*(x")(DF)(x"). (4.150)


To compute the second derivatives, we begin by observing that

(D2Fk)(x") = (D2f)(x")rk + (Df)(x")*(Drk)(x

") + (Drk)(x")*(Df)(x") (4.151)

and (Df)(x")(D")(x") = 0, so that

(D")(x")*(D2Fk)(x")(D")(x") = (D")(x")*(D2f)(x")(D")(x")rk (4.152)

and

(DF)("(x"))†(D2F)("(x"))2(Di")("(x")), (Dj")("(x

"))3

=1

*(x")(D2f)(x")2(Di")(x

"), (Dj")(x")3 r(x"). (4.153)

Proceeding as above, we also have

(D2F)(x")2r(x"), r(x")3 = (D2f)(x")2r(x"), r(x")3 r(x")+2*(x")(Dr)(x")r(x") (4.154)

and

(D2F)(x")2r(x"), ei3

= (D2f)(x")2r(x"), ei3 r(x") + *(x")(Dr)(x")ei + (!if)(x")(Dr)(x")r(x") (4.155)

whereas

(D2F)(x")2(DF)(x")†ei, (Dj")(x")3

=1

*(x")

!(Dif)(x")

*(x")(D2F)2r(x"), ej3+

(Dif)(x")(Djf)(x")

*2(x")(D2F)2r(x"), r(x")3

"

.

(4.156)


Finally, for all X % RK ,

(DF)(x")X = ((Df)(x") ·X) r(x") (4.157)

and thus

(DF)(x")nX = ((Df)(x") ·X)*n#1(x")r(x") (4.158)

and

e(DF)(x))tX =$

n=0

tn(DF)(x")nX

n!(4.159)

= X+ ((Df)(x") ·X) r(x")"$

n=1

*n#1(x")tn

n!(4.160)

=((Df)(x") ·X)

*(x")

(

e2(x))t " 1

)

r(x"). (4.161)

Further,

(D")(x")X = X"1

*(x")((Df)(x") ·X) r(x") (4.162)

so that(

e(DF)(x))t " (D")(x"))

X =((Df)(x") ·X)

*(x")e2(x

))tr(x"), (4.163)

and

H ij = limt%"

9 t

0

(D2F)(x")2(

e(DF)(x))s " (D")(x"))

ei,(

e(DF)(x))s " (D")(x"))

ej3 ds

= "(Dif)(x")(Djf)(x")

2*3(x")(D2F)2r(x"), r(x")3. (4.164)


Inserting the above terms into Lemma 4.2, with some calculations we find

(Dij")(x") =

1

*(x")(D2f)(x")2(Di")(x

"), (Dj")(x")3 r(x")

"1

*(x")(D")(x")(Dr)(x")

(!

(!if)(x")ej + (!jf)(x

")ei "(!if)(x")(!jf)(x")

*(x")r(x")

"

. (4.165)

Finally, we can use the derivatives (4.165) in (4.97) to obtain the following corollary

to Theorem 4.2, giving an explicit expression for the generator of Z(t) in co-dimension

one:

Corollary 4.4. Suppose F(x) = f(x)r(x) with (Df)(x) & 0 and r(x) > 0. Then,

C = f#1(0) and

(D")(x") = I"1

*(x")(DF)(x") = I"

1

*(x")(Df)(x")= r(x"). (4.166)

Recall that

a(x) :=$

n(ZK

nn**n(x). (4.167)

If we set

a(x) := (D")(x)a(x)(D")*(x) (4.168)

and

b(x) = (D")(x)#(x)"1

*(x)tr(

(D2f)(x)a(x))

r(x)

"1

*(x)(D")(x)(Dr)(x) (I+ (D")(x)) a(x)(Df)*(x), (4.169)


then Z has generator

(G&)(x) =K$

i=1

bi(x)!i&(x) +1

2

K$

i=1

K$

j=1

aij(x)!i!j&(x). (4.170)

In the next section, we will apply this to several density dependent birth-death-

mutation processes, but first we consider an alternative approach to computing the

derivatives of "(x), which is considerably more tractable, when the the Law of Large

numbers is of Lotka-Volterra-Gause type.

4.2.3 Energy-based Approaches

Consider the case when

Fi(x) = xi

#

bi "K$

j=1

aijxj

%

= xi(b"Ax)i, (4.171)

so the set of interior equilibria consists of the set of points C = {x %!

RK+ : Ax = b}. We

are thus in the weak selection regime if rank(A|b) = K " d for some d > 0. Thus, there

exist d linearly independent vectors c(j) % RK such that

(c(j))*A = (c(j))*b = 0. (4.172)

Let

Ej(x) =d$

i=1

c(j)i ln xi. (4.173)

Then, for all x %!

RK+

Ej(x) =d$

i=1

c(j)i

#

bi "K$

j=1

aijxj

%

= (c(j))*b" (c(j))*Ax = 0, (4.174)

so Ej is an invariant of motion for the system.


In particular, we have

Ej("(x)) = Ej(x), (4.175)

for j = 1, . . . , d, which, together with the rank K " d system

K$

j=1

aij4j(x) = bi, i = 1, . . . , K (4.176)

allows us to compute "(x) and it’s derivatives. For the latter, di!erentiating both sets

of equations with respect to xk yields

K$

i=1

c(j)i

4i(x)

!4i

!xk

=c(j)k

xk

, (4.177)

andK$

j=1

aij!4j

!xk

= 0, i = 1, . . . , K, (4.178)

which is generically a rank n system of linear equations that may be solved for the

unknown derivatives in terms of "(x). The second derivatives may be obtained in a

similar fashion.

Example 4.1 (The Co-dimension One Lotka-Volterra-Gause Model). In co-dimension

one, C is a hyperplane, and there exist scalars ", '1, . . . ,'K and ,1, . . . , ,K such that

bi = ,i" and aij = ,i'j (4.179)


for all i, so that ri(x) = ,ixi and f(x) = " "0K

j=1 'jxj . Thus,

(Dif)(x) = 'i, (Dijf)(x) = 0, *(x) =K$

l=1

,l'lxl,

(D*)(x) =

R

SSSST

,1'1

...

,K'K

U

VVVVW

,

and

(Dr)(x) =

R

SSSST

,1

. . .

,K

U

VVVVW

.

It may be readily verified that here we may take c(j) = ,1e1 " ,jej , so that

Ej(x) = ,1 ln x1 " ,j ln xj. (4.180)

We omit the calculations, but note that, together with the requirement that

K$

i=1

'i4i(x) = ", (4.181)

these invariants may be used to show that for x" % C,

(Di4k)(x") = %ik "

'i,kx"k

0Kl=1 ,l'lx

"l

(4.182)


and

(Dij4k)(x") =

,k0K

l=1 ,l'lx"l

("'i%jk " 'j%ik

+'i'jx

"k

0Kl=1 ,l'lx"

l

#

,i + ,j + ,k "0K

l=1 ,2l 'lx

"l

0Kl=1 ,l'lx"

l

%%

, (4.183)

which may be verified using the variational approach given previously.

We will return to this approach in the next chapter, where it will allow us to calculate

the first and second derivatives in an example where C has co-dimension 2.

4.3 Weakly Selected Density Dependent Birth-Death-

Mutation Processes

In this section, we apply the results of the previous sections to the Birth-Death-Mutation

processes discussed in Chapter 2. In particular, we consider the special case of weak

selection when C is a co-dimension one submanifold, C = f#1(0). We can then use

the explicit forms of the first and second partial derivatives of "(x) computed in Section

4.2.2 to state the generator of the limiting process. Unfortunately, even under this strong

assumption, the generator of the limiting process defies easy biological interpretation, so

we will take the approach of classical population genetics, and begin by analyzing the

exchangeable case, and then consider the consequences of relaxing exchangeability.

We begin by introducing some notation that will allow a more easy comparison to

classical population genetics. Let

"(N)i (x) :=

$

n(NK0

1n11 "(N)i,n (x)

"(N)i (x) :=

$

n(NK0

1n122 "(N)i,n (x),


and

µ(N)ij (x) :=

?

JJ@

JJA

1

%(N)i (x)

0

n(NK0nj"

(N)i,n (x) if i '= j

0 otherwise

"(N)i (x) is thus the expected total reproductive output of an individual of type i per unit

time in environment x, "(N)i (x)"

5

"(N)i (x)

62is the variance in total reproductive output

per unit time, and µ(N)ij (x) is the fraction of the expected number of o!spring which are

of type j. By assumption,

µ(N)ij (x) = O

(1N

)

. (4.184)

for i '= j. Let

$ij(x) := limN%"

Nµ(N)ij (x) (4.185)

be the rescaled rate of mutation,

s(N)i (x) =

5

"(N)i (x)" "i(x)

6

"5

%(N)i (x)" %i(x)

6

"(N)i (x)

, (4.186)

and let

'i(x) := limN%"

Ns(N)i (x), (4.187)

i.e.,'i(x) is the O(

1N

)

component of the net reproductive rate (that the limit exists is

guaranteed by Assumption (2.3). With this notation, we have

F(N)i (x)" Fi(x) = "

K$

j=1

µ(N)ij (x)"(N)

i (x)xi +K$

j=1

µ(N)ji (x)"(N)

j (x)xj + "(N)i (x)s(N)

i (x)xi.

(4.188)


so that

5i(x) = limN%"

N5

F(N)i (x)" Fi(x)

6

= "K$

j=1

$ij(x)"i(x)xi+K$

j=1

$ji(x)"j(x)xj+"i(x)'i(x)xi.

(4.189)

Further, by Assumptions 2.1 and 2.3, we have

"(N)i (x)# "i(x) (4.190)

and

"(N)i (x)# "i(x) :=

"$

n=1

n2"i(x) (4.191)

as N #$, uniformly on compact sets.

Thus, using the notation of Theorem 4.2,

#(x) = aij(x) =

?

JJ@

JJA

"i(x)xi + %i(x)xi if i = j,

0 otherwise.

(4.192)

In particular, at points x" % C, "i(x" = %i(x"), so aij(x") = %ij("i(x") + "i(x"))xi.

4.3.1 The Relative Frequency Process and Projection to the

Standard Simplex

In keeping with the convention in population genetics, we will focus on the relative

frequency of types, rather than their absolute density. As we discuss below, this will be

useful in interpreting the process, in particular in considering fixation probabilities. To

that end, we introduce a second projection map,

%(x) =1

0Ki=1 xi

x, (4.193)


with partial derivatives

(D%)(x) =1

0Ki=1 xi

(I" = %(x)) (4.194)

and

(Dij%)(x) = "1

50K

i=1 xi

62 (ei + ej " 2%(x)), (4.195)

where

=

L

PPPPM

1

...

1

N

QQQQO

(4.196)

is the vector with all entries equal to 1. % is thus a di!erentiable map from C to the

standard simplex #1 =+

p % RK+ :0K

i=1 pi = 1-

. In fact, if we assume that the origin is

a source, then % is a di!eomorphism. We begin with a lemma:

Lemma 4.3. If the dynamical system generated by the vector field F(x) = r(x)f(x) is

dissipative and totally competitive, then the set {x : f(x) , 0} is compact.

Proof. Suppose the contrary; then given any M > 0, there exist a point x0 such that

f(x) > 0 and 1x1 > M . Then, either f > 0 on the complement of the ball of radius

M , BM(0)c, in which case ,+(x) is unbounded for any x % BM(0)c, or there exists

x1 % BM(0)c such that f(x1) & 0. In the latter case, we must also have x" % BM(0)c

such that f(x") = 0 (indeed, there exists x" which is a convex combination of x0 and

x1) so that the dynamical system has an equilibrium point in BM(0)c. Since the system

is totally competitive, this fixed point must be a locally asymptotically stable attractor.

Since M was arbitrary, the dynamical system must have an attractor outside of any

compact set. Thus, in either case, the system fails to be dissipative.

Remark 4.8. We briefly remark that the analogous result doesn’t hold for the general

Kolmogorov system i.e.,we can have a system xi = fi(x)xi which is competitive and


dissipative, but for which the set {x : fi(x) > 0} need not be compact for all i. For

example,

x1 = (1" (1 + x1)x2)x1 (4.197)

x2 = (2" x1 " x2)x2 (4.198)

has saddle nodes at (0, 2) and (1, 0), and a unique fixed interior point,5

1+45

2 , 3#45

2

6

that

is globally attracting on!R2

+, whereas

{x : f1(x) > 0} =

2

x : x2 &1

1 + x1

4

(4.199)

is unbounded.

We now have

Proposition 4.2. Assume that the system with vector field F(x) = r(x)f(x) is dissipative

and totally competitive and that f(0) > 0. Then, % : C # #1 is a di"eomorphism.

Proof. First, note that p % %(C) if and only if there exists n(p) , 0 such that f(n(p)p) =

0. If p = %(x"), then

n(p) =K$

i=1

x"i . (4.200)

Let gp(t) := f(tp), so g is continuous, and by assumption gp(0) > 0. From the lemma, we

see that gp(t) < 0 for t su"ciently large. By the intermediate value theorem, gp(n(p)) = 0

for some n(p) > 0, so % is onto.

Now, suppose that there exist x"1 '= x"

2 % C such that %(x"1) = %(x"

2) = p. Then,

without loss of generality,

x"2 " x"

1 = cp (4.201)

for some c > 0. Let ,(t) be a geodesic in C such that there exist t2 > t1 with ,(ti) = xi.


Then, by the mean value theorem, there exists t3 % [t1, t2] such that

,(t3) =,(t2)" ,(t1)

t2 " t1=

c

t2 " t1p. (4.202)

On the other hand, ,(t) lies in C, so we must have f(,(t)) = 0 for all t and

0 =d

dt

,,,t=t3

= (Df)(,(t3)) · ,(t3) =c

t2 " t1(Df)(,(t3)) · p, (4.203)

i.e., (DF)(,(t3)) ·p = 0, contradicting the assumption of total competitivity. Hence % is

injective.

Lastly, let G(x, t) = (x, gx(t)), p % #1 and let n(p) be the unique t > 0 such that

tp % C. Then,

det ((DG)(x, t)) = (!tgx)(t) = (Df)(tx) · x, (4.204)

which, by assumption is strictly positive in a neighbourhood of (p, n(p)). By the inverse

function theorem, G#1 exists and is di!erentiable near (p, n(p)). In particular, g#1p (0)

exists and is singly valued – it is equal to n(p) – and di!erentiable in p, so p ># g#1p (0)x :

#1 # C is a di!erentiable injection.

We will henceforth refer to P(t) := %(Z(t)) as the relative frequency process corre-

sponding to Z(t). If Z(t) has generator

(G&)(x) =K$

i=1

bi(x)!i&(x) +1

2

K$

i=1

K$

j=1

aij(x)!i!j&(x), (4.205)


then, using Ito’s formula, it is readily shown that P(t) has generator

(GP&)(p) =1

n(p)

K$

i=1

.#

bi(n(p)p)"1

n(p)

K$

n=1

ain(n(p)p)

%

" pi

K$

j=1

#

bj(n(p)p)"1

n(p)

K$

m=1

amj(n(p)p)

%/

!i&(p)

+1

2n(p)

K$

i=1

K$

j=1

#

aij(n(p)p)%ij " pi

K$

m=1

amj(n(p)p)" pj

K$

n=1

ain(n(p)p)

+ pipj

K$

m=1

K$

n=1

amn(n(p)p)

%

!i!j&(p). (4.206)

In the next two sections, we will consider a few examples of density-dependent birth-

death-mutation processes in the weak selection regime and their corresponding limiting

relative frequency processes, with the intent of gleaning some biological insight and iden-

tify points of departure from the classical theory. As with Cannings’ approach, we start

with exchangeable birth-death-mutation processes, which have Kimura’s di!usion as their

invariance principle. We then consider a few biologically interesting examples of “nearby”

processes which depart from Kimura’s di!usion. In the final section, we revisit the Gen-

eralized Moran model as a means of understanding the population genetic consequences

of fecundity/mortality tradeo!s.

4.3.2 Exchangeable Processes, Neutrality and Kimura’s Di!u-

sion

We begin by considering the case when all types are exchangeable, so that "(N)i,n (x) =

"(N)j,n (x) and %

(N)j (x) = %

(N)j (x) for all i, j and all x. Then, "i(x) 4 "(x), %i(x) 4 %(x), and

we may thus take f(x) = "(x)" %(x) and r = x. Moreover, to ensure exchangeability, f

must be invariant under permutations of its arguments, so we must have !if = !jf for

all i, j. This largely characterizes f , as is shown by the following simple lemma.


Lemma 4.4. Suppose f : RK # R is di"erentiable. Then !if = !jf for all i, j if and

only if there exists a di"erentiable f such that f(x) = f(0K

i=1 xi).

Proof. If f(x) = f(0K

i=1 xi), then !if = f )(0K

i=1 xi) = !jf for all i, j.

Conversely, suppose f satisfies !if = !jf for all i, j. Given x % RK , let

g(t) = f(x1 + t

K$

i=2

xi, (1" t)x2, . . . , (1" t)xk) (4.207)

so that

g)(t) =K$

i=2

xi (!1f " !if) = 0, (4.208)

and

f(x) = g(0) = g(1) = f(K$

i=1

xi, 0, . . . , 0). (4.209)

The result follows taking f(t) = f(t, 0, . . . , 0).

We thus have f(x) = f(0K

i=1 xi). In a slight abuse of notation, we will use f and

f interchangeably. By assumption, our process is competitive, so !if < 0 for all i, and

thus f ) < 0. In particular, f can has exactly one zero, n", and C is the simplex

&

x :K$

i=1

xi = n"

'

. (4.210)

Moreover, for all x" % C, Tx)C is the hyperplane perpendicular to , *(x") = f )(n")n",

and (Df)(x) = f )(0K

i=1 xi) . Direct calculation then shows that on C the derivatives of

"(x), (4.150) and (4.165), simplify to

(D")(x") = I"1

n"= x" (4.211)

and

(Dij")(x") = "

1

n"

!

ei + ej "2

n"x"

"

, (4.212)


whilst P(t) = 1n)Z(t).

In this case, we have

G&(p) =K$

i=1

#

"K$

j=1

$ij(n"p)"(n"p)pi +

K$

j=1

$ji(n"p)"(n"p)pj

%

!i&(p)

+1

n"

K$

i=1

K$

j=1

"(n"p)pi(%ij " pj)!i!j&(p). (4.213)

Moreover, since " , 0,

,(t) := inf

2

u :

9 u

0

"(n"P(s)) ds > t

4

, (4.214)

is non-decreasing and we can apply Theorem 6.1.4 in [50] (Random Time Change), to

conclude that the time-rescaled process, P(t) := P(,(t)), satisfies

G&(p) =K$

i=1

#

"K$

j=1

$ij(n"p)pi +

K$

j=1

$ji(n"p)pj

%

!i&(p)

+1

n"

K$

i=1

K$

j=1

pi(%ij " pj)!i!j&(p). (4.215)

which is essentially the generator for Kimura’s di!usion, (1.12), except that we have

allowed for frequency-dependent mutation. Notice that the time rescaling is equivalent

to measuring time in units of 1%(x)

, the per-individual expected time to first reproduction.

In particular, the process P(t) is obtained from X(N)(t) via (among other things) a time

rescaling of N%(x)

, which is the generation time for the original process i.e., the expected

time for all individuals to have reproduced at least once. The census population size

at equilibrium is approximately n"N , so we may regard n" as analogous to an e!ective

population size.

Remark 4.9. In fact, to obtain Kimura’s di!usion, (1.12) as a limit, we did not require

the processes X(N)i (t) to be exchangeable, only the weaker conditions


(i) "i(x) 4 "(x) and %i(x) 4 %(x) for all i and all x,

(ii) "i(x) 4 "(x) for all i and all x,

(iii) !if(x) = !jf(x) for all i, j and all x, for f(x) = "(x)" %(x), and

(iv) 'i(x) = 0 for all x.

We investigate the the consequences of relaxing these assumptions in the sequel.

Classical Weak Selection

If we allow the individual total reproductive rate, "(N)i,n (x) and death rate %

(N)j (x) (and

thus expected total lifetime reproductive output) to di!er to order O(

1N

)

, then 'i(x) '4 0,

and we obtain a di!usion analogous to Kimura’s di!usion under weak selection. Rescaling

time, as above, by ,(t) = infG

u :K u

0 "(n"P(s)) ds > tH

, we have

G&(p) =K$

i=1

.

"K$

j=1

$ij(n"p)pi +

K$

j=1

$ji(n"p)pj

+ pi

#

'i(n"p)"

K$

j=1

'j(n"p)pj

%/

!i&(p)

+1

n"

K$

i=1

K$

j=1

pi(%ij " pj)!i!j&(p). (4.216)

Again, when the $ij and 'i are constant, we recover Kimura’s generator for processes

under weak selection. As in the classical case, if, without loss of generality, '1 > 'i for

all i > 1, then

Ep[P1(t)] = pi +

9 t

0

E

.

Pi(s)

#K$

j=1

('i " 'j)Pj(s)

%/

ds > pi, (4.217)


and there is selection favouring type 1. More generally, when the 'i are frequency depen-

dent, i.e.,depend on p, then

pi

#

'i(n"p)"

K$

j=1

'j(n"p)pj

%

(4.218)

is simply the projection onto #0 =+

x % RK :0K

i=1 xi = 0-

of the essentially arbitrary

vector field &(x), and we can only conclude that frequency dependent selection occurs:

that E[Pi(t)] will increase at those times when Pi(t) is non-zero and 'i(n"P(t)) is greater

than the population average value '(n"P(t)) =0K

j=1 'j(n"P(t))Pj(t).

Fecundity Variance Polymorphism

Among the most extensively studied departures from exchangeability is when "i 4 ",

%i 4 %, and !if(x) = !jf(x), but "i '4 "j for i '= j, so that all types have the same

mean reproductive output, but may di!er in the variance in the number of o!spring

per reproductive event. This is referred to as fecundity variance polymorphism (see

e.g., [66, 171, 67, 68, 61, 176] etc.). In this case, P(t) has generator

G&(p) =K$

i=1

.

"K$

j=1

$ij(n"p)"(n"p)pi +

K$

j=1

$ji(n"p)"(n"p)pj

"1

n"pi

#

"i(n"p)"

K$

j=1

"j(n"p)pj

%/

!i&(p) +1

n"

K$

i=1

K$

j=1

E

pi

5

("i + ")(n"p)%ij

"

#

("i + ")(n"p) + ("j + ")(n"p)"K$

n=1

("n + ")(n"p)pn

%

pj

%/

!i!j&(p) (4.219)

or, again, rescaling time by

,(t) := inf

2

u :

9 u

0

"(n"P(s)) ds > t

4

, (4.220)


the generator takes the form

G&(p) =K$

i=1

.

"K$

j=1

$ij(n"p)pi +

K$

j=1

$ji(n"p)pj

"1

n"pi

#

(2i (n

"p)"K$

j=1

(2j (n

"p)pj

%/

!i&(p)

+1

n"

K$

i=1

K$

j=1

.

pi

#

(2i (n

"p)%ij "

#

(2i (n

"p) + (2j (n

"p)"K$

n=1

(2i (n

"p)pn

%

pj

%/

!i!j&(p)

(4.221)

where

(2i (p) :=

"i(p)

"(p)+ 1 (4.222)

was shown in Corollary 2.1 to be the variance in total lifetime reproductive output for

type i when the population has reached the manifold of fixed points C. For K = 2

and constant (2i , this generator was derived via a heuristic argument in [66]. Under the

assumption of constant (2i , this shows that ceteris paribus, among types with the same

equilibrium expected lifetime reproductive success, the one with the lowest variance in

fecundity is selectively favoured.

Resource Competition

We now consider relaxing the third condition, !if(x) = !jf(x) for all i, j, while retaining

the first two. Unlike our previous examples, here we depart from the realm of classi-

cal population genetics to consider questions more traditionally within the purview of

population ecology.

As in the previous cases, we have r(x) = x, so that (Dr)(x) = I, *(x) = (Df)(x) · x,

(D")(x") = I"1

*(x")(Df)(x")= x" (4.223)


and

(Dij")(x") =

1

*(x")(D2f)(x")2(Di")(x

"), (Dj")(x")3x"

"1

*(x")(D")(x")

!

(!if)(x")ej + (!jf)(x

")ei "(!if)(x")(!jf)(x")

*(x")x"

"

. (4.224)

Moreover,

(D%)(x")(D")(x") = (D%)(x"), (4.225)

which greatly simplifies calculations. Here, we use the time rescaling

,(t) := inf

2

u :

9 u

0

(" + ")(n"P(s)) ds > t

4

, (4.226)

and under which the process P(t) := 2(Z(,(t))) has generator

G&(p) =K$

i=1

#

"K$

j=1

$ij(n"p)pi +

K$

j=1

$ji(n"p)pj

"pi

#

(!if)(n"p)"

K$

j=1

(!jf)(n"p)pj

%%

!i&(p)

+1

n"

K$

i=1

K$

j=1

pi(%ij " pj)!i!j&(p). (4.227)

Comparing with our expression for the generator under Classical Weak Selection, we see

that this is formally equivalent to local selection favouring any type for which !jf is

below the population mean.

This result has an interesting interpretation under the aegis of consumer resource

theory; f is asymptotically equal to the per-capita net growth rate of all individuals of all

types, which, under the assumption of total competition, is, at least in a neighbourhood

of C a strictly decreasing function of the density of all types. f can thus be considered

a measure of the degradation of the environment, by the depletion of essential resources


or the accumulation of wastes. !if is then the marginal degradation of the environment

with an increase of type i. We thus see that selection favours the type with the highest

marginal degradation per capita. This might be considered a “second-order” analogue

of a result of Volterra’s, popularized by Tilman as R" theory [177]: in R" theory, when

species compete for a single limiting resource, it is the type that can most deplete the

resource, and survive in the most degraded environment that will exclude the others,

here we see that among species which share a common lowest viable resource level, the

type that most e"ciently depletes the resource will exclude all others.

4.3.3 The Generalized Moran Model

To conclude this chapter, we return to our prototypical model, the Generalized Moran

Model, Example 2.1 which, at least when K = 2, is su"ciently specific that we may

obtain analytical results for the fixation probabilities, fixation and absorption times, and

for the quasi-stationary distribution. In this case we have

• "(N)i (x) = "i(x) = "i and "

(N)i (x) = "i(x) = "i,

• %i(x) = %i

5

1 +0K

j=1 xj

6

,

• µ(N)ij (x) = #ij

Nand $ij(x) = $ij ,

• and s(N)i (x) = 'i(x) = 0.

For the Generalized Moran model, the existence of a submanifold of attracting fixed

points is equivalent to the existence of multiplicative tradeo! between mortality and

fecundity, i.e.,when there exists a constant . such that

"i

%i= . > 1, (4.228)

independently of i. As we observed in Proposition 2.2, this corresponds to the case when

all types have the same expected lifetime reproductive success at equilibrium. Although


one type may have an elevated birth rate, this is compensated by a correspondingly

elevated death rate. Assuming (4.228),

C =

&

x % RK+ :

K$

i=1

xi = . " 1

'

(4.229)

is a linear submanifold, so as before, % is a bijection from C onto #1.

After rescaling time by ,(t) = *#12 t, the relative frequency process corresponding to

the Generalized Moran model has generator

G&(p) =K$

i=1

#

"K$

j=1

$ij"ipi +K$

j=1

$ji"jpj

+1

2

"ipi50K

k=1 "kpk

62

K$

k=1

("k " "i)"kpk

%

!i&(p)

+1

2

K#1$

i=1

K$

j=1

"ipi

#

%ij ""jpj

0Kk=1 "kpk

%

!i!j&(p). (4.230)

for K = 2, we have P(t) = (P1(t), 1" P1(t)), and the generator of P1 is

G&(p) =!

"$12"1p + $21"2(1" p) +1

2

"1"2("2 " "1)

"1p+ "2(1" p)p(1" p)

"

&)(p)

+1

2

"1"2

"1p+ "2(1" p)p(1" p)&))(p) (4.231)

Setting $12 = $21 = 0 and solving the Dirichlet problems

Gh = 0

h(0) = 0, h(1) = 1,

Ggabs = "1

tabs(0) = gabs(1) = 0,


and

Ggfix = "h

h(0)gfix(0) = h(1)gfix(1) = 0

give the fixation probability, absorption time, and fixation time for type 1 respectively,

while the gradient solution to G$5 = 0, gives, up to a normalizing constant, the density

for the stationary distribution of P1 when $12, $21 > 0. These are summarized in Table

4.1, along with the corresponding solutions (when closed forms are available) for the

Wright-Fisher model, with and without selection for the sake of contrast. The latter

processes have generator

G&(p) = ("$12p+ $21(1" p) + 'p(1" p))&)(p) +1

2p(1" p)&))(p) (4.232)

with ' = 0 giving the neutral model.

Under the Wright-Fisher model without mutation, a neutral allele will fix with prob-

ability given by its initial frequency. In the Generalized Moran model, however, the

fixation probability di!ers qualitatively from the Wright-Fisher expectation (Table 4.1).

Starting from a population at carrying capacity, the fixation probability of a mutant has

a quadratic dependence on the mutant’s initial frequency, and it favours fixation of the

type with low birth and death rates (Figure 4.1a). Di!erences in generation time induce

a form of apparent selection favouring one type over another – despite the fact that the

two types might superficially appear to be strictly neutral variants under a Wright-Fisher

model with Ne = N(. " 1).

A similar result holds for the mean time before a mutant allele fixes or goes extinct.

The behaviour of the Generalized Moran process is skewed relative to the neutral Wright-


Tab

le4.1:

Acomparison

oftheneu

tral

Wrigh

t-Fisher

mod

elan

dtheGen

eralized

Moran

mod

el.Thetable

summarizes

ouran

alytical,

asym

ptoticresultsfortheGen

eralized

Moran

mod

el,whicharederived

inthemaintext.Absorption

andfixa

tion

times

aregivenin

generations.

For

theGen

eralized

Moran

mod

elthegenerationtimeis

theaveragetimeto

reproduction,

1 2

51 %1+

1 %2

6

,an

dthecarrying

simplexis

C=+

x%RK +

:0

K i=1xi=

#"

1-.pis

theproportion

oftypeon

e.

Wrigh

t-Fisher

(neutral/w

ithselection)

Generalized

Moran

Fixationprobab

ility,h(p)

p/1"

e#2$

p

1"

e#2$

?p+

'p(1"

p)p+

!2"

!1

!1+

!2p(1"

p)

Absorption

time,

g abs(p)

plnp+

(1"

p)ln

(1"

p)/noclosed

form

"#"

1

!1+!2

:

(!1(1

+p)

+!2(1"

p))(1"

p)ln

(1"

p)+

(!1p+

!2(2"p))pln

p+

(!1"

!2)2

!1+

!2

p(1"

p)

;

Fixationtime,

g fix(p)

"1 p(1"

p)ln

(1"

p)/noclosed

form

"1

h(p)

#"1

!1+

!2

:

(!1(1

+p)

+!2(1"

p))(1"

p)ln

(1"

p)+

1 3

!1"

!2

(!1+

!2)2p(1"

p)(

2(!2 1"

!2 2)p

+(!

1+

2!2)(3!

1+

!2));

Station

ary

distribution

,5(p)

p#21#1(1"

p)# 1

2#1/p#

21#1(1"

p)# 1

2#1e2

$p

p# 21$2

$1#1(1"

p)# 1

2$1

$2#1e(%

1#%2)#

%12

$1+%21

$2

$

p


0 0.2 0.4 0.6 0.8 1

Initial proportion

0

0.2

0.4

0.6

0.8

1

Fix

atio

n p

rob

ab

ility

Quasi-neutral generalized Moran model (analytic)

Quasi-neutral generalized Moran model (numerical)

Neutral Wright-Fisher model

(a)

0 0.2 0.4 0.6 0.8 1

Initial proportion

0

100

200

300

400

500

600

700

Ab

sorp

tion

tim

e (

ge

ne

ratio

ns)

(b)

Figure 4.1: (a) The probability that type 1 fixes, as a function of its initial proportion, forthe Wright-Fisher model (blue) and the Generalized Moran model (exact numerical results,dashed black; analytical results from Table 4.1, solid red) starting at carrying capacity. (b)The absorption time of type one under the Wright-Fisher model (blue) and the GeneralizedMoran model (exact numerical results, solid red; analytical results from Table 4.1, dashed red). Parameter values: !1 = 2, !2 = 1, # = 2 and N = 1000 (reproduced from [151], originallyprepared by Christopher Quince).

Fisher prediction: the mean absorption time is longer than the classical expectation when

the type with higher birth rate forms a larger initial proportion of the population (Figure

4.1b). The conditional mean time before fixation of an allele is also skewed: the type with

low vital rates fixes more quickly than a type with high vital rates. Again, the Generalized

Moran model exhibits a form of selection that favours one type over another, despite the

fact that the two types would have the same expected lifetime reproductive output were

the population size held exogenously fixed.

When mutations are permitted between the two types, the stationary allele frequency

distribution of the Generalized Moran process is very similar to that of a Wright-Fisher

model with selection favouring the type with low birth rate. The results above potentially

have practical consequences for practitioners who wish to make inferences about natu-

ral selection based on samples from wild or laboratory populations. When life-history

traits are under genetic control, inference based on standard assumptions (i.e. Kimura’s

di!usion) may yield erroneous and even contradictory conclusions: at small population


densities one allele will appear superior, whereas at large densities the other allele will

dominate.

In Figure 4.2) we analyze polymorphism “data” generated by simulating the Gener-

alized Moran processes of competing, quasi-neutral types. We sampled 100 individuals

from a large population near carrying capacity, and we recorded the polymorphism fre-

quency spectrum across 1000 independent sites. The observed spectrum, which describes

the number of sites at which the type-one allele segregates at di!erent frequencies, is typ-

ical of the polymorphism data observed in field studies of microbes and higher eukaryotes

(e.g. [75, 76]). If we analyze the simulated polymorphism data using standard inference

techniques based on the Wright-Fisher di!usion [166, 22, 30], we would infer directional

selection favouring the type with low birth rate.

Alternative approaches to inferring selection pressures from data would reach the

opposite conclusion. Selection coe"cients are often quantified by competing two types

in a nutrient-rich laboratory environment, and measuring the di!erence in their initial

exponential growth rates [126, 125]. Under the Generalized Moran model starting from

a small population, type one will grow at exponential rate "1 " %1 = "1(1" ') and type

two at exponential rate "2 " %2 = "2(1" '), yielding an apparent selective advantage to

the type with high birth rate. Thus, the selection coe"cient inferred from a laboratory

competition experiment will have the opposite sign as the selection coe"cient inferred

from the stationary polymorphism frequency spectrum, as would be sampled in the field.

Thus, the interpretation of such a competition experiment depends on the trade-o!s that

occur between birth rate and other life history traits, and also on assumptions about the

underlying demographic process.


0.0 0.2 0.4 0.6 0.8 1.0

00.05

0.1

0.15

0.2

0.25

Frequency of type-one allele (%)

Pro

ba

bili

ty

Quasi-neutral generalized Moran (simulation)

Quasi-neutral generalized Moran (analytic)

Best-fit Wright-Fisher with selection

Figure 4.2: The stationary distribution of allele frequencies under reversible mutation. Barsshow results from Monte-Carlo simulations of the quasi-neutral Generalized Moran process(N = 1000, # = 2,!1 = 1.6,!2 = 2, µ = 0.0005), in frequency bins of size 10%. Dots show ouranalytic approximation to the Generalized Moran process, taken from the equation in Table4.1. The analytical approximation closely matches the numerical simulation. In addition,the quasi-neutral frequency spectrum is very similar to the spectrum produced by a Wright-Fisher model with selection (triangles). The parameters of a best-fit Wright-Fisher model(( = 0.512, Nes = 0.252) were obtained by applying standard maximum-likelihood inferencetechniques [166, 22, 30] to the simulated data, and they would indicate selection favoring thetype with low birth rate (reproduced from [151], originally prepared by Joshua B. Plotkin).

4.4 Summary

In this chapter, we considered the appropriate density-dependent analogue of the weak

selection regime of classical population genetics, in which the deterministic approxima-

tion to the birth-death mutation process given by the functional Law of Large Numbers

(Proposition 2.1) has an attracting submanifold of fixed points C; these points correspond

to equilibria at which two or more species can coexist deterministically. Theorems 4.1

and 4.2 tell us that the rescaled stochastic process Z(N)(t) = N#1X(N)(t) is also attracted


to C. Unlike the deterministic process, which arrives at some point x % C and then re-

mains there indefinitely, Z(N)(t) converges to a di!usion process Z(t) on C. Z(t) may be

explicitly characterized using the projection map

"(x) = limt%"

+tx,

which gives the equilibrium density of each type, assuming that type i initially has density

i, and the derivatives of "(x). Section 4.2 shows how these may be computed in any

specific example.

As in classical population genetics, we find that the Wright-Fisher di!usion is a ro-

bust description of neutral population dynamics (Section 4.3.2). To be precise, for any

population consisting of types that all have the same mean per-capita birth rates and

death rates, i.e., "i(x) = "j(x) and %i(x) = %j(x) for all i, j and all x, the same variance

in reproductive output, so "i(x) = "j(x) or all i, j and all x, and the same strength of

interspecific competition, i.e.,

!j(

"i(x)" %i(x))

= !l(

"k(x)" %k(x))

for all i, j, k, l and all x, the relative abundances of each type will, with an appropriate

rescaling of time, converge to the Wright-Fisher di!usion, which has generator

(Gf)(p) =1

2

K$

i,j=1

pi(%ij " pj)(!i!jf)(p).

Relaxing any of these assumptions leads to a qualitatively di!erent behaviours, which

can include frequency-dependent selection (Section 4.3.2), selection for minimal fecundity

variance (Section 4.3.2), selection for maximal e"ciency in resource consumption (Sec-

tion 4.3.2), or r vs. K selection in a model incorporating fecundity/mortality tradeo!s

(Section 4.3.3). The last example provides a cautionary tale for recent e!orts to infer the


presence of strength of selection from site frequency spectra, such as the Poisson Random

Field approach [166, 76, 22]: many qualitatively models can lead to an allele-frequency

distribution with the same functional form, but the parameters must be interpreted in a

model-specific manner. In particular, without a priori biological reasons to favour one

model over all others, inferred values of the selection coe"cient, e!ective population size,

or per-site mutation rates can have ambiguous, and even contradictory interpretations.

Chapter 5

An Application to Evolution in the

SIR Epidemic Model

In this chapter, we illustrate that the results of the previous chapters are applicable

beyond the scope of single population genetics. We will use a stochastic version of

a simple epidemic model, a multi-strain SIR model of host-parasitoid interactions, to

illustrate the use of our approach in a multi-trophic model involving interspecies inter-

actions. The crucial bifurcation parameter for the SIR and related epidemic model is

the basic reproduction number, R0,i, the expected total number of infections caused by

a single individual infected with strain i introduced into a population composed entirely

of susceptible individuals. Similar to our previous examples, we will find that strain 1 is

strongly selected if R0,1 > R0,j for all j > 1, whereas weak selection corresponds to the

case R0,i = R0,j for all i, j.

We will focus our e!orts on the case of two strains, K = 2. In the strong selection

case, we get a simple and appealing expression for the fixation probability of a novel strain

invading a population in which the ancestral strain is at its endemic equilibrium, while

the weak selection case will provides a biologically interesting example of a birth-death-

mutation process that concentrates on a co-dimension two submanifold. Our analysis

161

Chapter 5. Evolution in the SIR Epidemic Model 162

has some interesting implications: among strains with the same basic reproduction num-

ber, those that have the lowest virulence have a higher probability of fixation, and will

outcompete more virulent strains with the same R0.

The models in this chapter di!er slightly from those of the previous two, as we will

consider processes with immigration from an external source and processes in which

individuals may change their type during their lifetime, which we will call transmutation,

in honour of its Lamarckian historical use. We discuss how such processes fit into our

previous framework in the next section, before turning our focus to the SIR model for

the balance of the chapter.

5.1 Birth-Death-Mutation Processes with Immigra-

tion and Transmutation

As previously, immigration and transmutation events occur as counting processes with

Markovian rates that may depend on the current density of all species. Thus, we assume

that groups of immigrants, with ni , 0 individuals of type i, arrive according to a

counting process I(N)n (t) with intensity N6

(N)n (X(N)(t)), and that individuals of type i

change to type j according to the process T (N)ij (t) with rate N.

(N)ij (X(N)(t)). Thus

X(N)(t) = X(N)(0)+K$

i=1

$

n(NK0

nB(N)i,n (t)"

K$

i=1

eiD(N)i (t)+

$

n(NK0

nI(N)n (t)+

K$

j=1

(ej"ei)Tij(t),

(5.1)

We will assume that for all compact sets K that

limN%"

supx(K

$

n(NK0

1n12 6(N)n (x) <$ and lim

N%"supx(K

$

n(NK0

1n122 6(N)n (x) <$, (5.2)


and that there exists functions 6n(x) such that

limN%"

6(N)n (x) = 6n(x) and lim

N%".(N)ij (x) = .ij(x) (5.3)

uniformly on compacts, so that such birth-death-mutation processes with immigration or

transmutation are again a density dependent family with rates

*(N)n (x) =

K$

i=1

"(N)i,n (x)xi + 6(N)

n (x), *(N)#ei = %

(N)i (x)xi, and *

(N)ej#ei = .

(N)ij (x). (5.4)

Thus, the Law of Large Numbers, Proposition 2.1, continues to apply, as do Theorems

4.1 and 4.2 on the existence of di!usion limits under weak selection.

Definition 5.1. We will say immigration or transmutation is rare if 6n(x) 4 0 for all

n % NK0 , but

limN%"

N6(N)n (x) (5.5)

is nonzero for at least one n % NK0 and similarly for .ij(x) and .

(N)ij (x).

In the next sections, we will discuss an important example of a birth-death-mutation

processes with immigration and transmutation, applying our previous results to the study

of viral evolution in a two-strain SIR epidemic.

5.2 The SIR Model with Host Demography

In this section we consider a two-strain SIR epidemic model with host demography (see

e.g., [4] and references therein). We assume that the population is subdivided into groups

of susceptibles, infectives who may spread the disease to susceptible individuals, and

removed individuals who have either died or recovered from the illness and attained

immunity (for our purposes we will not distinguish between recovery and death, as in

either case the individual no longer a!ects the epidemic). Departing slightly from our


previous notational convention, we will write XN0 (t) for the number of susceptible hosts

at time t, and XNi (t) for the number of individuals infected with strain i.

We will consider a simple, Markovian version of the model, with a well-mixed pop-

ulation: each infective has contacts with other individuals a constant rate, with the

contacted individual chosen uniformly at random from the entire population. Encoun-

ters between a susceptible and an infective results in transmission (i.e., the transmutation

of a susceptible to an infective) with a fixed probability i.e., there exist contact rates "i

such that

.(N)0i (x) = "ix0xi. (5.6)

We will further assume that all individuals have intrinsic per-capita mortality rate %, and

individuals infected with strain i recover or experience excess mortality (virulence) at

rate 'i, so that

%(N)i (x) =

?

JJ@

JJA

% if i = 0,

% + 'i if i = 1, 2.

(5.7)

Finally, we assume that new individuals immigrate at rate N*, so that

6(N)ei

= *. (5.8)

These transitions are summarized in Figure 5.1.

5.2.1 Law of Large Numbers

As before, we let

Y(N)(t) =1

NX(N)(t) (5.10)


I1'+$1

!!!!!!!!!

2N "" S

$1N

X(N)1

##"""""""

$2N

X(N)2 !!!

!!!!!!

' "" R

I2

'+$2

##"""""""

(5.9)

Figure 5.1: Compartmental model of a two-strain SIR epidemic with demography. Arrowsindicate transitions between states and are labelled with the Markov transition rate.

and suppose that as N # $, Y(N)(0) # x. Then, by Proposition 2.1, Y(N) converges

almost surely to a deterministic limit Y satisfying

Y(t) = F(Y(t)) (5.11)

where

F(x0, x1, x2) =

L

PPPPM

*" ("1x1 + "2x2 + %)x0

("1x0 " (% + '1)) x1

("2x0 " (% + '2)) x2

N

QQQQO

. (5.12)

and Y(0) = x.

This deterministic dynamical system has equilibria at

x",0 :=

!*

%, 0, 0

"

, x",1 :=

!% + '1

"1,

*

% + '1"

%

"1, 0

"

, andx",2 :=

!% + '2

"2, 0,

*

% + '2"

%

"2

"

.

(5.13)

(DF)(x",0) has eigenvalues % and

"i

!*

%"

% + 'i

"i

"

(5.14)


for i = 1, 2. (DF)(x",1) has eigenvalues

"2

"1

#

% + '1

"1 " '+$2%2

%

(5.15)

and

"1

2

*"1

% + '1±

1

2

_!

*"1

% + '1

"2

+ 4*(% + '1)

!%

*"

"1

% + '1

"

, (5.16)

while the eigenvalues of (DF)(x",2) can be obtained by interchanging indices 1 and 2 in

those of (DF)(x",1). Thus the first of these equilibria is locally stable if and only if

"i

% + 'i

<%

*i = 1, 2, (5.17)

the second is locally stable if

%

*<

"1

% + '1and

"2

% + '2<

"1

% + '1, (5.18)

while the third is locally stable if the preceding inequalities hold with 1 and 2 inter-

changed.

Of particular interest to us is the degenerate case, where

%

*<

"1

% + '1=

"2

% + '2(5.19)

In this case, letting

x"0 =

% + '1

"1=

% + '2

"2, (5.20)

every point in the attracting submanifold

C := {(x"0, x

"1, x

"2) % R

3+ : "1x

"1 + "2x

"2 =

*

x"0

" %} (5.21)

is an equilibrium, i.e.,both strains can coexist stably, and we are in the weak selection


regime.

Recall that the basic reproduction number for strain i is the expected total number of

infections caused by a single individual infected with strain i introduced into a population

composed entirely of susceptible individuals. For our model, this is

R0,i :="i

% + 'i

, (5.22)

whereas the density of susceptibles at the fixed point x",i is x",i0 = 1

R0,i. The above

analysis shows tahat x",i is locally asymptotically stable if and only if R0,i > R0,j for all

j '= i.

In fact, the basic reproductive number is su"cient to determine global stability. In

the strong selection case, we have

Proposition 5.1 (Bremermann & Thieme [21]). (i) Suppose

%

*< R0,1 and R0,1 > R0,2, (5.23)

then x",1 is globally asymptotically stable.

(ii) If on the other hand,%

*< R0,2 and R0,1 < R0,2, (5.24)

then x",2 is globally asymptotically stable.

In the weakliy selected case we have

Proposition 5.2. If

%

*< R0 = R0,1 = R0,2. (5.25)

then the linear submanifold C is globally asymptotically stable.


Proof. We use a family of Lyapunov functions inspired by [113]. First, recall that

x"0 =

% + '1

"1=

% + '2

"2, (5.26)

and let

x&0 := (

% + '1

"1+ (1" ()

% + '2

"2= x"

0,

x&1 :=

(

"1

!*

x&0

" %

"

,

and

x&2 :=

1" (

"2

!*

x&0

" %

"

.

For 0 < ( < 1, let

V&(x0, x1, x2) = x&0

!x0

x&0

" x&0 ln

x0

x&0

"

+ x&1

!x1

x&1

" x&1 ln

x1

x&1

"

+ x&2

!x2

x&2

" x&2 ln

x2

x&2

"

. (5.27)

A calculation shows that

V&(x0, x1, x2) = "*x0

x"0

!

1"x"0

x0

"2

& 0. (5.28)

Hence V0(x0, x1, x2) & 0, and vanishes only at those points with x0 = x"0. The only

positive invariant subset of {x0 = x"0} for the dynamical system is C, so global asymptotic

stability then follows by LaSalle’s Invariance Principle [70].

Henceforth, we will restrict ourselves to the case when(2', 0, 0)

is unstable i.e.,when at

least one of the strains will persist indefinitely in the absence of demographic stochasticity.


5.2.2 Strong Results

Suppose that R0,1 > R0,2, so that the limiting deterministic system, Y(t), has a unique

global attractor at x",1 and a saddle node at x",2. In this case, a straightforward mod-

ification of Proposition 3.9 (we omit the details) allows us to consider a small number

of individuals infected with strain 1 entering a population where strain 2 is endemic, by

coupling with the non-autonomous branching process with transition rates

"i,ei(t) = "iY0(t) and %i(t) = (% + 'i). (5.29)

In particular, we can apply Corollary 3.1 to obtain an asymptotic estimate of the probabil-

ity that strain 1 eventually outcompetes strain 2. Unfortunately, unlike the Generalized

Moran/SIS model, Example 2.1, there is no simple closed form for the extinction proba-

bility across all initial conditions. We do, however, get an appealingly simple expression

for the fixation probability when the resident population starts in equilibrium:

Corollary 5.1. Consider a single individual infected with strain 1 entering a population

where strain 2 is endemic, i.e.,X1(0) = 1. Then the probability strain 1 eventually goes

extinct is

qx),2(t) =

K t

0 e#

! s0 %1Y0(u)#('+$1)) du(% + '1)) ds

1 +K t

0 e#

! s0 %1Y0(u)#('+$1)) du(% + '1)) ds

. (5.30)

When the resident population is initially at equilibrium, Y(t) = x",2 for all t, then

qx),2(t) =

K t

0 e#

! s0 %1x

),20 #('+$1)) du(% + '1)) ds

1 +K t

0 e#

! s0 %1x

),20 #('+$1)) du(% + '1)) ds

(5.31)

=

'+$1

%1x),20 #('+$1)

5

1" e#(%1x),20 #('+$1))t

6

1 + '+$1

%1x),20 #('+$1)

5

1" e#(%1x),20 #('+$1))t

6 . (5.32)

Recalling that x",20 = 1

R0,2, we see that assuming the resident is at equilibrium, the proba-

bility that strain 1 eventually fixes is asymptotically R0,2

R0,1if R0,1 > R0,2, and 0 otherwise.


5.2.3 Weak Results

Throughout this section, we assume that R0,1 = R0,2, so that the limiting dynamical

system (5.11) has a co-dimension two linear submanifold of fixed points, C. Here we

apply Theorem 4.2 from the previous chapter to conclude that the two-strain SIR model

converges weakly to a di!usion process on C, but first, we adapt the methods of §4.2.3

to compute the derivatives of the projection map 4(x).

The Projection Map and its Derivatives

Recall, if Y(t,x) is the solution to (5.11) with Y(0,x) = x, then

"(x) = limt%"

Y(t). (5.33)

From Section 5.2.1, we see that 40(x0, x1, x2) = x"0, independent of the initial condition,

so that

!i40 = !ij40 = 0 (5.34)

for all i and j. We are thus left with the task of determining the derivatives of 41 and

42.

We first observe that "(x) % C, so that

"141(x0, x1, x2) + "242(x0, x1, x2) =*

x"0

" %. (5.35)

.

We next turn to the task of determining an appropriate invariant of motion for the

dynamical system. Under our assumption (5.19), Fi(x), i = 1, 2, take the form

Yi(t) = "i(Y0(t)" x"0)Yi(t), (5.36)


so thatdY1

dY2=

"1Y1

"2Y2. (5.37)

Solving this givesY %12

Y %21

= C (5.38)

for a fixed constant C. Thus E(x) = x$12

x$21

is an invariant by which flows of the dynamical

system (5.11) may be characterized. In particular,

E(x) = E("(x)) (5.39)

Di!erentiating (5.35) yields

"1!41

!xi

= ""2!42

!xi

, (5.40)

while implicitly di!erentiating (5.39) with respect to x1 yields

"2x%2#11 4%1

2 + "1x%21 4%1#1

2

!42

!x1= "2x

%12 4%2#1

1

!41

!x1. (5.41)

Using (5.40), we can eliminate !31!x1

, and thus obtain

!42

!x1= "

"1"2x%2#11 4%1

2

"21x

%21 4%1#1

2 + "224

%2#11 x%1

2

. (5.42)

In a similar manner, we find

!42

!x2=

"214

%21 x%1#1

2

"21x

%21 4%1#1

2 + "224

%2#11 x%1

2

, (5.43)


!242

!x21

= "1

"21x

%21 4%1#1

2 + "224

%2#11 x%1

2

.

"1"2("2" 1)x%2#21 4%1

2 +2"21"2x

%2#11 (42)

%1#1

!!42

!x1

"

+1

"1

5

"31("1 " 1)x%2

1 4%1#22 " "3

2("2 " 1)4%2#21 x%1

2

6!!42

!x1

"2/

, (5.44)

!242

!x1!x2= "

1

"21x

%21 4%1#1

2 + "224

%2#11 x%1

2

.

"1"224

%2#11 x%1#1

2

!!42

!x1

"

+"21"2x

%2#11 4%1#1

2

!!42

!x2

"

+1

"1

5

"31("1 " 1)x%2

1 4%1#22 " "3

2("2 " 1)4%2#21 x%1

2

6!!42

!x1

"!!42

!x2

"/

, (5.45)

and

!242

!x22

=1

"21x

%21 4%1#1

2 + "224

%2#11 x%1

2

.

"21("1 " 1)4%2

1 x%1#22 " 2"1"

224

%2#11 x%1#1

2

!!42

!x2

"

"1

"1

5

"31("1 " 1)x%2

1 4%1#22 " "3

2("2 " 1)4%2#21 x%1

2

6!!42

!x2

"2/

. (5.46)

The corresponding derivatives for 41 can be obtained from these via (5.40) and its higher

derivatives.

At points (x"0, x

"1, x

"2) % C, these simplify to

!42

!x1

,,,(x0,x1,x2)=(x)0,x

)1,x

)2)= "

"1"2x"2

"21x

"1 + "2

2x"2

,

!42

!x1

,,,(x0,x1,x2)=(x)0 ,x

)1,x

)2)=

"21x

"1

"21x

"1 + "2

2x"2

,

!242

!x21

,,,(x0,x1,x2)=(x)0,x

)1,x

)2)= "

"1"2x"2

"21x

"1 + "2

2x"2

.

"2 " 1

x"1

" 2"21"2

"21x

"1 + "2

2x"2

+

!

"31("1 " 1)

x"1

x"2

" "32("2 " 1)

x"2

x"1

""2x

"2

"21x

"1 + "2

2x"2

/

, (5.47)


!242

!x1!x2

,,,(x0,x1,x2)=(x)0 ,x

)1,x

)2)=

"21"2

"21x

"1 + "2

2x"2

.

"22x

"2 " "2

1x"1

+

!

"31("1 " 1)

x"1

x"2

" "32("2 " 1)

x"2

x"1

"x"1x

"2

"21x

"1 + "2

2x"2

/

, (5.48)

and

!242

!x22

,,,(x0,x1,x2)=(x)0,x

)1,x

)2)=

"21x

"1

"21x

"1 + "2

2x"2

.

"1 " 1

x"2

" 2"1"

22

"21x

"1 + "2

2x"2

"!

"31("1 " 1)

x"1

x"2

" "32("2 " 1)

x"2

x"1

""1x

"1

"21x

"1 + "2

2x"2

/

. (5.49)

Di!usion Limit

As before, we will be interested in the time rescaled process

Z(N)(t) :=1

NX(N)(Nt) (5.50)

and its modification Z(N)(t)), which coincides with Z(N)(t) whenever the latter begins in

C.

Applying Theorem 4.2, using the derivatives calculated above, we have

dZ2(t) = x"0

"21"2Z1Z2

("21Z1 + "2

2Z2)2("1 " "2)

.

"1 + "2 +"1"2("1Z1 + "2Z2)

"21Z1 + "2

2Z2

/

dt

+]

x"0

"1

("21Z1 + "2

2Z2)

:

""2Z2

^

"1Z1 dB1(t) + "1Z1

^

"2Z2 dB2(t)

;

(5.51)

Finally, letting

P (Z1, Z2) =Z1

Z1 + Z2

, (5.52)


we have, via Ito’s formula,

dP (t) = x"0

"1"2

ne(P )

"1P + "2(1" P )

("21P + "2

2(1" P ))2P (1" P )("2 " "1)

(:

"1(1" P ) + "2P +"1"2 ("1P + "2(1" P ))

"21P + "2

2(1" P )

;

dt

"@x"0

]

ne(P )

"1P + "2(1" P )

"21P + "2

2(1" P )

E

"2(1" P )]

"1P dB1(t) + "1P]

"2(1" P ) dB2(t)F

(5.53)

where

ne(P ) = 41 + 42, (5.54)

which, given that "141 + "242 =2x)0" %, may be written

ne(P ) =

2x)0" %

"1P + "2(1" P ). (5.55)

We adopt the notation ne(P ) in analogy with the e!ective population size common in

population genetics, noting that Ne(P ) = Nne(P ) is the actual number of individuals of

all types when type 1 has frequency P .

The corresponding infinitesimal generator takes the form

G&(p) = b(p)&)(p) +1

2a(p)&))(p), (5.56)

where

b(p) = x"0

"1"2

ne(p)

"1p+ "2(1" p)

("21p+ "2

2(1" p))2p(1" p)("2 " "1)

(!

"1(1" p) + "2p+"1"2 ("1p + "2(1" p))

"21p+ "2

2(1" p)

"

(5.57)


and

a(p) = 2x"0

"1"2

ne(p)

("1p+ "2(1" p))3

("21p+ "2

2(1" p))2p(1" p) (5.58)

Thus, there is a drift towards the type with lower contact rate, in proportion to the

di!erence in contact rates. Under our assumption of equal basic reproductive number

R0, this is equivalent to a drift favouring the type with longer infective period, which

may be achieved by reducing either the excess mortality caused by the pathogen, or by

a reduced recovery rate.

In particular, writing

s(p) = e#2! b(p)

a(p) dp (5.59)

="21p+ "2

2(1" p)

("1p+ "2(1" p))2e# $1+$2$1p+$2(1"p) , (5.60)

the probability that strain 1 out-competes strain 2, given an initial frequency of p is [53]

K p

0 s(z) dzK 10 s(z) dz

. (5.61)

Unfortunately, no simple closed form is available for the above integrals. We observe,

however, that writing "2 = "1(1 + ') the above probability admits an asymptotic series

expansion

p+ 'p(1" p) +'2

3p(1" p)(2" p) +O('3) (5.62)

for small values of '. The expected time to absorption and fixation may be computed

similarly.

This result, and Corollary 5.1 above have an interesting context within the develop-

ment of the theory of infectious diseases. Until fairly recently, the “conventional wisdom”

[145, 1, 135], inspired by such examples as myxomatosis in Australian rabbits [58], or,

more recently simian immunodeficiency virus (SIV), was that pathogens evolved towards

reduced virulence, supported by such verbal arguments as, without “the early appear-


ance and dominance of strains of virus which caused a lower mortality”, “rabbits would

have been eradicated or greatly reduced in numbers, and the rabbit itself would have

disappeared from such localities” ([58] p. 494, as quoted in [127]). A series of theoret-

ical studies using ordinary di!erential equations or game-theoretic invasion arguments

(e.g., [127, 2, 3, 20, 136, 110, 137]) indicated that tradeo!s between virulence and the

length of the infectious period could support intermediate levels of virulence, and that

evolution tended to maximize the basic reproduction number, R0; in Corollary 5.1 we

quantified the probability that a strain with a higher value of R0 would successfully

invade, and found that it could be expressed simply in terms of the ratio of basic repro-

ductive numbers for the two strains, which is perhaps unsurprising, as R0 is essentially

the expected lifetime reproductive success rate or absolute fitness of the pathogen, and,

as we observed in the introduction, Haldane first showed that the fixation probability of

an invading mutant is simply its relative fitness [71].

More interesting is the fixation probability for the di!usive limit, when the two strains

have the same R0: here, we see that “conventional wisdom” is not completely lacking,

and that if we account for the common association between increased pathogenicity

and enhanced rates of production for the pathogen [19, 135, 62] as a tradeo! holding

R0 asymptotically fixed (a mathematical idealization, but one that may approximate

the very long term evolutionary dynamics of pathogens which have exhausted most big

benefit mutations) then we do see evolution towards reduced virulence. Recall that the

basic reproduction number for strain i is

R0,i ="i

% + 'i

, (5.63)

where the virulence (the rate excess host mortality) was incorporated together with the

rate of host recovery into the parameter 'i. The expression for the approximate fixation

probability (5.62) shows selection favouring types with higher transmissibility, which


must be balanced either by an increase in the duration of the hosts’ infectious period

or a decrease in virulence. We thus recover the original hypothesis on the evolution

of pathogenicity as a “second–order” consequence of stochastic dynamics (see also [173]

which obtained a similar finding using numerical and heuristic analyses of a related model

for meningococcal disease in which pathogenic strains competed against a benign resident

strain) which may be relevant to long persisting viral pathogens like myxomatosis or SIV.

5.3 Summary

Here, we saw that the results of the previous chapters have applications beyond tradi-

tional population genetics, as illustrated by a study of viral evolution in a multi-strain

SIR epidemic with host demography. Interestingly, the introduction of stochasticity res-

urrected the out-of-favour hypothesis that pathogens become less virulent over time. This

is a small e!ect relative to the advantage of a higher basic reproductive number, but we

believe it merits further study in more realistic epidemic models.

Most significantly, the SIR model here shows that our methods have applicability well

beyond the purview of classical population genetics, which tends to consider populations

and their evolution in isolation, and allow us to consider co-evolution between hosts

and parasitoids, hosts and prey, and similar collections of species interacting at multiple

trophic levels. As such, we believe it is an interesting first step from population genetics

to a theory of “community genetics” that considers evolution in the context of a plurality

of interacting species.

Chapter 6

Conclusion: Towards a

Mathematical Framework for

Density Dependent Population

Genetics

It borders upon absurdity to attempt to write a conclusion when we have only obtained

the most preliminary results on the density-dependent birth-death-mutation processes

that have been the object of study. This chapter then, rather than reflecting upon what

has been done, will identify some of the many tasks that remain to be done.

Thus far, all results we have obtained have been in the context of specific asymptotic

dynamics in the associated functional law of large numbers, Proposition 2.1. Depend-

ing on one’s perspective, this is either a blessing or curse – as we saw in Example 1.1,

there are an unlimited supply of limiting dynamics, and thus no end of questions to

ponder. With no other, more general, strategy apparent, we began by considering two

examples with obvious precedents in classical population genetics, and for which previ-

ously applied strategies could be adapted. In Chapters 3 and 4, we set about generaliz-

178

Chapter 6. Conclusion 179

ing the classical single-population, single-locus theory of population genetics to include

more mechanistic population dynamics, in particular, allowing variable population sizes

and density-dependent regulation. In the first of these chapters, we considered a novel

mutant successfully invading an established wild-type population and found that birth-

death-mutation processes could be coupled to time-inhomogeneous branching processes

(Propostion 3.9) in such a way that a successful invasion was equivalent to non-extinction

of the approximating branching process. In particular, under the assumption of strong se-

lection – that is, deterministic fixation in the functional Law of Large numbers – we could

explicitly compute fixation probabilities (Corollary 3.4) and fixation times (Proposition

3.11). In Chapter 4, we showed that when the deterministic dynamics of the functional

Law of Large Numbers allow stationary coexistence of multiple types at all points of

an attracting submanifold of phase space, we obtained a generalization of the weakly

selected population genetics. Under this assumption, the abundances of types could be

described by a di!usion (Theorem 4.2) that included the Wright-Fisher di!usion as a

special case (Section 4.3.2). Moreover, our results also apply to models incorporating

frequency-dependent selection, fecundity variance polymorphism, resource-ratio theory,

and life-history strategies, such as fecundity/mortality tradeo!s. At least in the case

of two alleotypes, we could obtain explicit expressions for the fixation probabilities and

times, and thus explore which strategies were selectively favoured and thus expected to

persist. Finally, in Chapter 5, we took a tentative step beyond single-population genetics

into the realm of multi-trophic interactions, analyzing simple host-parasitoid dynamics

and finding new theoretical support for selection favouring reduced virulence (Equation

5.62).

Given our stated goal of building bridges between well established traditions of math-

ematical population genetics and population ecology, the obvious next step would be a

more profound interaction with mathematical ecology. In particular, there are a number

of well-established models – with well-understood deterministic dynamics – for which a


stochastic analogue should be available.

The most obvious example is one that should already be here, but for the limits of time

and the interests of brevity, which is the case of a single asymptotically stable or unstable

interior fixed point in RK+ or, more generally, a heteroclinic cycle consisting of several

hyperbolic fixed points. Such dynamics have been frequently proposed as a mechanism

underlying stable coexistence in the ecological literature, but have been largely neglected

in population genetics, where the prevailing model for coexistence is neutrality or near

neutrality. Given su"cient care to deal with degeneracies at the boundary, these cases

are essentially the classical exit problem, and should be easily approached by the large

deviation principle of Freıdlin and Wentzell [63] or the more recent “path-based” approach

of Bakhtin [11, 12, 13], which is closer in spirit to the approach taken here. The latter

also suggest the possibility of obtaining results that depend only on the topological

equivalence class of the law of large numbers.

The other obvious, and perhaps unforgivable lacuna is the complete absence of coa-

lescent theory in the preceding chapters. Just as the “forward-time” theory of population

genetics considerably preceded Kingman’s coalescent, similarly coalescents for density-

dependent birth-death-mutation processes remain work in progress. A preliminary in-

vestigation with Jesse Taylor suggests that the approaches in [15, 176] may be adapted

to obtain the genealogy of a neutral locus linked to one that is weakly selected. The

analogous case of a neutral allele linked to a strongly selected allele should be able to

be approached similarly to [40, 41, 167]. The true analogue of Kingman’s coalescent and

the corresponding infinite-sites or infinite alleles model, describing the genealogy of an

arbitrary collection of linked alleles, each potentially under selection, remains an elusive

goal.

Other extensions that would greatly extend the population-genetic relevance of our

approach would involve moving beyond populations of haploid organisms to diploid or-

ganisms, and from single to multiple loci, and with these the complications of recombi-


nation, linkage and epistasis.

From a more ecologically inspired perspective, natural next steps would be to con-

sider explicit consumer resource models, with one or more limiting resources and explicit

functional responses, which could be more readily compared to experiments in the chemo-

stat. Preliminary work with Christopher Quince suggests this is a profitable direction –

we have been particularly interested to discover competition experiments between strains

of E. coli by Hansen and Hubbell [72] in which they observed qualitative dynamics very

similar to those exhibited by our Generalized Moran model, (Example 2.1). We hope to

have results available soon.

Further afield, but much more interesting, is to consider more complicated food webs

with more trophic levels (e.g. competition among predators for prey which in turn forage

on a basal species, or between prey subject to top-down control by a predator). The

dynamics of such systems are increasingly complicated, with periodic or even chaotic

attractors, and as such will demand considerably more mathematical sophistication than

has been presented here, but would be essential steps in any honest e!ort at bridging

population ecology and population genetics.

Finally, we note that the work presented here has largely attempted to incorporate

insights from population ecology into population genetics, to understand how more bio-

logically realistic dynamics would alter fixation probabilities and gene frequencies, and

how these might complicate the detection and understanding of selection pressures. It is

equally important, however, that the exchange flow in both directions, and this opens up

entirely new avenues of research. Among the most significant of these will be to intro-

duce the finite population perspective of population genetics, with its awareness of drift,

together with the density dependent dynamics considered herein, into the framework of

conservation biology. Current methods of population viability analysis, an essential tool

in species preservation, often neglect density dependence and are limited in their consid-

eration of small e!ective population sizes. An extension of the approach to population


dynamics and population genetics presented herein could provide a valuable correction

to current methods.

Almost one hundred years ago, R. A. Fisher, J. B. S. Haldane, and Sewall Wright

provided mathematical insights invaluable to the creation of the modern evolutionary

synthesis. In doing so, they opened up new directions of mathematical inquiry which

would influence the study of stochastic processes. From our present vantage point, it

is exciting to realize that they were making the first forays into what remains largely

uncharted territory, and that there is still a significant role for mathematics to play in

forging an “extended evolutionary synthesis” [157, 158, 159] that will bring together in-

sights from population genetics, population ecology, evolutionary developmental biology,

genomics, and beyond. It is equally exciting to imagine how participating in a new syn-

thesis will inform new developments in probability theory. In the preceding pages, we

have attempted to make a small contribution at the interface between population dy-

namics and population genetics. With luck, they have also been the point of departure

for more significant contributions to both biology and mathematics.

Appendix A

Probability Glossary

Here, we give a quick and informal survey – without proofs – of the essential definitions

and results we shall need from the theory of probability and stochastic processes. De-

tails on random variables and stochastic processes can be found in any standard text

on measure-theoretic probability, e.g. [18, 37] ([164] is an excellent introductory text).

Results on convergence of random variables may be found in [17, 50, 86] . Results on

martingales, semi-martingales and stochastic integration can be found in [140, 161]. [95]

includes a particularly readable account of Markov processes and di!usions for the applied

practitioner. [129] is an excellent text on coupling, whilst [14] is a very good introduction

for those interested in biological applications.

Throughout, we will implicitly assume a probability space (%,F ,P): P is a probability

measure on an underlying space of outcomes % i.e. P(%) = 1, and F is a (-algebra

of events i.e. measurable subsets of %. We will assume throughout that (%,F ,P) is

complete, i.e. if B % F , A . B, and P(B) = 0, then A % F (and, necessarily, P(A) = 0

also) .

We will use E for the expectation (i.e. the integral) with respect to P.

183

Appendix A. Probability Glossary 184

A.1 Random Variables and Stochastic Processes

Let (S,S) be a measurable space, i.e. S is a (-algebra of measurable subsets of S. A

random variable X with state space S is a measurable function X : % # S.

A random variable X is uniformly integrable if for all ) > 0, there exists % > 0 such

that if P(A) < ) for some A % F , then E [X A] < %. Here, A is the indicator function

of the set A, so

A(x) :=

?

JJ@

JJA

1 if x % A, and

0 otherwise

(A.1)

There are a number of notions of equivalence for random variables. We give them in

order of increasing stringency:

(i) We say that two random variables X and Y taking values in S are equal in distri-

bution if they have the same distribution, i.e.,

E[f(X)] = E[f(Y )]. (A.2)

for all continuous and bounded functions f on S. We write this as XD= Y .

(ii) They are equal almost surely if

P{X = Y } = 1. (A.3)

(iii) X and Y are equal if they are equal as functions on %, i.e.,X(-) = Y (-) for all

- % %.

A stochastic process is a collection of S-valued random variables, {X(t)}t(T , indexed

by a totally ordered set T – for our purposes, T is time, and will either be a compact

set [0, T ] for some fixed T or [0,$). For ease of notation, we will frequently abbreviate

{X(t)}t(T by X(t) or X when the set T is clear.


We will regard stochastic processes taking values in a metric space (S, d) as random

variables taking values in the set of cadlag functions – functions that are continuous from

the right and have left-hand limits – endowed with the Skorokhod topology. With this

topology the cadlag functions form a complete, separable metric space, which we denote

DS[0, T ] or DS[0,$) according to our set of times, T . While the Skorokhod topology can

be described via an explicit metric, more insight is gained from defining the topology by

sequential convergence: we say that X(N)(t) converges to X(t) in the Skorokhod topology

if there exists a sequence of strictly increasing continuous functions *(N) : R+ # R+ such

that

• *(N)(0) = 0,

• limt%"

*(N)(t) =$,

• For all T > 0, limN%"

sup0&s&t

,,,,ln

*(N)(t)" *(N)(s)

t" s

,,,,= 0 (N. B. this is a technical con-

dition to require that the *(N)(t) not change too rapidly; a consequence of this

condition is that limN%"

sup0&t&T

,,*(N)(t)" t

,, = 0), and

• limN%"

sup0&t&T

d(X(N)(*(N)(t)), X(t)) = 0.

Intuitively, for two processes to be close in the Skorokhod topology, we require them to

be close in the topology on S at all points of continuity, as in the usual topology on

continuous functions, but also that the times at which their discontinuities occur must

also be close. Standard references on convergence in DS are [17, 50, 86].

A filtration, {Ft}t(T , is an increasing set of sub-(-algebras of F indexed by a totally

ordered set T , i.e. if s, t % T and s & t, then Fs . Ft . F . A process X is adapted

with respect to F (F -adapted) if X(t) is a Ft-measurable function for all t. The natural

filtration associated to a stochastic process {X(t)}t(T is the filtration

Ft = ((X(t)) := ((

{X(t)#1(B) : B % S})

, (A.4)


i.e. the (-algebra generated by the pre-image of all measurable sets in S under X(t)

(since X(t) is assumed to be measurable for all t % T , X(t)#1(B) is a measurable set

for all B % S). Intuitively, Ft is the available information up to time t. Whenever

a filtration is not explicitly given, we will adopt the convention that we are using the

natural filtration.

A random variable T taking values in T 7 {$} is a stopping time with respect to a

filtration {Ft}t(T if {T < t} % Ft for all t % T . We explain the origin of the term in the

discussion of martingales below. Note that a fixed time is trivially a stopping time.

A.2 Coupling

We say that random variablesX1 andX2 on probability spaces (%1,F1,P1) and (%2,F2,P2)

are coupled if there exists a third probability space (%,F ,P) and random variables X1

and X2 both defined on (%,F ,P) such that X1D= X1 and X2

D= X2.

Suppose S is a totally ordered set with order relation >. If X1 and X2 are random

variables taking values in S, we say that X stochastically dominates Y if for all x % S,

P{X1 > x} , P{X2 > x}. (A.5)

We write this as X1D

> X2.

We say that there is a monotone coupling between X1 and X2 if there exists a prob-

ability space (%,F ,P) and random variables X1 and X2 both defined on (%,F ,P) such

that X1D= X2 and X2

D= X2 and X1 > X2 almost surely.

Strassen’s theorem states that X1D

> X2 if and only if there is a monotone coupling

between X1 and X2. We will use Strassen’s theorem extensively, albeit implicitly – on

several occasions, we will construct a monotone coupling between random variables and

use this to conclude that one stochastically dominates the other.


A.3 Convergence of Random Variables

Just as there are multiple notions of equivalence for random variables, there are also

a number notions of convergence considered for random variables. We shall make use

of four common topologies of convergence, which are described below. For the latter

three, we require that the state space S to be a metric space, with metric d. To avoid

technicalities, we assume that (S, d) is complete and separable i.e. a Polish space. This

will be su"cient for all spaces that we consider.

Almost sure convergence a collection of random variables {X(N)}N(N converges to

X almost surely (abbreviated a.s.) if

P

+

limN%"

X(N) = X-

= 1. (A.6)

We will sometimes write this as X(N) a.s."# X .

Convergence in probability X(N) converges to X in probability (X(N) P"# X) if for

all ) > 0,

PG

d(X(N), X) > )H

# 0 (A.7)

as N #$.

Convergence in distribution X(N) converges to X in distribution (also referred to as

weakly or in law) if for all continuous, bounded functions f : S # R,

E=

f(X(N))>

# E [f(X)] (A.8)

as N #$. We will write this as X(N) D"# X .

Lp convergence X(N) converges to X in Lp (p , 1) if

E=

d(X(N), X)p>

# 0 (A.9)


as N #$. We will sometimes write this as X(N) Lp

"# X .

The first of these topologies are relatively coarser:

X(N) a.s."# X A X(N) P"# X A X(N) D"# X. (A.10)

Almost sure convergence and convergence in Lp are not directly comparable, but

X(N) Lp

"# X A X(N) P"# X. (A.11)

Finally, % is a finite measure space, so

X(N) Lr

"# X A X(N) Lp

"# X (A.12)

for r , p , 1.

A.4 Markov Processes

A stochastic process {X(t)}t(T is a Markov process if

E [X(t)|Fs] = E [X(t)|X(s)] (A.13)

for all s & t i.e. the process is “memoryless”, so that knowing X(s) is equivalent to

knowing the entire history up to time s, Fs.

When considering Markov processes, we will frequently use the notational shorthands

Px {·} := P {·|X0 = x} , (A.14)

Ex [·] := E [·|X0 = x] . (A.15)

The infinitesimal generator (generator, for short) of a Markov process is the linear


operator G defined by

(G&)(x) = limh10

Ex [&(X(h))]" &(x)

h. (A.16)

for the set of functions, D(G), for which this limit exists. D(G) is the domain of G.

Dynkin’s formula states that if T is a stopping time, then

Ex [&(X(T ))] = &(x) + Ex

:9 T

0

(G&)(X(s)) ds

;

. (A.17)

We can use Dynkin’s formula to analyze quantities related to the exit distribution. Let

X(t) be a Markov process with state space S, and let A . S. Then,

Texit := inf{t , 0 : X(t) '% A} (A.18)

is the time of first exit from A. Suppose X(t) is a Markov process with continuous sample

paths. Let $ . !A, and suppose that we have a solution to the Dirichlet problem

(Gh!)(x) = 0 (A.19)

h!(x)|!A =

?

JJ@

JJA

1 if x % $

0 otherwise.

(A.20)

Then

h!(x) = Ex [f(X(Texit))] = Px {X(Texit) % $} (A.21)

gives the probability of exit through $. Similarly, if gexit is the solution to

(Ggexit)(x) = "1 (A.22)

gexit(x)|!A 4 0, (A.23)


then

gexit(x) = Ex [Texit] . (A.24)

Let h be a harmonic function with respect to G (i.e. Gh = 0). The Doob h-transform

of P is the measure P defined by

Px(A) := Ex

:

Ah(X(t))

h(x)

;

, (A.25)

with corresponding generator

(Gf)(x) =1

h(x)G(hf)(x). (A.26)

In particular, if h! solves (A.19), then h! is harmonic, and P is the probability conditioned

on the first exit through $. Similarly to (A.22), if f solves

(Gh!g!)(x) = "h(x) (A.27)

(h!g!)(x)|!A 4 0, (A.28)

then

g!(x) = Ex [Texit|X(Texit) % $] . (A.29)

We will use this to compute expected times to fixation.

We can also use the infinitesimal generator to find the stationary distribution for a

Markov process X(t). Suppose that a stationary measure µ exists. Then,

d

dt

9

S

Ex [f(X(t))] µ(dx) = 0, (A.30)

so that9

S

(Gf)(x), µ(dx) = 0 (A.31)


also. Using the duality between continuous bounded functions and measures, we can

write this as9

S

f(x)(G"µ)(dx) = 0, (A.32)

where G" is the adjoint of G. Now, f is an arbitrary function, so we must have (G"µ)(dx) 4

0. In particular, if µ has a density u, then G" may be regarded as a linear operator acting

on u and G"u = 0.

We will be particularly interested in the case for a K-dimensional di!usion process,

given by

X(t) = X(0) +

9 t

0

b(X(s)) ds+

9 t

0

'(X(s)) dW(s), (A.33)

where b : RK # RK , ' : RK # MK2K(R), and W is K-dimensional Brownian motion.

The generator of X is

(Gf)(x) =K$

i=1

bi(x)(!if)(x) +1

2

K$

i=1

K$

j=1

aij(x)(!i!jf)(x), (A.34)

where a := ''*. The adjoint of G is

(G"f)(x) = "K$

i=1

!i(bif)(x) +1

2

K$

i=1

K$

j=1

!i!j(aijf)(x). (A.35)

In particular, if X(t) is a one dimensional di!usion on an interval [l, r], with generator

(Gf)(x) = b(x)f )(x) +1

2a(x)f ))(x), (A.36)

the solution to the Dirichlet problems Gf = 0 and G"f = 0 – and thus the exit probabil-

ities and stationary distributions – can be explicitly solved. Let

s(x) := e#

! xx0

2b(&)a(&) d+ (A.37)


for some arbitrary x0 % [l, r],

S(x) :=

9 x

x0

s(/) d/, (A.38)

and

m(x) :=1

a(x)s(x). (A.39)

S(x) and m(x) are referred to as the scale function and speed density respectively, and

solve GS = 0 and G"m = 0. In particular, when the denominator is non-zero and finite,

m(x)K r

lm(x) dx

(A.40)

is the stationary distribution for X(t), and if the integrals

9 x0

l

(S(x)" S(l))m(x) dx (A.41)

and9 r

x0

(S(r)" S(x))m(x) dx (A.42)

converge, then X(t) can reach the boundary points l and r in finite time, and

S(x)" S(r)

S(l)" S(r)(A.43)

is the probability that X(t), started from X(0) = x, hits the left endpoint l before the

right endpoint r.

If the probability distribution

P {X(t) % A|X(s) = y} (A.44)


has a density function f , i.e.,

P {X(t) % A|X(s) = y} =

9

A

f(t,x, s,y) dx, (A.45)

then !tf = G"f and !sf = Gf . In the context of di!usions, these two partial di!er-

ential equations are commonly referred to the Fokker-Planck or forward equation and

the Kolmogorov backward equation, respectively, as they describe the forward-time and

backward-time evolution of the process X(t).

A.5 Martingales

A stochastic process {M(t)}t(T is a martingale/sub-martingale/super-martingale with

respect to the filtration F (F -martingale) if

E [M(t)|Fs] = M(s) (A.46)

E [M(t)|Fs] ,M(s) (A.47)

E [M(t)|Fs] &M(s) (A.48)

for all s & t, s, t % T . In particular, if {M(t)}t(T is a Markov martingale, then

E [M(t)|M(s)] = M(s), (A.49)

etc.

The notion of martingale and stopping time arise from probability’s origin in gam-

bling: martingales were coin-tossing games that were popular in 18th century France.

The game was fair if one’s expected holdings after a period of gambling was equal to

the amount with which the player began. Stopping times emerge from betting strategies

(“know when to fold ’em”): non-clairvoyant players can only rely on information up to


the present when deciding when to stop.

The optional sampling theorem or optional stopping theorem states that if S & T are

stopping times with respect to the natural filtration for a super-martingale {M(t)}t(T ,

then

E [M(T )|FS] &M(S). (A.50)

If {M(t)}t(T is a uniformly integrable martingale, then

E [M(T )|FS] = M(S). (A.51)

The optional sampling theorem tells us that in a fair game, there is no gambling strategy

that improves the players odds beyond random chance.

Doob’s martingale inequalities tell us that if M is a non-negative real-valued sub-

martingale then

P

2

supt&T

M(t) , C

4

&E [M(T )]

C, (A.52)

and, if p > 1,

E

:

supt&T

M(t)p;

&!

p

p" 1

"p

E [M(T )p] . (A.53)

Note that if M(t) is a martingale and 3 is a convex function, then 3(M(t)) is a sub-

martingale. In particular, if S is a normed linear space, its norm 1·1 is a convex function,

so

P

2

supt&T

1M(t)1 , C

4

&E [1M(T )1]

C, (A.54)

and, for p > 1,

E

:

supt&T

1M(t)12;

& 4E=

1M(T )12>

. (A.55)

We shall make considerable use of these last two inequalities.

For simplicity, we will henceforth assume that (S, 1·1) is a complete normed linear

space, which is adequate for all random variables and stochastic processes contained


herein.

Doob’s martingale convergence theorem states that if M(t) is an L1-martingale, so

supt([0,") E [1M(t)1] <$, then

limt%"

M(t) (A.56)

exists almost surely. If in addition, M(t) is a square integrable martingale (L2-martingale),

i.e. supt([0,") E=

1M(t)12>

<$, then

E

E

limt%"

M(t)F

= limt%"

E [M(t)] . (A.57)

In general, this need not hold, and one can construct examples using branching processes

for which limt%"M(t) 4 0 whilst E [M(t)] = 1 for all t.

A.6 Quadratic Variation and Covariation

The quadratic covariation (or simply covariation) of two stochastic processes X1 and X2

is

[X1, X2](t) = limP

$

ti(P

|X1(ti+1)"X1(ti)| |X2(ti+1)"X2(ti)| (A.58)

where the limit is over all partitions P = {t0 = 0 < t1 < · · · < tn#1 < tn = t} of [0, t]

with maxi |ti+1 " ti|# 0 and is taken in probability, provided this limit exists.

The covariation of a process X with itself is referred to as its quadratic variation and

is denoted [X ]. If B is Brownian motion, then [B](t) = t.

If M is a local martingale, then its quadratic variation exists and is right-continuous.

If, in addition, M is locally square integrable, then the limit defining the quadratic

variation converges in L1 and

E=

M(t)2>

= E [[M ](t)] . (A.59)


Let {M (N)} be a sequence of martingales such that

limN%"

E

:

supt&T

,,M (N)(t)"M (N)(t")

,,

;

= 0 (A.60)

for all T > 0, i.e., such that the jump sizes converge to 0, and suppose that

limN%"

E=,,[M (N)](t)" t

,,>

= 0. (A.61)

for all t. Then, the martingale Central Limit Theorem states that M (N) D"# B, where B

is Brownian motion.

A.7 Semimartingales

Recall that x 5 y := max{x, y}. A process M(t) is a local martingale (resp. local

square integrable martingale) if there exists a sequence of stopping times T (N) such that

T (N) a.s."# $ and M(t5 T (N)) is a martingale (resp. square integrable martingale) for all

N .

The total variation of a process V up to time t is

Tt(V ) = supP

$

ti(P

|V (ti+1)" V (ti)| (A.62)

where the supremum is over all partitions P = {t0 = 0 < t1 < · · · < tn#1 < tn = t} of

[0, t]. V is a finite variation process if Tt(V ) <$ for all t , 0.

A stochastic process X is a semimartingale with respect to a filtration F if and only

if X = M + V for a local F -martingale M and a F -adapted finite variation process V .

M and V may always be chosen such that M is a local square integrable martingale.

Moreover, if we require V (0) = 0, this decomposition is unique.

If M is a local martingale, then its quadratic variation [M ] is a submartingale and


a semimartingale, and can be decomposed as [M ] = M ) + 2M3, where M ) is a local

martingale and 2M3 is the unique finite variation process with 2M3(0) = 0. 2M3 is called

the predictable quadratic variation or Meyer process associated with M . Note that

E=

M(t)2>

= E [[M ](t)] = E [2M3(t)] .

Given martingales M and N , we may similarly define 2M,N3 to be the finite variation

process with 2M,N3(0) = 0 such that MN " 2M,N3 is a martingale.

If M % RK is a vector-valued martingale, the tensor quadratic variation of M is the

MK2K(R) valued process !M" with

!M"ij = [Mi,Mj]. (A.63)

Similarly,

22M33ij = 2Mi,Mj3. (A.64)

Semimartingales are the class of “good integrators” [161] in that if Y is a cadlag

process and X is a semimartingale, then we can define a stochastic integral

9 t

0

Y (s") dX(s) =

9 t

0

Y (s") dM(s) +

9 t

0

Y (s") dV (s), (A.65)

such thatK t

0 Y (s") dV (s) is a finite variation process. Moreover, if

E

:9 t

0

Y (s")2 d[M ](s)

;

<$, (A.66)

thenK t

0 Y (s") dM(s) is a local square integrable martingale such that

E

.!9 t

0

Y (s") dM(s)

"2/

= E

:9 t

0

Y (s")2 d[M ](s)

;

. (A.67)


When M is Brownian motion, his equality is sometimes referred to as the Ito isometry.

Let X = (X1, . . . , XK) be a RK-valued semimartingale. Ito’s formula states that for

any f % C2(RK),

f(X(t)) = f(X(0)) +K$

i=1

(!if)(X(s")) dXi(s) +1

2

K$

i=1

K$

j=1

(!i!jf)(X(s")) d[Xi, Xj ](s)

+$

s&t

&

f(X(s"))" f(X(s))"K$

i=1

(!if)(X(s"))#Xi(s)

"1

2

K$

i=1

K$

j=1

(!i!jf)(X(s"))#Xi(s)#Xj(s)

'

, (A.68)

where

#Xi(s) := Xi(s)"Xi(s"). (A.69)

Ito’s formula is the analogue of the chain rule for a semimartingale.

A.8 Counting Processes

A random variable N(t) taking values in N0 is a counting process if

(i) N is increasing: N(s) & N(t) for s < t,

(ii) #N(t) = N(t)"N(t") is equal to 0 or 1, and

(iii) E [N(t)] <$ for all t > 0.

A counting process is a semimartingale, and can be decomposed into local martingale

and finite variation components: N(t) = M(t) + V (t). V is always be an increasing

process, and is commonly referred to as the compensator or cumulative intensity of N ,

whilst M is referred to as the compensated process, and is sometimes written N . V (t) has

finite variation, so its derivative V )(t) = *(t) exists almost everywhere. *(t) is referred

to as the intensity of the counting process. Intuitively, N(t) " N(s) is the number of


“events” occurring in the interval (s, t], whilst *(t)#t is the probability an event occurs

in [t, t+#t).

Poisson processes and renewal processes are examples of counting processes. In par-

ticular, given a Markov counting process with intensity *, there exists a rate-one poisson

process P (t) such that

N(t)D= P

!9 t

0

*(s) ds.

"

. (A.70)

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