Coalescence and sampling distributions for Feller diffusions

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Coalescence and sampling distributions for Feller

diﬀusions

Conrad J. Burdena, Robert C. Griﬃthsb

aMathematical Sciences Institute, Australian National University, Canberra, Australia

bSchool of Mathematics, Monash University, Australia

Abstract

Consider the diﬀusion process deﬁned by the forward equation ut(t, x) =

2{xu(t, x)}xx −α{xu(t, x)}xfor t, x ≥0 and −∞ < α < ∞, with an initial

condition u(0, x) = δ(x−x0). This equation was introduced and solved by

Feller to model the growth of a population of independently reproducing

individuals. We explore important coalescent processes related to Feller’s

solution. For any αand x0>0 we calculate the distribution of the random

variable An(s;t), deﬁned as the ﬁnite number of ancestors at a time sin

the past of a sample of size ntaken from the inﬁnite population of a Feller

diﬀusion at a time tsince since its initiation. In a subcritical diﬀusion we

ﬁnd the distribution of population and sample coalescent trees from time t

back, conditional on non-extinction as t→ ∞. In a supercritical diﬀusion

we construct a coalescent tree which has a single founder and derive the

distribution of coalescent times.

Keywords: Coalescent, Diﬀusion process, Branching process, Feller

diﬀusion, Sampling distributions

1. Introduction

Feller (1939, 1951a, Section 5) introduced the process governed by the

diﬀusion equation

ut(t, x) = 1

2{xu(t, x)}xx −α{xu(t, x)}x,0≤x < ∞, α ∈R,(1)

Email addresses: conrad.burden@anu.edu.au (Conrad J. Burden),

Bob.Griffiths@Monash.edu (Robert C. Griﬃths)

Preprint submitted to Theoretical Population Biology September 13, 2023

arXiv:2210.12894v2 [math.PR] 12 Sep 2023

as the continuum analogue of a classical branching process, primarily with

the intention of modelling the growth of a population of statistically inde-

pendently reproducing individuals. This stochastic process has subsequently

found broader applications in biology, genetics, ecology, nuclear physics, sta-

tistical physics, seismology and ﬁnance (see Gan and Waxman (2015, Section

I) for a detailed list of references).

In the present paper we are interested in the Feller diﬀusion as the contin-

uum limit of a Bienaym´e-Galton-Watson (BGW) process, and applications

to population genetics including sampling distributions of populations under-

going neutral mutations between multiple types. Our approach is based on

the coalescent. In common with the diﬀusion limit of the Wright-Fisher and

Moran coalescent trees, the coalescent of a Feller diﬀusion has the property

that it ‘comes down from inﬁnity’. That is to say, the entire, uncountably

inﬁnite, population at any time chosen as the present has a ﬁnite number

of ancestral lineages at any positive time in the past (see Fig. 1). The key

step in our analysis is Theorem 1 in which we calculate the distribution of

the discrete random variable A∞(s;t), deﬁned to be the number of ancestors

at an earlier time t−sof the entire population at the current time tsince

initiation of the process.

Starting from this theorem we follow two paths. In the ﬁrst we consider

the limit t→ ∞ at ﬁxed sof a subcritical Feller diﬀusion conditioned on sur-

vival of the population. Results for expected inter-coalescent waiting times

for a random sample agree with a recent, less straightforward derivation using

the Laplace transform by Burden and Griﬃths (2023).

The second path involves taking the limit s→0 at ﬁxed t. At any

s > 0 we ﬁnd that the ancestry of the process over the interval [0, t −s] is

equivalent to the ancestry of a homogeneous birth-death (BD) process with

s-dependent birth and death rates. Both of these rates become inﬁnite as

s→0 while their diﬀerence remains constant and equal to the parameter

α. Useful tools for characterising the properties of coalescent trees generated

by a homogeneous birth-death (BD) process are the reconstructed process

(RP) (Nee et al., 1994) and the reversed-reconstructed process (RRP) (Al-

dous and Popovic, 2005; Gernhard, 2008; Stadler, 2009; Wiuf, 2018; Ignatieva

et al., 2020), which assumes an improper prior on the time since initiation

of the process, starting with a single ancestor. We construct the coalescent

tree for a Feller diﬀusion as the limit of a RP from a linear BD process, and

from this construct a RRP for the supercritical process and hence calculate

the distribution of coalescent times.

time

xx x x xx

ℓfounders at time 0

ℓexponentially distributed families at time t

X(0) = x0

X(t)

Figure 1: Structure of the ancestral tree of a Feller diﬀusion conditioned on X(0) = x0.

The ancestry of the population X(t) “comes down from inﬁnity”, that is, the number of

ancestors (black lines) of the currently observed inﬁnite population at any ﬁnite time in the

past is ﬁnite. The blue lines either side represent the entire, inﬁnite population at earlier

times, most of whose descendant lines become extinct by time t. The ﬁnal population is

the sum of a Poisson number of independent exponentially distributed family sizes (see

Eq. (7)).

Previously Burden and Simon (2016) and Burden and Soewongsono (2019)

have studied coalescence of the supercritical Feller diﬀusion from a frequen-

tist point of view, in the sense that they calculate conﬁdence intervals on the

time since initiation and initial parameters of the Feller diﬀusion in terms of

a later observation. The current paper is complementary in that the analysis

of the supercritical diﬀusion is Bayesian in that the RRP requires a uniform

prior on the time since initiation of the ancestral tree.

The Feller diﬀusion is implicit within the near-critical continuum limit of a

continuous-time BGW process studied by O’Connell (1995) and Harris et al.

(2020), who also calculate distributions of coalescent times. However their

analysis diﬀers in that it assumes a process initiated by a single progenitor,

their near-critical limit consists of taking the time since initiation to inﬁnity

while keeping the branching time and reproduction rate constant, and results

are stated in terms of the fraction of time since initiation. These diﬀerences

make direct comparison with the current work problematic, though within

the text of this paper we will note connections with O’Connell (1995) where

appropriate.

Crespo et al. (2021) analyse coalescence in the inﬁnite population limit

of a RRP constructed from a BD process. Their limiting process diﬀers from

the s→0 diﬀusion limit in the current paper in that the time axis is rescaled.

Numerical simulations of their coalescent model employ further approxima-

tions which amount to setting the population size as a deterministic function

rather a stochastic process.

The layout of the paper is as follows. In Section 2 we summarise prop-

erties of the Feller diﬀusion including its connection with continuum limits

of BD and BGW processes and the interpretation of Feller’s solution. This

establishes known results and a notation required for subsequent sections. In

Section 3 distributions are derived for An(s, t), deﬁned to be the number of

ancestors at an earlier time t−sof the entire population at the current time

tsince initiation of the process, and for the corresponding count A∞(s;t) of

the number of ancestors of the entire population at time t. The limit t→ ∞

at ﬁxed sof a sub-critical Feller diﬀusion conditioned on survival and the

quasi-stationary sampling distribution are dealt with in Section 4. The limit

s→0 at ﬁxed tand the connection between the RP of a BD process and the

coalescent tree of a Feller diﬀusion are dealt with in Section 5. In Section 6

the coalescent tree for the entire population generated by a a supercritical

Feller diﬀusion is constructed as a RRP, from which the distributions of co-

alescent times for the population and for a ﬁnite sample are calculated in

Section 7. Results are summarised in Section 8.

2. Properties of the Feller Diﬀusion

For any bounded continuous function gwith second derivatives existing,

a standard backward Kolmogorov equation for a stochastic process X(t) is

dtE[g(X(t))] = E[Lg(X(t))] ,

where the right side is the expectation of the function hdeﬁned by h=Lg.

The operator Lis referred to as the generator of the process. In general, if a

continuous random variable X(t) evolves so that

E[X(t+δt)−X(t)|X(t) = x] = a(x)δt +o(δt),

E(X(t+δt)−X(t))2|X(t) = x=b(x)δt +o(δt),

E(X(t+δt)−X(t))k|X(t) = x=o(δt), k ≥3,

(2)

as δt →0, then the generator takes the form (Karlin and Taylor, 1981, p214;

Ewens, 2004, Chapter 4)

L=1

2b(x)∂2

∂x2+a(x)∂

∂x.

For the particular case a(x) = αx,b(x) = xwe refer to the process with

generator

L=1

2x∂2

∂x2+αx ∂

∂x,(3)

as a Feller diﬀusion (Feller, 1951a,b). The process is said to be sub-critical,

critical or super-critical if α < 0, α= 0 or α > 0 respectively. A Feller

diﬀusion can arise as a limit of a BD process or as a limit of a BGW process.

2.1. Feller Diﬀusion as the limit of a BD process

Baake and Wakolbinger (2015) argue that Feller (1951a) regarded Eq. (1)

primarily in terms of the large-population, near-critical limit of a continuous-

time BD process. In Proposition 1 we set out details of how this limit scales,

using notation which will prove useful in Section 5.

Consider a linear BD process with birth rate ˆ

λ(ϵ) and death rate ˆµ(ϵ)

speciﬁed as functions of a parameter ϵ∈R≥0with the property

λ(ϵ) = 1

2ϵ−1+1

2α+O(ϵ),

ˆµ(ϵ) = 1

2ϵ−1−1

2α+O(ϵ),

as ϵ→0. If Mϵ(t) is the number of particles alive at time t, then for

n, y = 0,1,2, . . .,

pn,y := P(Mϵ(t+δt) = y|Mϵ(t) = n)

=









nˆ

λδt +O(δt2), y =n+ 1;

1−n(ˆ

λ+ ˆµ)δt +O(δt2), y =n;

nˆµδt +O(δt2), y =n−1;

O(δt2),otherwise.

(4)

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CoalescenceandsamplingdistributionsforFellerdiffusionsConradJ.Burdena,RobertC.GriffithsbaMathematicalSciencesInstitute,AustralianNationalUniversity,Canberra,AustraliabSchoolofMathematics,MonashUniversity,AustraliaAbstractConsiderthediffusionprocessdefinedbytheforwardequationut(t,x)=12{xu(t,x)}xx−α{x...

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Coalescence and sampling distributions for Feller diffusions

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