Some Mathematical Models from Population Genetics 4: Spatial Models

Some mathematical models from population genetics 4: Spatial models

Alison Etheridge

University of Oxford joint work with Jochen Blath (TU Berlin), Mark Meredith (Oxford) Nick Barton (Edinburgh), Frantz Depaulis (Paris VI)

New York, Sept. 07 – p.1 Lessons learned so far

The key message of our balancing selection example is that if we want to study the genealogy of a sample from a structured population, then ﬂuctuations in background frequencies matter.

A central question then is how should we model ﬂuctuations of spatially distributed populations.

New York, Sept. 07 – p.2 Large population, long timescales, measured in units of size N.

Write Zn for the population size after n generations.

0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N .

If Z0 converges, so does ZbNtc . N N≥1 N N≥1

Limit process: dXt = aXtdt + √γXtdBt.

Reminder: Feller’s rescaling

Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . ∞

New York, Sept. 07 – p.3 0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N .

If Z0 converges, so does ZbNtc . N N≥1 N N≥1

Limit process: dXt = aXtdt + √γXtdBt.

Reminder: Feller’s rescaling

Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . ∞ Large population, long timescales, measured in units of size N.

Write Zn for the population size after n generations.

New York, Sept. 07 – p.3 If Z0 converges, so does ZbNtc . N N≥1 N N≥1

Limit process: dXt = aXtdt + √γXtdBt.

Reminder: Feller’s rescaling

Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . ∞ Large population, long timescales, measured in units of size N.

Write Zn for the population size after n generations.

0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N .

New York, Sept. 07 – p.3 Reminder: Feller’s rescaling

Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . ∞ Large population, long timescales, measured in units of size N.

Write Zn for the population size after n generations.

0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N .

If Z0 converges, so does ZbNtc . N N≥1 N N≥1

Limit process: dXt = aXtdt + √γXtdBt.

New York, Sept. 07 – p.3 Galton-Watson branching process branching Brownian motion/ branching random walk.

Offspring born where parent died.

1 Feller rescaling: individual represented by atom of mass N , time in units of size N.

Spatially distributed populations

Populations dispersed in Rd or Zd,

New York, Sept. 07 – p.4 1 Feller rescaling: individual represented by atom of mass N , time in units of size N.

Spatially distributed populations

Populations dispersed in Rd or Zd,

Galton-Watson branching process branching Brownian motion/ branching random walk.

Offspring born where parent died.

New York, Sept. 07 – p.4 Spatially distributed populations

Populations dispersed in Rd or Zd,

Galton-Watson branching process branching Brownian motion/ branching random walk.

Offspring born where parent died.

1 Feller rescaling: individual represented by atom of mass N , time in units of size N.

New York, Sept. 07 – p.4 Super-random walk: dX (t) = m (X (t) X (t)) dt+aX (t)dt+ γX (t)dW (t), i Zd, i ij j − i i i i ∈ Xj p where W (t), t 0 Zd is a collection of independent Brownian mo- { i ≥ }i∈ tions and Xi(t) is the size of the population in deme i at time t.

The limiting processes

The Dawson-Watanabe superprocess: For positive, twice differentiable test functions φ,

t t φ, Xt φ, X0 D∆φ, Xs ds aφ, Xs ds h i − h i − Z0 h i − Z0 h i

is a martingale with quadratic variation t γφ2, X ds. 0 h si R

New York, Sept. 07 – p.5 The limiting processes

The Dawson-Watanabe superprocess: For positive, twice differentiable test functions φ,

t t φ, Xt φ, X0 D∆φ, Xs ds aφ, Xs ds h i − h i − Z0 h i − Z0 h i

is a martingale with quadratic variation t γφ2, X ds. 0 h si Super-random walk: R

dX (t) = m (X (t) X (t)) dt+aX (t)dt+ γX (t)dW (t), i Zd, i ij j − i i i i ∈ Xj p

where W (t), t 0 Zd is a collection of independent Brownian mo- { i ≥ }i∈

tions and Xi(t) is the size of the population in deme i at time t.

New York, Sept. 07 – p.5 Clumping and extinction.

E [ φ, X ] = eat T φ, X , h ti h t 0i Take a = 0.

t 2 var ( φ, Xt ) = γTt−s (Tsφ) , X0 ds. h i Z h i 0 In one and two dimensions grows without bound.

New York, Sept. 07 – p.6 As we saw in the ﬁrst lecture, in Zd can specify the population size locally The Classical Stepping Stone Model.

Populations should be regulated by local rules.

Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions.

Controlling the population

Exogenously specify total population size Fleming-Viot superprocess.

New York, Sept. 07 – p.7 Populations should be regulated by local rules.

Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions.

Controlling the population

Exogenously specify total population size Fleming-Viot superprocess.

As we saw in the ﬁrst lecture, in Zd can specify the population size locally The Classical Stepping Stone Model.

New York, Sept. 07 – p.7 Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions.

Controlling the population

Exogenously specify total population size Fleming-Viot superprocess.

As we saw in the ﬁrst lecture, in Zd can specify the population size locally The Classical Stepping Stone Model.

Populations should be regulated by local rules.

New York, Sept. 07 – p.7 Controlling the population

Exogenously specify total population size Fleming-Viot superprocess.

As we saw in the ﬁrst lecture, in Zd can specify the population size locally The Classical Stepping Stone Model.

Populations should be regulated by local rules.

Individuals living in locally crowded regions will have a lower reproduc-

tive success than those living in sparsely populated regions.

New York, Sept. 07 – p.7 For inﬁnite initial measures, to prevent immediate catastrophe, ∞ h(r)rd−1dr < . 0 ∞ R The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is

dXt(i) = mij (Xt(j) Xt(i)) dt + α M λij Xt(j) Xt(i)dt − − Xj Xj   (i) + γXt(i)dBt . p Note that moment equations not closed.

Locally regulated populations

a(s, x) = α M h(x, y), X (dy) . − h s i For simplicity h(x, y) = h( x y ). k − k

New York, Sept. 07 – p.8 The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is

dXt(i) = mij (Xt(j) Xt(i)) dt + α M λij Xt(j) Xt(i)dt − − Xj Xj   (i) + γXt(i)dBt . p Note that moment equations not closed.

Locally regulated populations

a(s, x) = α M h(x, y), X (dy) . − h s i For simplicity h(x, y) = h( x y ). For inﬁnite initial measures, to k − k prevent immediate catastrophe, ∞ h(r)rd−1dr < . 0 ∞ R

New York, Sept. 07 – p.8 Note that moment equations not closed.

Locally regulated populations

a(s, x) = α M h(x, y), X (dy) . − h s i For simplicity h(x, y) = h( x y ). For inﬁnite initial measures, to k − k prevent immediate catastrophe, ∞ h(r)rd−1dr < . 0 ∞ R The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is

dXt(i) = mij (Xt(j) Xt(i)) dt + α M λij Xt(j) Xt(i)dt − − Xj Xj   (i) + γXt(i)dBt . p

New York, Sept. 07 – p.8 Locally regulated populations

dXt(i) = mij (Xt(j) Xt(i)) dt + α M λij Xt(j) Xt(i)dt − − Xj Xj   (i) + γXt(i)dBt . Note that moment equations not closed. p

New York, Sept. 07 – p.8 Survival and Extinction

Theorem For each ﬁxed interaction kernel h and γ, K > 0 there exists

α0 = α0(K, γ, h) such that for α > α0, the superprocess version of the Bolker-Pacala model with parameters (h, K/α, α, γ) started from any finite initial measure dies out in finite time. If h also satisfies h(r)rd−1dr < , then when started from any tempered initial ∞ measureR (with p > d) the process with these parameters suffers local extinction.

New York, Sept. 07 – p.9 Let α > 0 be fixed. If r2−δh(r) is unbounded for some δ > 0, then for each fixed • γ > 0, there is an M0 > 0 such that for M < M0 the superprocess version of the Bolker-Pacala model with parameters (h, M, α, γ) started from any finite initial measure dies out in finite time. If also h(r)rd−1dr < , so that in ∞ particular d = 1, then when starR ted from any tempered initial measure (with p > 1) the process with these parameters suffers local extinction.

New York, Sept. 07 – p.10 Hutzenthaler & Wakolbinger prove an ergodic theorem and also show that if M is too small, the process dies out.

Suppose that the population X (i) Zd evolves according to • { t }i∈ ,t≥0 the stepping stone version of the Bolker-Pacala model, then if

mij > cλij, for some c > 0, then there exists M1 > 0 such that for

M > M1 the process survives for all time with (strictly) positive probability (started from any non-trivial initial condition).

New York, Sept. 07 – p.11 Suppose that the population X (i) Zd evolves according to • { t }i∈ ,t≥0 the stepping stone version of the Bolker-Pacala model, then if

mij > cλij, for some c > 0, then there exists M1 > 0 such that for

M > M1 the process survives for all time with (strictly) positive probability (started from any non-trivial initial condition).

Hutzenthaler & Wakolbinger prove an ergodic theorem and also show that if M is too small, the process dies out.

New York, Sept. 07 – p.11 t θ θ θ φ, Xt φ, X0 D∆φ, Xs ds h i − h i − Z0 h i t θ2α M hθ( x y ), Xθ(dy) φ(x), Xθ(dx) ds − Z − h k − k s i s 0 D E a martingale with quadratic variation t 2 θ γφ , Xs ds. Z0 h i

Rescaling:

Take d = 2. Deﬁne Xθ by

1 x θ 2 φ, Xt = 2 φ , Xθ t(dx) . h i θ θ

Notation hθ(r) = θ2h(θr).

New York, Sept. 07 – p.12 Rescaling:

Take d = 2. Deﬁne Xθ by

1 x θ 2 φ, Xt = 2 φ , Xθ t(dx) . h i θ θ

Notation hθ(r) = θ2h(θr).

t θ θ θ φ, Xt φ, X0 D∆φ, Xs ds h i − h i − Z0 h i t θ2α M hθ( x y ), Xθ(dy) φ(x), Xθ(dx) ds − Z − h k − k s i s 0 D E a martingale with quadratic variation

t 2 θ γφ , Xs ds. Z0 h i New York, Sept. 07 – p.12 In d = 2, for classical Dawson-Watanabe superprocess, if x is typical point in support of Xt, then

χ , Xt lim E(x) h B(x,r) i = k r↓0 r2 log(1/r) for a constant k (independent of x and t). hθ( x y ), X (dy) log θ. h k − k s i ∼

Survival in two dimensions reﬂects successful eradication of clumping.

Notes:

If r2h(r) as r , then hθ grows without bound as θ , → ∞ → ∞ → ∞ suggesting extinction.

New York, Sept. 07 – p.13 hθ( x y ), X (dy) log θ. h k − k s i ∼

Survival in two dimensions reﬂects successful eradication of clumping.

Notes:

If r2h(r) as r , then hθ grows without bound as θ , → ∞ → ∞ → ∞ suggesting extinction. In d = 2, for classical Dawson-Watanabe superprocess, if x is typical

point in support of Xt, then

χ , Xt lim E(x) h B(x,r) i = k r↓0 r2 log(1/r)

for a constant k (independent of x and t).

New York, Sept. 07 – p.13 Notes:

If r2h(r) as r , then hθ grows without bound as θ , → ∞ → ∞ → ∞ suggesting extinction. In d = 2, for classical Dawson-Watanabe superprocess, if x is typical

point in support of Xt, then

χ , Xt lim E(x) h B(x,r) i = k r↓0 r2 log(1/r)

for a constant k (independent of x and t).

hθ( x y ), X (dy) log θ. h k − k s i ∼

Survival in two dimensions reﬂects successful eradication of clumping.

New York, Sept. 07 – p.13 colonise relatively unpopulated areas quickly, • quickly exploit resources in those areas, • tolerate local competition. •

Competing species

Strategies for survival:

New York, Sept. 07 – p.14 quickly exploit resources in those areas, • tolerate local competition. •

Competing species

Strategies for survival: colonise relatively unpopulated areas quickly, •

New York, Sept. 07 – p.14 tolerate local competition. •

Competing species

Strategies for survival: colonise relatively unpopulated areas quickly, • quickly exploit resources in those areas, •

New York, Sept. 07 – p.14 Competing species

Strategies for survival: colonise relatively unpopulated areas quickly, • quickly exploit resources in those areas, • tolerate local competition. •

New York, Sept. 07 – p.14 dY (t) = m0 (Y (t) Y (t)) dt i ij j − i jX∈Zd

0 0 0 0 + α M λ Yj(t) γ Xj (t) Yi(t)dt + Yi(t)dB˜i(t). − ij − ij jX∈Zd jX∈Zd p  

Model I

dX (t) = m (X (t) X (t)) dt i ij j − i jX∈Zd

+ α M λijXj (t) γij Yj(t) Xi(t)dt + Xi(t)dBi(t), − − jX∈Zd jX∈Zd p  

New York, Sept. 07 – p.15 Model I

dX (t) = m (X (t) X (t)) dt i ij j − i jX∈Zd

+ α M λijXj (t) γij Yj(t) Xi(t)dt + Xi(t)dBi(t), − − jX∈Zd jX∈Zd p  

dY (t) = m0 (Y (t) Y (t)) dt i ij j − i jX∈Zd

0 0 0 0 + α M λ Yj(t) γ Xj (t) Yi(t)dt + Yi(t)dB˜i(t). − ij − ij jX∈Zd jX∈Zd p  

New York, Sept. 07 – p.15 competition is only within-site • migration mechanism is the same for both populations • total population size in each site is ﬁxed • Write Xi(t) Xi(t) pi(t) = = . (Xi(t) + Yi(t)) N

Model II

Simplify our previous model:

New York, Sept. 07 – p.16 migration mechanism is the same for both populations • total population size in each site is ﬁxed • Write Xi(t) Xi(t) pi(t) = = . (Xi(t) + Yi(t)) N

Model II

Simplify our previous model: competition is only within-site •

New York, Sept. 07 – p.16 total population size in each site is ﬁxed • Write Xi(t) Xi(t) pi(t) = = . (Xi(t) + Yi(t)) N

Model II

Simplify our previous model: competition is only within-site • migration mechanism is the same for both populations •

New York, Sept. 07 – p.16 Write Xi(t) Xi(t) pi(t) = = . (Xi(t) + Yi(t)) N

Model II

Simplify our previous model: competition is only within-site • migration mechanism is the same for both populations • total population size in each site is ﬁxed •

New York, Sept. 07 – p.16 Model II

Simplify our previous model: competition is only within-site • migration mechanism is the same for both populations • total population size in each site is ﬁxed • Write Xi(t) Xi(t) pi(t) = = . (Xi(t) + Yi(t)) N

New York, Sept. 07 – p.16 `Selection in favour of heterozygosity' when µ > 1, s > 0,

(αλ α0γ0 ) N > αM α0M 0, and (α0λ0 αγ ) N > α0M 0 αM. ii − ii − ii − ii −

dp (t) = m (p (t) p (t)) dt i ij j − i jX∈Zd 1 + sp (t) (1 p (t)) (1 µp (t)) dt + p (t) (1 p (t)) dW (t), i − i − i rN i − i i

where s = αM α0M 0 + (α0λ0 αγ ) N, − ii − ii (α0λ0 αγ ) N + (αλ α0γ0 ) N µ = ii − ii ii − ii αM α0M 0 + (α0λ0 αγ ) N − ii − ii

New York, Sept. 07 – p.17 dp (t) = m (p (t) p (t)) dt i ij j − i jX∈Zd 1 + sp (t) (1 p (t)) (1 µp (t)) dt + p (t) (1 p (t)) dW (t), i − i − i rN i − i i where s = αM α0M 0 + (α0λ0 αγ ) N, − ii − ii (α0λ0 αγ ) N + (αλ α0γ0 ) N µ = ii − ii ii − ii αM α0M 0 + (α0λ0 αγ ) N − ii − ii `Selection in favour of heterozygosity' when µ > 1, s > 0,

(αλ α0γ0 ) N > αM α0M 0, and (α0λ0 αγ ) N > α0M 0 αM. ii − ii − ii − ii −

New York, Sept. 07 – p.17 For general s there is no convenient moment dual, but we ﬁnd an al- ternative duality with a system of branching annihilating random walks.

The symmetric case

In the case when the two populations evolve symmetrically, Model II reduces to

dp (t) = m (p (t) p (t)) dt i ij j − i Xj 1 + sp (t) (1 p (t)) (1 2p (t)) dt + p (t) (1 p (t)) dW (t), i − i − i rN i − i i

New York, Sept. 07 – p.18 The symmetric case

In the case when the two populations evolve symmetrically, Model II reduces to

dp (t) = m (p (t) p (t)) dt i ij j − i Xj 1 + sp (t) (1 p (t)) (1 2p (t)) dt + p (t) (1 p (t)) dW (t), i − i − i rN i − i i

For general s there is no convenient moment dual, but we ﬁnd an al-

ternative duality with a system of branching annihilating random walks.

New York, Sept. 07 – p.18 Duality: Set w = 1 2p and let n be branching annihilating random i − i t walk with offspring number two, then for s > 0

E w(t)n(0) = E w(0)n(t) . h i h i

Branching annihilating random walk

The Markov process n (t), i Zd , in which n (t) Z , with { i ∈ }t≥0 i ∈ + dynamics

New York, Sept. 07 – p.19 Branching annihilating random walk

The Markov process n (t), i Zd , in which n (t) Z , with { i ∈ }t≥0 i ∈ + dynamics

ni ni 1,  7→ − at rate nimij  n n + 1 j 7→ j n n + m at rate sn i 7→ i i n n 2 at rate 1 n (n 1) i 7→ i − 2 i i − is called a branching annihilating random walk with offspring number m and branching rate s. Duality: Set w = 1 2p and let n be branching annihilating random i − i t walk with offspring number two, then for s > 0

E w(t)n(0) = E w(0)n(t) . h i h i New York, Sept. 07 – p.19 Conjectures for Model II

Based on results of Cardy and Täuber, we conjecture:

For Model II with µ = 2 In d = 1, there is a critical value s > 0 such that the populations • 0 will both persist for all time with positive probability if and only if

s > s0, In d = 2, there is positive probability that both populations will • persist for all time if and only if s > 0, In d 3, this probability is positive if and only if s 0. • ≥ ≥ It would be odd if the case µ = 2 were pathological.

New York, Sept. 07 – p.20 Conjectures for Model I

0 0 0 0 0 Let mij = mij, α = α , M = M ﬁxed, and λij = λij , γij = γij. Parameters such that each population can survive in absence of the other.

1. If λij < γij for all j, then eventually only one population will be present. 2. If λ > γ for all j, then if d 2, with positive probability both ij ij ≥ populations will exihibit longterm coexistence. In one dimension the same result will hold true provided that λ γ is sufﬁciently large. ij − ij 3. If λ = γ and d 3 positive probability coexistence. ij ij ≥ If d 2 then with probability one, one population will die out. ≤

New York, Sept. 07 – p.21 Murrell & Law 2003: asymmetry in interaction. Overall strength of interspecific and intraspecific competition is • the same ( j λij = j γij) but distance over which sense heterospecificP neighboursP (competitors) is shorter than that over which sense conspecific neighbours.

Analogue in our setting: symmetric version of Model I with • λ = λ ( i j ), γ = γ ( i j ), where the functions λ and γ ij k − k ij k − k are monotone decreasing and j λij = j γij, but the range of λij is greater than that of γij. P P Small scales homozygous advantage.

Larger scales heterozygous advantage.

Heteromyopia

Does space promote coexistence?

New York, Sept. 07 – p.22 Analogue in our setting: symmetric version of Model I with • λ = λ ( i j ), γ = γ ( i j ), where the functions λ and γ ij k − k ij k − k are monotone decreasing and j λij = j γij, but the range of λij is greater than that of γij. P P Small scales homozygous advantage.

Larger scales heterozygous advantage.

Heteromyopia

Does space promote coexistence? Murrell & Law 2003: asymmetry in interaction. Overall strength of interspecific and intraspecific competition is • the same ( j λij = j γij) but distance over which sense heterospecificP neighboursP (competitors) is shorter than that over which sense conspecific neighbours.

New York, Sept. 07 – p.22 Small scales homozygous advantage.

Larger scales heterozygous advantage.

Heteromyopia

New York, Sept. 07 – p.22 Heteromyopia

Larger scales heterozygous advantage.

New York, Sept. 07 – p.22 The genealogy of a sample from the population is given by a system of coalescing random walks in a random environment. What do we need to know about N (or its continuous counterpart) to make good { i} approximations to the genealogy?

Hidden assumption: the population size in each deme is large.

What about genetics?

Recall from 1st lecture that for a neutral subdivided population with allelic types a, A, the proportion of type a alleles is

N γ dp (t) = j m (p (t) p (t)) dt + p (t) (1 p (t))dW (t). i N ij j − i rN i − i i Xj i i

New York, Sept. 07 – p.23 What do we need to know about N (or its continuous counterpart) to make good { i} approximations to the genealogy?

Hidden assumption: the population size in each deme is large.

What about genetics?

Recall from 1st lecture that for a neutral subdivided population with allelic types a, A, the proportion of type a alleles is

N γ dp (t) = j m (p (t) p (t)) dt + p (t) (1 p (t))dW (t). i N ij j − i rN i − i i Xj i i

The genealogy of a sample from the population is given by a system of coalescing random walks in a random environment.

New York, Sept. 07 – p.23 Hidden assumption: the population size in each deme is large.

What about genetics?

Recall from 1st lecture that for a neutral subdivided population with allelic types a, A, the proportion of type a alleles is

N γ dp (t) = j m (p (t) p (t)) dt + p (t) (1 p (t))dW (t). i N ij j − i rN i − i i Xj i i

The genealogy of a sample from the population is given by a system of coalescing random walks in a random environment. What do we need to know about N (or its continuous counterpart) to make good { i} approximations to the genealogy?

New York, Sept. 07 – p.23 What about genetics?

Recall from 1st lecture that for a neutral subdivided population with allelic types a, A, the proportion of type a alleles is

N γ dp (t) = j m (p (t) p (t)) dt + p (t) (1 p (t))dW (t). i N ij j − i rN i − i i Xj i i

Hidden assumption: the population size in each deme is large.

New York, Sept. 07 – p.23 An approach of Malécot

Discrete time: Inﬁnite alleles model, write F (y) for the probability of identity in state of two genes separated by y.

Ancestral lineages follow independent Brownian motions, • local population density, δ, a constant, • probability two lineages currently at separation y (a vector in R2 • in the most interesting setting) have a common ancestor in the previous generation is 1 g (y z)g (z)dz, where g is a δ 1 − 1 1 Gaussian density. R

New York, Sept. 07 – p.24 Continuous time Many authors: lineages currently at separation y coalesce at instantaneous rate γgα(y). Problem: There is no consistent forwards in time population model. No sampling consistency •

A recursion for identity

Writing k for the mutation probability

2 1 F (0) F (y) = (1 k) − g1(y z)g1(z)dz − δ Z − 0 0 0 + g1(x)g1(x )F (y + x x)dxdx . Z −

New York, Sept. 07 – p.25 Problem: There is no consistent forwards in time population model. No sampling consistency •

A recursion for identity

Writing k for the mutation probability

2 1 F (0) F (y) = (1 k) − g1(y z)g1(z)dz − δ Z − 0 0 0 + g1(x)g1(x )F (y + x x)dxdx . Z − Continuous time Many authors: lineages currently at separation y coalesce at

instantaneous rate γgα(y).

New York, Sept. 07 – p.25 A recursion for identity

Writing k for the mutation probability

2 1 F (0) F (y) = (1 k) − g1(y z)g1(z)dz − δ Z − 0 0 0 + g1(x)g1(x )F (y + x x)dxdx . Z − Continuous time Many authors: lineages currently at separation y coalesce at

instantaneous rate γgα(y). Problem: There is no consistent forwards in time population model. No sampling consistency •

New York, Sept. 07 – p.25 Fourier transform

2 1 F (0) F (y) = (1 k) − g1(y z)g1(z)dz − δ Z − 0 0 0 + g1(x)g1(x )F (y + x x)dxdx . Z −

Writing f(y) = 1−F (0) g (y z)g (z)dz, δ 1 − 1 R 1 F˜(y˜) = eiy.y˜F (y)dy. 2π Z

2 2 F˜(y˜) = (1 k) f˜(y˜) + (2π) g˜1( y˜)g˜1(y˜)F˜(y˜) − −

New York, Sept. 07 – p.26 Rearranging,

(1 k)2f˜(y˜) F˜(y˜) = − . 1 (1 k)2(2π)2g˜ (y˜)g˜ ( y˜) − − 1 1 − For y = 0 6

∞ 2 2 (1 F (0) 1 2t −kyk /(4σ t) F (y) = − 2 (1 k) e dt, δ Z0 4πσ t −

2 where σ is the variance of the dispersal distribution g1.

New York, Sept. 07 – p.27 Write (1 k) = e−µ. Since −

∞ ν/2 α 1 α −pt ν−1 − 4t e t e dt = 2 Kν (√αp), Reα > 0, Rep > 0, Z0 4 p

(1 F (0)) y F (y) = − K k k 2µ . δ2πσ2 0 σ p Now assume a local scale κ over which F (κ) F (0). Using ≈ K (z) log(1/z) as z 0 0 ∼ → 1 F (y) K ( 2µ y /σ), y > κ. ≈ + log(σ/κ√2µ) 0 k k k k N p = 2πδσ2 is Wright’s neighbourhood size. N

New York, Sept. 07 – p.28 Assumptions: For large timesteps, temporal correlations are negligible. • For well separated lineages: • the chances of coancestry are negligible, • movements of lineages uncorrelated, • Then over all but small scales, Malécot’s formula remains valid if parameters replaced by effective parameters (dispersal rate, neighbourhood size and local scale).

Extending Malécot’s formula

Over sufﬁciently large scales populations may look approximately homogeneous.

New York, Sept. 07 – p.29 Then over all but small scales, Malécot’s formula remains valid if parameters replaced by effective parameters (dispersal rate, neighbourhood size and local scale).

Extending Malécot’s formula

Over sufﬁciently large scales populations may look approximately homogeneous. Assumptions: For large timesteps, temporal correlations are negligible. • For well separated lineages: • the chances of coancestry are negligible, • movements of lineages uncorrelated, •

New York, Sept. 07 – p.29 Extending Malécot’s formula

eters replaced by effective parameters (dispersal rate, neighbourhood

size and local scale).

New York, Sept. 07 – p.29 No explicit models for which we can calculate the parameters. No extension to a sample of size n > 2.

. . . but anyway in a spatial continuum, neighbourhood size could be small and then pairwise coalescences may not dominate.

Some comments

Effective parameters may be hard to ﬁnd. 20 x 2

0 x 1 0 5 10 15 20

New York, Sept. 07 – p.30 No extension to a sample of size n > 2.

. . . but anyway in a spatial continuum, neighbourhood size could be small and then pairwise coalescences may not dominate.

Some comments

Effective parameters may be hard to ﬁnd. 20 x 2

0 x 1 0 5 10 15 20

No explicit models for which we can calculate the parameters.

New York, Sept. 07 – p.30 . . . but anyway in a spatial continuum, neighbourhood size could be small and then pairwise coalescences may not dominate.

Some comments

Effective parameters may be hard to ﬁnd. 20 x 2

0 x 1 0 5 10 15 20

No explicit models for which we can calculate the parameters. No extension to a sample of size n > 2.

New York, Sept. 07 – p.30 Some comments

Effective parameters may be hard to ﬁnd. 20 x 2

0 x 1 0 5 10 15 20

No explicit models for which we can calculate the parameters. No extension to a sample of size n > 2.

. . . but anyway in a spatial continuum, neighbourhood size could be

small and then pairwise coalescences may not dominate.

New York, Sept. 07 – p.30