Download A Mathematical Framework for Density Dependent Population

Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A Mathematical Framework for Density
Dependent Population Genetics
Todd L. Parsons
Department of Biology
University of Pennsylvania
[email protected]
Applied Mathematics and Statistics Seminar
University of California, Santa Cruz
April 10, 2011
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE G OLDEN A GE OF B IOMATHEMATICS
Georgii Frantsevich
Gause
(1910–1986)
The Struggle For Existence
Williams & Wilkins Co.,
Baltimore, 1934
Alfred James Lotka
(1880–1949)
Vito Volterra
(1860–1940)
Elements of physical biology
Williams & Wilkins Co.,
Baltimore, 1925
Fluctuations in the
abundance of a species
considered mathematically
Nature, (1926) 118: 558–60.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
T HE G AUSE -L OTKA -V OLTERRA M ODEL
◮
Deterministic model of dynamics of
interacting populations; the density
of species i, Yi (t), satisfies


K
X
Ẏi (t) = bi +
aij Yj (t) Yi (t).
j=1
◮
Population kept finite by density
dependent regulation.
◮
Empirical studies with
micro-organisms show excellent fits.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
T HE G AUSE -L OTKA -V OLTERRA M ODEL
◮
Deterministic model of dynamics of
interacting populations; the density
of species i, Yi (t), satisfies


K
X
Ẏi (t) = bi +
aij Yj (t) Yi (t).
j=1
◮
Population kept finite by density
dependent regulation.
◮
Empirical studies with
micro-organisms show excellent fits.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
T HE G AUSE -L OTKA -V OLTERRA M ODEL
◮
Deterministic model of dynamics of
interacting populations; the density
of species i, Yi (t), satisfies


K
X
Ẏi (t) = bi +
aij Yj (t) Yi (t).
j=1
◮
Population kept finite by density
dependent regulation.
◮
Empirical studies with
micro-organisms show excellent fits.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
K OLMOGOROV ’ S E QUATIONS
Kolmogorov proposed
considering the class of
dissipative dynamical systems
such that
Ẏi = Mi (Y(t))Yi (t).
◮
Species i competes
(cooperates) with species j if
∂Mi
∂xj ≤ 0 (≥ 0)
◮
Rich enough to include
many ecological models
and be topologically closed.
However, arbtitrary
dynamics are possible
under these axioms! (Smale,
1975)
◮
Andrey Nikolaevich
Kolmogorov
(1903–1987)
Sulla teoria di Volterra della lotta per
l’esistenza
Giorn. Instituto Ital. Attuari. (1936) 7:
74–80
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
K OLMOGOROV ’ S E QUATIONS
Kolmogorov proposed
considering the class of
dissipative dynamical systems
such that
Ẏi = Mi (Y(t))Yi (t).
◮
Species i competes
(cooperates) with species j if
∂Mi
∂xj ≤ 0 (≥ 0)
◮
Rich enough to include
many ecological models
and be topologically closed.
However, arbtitrary
dynamics are possible
under these axioms! (Smale,
1975)
◮
Andrey Nikolaevich
Kolmogorov
(1903–1987)
Sulla teoria di Volterra della lotta per
l’esistenza
Giorn. Instituto Ital. Attuari. (1936) 7:
74–80
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
K OLMOGOROV ’ S E QUATIONS
Kolmogorov proposed
considering the class of
dissipative dynamical systems
such that
Ẏi = Mi (Y(t))Yi (t).
◮
Species i competes
(cooperates) with species j if
∂Mi
∂xj ≤ 0 (≥ 0)
◮
Rich enough to include
many ecological models
and be topologically closed.
However, arbtitrary
dynamics are possible
under these axioms! (Smale,
1975)
◮
Andrey Nikolaevich
Kolmogorov
(1903–1987)
Sulla teoria di Volterra della lotta per
l’esistenza
Giorn. Instituto Ital. Attuari. (1936) 7:
74–80
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
K OLMOGOROV ’ S E QUATIONS
Kolmogorov proposed
considering the class of
dissipative dynamical systems
such that
Ẏi = Mi (Y(t))Yi (t).
◮
Species i competes
(cooperates) with species j if
∂Mi
∂xj ≤ 0 (≥ 0)
◮
Rich enough to include
many ecological models
and be topologically closed.
However, arbtitrary
dynamics are possible
under these axioms! (Smale,
1975)
◮
Andrey Nikolaevich
Kolmogorov
(1903–1987)
Sulla teoria di Volterra della lotta per
l’esistenza
Giorn. Instituto Ital. Attuari. (1936) 7:
74–80
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
C LASSICAL P OPULATION G ENETICS
Sewall Wright
(1889–1988)
Evolution in Mendelian populations
Genetics 16, 1931, 97–159
Sir Ronald Aylmer Fisher
(1890–1962)
The Genetical Theory of Natural
Selection
Clarendon Press, Oxford, 1930
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
•
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..
.
◮
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Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
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.
◮
◮
◮
Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
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..
.
◮
◮
◮
Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
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.
◮
◮
◮
Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
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.
◮
◮
◮
Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
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.
◮
◮
◮
Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE W RIGHT-F ISHER M ODEL
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•
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2
3
4
..
.
◮
◮
◮
Assume a population with exactly N individuals of a possible K
types, with discrete generations.
Each generation “looks back” to the previous and “picks” a
parent.
Offspring have the same type as their parent.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
G ENERALIZATIONS
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◮
When all types have the same expected number of offspring, we
have the neutral model.
◮
When type i has 1 + si offspring on average, it is selected with
coefficient si
We can also include mutation, by allowing individuals to
randomly switching from type i to type j, with probability µij per
generation.
◮
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
G ENERALIZATIONS
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2
3
4
◮
When all types have the same expected number of offspring, we
have the neutral model.
◮
When type i has 1 + si offspring on average, it is selected with
coefficient si
We can also include mutation, by allowing individuals to
randomly switching from type i to type j, with probability µij per
generation.
◮
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
G ENERALIZATIONS
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@@
•
•
•
1
2
3
4
◮
When all types have the same expected number of offspring, we
have the neutral model.
◮
When type i has 1 + si offspring on average, it is selected with
coefficient si
We can also include mutation, by allowing individuals to
randomly switching from type i to type j, with probability µij per
generation.
◮
Outline
Foundational Models
Unfortunately, it is exceedingly
difficult to obtain analytical
results regarding the
Wright-Fisher model. This was
resolved in 1955 when Kimura
proposed an asymptotic
approximation to the
Wright-Fisher model for large
values of N.
Generalized Birth-Death-Mutation Processes
Applications
Motoo Kimura
(1924–1994)
Solution of a process of random genetic
drift with a continuous model
PNAS (1955), 41: 144–150
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
K IMURA’ S D IFFUSION E QUATION
Let XiN (t) be the number of individuals of type i at time t and let
PN
i (t) =
1 N
X (Nt).
N i
Suppose θij = 2Nµij and σi = 2Nsi are constant. As N → ∞ is large,
D
PN
2 (t) −→ P(t), a diffusion process with probability density f (t, p)
satisfying
K
1X ∂
∂f
=−
∂t
2
∂pi
i=1

"
+ pi σi −

−
K
X
j=1
K
X
j=1

θij  pk +
K
X
θji pj
j=1
! #
"
#
K X
K
2
X
∂
1
pi δij − pj f .
σj pj  f +
2
∂pi ∂pj
i=1 j=1
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
A PPLICATIONS OF K IMURA’ S D IFFUSION , K = 2,
With this, Kimura obtained expressions for the fixation probability of
the mutant: suppose µ12 = µ21 = 0 and let
T = inf {t > 0 : P(t) ∈ {0, 1}}
1 − e−σp
1 − e−σ
= p + σp(1 − p) + O(σ 2 )
P{P(T) = 1|P(0) = p} =
If the mutation rate is nonzero, the diffusion approaches a
steady-state, for which the frequency spectrum of the two alleles is:
lim P {P(t) = p} = pθ12−1 (1 − p)θ21 −1 eσp
t→∞
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
A N U NBRIDGEABLE C HASM ?
Ecology
◮
Density
dependence
◮
Resource
limitation
◮
◮
◮
◮
◮
Multi-species
Competition,
mutualism,
Allee effects
Continuous
Deterministic
ODEs, PDEs
Population Genetics
◮
Fixed
population
sizes
◮
Sampling
arguments
◮
Single species
Discrete
◮
◮
◮
Stochastic
Branching
processes,
Markov chains,
diffusions
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
A N U NBRIDGEABLE C HASM ?
Ecology
◮
Density
dependence
◮
Resource
limitation
◮
◮
◮
◮
◮
Multi-species
Competition,
mutualism,
Allee effects
Continuous
Deterministic
ODEs, PDEs
Population Genetics
◮
Fixed
population
sizes
◮
Sampling
arguments
◮
Single species
Discrete
◮
◮
◮
Stochastic
Branching
processes,
Markov chains,
diffusions
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
A N U NBRIDGEABLE C HASM ?
Ecology
◮
Density
dependence
◮
Resource
limitation
◮
◮
◮
◮
◮
Multi-species
Competition,
mutualism,
Allee effects
Continuous
Deterministic
ODEs, PDEs
Population Genetics
◮
Fixed
population
sizes
◮
Sampling
arguments
◮
Single species
Discrete
◮
◮
◮
Stochastic
Branching
processes,
Markov chains,
diffusions
Outline
Foundational Models
F OOLS RUSH IN . . .
Generalized Birth-Death-Mutation Processes
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
G ENERALIZED B IRTH -D EATH -M UTATION P ROCESSES
XiN (t) = the number of individuals of type i at time t
Event
XN (t) → XN (t) + l
Rate
PK
N
N
N X (t)
i=1 βi,n ( N )Xi (t)
N
δi ( X N(t) )XiN (t)
XiN (t) → XiN (t) − 1
◮
Produce a clutch of n offspring, ni of type i, at per capita rate
N
N X (t)
βi,n
( N ).
N
◮
◮
Per capita mortality rate δi ( X N(t) ), and increasing mortality due
to competition.
Density dependence arises via a “system size” N, which we’ll see
is proportional to the equilibrium carrying capacity.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
G ENERALIZED B IRTH -D EATH -M UTATION P ROCESSES
XiN (t) = the number of individuals of type i at time t
Event
XN (t) → XN (t) + l
Rate
PK
N
N
N X (t)
i=1 βi,n ( N )Xi (t)
N
δi ( X N(t) )XiN (t)
XiN (t) → XiN (t) − 1
◮
Produce a clutch of n offspring, ni of type i, at per capita rate
N
N X (t)
βi,n
( N ).
N
◮
◮
Per capita mortality rate δi ( X N(t) ), and increasing mortality due
to competition.
Density dependence arises via a “system size” N, which we’ll see
is proportional to the equilibrium carrying capacity.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
G ENERALIZED B IRTH -D EATH -M UTATION P ROCESSES
XiN (t) = the number of individuals of type i at time t
Event
XN (t) → XN (t) + l
Rate
PK
N
N
N X (t)
i=1 βi,n ( N )Xi (t)
N
δi ( X N(t) )XiN (t)
XiN (t) → XiN (t) − 1
◮
Produce a clutch of n offspring, ni of type i, at per capita rate
N
N X (t)
βi,n
( N ).
N
◮
◮
Per capita mortality rate δi ( X N(t) ), and increasing mortality due
to competition.
Density dependence arises via a “system size” N, which we’ll see
is proportional to the equilibrium carrying capacity.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A DDITIONAL A SSUMPTIONS
◮
Weak mutation:
◮
◮
◮
◮
◮
N
nj βi,n
(x) < ∞.
N
There exists βin (x) such that limN→∞ N βi,ne
(x) − βin (x) < ∞
i
uniformly on compact sets.
For i 6= j limN→∞ N
P
n∈NK
0
There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞
uniformly on compact sets.
P
Finite mean clutch size: β̄i (x) := ∞
n=1 nβi,n (x) converges
uniformly on compact sets.
P∞
FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges
uniformly on compact sets.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A DDITIONAL A SSUMPTIONS
◮
Weak mutation:
◮
◮
◮
◮
◮
N
nj βi,n
(x) < ∞.
N
There exists βin (x) such that limN→∞ N βi,ne
(x) − βin (x) < ∞
i
uniformly on compact sets.
For i 6= j limN→∞ N
P
n∈NK
0
There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞
uniformly on compact sets.
P
Finite mean clutch size: β̄i (x) := ∞
n=1 nβi,n (x) converges
uniformly on compact sets.
P∞
FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges
uniformly on compact sets.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A DDITIONAL A SSUMPTIONS
◮
Weak mutation:
◮
◮
◮
◮
◮
N
nj βi,n
(x) < ∞.
N
There exists βin (x) such that limN→∞ N βi,ne
(x) − βin (x) < ∞
i
uniformly on compact sets.
For i 6= j limN→∞ N
P
n∈NK
0
There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞
uniformly on compact sets.
P
Finite mean clutch size: β̄i (x) := ∞
n=1 nβi,n (x) converges
uniformly on compact sets.
P∞
FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges
uniformly on compact sets.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A DDITIONAL A SSUMPTIONS
◮
Weak mutation:
◮
◮
◮
◮
◮
N
nj βi,n
(x) < ∞.
N
There exists βin (x) such that limN→∞ N βi,ne
(x) − βin (x) < ∞
i
uniformly on compact sets.
For i 6= j limN→∞ N
P
n∈NK
0
There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞
uniformly on compact sets.
P
Finite mean clutch size: β̄i (x) := ∞
n=1 nβi,n (x) converges
uniformly on compact sets.
P∞
FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges
uniformly on compact sets.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A DDITIONAL A SSUMPTIONS
◮
Weak mutation:
◮
◮
◮
◮
◮
N
nj βi,n
(x) < ∞.
N
There exists βin (x) such that limN→∞ N βi,ne
(x) − βin (x) < ∞
i
uniformly on compact sets.
For i 6= j limN→∞ N
P
n∈NK
0
There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞
uniformly on compact sets.
P
Finite mean clutch size: β̄i (x) := ∞
n=1 nβi,n (x) converges
uniformly on compact sets.
P∞
FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges
uniformly on compact sets.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
A DDITIONAL A SSUMPTIONS
◮
Weak mutation:
◮
◮
◮
◮
◮
N
nj βi,n
(x) < ∞.
N
There exists βin (x) such that limN→∞ N βi,ne
(x) − βin (x) < ∞
i
uniformly on compact sets.
For i 6= j limN→∞ N
P
n∈NK
0
There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞
uniformly on compact sets.
P
Finite mean clutch size: β̄i (x) := ∞
n=1 nβi,n (x) converges
uniformly on compact sets.
P∞
FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges
uniformly on compact sets.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
L AW OF L ARGE N UMBERS
Let YN (t) =
1 N
N X (t),
and let Y(t, x) be the solution of
Ẏi (t) = F(Y(t)) := β̄i (Y(t)) − δi (Y(t)) Yi (t)
with Y(0) = x0 .
Theorem
If YN (0) → x0 as N → ∞ and r < 1, there exists a constant Cr,x0 < 1 such
that
(
)
N
−r
P sup Y (t) − Y(t, x0 ) > N
< N−r
t≤Cr,x0
I’ll call a generalized birth-death-mutation process density-dependent
if this dynamical system is dissipative. Then, the Law of Large
Numbers is of Kolmogorov type.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
L AW OF L ARGE N UMBERS
Let YN (t) =
1 N
N X (t),
and let Y(t, x) be the solution of
Ẏi (t) = F(Y(t)) := β̄i (Y(t)) − δi (Y(t)) Yi (t)
with Y(0) = x0 .
Theorem
If YN (0) → x0 as N → ∞ and r < 1, there exists a constant Cr,x0 < 1 such
that
(
)
N
−r
P sup Y (t) − Y(t, x0 ) > N
< N−r
t≤Cr,x0
I’ll call a generalized birth-death-mutation process density-dependent
if this dynamical system is dissipative. Then, the Law of Large
Numbers is of Kolmogorov type.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRATEGY
◮
◮
We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics
are effectively deterministic, at least for t = O (ln N).
In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value.
Thus, we are left with two tasks:
◮
◮
◮
◮
understanding the dynamics when XiN (t) ≪ N, and
understanding the dynamics in the vicinity of the attractor.
For the former, the picture is fairly complete: XiN (t) ≪ N is a
branching process.
For the latter, I’ll discuss some special cases. Remember, as
Smale as shown, anything is possible!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRATEGY
◮
◮
We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics
are effectively deterministic, at least for t = O (ln N).
In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value.
Thus, we are left with two tasks:
◮
◮
◮
◮
understanding the dynamics when XiN (t) ≪ N, and
understanding the dynamics in the vicinity of the attractor.
For the former, the picture is fairly complete: XiN (t) ≪ N is a
branching process.
For the latter, I’ll discuss some special cases. Remember, as
Smale as shown, anything is possible!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRATEGY
◮
◮
We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics
are effectively deterministic, at least for t = O (ln N).
In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value.
Thus, we are left with two tasks:
◮
◮
◮
◮
understanding the dynamics when XiN (t) ≪ N, and
understanding the dynamics in the vicinity of the attractor.
For the former, the picture is fairly complete: XiN (t) ≪ N is a
branching process.
For the latter, I’ll discuss some special cases. Remember, as
Smale as shown, anything is possible!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRATEGY
◮
◮
We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics
are effectively deterministic, at least for t = O (ln N).
In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value.
Thus, we are left with two tasks:
◮
◮
◮
◮
understanding the dynamics when XiN (t) ≪ N, and
understanding the dynamics in the vicinity of the attractor.
For the former, the picture is fairly complete: XiN (t) ≪ N is a
branching process.
For the latter, I’ll discuss some special cases. Remember, as
Smale as shown, anything is possible!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRATEGY
◮
◮
We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics
are effectively deterministic, at least for t = O (ln N).
In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value.
Thus, we are left with two tasks:
◮
◮
◮
◮
understanding the dynamics when XiN (t) ≪ N, and
understanding the dynamics in the vicinity of the attractor.
For the former, the picture is fairly complete: XiN (t) ≪ N is a
branching process.
For the latter, I’ll discuss some special cases. Remember, as
Smale as shown, anything is possible!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRATEGY
◮
◮
We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics
are effectively deterministic, at least for t = O (ln N).
In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value.
Thus, we are left with two tasks:
◮
◮
◮
◮
understanding the dynamics when XiN (t) ≪ N, and
understanding the dynamics in the vicinity of the attractor.
For the former, the picture is fairly complete: XiN (t) ≪ N is a
branching process.
For the latter, I’ll discuss some special cases. Remember, as
Smale as shown, anything is possible!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
T HE T HREE R EGIONS OF I NTEREST
Phase 2: Logistic
Phase 3: Diffusion
Type 1 (high birth)
200
400
600
Type 2 (low birth)
0
Number
800
Phase 1: Birth-Death
0
5
10
Time (generations)
15
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I NVASION D YNAMICS : Xi (0) ≪ N
First assume no mutation. Fix ε > 0 and let
Kx,ε (t) := {x ∈ RK+ : kx − Y(t, x0 )k < ε},
Tε,N := sup{t : XiN (t) < εN},
+ε
βi,n
(t) :=
−ε
(t) :=
βi,n
δi+ε (t) :=
sup
βi,n (x) + ε,
x∈Kx0 ,ε (t)
x∈Kx0 ,ε (t)
inf
βi,n (x) − ε,
sup
δi,n (x) + ε,
x∈Kx0 ,ε (t)
and
δi−ε (t) :=
inf
x∈Kx0 ,ε (t)
δi,n (x) − ε.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Then we have:
Theorem
−ε
There exist branching processes Xi−ε (t), Xi (t) and Xi+ε (t) with rates βi,n
(t)
+ε
+ε
−ε
and δi (t), βi,n (Y(t, x0 )) and δi (Y(t, x0 )), and βi,n (t) and δi (t) such that
Xi−ε (t) ≤ Xi (t), XiN (t) ≤ Xi+ε (t)
for all t < Tε,N and N suffiiciently large. Moreover,
limε→0 Xi−ε (t) = limε→0 Xi−ε (t) = Xi (t).
More generally, get a branching process with immigration.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Then we have:
Theorem
−ε
There exist branching processes Xi−ε (t), Xi (t) and Xi+ε (t) with rates βi,n
(t)
+ε
+ε
−ε
and δi (t), βi,n (Y(t, x0 )) and δi (Y(t, x0 )), and βi,n (t) and δi (t) such that
Xi−ε (t) ≤ Xi (t), XiN (t) ≤ Xi+ε (t)
for all t < Tε,N and N suffiiciently large. Moreover,
limε→0 Xi−ε (t) = limε→0 Xi−ε (t) = Xi (t).
More generally, get a branching process with immigration.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Corollary
lim P XN (t) = 0 = q(x0 , t) := P{Xi (t) = 0}
N→∞
Moreover,
lim P {Xi (t) > εN} = 1 − q(x0 , +∞).
N→∞
and
Rs
e− 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du δi (Y(s, x0 )) ds
≤ q(x, t)
R t − R s β̄ (Y(u,x ))−δ (Y(u,x )) du
0
0
i
1+ 0 e 0 i
δi (Y(s, x0 )) ds
R t − R s β̄ (Y(u,x ))−δ (Y(u,x )) du
β̄ (Y(s,x0 ))−β̂(Y(s,x0 ))
0
0
i
δi (Y(s, x0 )) + i
e 0 i
ds,
2
0
,
≤
R t − R s β̄ (Y(u,x ))−δ (Y(u,x )) du
β̄ (Y(s,x0 ))−β̂(Y(s,x0 ))
0
0
i
1+ 0 e 0 i
δi (Y(s, x0 )) + i
ds
2
Rt
0
and β̄i (x) = β̂i (x) if and only if βi,n (x) ≡ 0 for n > 1.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I MPLICATIONS I: F ECUNDITY VARIANCE
◮
◮
◮
Rt
The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du
and variance Var(X(t)) =
R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0
0
s
e
β̂(Y(s,
x
))
+
δ(Y(s,
x
))
ds.
0
0
0
β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1.
The probability of extinction for a birth and death process with
birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is
exactly
Rt
◮
Rs
e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds
R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du
0
0
1+ 0e 0
δ(Y(s, x0 )) ds
0
Thus, among all processes with the same mean clutch size and
the same death rate, the one with minimal variance has the
highest invasion probability!
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I MPLICATIONS I: F ECUNDITY VARIANCE
◮
◮
◮
Rt
The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du
and variance Var(X(t)) =
R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0
0
s
e
β̂(Y(s,
x
))
+
δ(Y(s,
x
))
ds.
0
0
0
β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1.
The probability of extinction for a birth and death process with
birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is
exactly
Rt
◮
Rs
e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds
R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du
0
0
1+ 0e 0
δ(Y(s, x0 )) ds
0
Thus, among all processes with the same mean clutch size and
the same death rate, the one with minimal variance has the
highest invasion probability!
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I MPLICATIONS I: F ECUNDITY VARIANCE
◮
◮
◮
Rt
The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du
and variance Var(X(t)) =
R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0
0
s
e
β̂(Y(s,
x
))
+
δ(Y(s,
x
))
ds.
0
0
0
β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1.
The probability of extinction for a birth and death process with
birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is
exactly
Rt
◮
Rs
e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds
R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du
0
0
1+ 0e 0
δ(Y(s, x0 )) ds
0
Thus, among all processes with the same mean clutch size and
the same death rate, the one with minimal variance has the
highest invasion probability!
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I MPLICATIONS I: F ECUNDITY VARIANCE
◮
◮
◮
Rt
The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du
and variance Var(X(t)) =
R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0
0
s
e
β̂(Y(s,
x
))
+
δ(Y(s,
x
))
ds.
0
0
0
β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1.
The probability of extinction for a birth and death process with
birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is
exactly
Rt
◮
Rs
e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds
R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du
0
0
1+ 0e 0
δ(Y(s, x0 )) ds
0
Thus, among all processes with the same mean clutch size and
the same death rate, the one with minimal variance has the
highest invasion probability!
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Today, I’ll focus on two types of long-term (deterministic) dynamics:
◮ Strong selection: the dynamics have a unique globally stable axial
fixed point
lim Y(t, x0 ) = x⋆
t→∞
◮
◮
wtih x⋆ = x⋆i ei .
In this case, invasion ⇒ fixation; from previous results, the
probability that type i fixes is 1 − qi (x0 , +∞).
Quasi-neutrality:
(i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and
(ii) DF(x⋆ ) has no eigenvalue with positive real part.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Today, I’ll focus on two types of long-term (deterministic) dynamics:
◮ Strong selection: the dynamics have a unique globally stable axial
fixed point
lim Y(t, x0 ) = x⋆
t→∞
◮
◮
wtih x⋆ = x⋆i ei .
In this case, invasion ⇒ fixation; from previous results, the
probability that type i fixes is 1 − qi (x0 , +∞).
Quasi-neutrality:
(i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and
(ii) DF(x⋆ ) has no eigenvalue with positive real part.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Today, I’ll focus on two types of long-term (deterministic) dynamics:
◮ Strong selection: the dynamics have a unique globally stable axial
fixed point
lim Y(t, x0 ) = x⋆
t→∞
◮
◮
wtih x⋆ = x⋆i ei .
In this case, invasion ⇒ fixation; from previous results, the
probability that type i fixes is 1 − qi (x0 , +∞).
Quasi-neutrality:
(i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and
(ii) DF(x⋆ ) has no eigenvalue with positive real part.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Today, I’ll focus on two types of long-term (deterministic) dynamics:
◮ Strong selection: the dynamics have a unique globally stable axial
fixed point
lim Y(t, x0 ) = x⋆
t→∞
◮
◮
wtih x⋆ = x⋆i ei .
In this case, invasion ⇒ fixation; from previous results, the
probability that type i fixes is 1 − qi (x0 , +∞).
Quasi-neutrality:
(i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and
(ii) DF(x⋆ ) has no eigenvalue with positive real part.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
P ROJECTION . . .
Let ZN (t) =
1 N
N X (Nt).
Theorem
There exists a generalized Lyapunov function V(x) such that
(i) V(x⋆ ) = 0 if and only if x⋆ ∈ Ω,
(ii) there exists λ > 0 such that V̇(x) := ∇V(x) · F(x) < −λV(x),
and
lim V(ZN (t)) = 0
N→∞
in distribution for all t > 0.
i.e. the rescaled process is eventually confined to Ω.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
P ROJECTION . . .
Let ZN (t) =
1 N
N X (Nt).
Theorem
There exists a generalized Lyapunov function V(x) such that
(i) V(x⋆ ) = 0 if and only if x⋆ ∈ Ω,
(ii) there exists λ > 0 such that V̇(x) := ∇V(x) · F(x) < −λV(x),
and
lim V(ZN (t)) = 0
N→∞
in distribution for all t > 0.
i.e. the rescaled process is eventually confined to Ω.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
. . . AND D IFFUSION
Let
π(x) := lim Y(t, x).
t→∞
Theorem
Let π0 = π(x0 ). Then,
Π(t) = weak − lim π(ZN (t)) = weak − lim ZN (t)
N→∞
N→∞
is a diffusion process on Ω with Π(0+) = π 0 .
The proof also shows how to construct the Kolmogorov equation for
the diffusion, but it’s a mess in general. . .
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
. . . AND D IFFUSION
Let
π(x) := lim Y(t, x).
t→∞
Theorem
Let π0 = π(x0 ). Then,
Π(t) = weak − lim π(ZN (t)) = weak − lim ZN (t)
N→∞
N→∞
is a diffusion process on Ω with Π(0+) = π 0 .
The proof also shows how to construct the Kolmogorov equation for
the diffusion, but it’s a mess in general. . .
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE G ENERALIZED M ORAN M ODEL
Joint work with
Christopher Quince
University of Glasgow
Joshua B. Plotkin
University of Pennsylvania
◮ Parsons, T. L., Quince, C., and Plotkin, J. B. (2010) Some Consequences of demographic
stochasticity in population genetics. Genetics. 185 (4): 1345–1354.
◮ Parsons, T. L., Quince, C., and Plotkin, J. B. (2008) Expected times to absorption and fixation
for neutral and quasi-neutral haploid populations with density dependence. Theor. Pop. Biol.:
74 (4), 302–310.
◮ Parsons, T. L. and Quince, C. (2007b) Fixation in haploid populations exhibiting density
dependence II: The quasi-neutral case. Theor. Pop. Biol. 72 (4): 468–479.
◮ Parsons, T. L. and Quince, C. (2007a) Fixation in haploid populations exhibiting density
dependence I: The non-neutral case. Theor. Pop. Biol. 72 (1): 121–135.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Consider the Generalized Birth-Death-Mutation Process with

P

(1 − j6=i µij )βi xi if n = ei ,
βi,n (x) = µji βj xj
if n = ej ,


0
otherwise
and
δi (x) = δi
◮
◮
◮
1+
PK
j=1
xj (t)
N
!
xi
Reproduction by binary fission, with per-capita birth rate βi .
Intrinsic death rate δi , and increasing mortality due to
competition,
θ
Additionally, with probability µij , µij = Nij , a type i individual
gives birth to an individual of type j 6= i.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Consider the Generalized Birth-Death-Mutation Process with

P

(1 − j6=i µij )βi xi if n = ei ,
βi,n (x) = µji βj xj
if n = ej ,


0
otherwise
and
δi (x) = δi
◮
◮
◮
1+
PK
j=1
xj (t)
N
!
xi
Reproduction by binary fission, with per-capita birth rate βi .
Intrinsic death rate δi , and increasing mortality due to
competition,
θ
Additionally, with probability µij , µij = Nij , a type i individual
gives birth to an individual of type j 6= i.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Consider the Generalized Birth-Death-Mutation Process with

P

(1 − j6=i µij )βi xi if n = ei ,
βi,n (x) = µji βj xj
if n = ej ,


0
otherwise
and
δi (x) = δi
◮
◮
◮
1+
PK
j=1
xj (t)
N
!
xi
Reproduction by binary fission, with per-capita birth rate βi .
Intrinsic death rate δi , and increasing mortality due to
competition,
θ
Additionally, with probability µij , µij = Nij , a type i individual
gives birth to an individual of type j 6= i.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Consider the Generalized Birth-Death-Mutation Process with

P

(1 − j6=i µij )βi xi if n = ei ,
βi,n (x) = µji βj xj
if n = ej ,


0
otherwise
and
δi (x) = δi
◮
◮
◮
1+
PK
j=1
xj (t)
N
!
xi
Reproduction by binary fission, with per-capita birth rate βi .
Intrinsic death rate δi , and increasing mortality due to
competition,
θ
Additionally, with probability µij , µij = Nij , a type i individual
gives birth to an individual of type j 6= i.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
The Law of Large Numbers is given by

Ẏi (t) = βi Yi (t) − δi Yi (t) 1 +
K
X
j=1
Applications

Yj (t) .
This dynamical system has an unstable equilibrium at 0 and stable
equilibria at all points x⋆ such that
β
Strong Selection x⋆i = βδii − 1, x⋆j = 0 if δjj < βδii for all j 6= i (here, x⋆i N is
the carrying capacity of type i
PK ⋆
βi
1
1
Quasi-Neutrality
i=1 xi = α − 1 if δi ≡ α for i = 1, . . . , K i.e. to
order O (N), all types have the same carrying capacity.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
The Law of Large Numbers is given by

Ẏi (t) = βi Yi (t) − δi Yi (t) 1 +
K
X
j=1
Applications

Yj (t) .
This dynamical system has an unstable equilibrium at 0 and stable
equilibria at all points x⋆ such that
β
Strong Selection x⋆i = βδii − 1, x⋆j = 0 if δjj < βδii for all j 6= i (here, x⋆i N is
the carrying capacity of type i
PK ⋆
βi
1
1
Quasi-Neutrality
i=1 xi = α − 1 if δi ≡ α for i = 1, . . . , K i.e. to
order O (N), all types have the same carrying capacity.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
The Law of Large Numbers is given by

Ẏi (t) = βi Yi (t) − δi Yi (t) 1 +
K
X
j=1
Applications

Yj (t) .
This dynamical system has an unstable equilibrium at 0 and stable
equilibria at all points x⋆ such that
β
Strong Selection x⋆i = βδii − 1, x⋆j = 0 if δjj < βδii for all j 6= i (here, x⋆i N is
the carrying capacity of type i
PK ⋆
βi
1
1
Quasi-Neutrality
i=1 xi = α − 1 if δi ≡ α for i = 1, . . . , K i.e. to
order O (N), all types have the same carrying capacity.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRONG S ELECTION
β1
δ1
>
βj
δj
for all j 6= 1.
◮
Without loss of generality,
◮
Assume no mutation or migration: the Law of Large Numbers
tells us that if X1N (t) = εN at some t, then if N is sufficiently large,
type 1 will almost surely exclude all other types.
◮
i.e. a selective sweep occurs.
To find the fixation probability, it suffices to consider the
probability that, starting from X1N (0) << N, type 1 reaches εN
before going extinct.
From our general result, we have
◮
◮
lim P
N→∞
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




R t − 0 β1 −δ1 1+ j=2 Yj (u) du
PK
Yj (s) ds
e
δ1 1 +
0
j=2
0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRONG S ELECTION
β1
δ1
>
βj
δj
for all j 6= 1.
◮
Without loss of generality,
◮
Assume no mutation or migration: the Law of Large Numbers
tells us that if X1N (t) = εN at some t, then if N is sufficiently large,
type 1 will almost surely exclude all other types.
◮
i.e. a selective sweep occurs.
To find the fixation probability, it suffices to consider the
probability that, starting from X1N (0) << N, type 1 reaches εN
before going extinct.
From our general result, we have
◮
◮
lim P
N→∞
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




R t − 0 β1 −δ1 1+ j=2 Yj (u) du
PK
Yj (s) ds
e
δ1 1 +
0
j=2
0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRONG S ELECTION
β1
δ1
>
βj
δj
for all j 6= 1.
◮
Without loss of generality,
◮
Assume no mutation or migration: the Law of Large Numbers
tells us that if X1N (t) = εN at some t, then if N is sufficiently large,
type 1 will almost surely exclude all other types.
◮
i.e. a selective sweep occurs.
To find the fixation probability, it suffices to consider the
probability that, starting from X1N (0) << N, type 1 reaches εN
before going extinct.
From our general result, we have
◮
◮
lim P
N→∞
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




R t − 0 β1 −δ1 1+ j=2 Yj (u) du
PK
Yj (s) ds
e
δ1 1 +
0
j=2
0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRONG S ELECTION
β1
δ1
>
βj
δj
for all j 6= 1.
◮
Without loss of generality,
◮
Assume no mutation or migration: the Law of Large Numbers
tells us that if X1N (t) = εN at some t, then if N is sufficiently large,
type 1 will almost surely exclude all other types.
◮
i.e. a selective sweep occurs.
To find the fixation probability, it suffices to consider the
probability that, starting from X1N (0) << N, type 1 reaches εN
before going extinct.
From our general result, we have
◮
◮
lim P
N→∞
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




R t − 0 β1 −δ1 1+ j=2 Yj (u) du
PK
Yj (s) ds
e
δ1 1 +
0
j=2
0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S TRONG S ELECTION
β1
δ1
>
βj
δj
for all j 6= 1.
◮
Without loss of generality,
◮
Assume no mutation or migration: the Law of Large Numbers
tells us that if X1N (t) = εN at some t, then if N is sufficiently large,
type 1 will almost surely exclude all other types.
◮
i.e. a selective sweep occurs.
To find the fixation probability, it suffices to consider the
probability that, starting from X1N (0) << N, type 1 reaches εN
before going extinct.
From our general result, we have
◮
◮
lim P
N→∞
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




R t − 0 β1 −δ1 1+ j=2 Yj (u) du
PK
Yj (s) ds
e
δ1 1 +
0
j=2
0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S PECIAL CASE : K = 2
◮
Can explicitly evaluate Y2 (t):
Y2 (t) =
◮
The extinction probability of type 1 satisfies
lim P
N→∞
◮
β2 − δ 2
.
2
−(β2 −δ2 )t
e
δ2 + βY22−δ
(0)
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




−
β
−δ
1+
Y
(u)
du
R∞
PK
1
0 1
j=2 j
Y (s) ds
e
δ1 1 +
0
j=2 j
Alternatively, if x2 is the initial frequency of type 2, and
x⋆2 = βδ22 − 1 is its (asymptotic) carrying capacity in the absence of
type 1, a change of variable gives
−X
0




1


.
1 +



β1 −δ1
β1 δ2 −δ1 β2 

R x⋆
2 (β2 −δ2 (1+ξ)) β (β −δ2 ) ξ β2 −δ2 dξ
x2 (β −δ (1+x )) 2 2
x 2
2
2
2

0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S PECIAL CASE : K = 2
◮
Can explicitly evaluate Y2 (t):
Y2 (t) =
◮
The extinction probability of type 1 satisfies
lim P
N→∞
◮
β2 − δ 2
.
2
−(β2 −δ2 )t
e
δ2 + βY22−δ
(0)
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




−
β
−δ
1+
Y
(u)
du
R∞
PK
1
0 1
j=2 j
Y (s) ds
e
δ1 1 +
0
j=2 j
Alternatively, if x2 is the initial frequency of type 2, and
x⋆2 = βδ22 − 1 is its (asymptotic) carrying capacity in the absence of
type 1, a change of variable gives
−X
0




1


.
1 +



β1 −δ1
β1 δ2 −δ1 β2 

R x⋆
2 (β2 −δ2 (1+ξ)) β (β −δ2 ) ξ β2 −δ2 dξ
x2 (β −δ (1+x )) 2 2
x 2
2
2
2

0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S PECIAL CASE : K = 2
◮
Can explicitly evaluate Y2 (t):
Y2 (t) =
◮
The extinction probability of type 1 satisfies
lim P
N→∞
◮
β2 − δ 2
.
2
−(β2 −δ2 )t
e
δ2 + βY22−δ
(0)
n
N
N
X1 (t) = 0|Xi (0) = X0
o

−X




1


= 1 +

Rs
PK




−
β
−δ
1+
Y
(u)
du
R∞
PK
1
0 1
j=2 j
Y (s) ds
e
δ1 1 +
0
j=2 j
Alternatively, if x2 is the initial frequency of type 2, and
x⋆2 = βδ22 − 1 is its (asymptotic) carrying capacity in the absence of
type 1, a change of variable gives
−X
0




1


.
1 +



β1 −δ1
β1 δ2 −δ1 β2 

R x⋆
2 (β2 −δ2 (1+ξ)) β (β −δ2 ) ξ β2 −δ2 dξ
x2 (β −δ (1+x )) 2 2
x 2
2
2
2

0
.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
K = 2, ctd.
◮
In particular, taking x2 → x⋆2 and writing the relative fitness of
type 2 as ν = βδ22 / βδ11 , so ν1 = 1 + s, we get
where p =
◮
X0
N
1+2
and Ne =
1−ν ⋆
x
1+ν 2
−X0
≈ e−2Ne sp ,
Nx⋆
2 .
Thus, when type 1 invades a type 2 population at equilibrium,
we have approximately Kimura’s expresson.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
K = 2, ctd.
◮
In particular, taking x2 → x⋆2 and writing the relative fitness of
type 2 as ν = βδ22 / βδ11 , so ν1 = 1 + s, we get
where p =
◮
X0
N
1+2
and Ne =
1−ν ⋆
x
1+ν 2
−X0
≈ e−2Ne sp ,
Nx⋆
2 .
Thus, when type 1 invades a type 2 population at equilibrium,
we have approximately Kimura’s expresson.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
F REQUENCY D EPENDENT Ne
◮
◮
If invasion occurs away from equilibrium, for example, after a
bottleneck, there is no analogue for the selection coefficient s.
In general, we have a frequency dependent effective population
size:
Ne (x2 ) =
N
β1 δ2 −δ1 β2 β1 −δ1
R x⋆
2 (β2 −δ2 (1+ξ)) β2 (β2 −δ2 ) ξ β2 −δ2 dξ
x x2 (β −δ (1+x )) 2
◮
2
2
2
This leads to r vs. K selection: fixation probability of beneficial
alleles are increased in growing populations and decreased in
shrinking populations (Fisher 1930).
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
F REQUENCY D EPENDENT Ne
◮
◮
If invasion occurs away from equilibrium, for example, after a
bottleneck, there is no analogue for the selection coefficient s.
In general, we have a frequency dependent effective population
size:
Ne (x2 ) =
N
β1 δ2 −δ1 β2 β1 −δ1
R x⋆
2 (β2 −δ2 (1+ξ)) β2 (β2 −δ2 ) ξ β2 −δ2 dξ
x x2 (β −δ (1+x )) 2
◮
2
2
2
This leads to r vs. K selection: fixation probability of beneficial
alleles are increased in growing populations and decreased in
shrinking populations (Fisher 1930).
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
F REQUENCY D EPENDENT Ne
◮
◮
If invasion occurs away from equilibrium, for example, after a
bottleneck, there is no analogue for the selection coefficient s.
In general, we have a frequency dependent effective population
size:
Ne (x2 ) =
N
β1 δ2 −δ1 β2 β1 −δ1
R x⋆
2 (β2 −δ2 (1+ξ)) β2 (β2 −δ2 ) ξ β2 −δ2 dξ
x x2 (β −δ (1+x )) 2
◮
2
2
2
This leads to r vs. K selection: fixation probability of beneficial
alleles are increased in growing populations and decreased in
shrinking populations (Fisher 1930).
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
B ENEFICIAL M UTATIONS
1
2
u = 0.2
u = 0.4
u = 0.6
0.8
2
ψ(z )
Fixation probability
1.5
0.6
1
β1 = 1, δ 1 = 9/10
0.4
β1 = 1, δ 1 = 7/10
β1 = 10/9, δ 1 = 1
0.2
0
β1 = 10/7, δ 1 = 1
0.5
0
0.05
Frequency of mutant allele p
0.1
0
0
0.2
0.4
z2
0.6
0.8
1
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
D ELETERIOUS M UTATIONS
0.1
β2 varied
δ 2 varied
Fixed population size N = 93.14
Fixed population size N = 100
0.001
Relative deviation
Mutant fixation probability
0.01
0.0001
1e-05
0.2
0
1
0.95
1e-06
0.9
1
0.95
1.05
1.05
Relative fitness ν
Deleterious mutations have much higher fixation probabilities than
would be predicted under the equivalent Wright-Fisher/Moran
model.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Now consider the case
1
βi
≡
δi
α
for i = 1, . . . , K
◮
Trade-off: an increased birth rate is compensated by an increased
death rate.
◮
In deterministic
limit, all types have the same carrying capacity,
N⋆ = α1 − 1 N,
◮
◮
...and they coexist stably.
If the population size were fixed, quasi-neutral types would be
competitively neutral in the sense of Kimura.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Now consider the case
1
βi
≡
δi
α
for i = 1, . . . , K
◮
Trade-off: an increased birth rate is compensated by an increased
death rate.
◮
In deterministic
limit, all types have the same carrying capacity,
N⋆ = α1 − 1 N,
◮
◮
...and they coexist stably.
If the population size were fixed, quasi-neutral types would be
competitively neutral in the sense of Kimura.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Now consider the case
1
βi
≡
δi
α
for i = 1, . . . , K
◮
Trade-off: an increased birth rate is compensated by an increased
death rate.
◮
In deterministic
limit, all types have the same carrying capacity,
N⋆ = α1 − 1 N,
◮
◮
...and they coexist stably.
If the population size were fixed, quasi-neutral types would be
competitively neutral in the sense of Kimura.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Now consider the case
1
βi
≡
δi
α
for i = 1, . . . , K
◮
Trade-off: an increased birth rate is compensated by an increased
death rate.
◮
In deterministic
limit, all types have the same carrying capacity,
N⋆ = α1 − 1 N,
◮
◮
...and they coexist stably.
If the population size were fixed, quasi-neutral types would be
competitively neutral in the sense of Kimura.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -N EUTRALITY
Now consider the case
1
βi
≡
δi
α
for i = 1, . . . , K
◮
Trade-off: an increased birth rate is compensated by an increased
death rate.
◮
In deterministic
limit, all types have the same carrying capacity,
N⋆ = α1 − 1 N,
◮
◮
...and they coexist stably.
If the population size were fixed, quasi-neutral types would be
competitively neutral in the sense of Kimura.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
S AMPLE PATHS FOR THE Q UASI -N EUTRAL P ROCESS
K=2
2
y2
1
0
0
1
2
y1
stochastic process —
deterministic limit —
Two sample paths, along with the deterministic trajectories.
β2 /β1 = 2.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
K OLMOGOROV E QUATION FOR P(t)
Let P(t) = limN→∞
PK 1 N ZN (t).
i=1 Zi (t)
P(t) is a diffusion process with
Kolmogorov equation
K−1
∂f
1X ∂
=−
∂t
2
∂pi
i=1
"

−
+
PK
K
X
j=1
βi p i
k=1
βk p k

θij  βi pi +
2
K
X
k=1
K
X
for θij = Nµij .
! #
(βk − βi )βk pk f
"
K−1 K−1
1 X X ∂2
+
βi p i
2
∂pi ∂pj
i=1 j=1
θji βj pj
j=1
βj p j
δij − PK
k=1
βk p k
! #
f ,
Unlike Kimura’s diffusion, we have drift terms even when all types
have the same expected lifetime reproductive success!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
K OLMOGOROV E QUATION FOR P(t)
Let P(t) = limN→∞
PK 1 N ZN (t).
i=1 Zi (t)
P(t) is a diffusion process with
Kolmogorov equation
K−1
∂f
1X ∂
=−
∂t
2
∂pi
i=1
"

−
+
PK
K
X
j=1
βi p i
k=1
βk p k

θij  βi pi +
2
K
X
k=1
K
X
for θij = Nµij .
! #
(βk − βi )βk pk f
"
K−1 K−1
1 X X ∂2
+
βi p i
2
∂pi ∂pj
i=1 j=1
θji βj pj
j=1
βj p j
δij − PK
k=1
βk p k
! #
f ,
Unlike Kimura’s diffusion, we have drift terms even when all types
have the same expected lifetime reproductive success!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
I NTERPRETING THE D RIFT T ERMS
◮
◮
Recall that the Law of Large Numbers was given by
Ẏ = F(Y(t)),
P
where Fi (x) = βi xi − δi xi 1 + Kj=1 xj .
The Jacobian, DF, evaluated at a point α1 − 1 p on the carrying
simplex has eigenvalues 0 (multiplicity K − 1) and
λ(p) = −
K
X
βi p i
i=1
with eigenvector tangent to the stable manifold.
◮
The drift terms are
b(p) = grad(aij (p)) (ln |λ(p)|)
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
I NTERPRETING THE D RIFT T ERMS
◮
◮
Recall that the Law of Large Numbers was given by
Ẏ = F(Y(t)),
P
where Fi (x) = βi xi − δi xi 1 + Kj=1 xj .
The Jacobian, DF, evaluated at a point α1 − 1 p on the carrying
simplex has eigenvalues 0 (multiplicity K − 1) and
λ(p) = −
K
X
βi p i
i=1
with eigenvector tangent to the stable manifold.
◮
The drift terms are
b(p) = grad(aij (p)) (ln |λ(p)|)
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
I NTERPRETING THE D RIFT T ERMS
◮
◮
Recall that the Law of Large Numbers was given by
Ẏ = F(Y(t)),
P
where Fi (x) = βi xi − δi xi 1 + Kj=1 xj .
The Jacobian, DF, evaluated at a point α1 − 1 p on the carrying
simplex has eigenvalues 0 (multiplicity K − 1) and
λ(p) = −
K
X
βi p i
i=1
with eigenvector tangent to the stable manifold.
◮
The drift terms are
b(p) = grad(aij (p)) (ln |λ(p)|)
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
I NTERPRETING THE D RIFT T ERMS
◮
The drift is in the direction transverse the carrying simplex in
which this eigenvalue increases most rapidly, with respect to the
Riemannian metric induced by the diffusion terms (aij )(p).
◮
This provides a stochastic second order effect to Tilman’s
R⋆ -theory of competitive exclusion in ecology: the species with
greatest equilibrium density (and thus can survive at the lowest
resource density) will exclude all others.
Here, we see that when two species can both exist at this lowest
resource density, the type that can minimize the size of stochastic
fluctuations from this minimum level has a competitive
advantage.
◮
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
I NTERPRETING THE D RIFT T ERMS
◮
The drift is in the direction transverse the carrying simplex in
which this eigenvalue increases most rapidly, with respect to the
Riemannian metric induced by the diffusion terms (aij )(p).
◮
This provides a stochastic second order effect to Tilman’s
R⋆ -theory of competitive exclusion in ecology: the species with
greatest equilibrium density (and thus can survive at the lowest
resource density) will exclude all others.
Here, we see that when two species can both exist at this lowest
resource density, the type that can minimize the size of stochastic
fluctuations from this minimum level has a competitive
advantage.
◮
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
I NTERPRETING THE D RIFT T ERMS
◮
The drift is in the direction transverse the carrying simplex in
which this eigenvalue increases most rapidly, with respect to the
Riemannian metric induced by the diffusion terms (aij )(p).
◮
This provides a stochastic second order effect to Tilman’s
R⋆ -theory of competitive exclusion in ecology: the species with
greatest equilibrium density (and thus can survive at the lowest
resource density) will exclude all others.
Here, we see that when two species can both exist at this lowest
resource density, the type that can minimize the size of stochastic
fluctuations from this minimum level has a competitive
advantage.
◮
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
F IXATION P ROBABILITY, K = 2,
Suppose µ12 = µ21 = 0, and let
T = inf {t > 0 : P(t) ∈ {0, 1}} .
P {P(T) = 1|P(0) = p} = p +
β1 − β2
p(1 − p),
β1 + β2
This is (formally) equivalent to selection favouring a lower birth rate:
2
identical to Kimura’s approximation for σ = ββ11 −β
+β2 .
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
F IXATION P ROBABILITY, K = 2,
Suppose µ12 = µ21 = 0, and let
T = inf {t > 0 : P(t) ∈ {0, 1}} .
P {P(T) = 1|P(0) = p} = p +
β1 − β2
p(1 − p),
β1 + β2
This is (formally) equivalent to selection favouring a lower birth rate:
2
identical to Kimura’s approximation for σ = ββ11 −β
+β2 .
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -S TATIONARY D ISTRIBUTION , K = 2
Now suppose µ12 > 0, µ21 > 0.
β
lim P {P(t) = p} = p
t→∞
θ12 β1 −1
2
β
θ21 β2 −1
(1 − p)
1
e
θ12
β1
+
θ21
β2
(β1 −β2 )p
,
Again, this resembles Kimura’s expression, with
β2
θ21
θ12
β1
(β1 − β2 ).
θ̃21 = θ21 − 1
σ=
+
θ̃12 = θ12 − 1
β2
β1
β1
β2
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -S TATIONARY D ISTRIBUTION , K = 2
Now suppose µ12 > 0, µ21 > 0.
β
lim P {P(t) = p} = p
t→∞
θ12 β1 −1
2
β
θ21 β2 −1
(1 − p)
1
e
θ12
β1
+
θ21
β2
(β1 −β2 )p
,
Again, this resembles Kimura’s expression, with
β2
θ21
θ12
β1
(β1 − β2 ).
θ̃21 = θ21 − 1
σ=
+
θ̃12 = θ12 − 1
β2
β1
β1
β2
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
F IXATION P ROBABILITY, K = 2
1
Analytic approximation
Numerical
Probability of mutant fixing
0.8
0.6
0.4
0.2
0
0
200
400
600
Initial total population size
Fixation favours high birth rate at low density
(N = 1000, β1 = 1, β2 = 2).
800
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Q UASI -S TATIONARY F REQUENCY S PECTRUM , K = 2
Quasi-neutral birth-death model
WF model with selection (γ=-0.6)
Probability density
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Frequency of mutant (type 2)
The frequency spectrum is almost identical to the Wright-Fisher
frequency spectrum with selection. The type with low birth rate
appears to be advantageous (β1 = 2 and β1 = 1).
Foundational Models
Generalized Birth-Death-Mutation Processes
◮
However, had selection been
inferred from a competition assay,
only the initial birth-death phase
would have been seen.
◮
Yields ŝ = (β1 − β2 )(1 − α) = 0.5.
i.e. would have predicted type with
high birth rate favoured.
◮
Number of individuals
Outline
Time (1000’s of generations)
Two standard methods of inference yield contradictory results.
Applications
Foundational Models
Generalized Birth-Death-Mutation Processes
◮
However, had selection been
inferred from a competition assay,
only the initial birth-death phase
would have been seen.
◮
Yields ŝ = (β1 − β2 )(1 − α) = 0.5.
i.e. would have predicted type with
high birth rate favoured.
◮
Number of individuals
Outline
Time (1000’s of generations)
Two standard methods of inference yield contradictory results.
Applications
Foundational Models
Generalized Birth-Death-Mutation Processes
◮
However, had selection been
inferred from a competition assay,
only the initial birth-death phase
would have been seen.
◮
Yields ŝ = (β1 − β2 )(1 − α) = 0.5.
i.e. would have predicted type with
high birth rate favoured.
◮
Number of individuals
Outline
Time (1000’s of generations)
Two standard methods of inference yield contradictory results.
Applications
Foundational Models
Generalized Birth-Death-Mutation Processes
◮
However, had selection been
inferred from a competition assay,
only the initial birth-death phase
would have been seen.
◮
Yields ŝ = (β1 − β2 )(1 − α) = 0.5.
i.e. would have predicted type with
high birth rate favoured.
◮
Number of individuals
Outline
Time (1000’s of generations)
Two standard methods of inference yield contradictory results.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
E MPIRICAL S UPPORT ?
Phase 2: Logistic
Phase 3: Diffusion
Type 1 (high birth)
400
600
Type 2 (low birth)
0
200
Number
800
Phase 1: Birth-Death
0
5
10
Time (generations)
(Hansen & Hubbell, Science 207, 1980, 1491–1493)
15
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
E PISTATICALLY I NTERACTING N EUTRAL M UTATIONS
Joint work with:
Jeremy Draghi
University of
Pennsylvania
Günter Wagner
Yale University
Joshua Plotkin
University of
Pennsylvania
◮ Draghi J. A., Parsons, T. L. and Plotkin, J. B. Epistasis increases the rate of conditionally
neutral substitution in an adapting population. Genetics 187 (4): 1139–1152.
◮ Draghi J. A., Parsons, T. L., Wagner, G. P., and Plotkin, J. B. (2010) Mutational robustness can
facilitate adaptation. Nature 463 (7279): 353–355.
(Just a few words to avoid any spoliers. . . )
Outline
Foundational Models
◮
◮
◮
◮
◮
Generalized Birth-Death-Mutation Processes
Applications
Consider a population that has reached a local fitness peak, i.e.
all subsequent mutations are neutral or deleterious.
Some neutral mutations, however, may interact to facilitate
subsequent beneficial mutations.
Suppose neutral mutations happen at rate µn = θNn , and of them,
a fraction pne act epistatically. Further, individuals with an
epistatic mutation experience beneficial mutations at rate
µb = θNb .
For simplicity, assume all other mutations are deleterious.
What is the waiting time until the first beneficial mutation
appears?
Outline
Foundational Models
◮
◮
◮
◮
◮
Generalized Birth-Death-Mutation Processes
Applications
Consider a population that has reached a local fitness peak, i.e.
all subsequent mutations are neutral or deleterious.
Some neutral mutations, however, may interact to facilitate
subsequent beneficial mutations.
Suppose neutral mutations happen at rate µn = θNn , and of them,
a fraction pne act epistatically. Further, individuals with an
epistatic mutation experience beneficial mutations at rate
µb = θNb .
For simplicity, assume all other mutations are deleterious.
What is the waiting time until the first beneficial mutation
appears?
Outline
Foundational Models
◮
◮
◮
◮
◮
Generalized Birth-Death-Mutation Processes
Applications
Consider a population that has reached a local fitness peak, i.e.
all subsequent mutations are neutral or deleterious.
Some neutral mutations, however, may interact to facilitate
subsequent beneficial mutations.
Suppose neutral mutations happen at rate µn = θNn , and of them,
a fraction pne act epistatically. Further, individuals with an
epistatic mutation experience beneficial mutations at rate
µb = θNb .
For simplicity, assume all other mutations are deleterious.
What is the waiting time until the first beneficial mutation
appears?
Outline
Foundational Models
◮
◮
◮
◮
◮
Generalized Birth-Death-Mutation Processes
Applications
Consider a population that has reached a local fitness peak, i.e.
all subsequent mutations are neutral or deleterious.
Some neutral mutations, however, may interact to facilitate
subsequent beneficial mutations.
Suppose neutral mutations happen at rate µn = θNn , and of them,
a fraction pne act epistatically. Further, individuals with an
epistatic mutation experience beneficial mutations at rate
µb = θNb .
For simplicity, assume all other mutations are deleterious.
What is the waiting time until the first beneficial mutation
appears?
Outline
Foundational Models
◮
◮
◮
◮
◮
Generalized Birth-Death-Mutation Processes
Applications
Consider a population that has reached a local fitness peak, i.e.
all subsequent mutations are neutral or deleterious.
Some neutral mutations, however, may interact to facilitate
subsequent beneficial mutations.
Suppose neutral mutations happen at rate µn = θNn , and of them,
a fraction pne act epistatically. Further, individuals with an
epistatic mutation experience beneficial mutations at rate
µb = θNb .
For simplicity, assume all other mutations are deleterious.
What is the waiting time until the first beneficial mutation
appears?
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
M UTATIONAL C LASSES
We can group the individuals into three classes, A’s, which cannot
access beneficial mutations, B’s that can, and C’s, which have
beneficial mutations:
µn pne
µn (1−pne )
5 A o
µn (1−pne )
/
µb
B _ _ _ _ _ _/ C
U
µn pne
All other mutations are lethal and no replacement occurs.
Applications
Outline
Foundational Models
◮
◮
Generalized Birth-Death-Mutation Processes
For simplicity, assume that the ecological dynamics of the A’s,
have reached an equilibrium, x⋆A .
Starting from a single individual, the number of B’s, XB (t) grow
as a critical Markov branching process with immigration:
Transition
XB → XB + n
XB → XB + 1
XB → XB − 1
◮
Applications
Rate
βB,n (x∗
A )XB
θn pne x⋆
A
δB (x∗
A )XB
h θ Rt
i
b
The waiting time until the first C is E e− N 0 XB (s) ds .
Outline
Foundational Models
◮
◮
Generalized Birth-Death-Mutation Processes
For simplicity, assume that the ecological dynamics of the A’s,
have reached an equilibrium, x⋆A .
Starting from a single individual, the number of B’s, XB (t) grow
as a critical Markov branching process with immigration:
Transition
XB → XB + n
XB → XB + 1
XB → XB − 1
◮
Applications
Rate
βB,n (x∗
A )XB
θn pne x⋆
A
δB (x∗
A )XB
h θ Rt
i
b
The waiting time until the first C is E e− N 0 XB (s) ds .
Outline
Foundational Models
◮
◮
Generalized Birth-Death-Mutation Processes
For simplicity, assume that the ecological dynamics of the A’s,
have reached an equilibrium, x⋆A .
Starting from a single individual, the number of B’s, XB (t) grow
as a critical Markov branching process with immigration:
Transition
XB → XB + n
XB → XB + 1
XB → XB − 1
◮
Applications
Rate
βB,n (x∗
A )XB
θn pne x⋆
A
δB (x∗
A )XB
h θ Rt
i
b
The waiting time until the first C is E e− N 0 XB (s) ds .
Outline
Foundational Models
◮
◮
◮
◮
Applications
Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process
with Immigration.
Rt
P⌊t/δ⌋
X (s) ds ≈ n=0 XB (nδ)δ.
0 B
Taking δ =
√1 ,
N
√
2
and replace t with Nt, get Ee−θb t
Pn
P
√
Nt XN
k=0 k
Nt2
.
X
k
Lastly, use known results for the GWI: k=0
where Y has
n2 → Y,
q θn (1−pne )
−θ Y θb
Laplace Transform E e b = sech2 γ
2 , where
γ=
◮
Generalized Birth-Death-Mutation Processes
P∞
⋆
n=2 n(n−1)βBl (xA )
P∞
⋆ )+δ(x⋆ ) .
β
(x
Bl
n=2
A
A
For the implications, go to Jeremy Draghi’s talk in EEB on
Wednesday!
Outline
Foundational Models
◮
◮
◮
◮
Applications
Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process
with Immigration.
Rt
P⌊t/δ⌋
X (s) ds ≈ n=0 XB (nδ)δ.
0 B
Taking δ =
√1 ,
N
√
2
and replace t with Nt, get Ee−θb t
Pn
P
√
Nt XN
k=0 k
Nt2
.
X
k
Lastly, use known results for the GWI: k=0
where Y has
n2 → Y,
q θn (1−pne )
−θ Y θb
Laplace Transform E e b = sech2 γ
2 , where
γ=
◮
Generalized Birth-Death-Mutation Processes
P∞
⋆
n=2 n(n−1)βBl (xA )
P∞
⋆ )+δ(x⋆ ) .
β
(x
Bl
n=2
A
A
For the implications, go to Jeremy Draghi’s talk in EEB on
Wednesday!
Outline
Foundational Models
◮
◮
◮
◮
Applications
Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process
with Immigration.
Rt
P⌊t/δ⌋
X (s) ds ≈ n=0 XB (nδ)δ.
0 B
Taking δ =
√1 ,
N
√
2
and replace t with Nt, get Ee−θb t
Pn
P
√
Nt XN
k=0 k
Nt2
.
X
k
where Y has
Lastly, use known results for the GWI: k=0
n2 → Y,
q θn (1−pne )
−θ Y θb
Laplace Transform E e b = sech2 γ
2 , where
γ=
◮
Generalized Birth-Death-Mutation Processes
P∞
⋆
n=2 n(n−1)βBl (xA )
P∞
⋆ )+δ(x⋆ ) .
β
(x
Bl
n=2
A
A
For the implications, go to Jeremy Draghi’s talk in EEB on
Wednesday!
Outline
Foundational Models
◮
◮
◮
◮
Applications
Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process
with Immigration.
Rt
P⌊t/δ⌋
X (s) ds ≈ n=0 XB (nδ)δ.
0 B
Taking δ =
√1 ,
N
√
2
and replace t with Nt, get Ee−θb t
Pn
P
√
Nt XN
k=0 k
Nt2
.
X
k
Lastly, use known results for the GWI: k=0
where Y has
n2 → Y,
q θn (1−pne )
−θ Y θb
Laplace Transform E e b = sech2 γ
2 , where
γ=
◮
Generalized Birth-Death-Mutation Processes
P∞
⋆
n=2 n(n−1)βBl (xA )
P∞
⋆ )+δ(x⋆ ) .
β
(x
Bl
n=2
A
A
For the implications, go to Jeremy Draghi’s talk in EEB on
Wednesday!
Outline
Foundational Models
◮
◮
◮
◮
Applications
Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process
with Immigration.
Rt
P⌊t/δ⌋
X (s) ds ≈ n=0 XB (nδ)δ.
0 B
Taking δ =
√1 ,
N
√
2
and replace t with Nt, get Ee−θb t
Pn
P
√
Nt XN
k=0 k
Nt2
.
X
k
Lastly, use known results for the GWI: k=0
where Y has
n2 → Y,
q θn (1−pne )
−θ Y θb
Laplace Transform E e b = sech2 γ
2 , where
γ=
◮
Generalized Birth-Death-Mutation Processes
P∞
⋆
n=2 n(n−1)βBl (xA )
P∞
⋆ )+δ(x⋆ ) .
β
(x
Bl
n=2
A
A
For the implications, go to Jeremy Draghi’s talk in EEB on
Wednesday!
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
E VOLUTION IN M ULTI - STRAIN E PIDEMICS
Joint work with
Amaury Lambert
Université Paris 06
In preparation!
Sylvain Gandon
CNRS Montpellier
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
A T WO S TRAIN SIR M ODEL WITH D EMOGRAPHY
β1 I1
N
λN
/
?
I1
S
δ
??
??α1
δ
??
?
?




>> β I
>> N2 2
>>
>
α2
I2
δ
Transition
SN → SN + 1
SN → SN − 1
R
δ
(SN , IiN ) → (SN − 1, IiN + 1)
IiN → IiN − 1
Rate
λN
δSN
βi SN IN
i
N
(δ + αi )IiN
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
L AW OF L ARGE N UMBERS
Let
XN (t) =
1 N
S (t)
N
YiN (t) =
1 N
I (t),
N i
and suppose that as N → ∞, XN (0) → x and YiN (0) → yi . Then, XN
and YiN converge almost surely to deterministic limits X(t) and Yi (t)
satisfying
Ẋ(t) =
Ẏi (t) =
λ − (β1 Y1 (t) + β2 Y2 (t) + δ) X(t)
(βi X(t) − (δ + αi )) Yi (t)
with X(0) = x and Y(0) = yi .
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE B ASIC R EPRODUCTION N UMBER
◮
This deterministic dynamical system has equilibria at the points
(x⋆ , y⋆1 , y⋆2 ),
◮
◮
◮
λ
δ
λ
δ
δ + α1
δ + α2
λ
.
, 0, 0 ,
,
−
, 0 , and
, 0,
−
δ
β1
δ + α1
β1
β2
δ + α2
β2
The first is stable if
βi
δ+αi
< λδ , i = 1, 2.
β1
β2
β1
The second (third) is stable if λδ < δ+α
and δ+α
(resp. >
< δ+α
2
1
1
).
this leads us to define the basic reproduction numbers,
βi
.
Ri0 = δ+α
i
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE B ASIC R EPRODUCTION N UMBER
◮
This deterministic dynamical system has equilibria at the points
(x⋆ , y⋆1 , y⋆2 ),
◮
◮
◮
λ
δ
λ
δ
δ + α1
δ + α2
λ
.
, 0, 0 ,
,
−
, 0 , and
, 0,
−
δ
β1
δ + α1
β1
β2
δ + α2
β2
The first is stable if
βi
δ+αi
< λδ , i = 1, 2.
β1
β2
β1
The second (third) is stable if λδ < δ+α
and δ+α
(resp. >
< δ+α
2
1
1
).
this leads us to define the basic reproduction numbers,
βi
.
Ri0 = δ+α
i
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE B ASIC R EPRODUCTION N UMBER
◮
This deterministic dynamical system has equilibria at the points
(x⋆ , y⋆1 , y⋆2 ),
◮
◮
◮
λ
δ
λ
δ
δ + α1
δ + α2
λ
.
, 0, 0 ,
,
−
, 0 , and
, 0,
−
δ
β1
δ + α1
β1
β2
δ + α2
β2
The first is stable if
βi
δ+αi
< λδ , i = 1, 2.
β1
β2
β1
The second (third) is stable if λδ < δ+α
and δ+α
(resp. >
< δ+α
2
1
1
).
this leads us to define the basic reproduction numbers,
βi
Ri0 = δ+α
.
i
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HE B ASIC R EPRODUCTION N UMBER
◮
This deterministic dynamical system has equilibria at the points
(x⋆ , y⋆1 , y⋆2 ),
◮
◮
◮
λ
δ
λ
δ
δ + α1
δ + α2
λ
.
, 0, 0 ,
,
−
, 0 , and
, 0,
−
δ
β1
δ + α1
β1
β2
δ + α2
β2
The first is stable if
βi
δ+αi
< λδ , i = 1, 2.
β1
β2
β1
The second (third) is stable if λδ < δ+α
and δ+α
(resp. >
< δ+α
2
1
1
).
this leads us to define the basic reproduction numbers,
βi
.
Ri0 = δ+α
i
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I NVASION OF A N EW S TRAIN
Consider a single individual infected with strain 1 entering a
population where strain 2 is endemic.
Then
λ
2
(X(t), Y1 (t), Y2 (t)) = (x⋆ , 0, y⋆2 ) = δ+α
β2 , 0, δ+α2 −
the invasion probability reduces to
δ
β2
for all t, and
1
1+
Rt
0
e−
Rs
β̄ (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ ) du
2
2
0 1
δ1 (x⋆ , 0, y⋆
2 ) ds
=
1−
δ1 (x⋆ ,0,y⋆ )
2
β̄1 (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ )
2
2
As t → ∞, this tends to 1 −
R20
R10
1
⋆
⋆
⋆
⋆
e−(β̄1 (x ,0,y2 )−δ1 (x ,0,y2 ))t − 1
if R10 > R20 , and 0 otherwise.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
I NVASION OF A N EW S TRAIN
Consider a single individual infected with strain 1 entering a
population where strain 2 is endemic.
Then
λ
2
(X(t), Y1 (t), Y2 (t)) = (x⋆ , 0, y⋆2 ) = δ+α
β2 , 0, δ+α2 −
the invasion probability reduces to
δ
β2
for all t, and
1
1+
Rt
0
e−
Rs
β̄ (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ ) du
2
2
0 1
δ1 (x⋆ , 0, y⋆
2 ) ds
=
1−
δ1 (x⋆ ,0,y⋆ )
2
β̄1 (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ )
2
2
As t → ∞, this tends to 1 −
R20
R10
1
⋆
⋆
⋆
⋆
e−(β̄1 (x ,0,y2 )−δ1 (x ,0,y2 ))t − 1
if R10 > R20 , and 0 otherwise.
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
T HE Q UASI -N EUTRAL E PIDEMIC
The two strains are quasi-neutral if both have the same R0 :
β1
β2
=
,
δ + α2
δ + α1
in which case
Ω = {(x∗ , y∗1 , y∗2 ) ∈ R3+ : β1 y∗1 + β2 y∗2 =
λ
− δ}.
x∗
Applications
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Again, we have a Kolmogorov equation for the frequency of the first
strain:
∂f
d
1 d2
[a(p)f (p, t)] ,
= − [b(p)f (p, t)] +
∂p
dp
2 dp2
where
ne (P) =
b(p) = x∗
λ
x∗
−δ
β1 P + β2 (1 − P)
β1 β2 β1 p + β2 (1 − p)
p(1 − p)(β2 − β1 )
ne (p) β12 p + β22 (1 − p) 2
β1 (1 − p) + β2 p +
β1 β2 (β1 p + β2 (1 − p))
β12 p + β22 (1 − p)
and
a(p) = 2x∗
β1 β2 (β1 p + β2 (1 − p))3
p(1 − p)
ne (p) β12 p + β22 (1 − p) 2
There is a drift towards the type with lower contact rate. Under our
assumption of equal 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.
!
,
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
Again, we have a Kolmogorov equation for the frequency of the first
strain:
∂f
d
1 d2
[a(p)f (p, t)] ,
= − [b(p)f (p, t)] +
∂p
dp
2 dp2
where
ne (P) =
b(p) = x∗
λ
x∗
−δ
β1 P + β2 (1 − P)
β1 β2 β1 p + β2 (1 − p)
p(1 − p)(β2 − β1 )
ne (p) β12 p + β22 (1 − p) 2
β1 (1 − p) + β2 p +
β1 β2 (β1 p + β2 (1 − p))
β12 p + β22 (1 − p)
and
a(p) = 2x∗
β1 β2 (β1 p + β2 (1 − p))3
p(1 − p)
ne (p) β12 p + β22 (1 − p) 2
There is a drift towards the type with lower contact rate. Under our
assumption of equal 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.
!
,
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
T HANKS . . .
◮
. . . to you all for listening!
◮
. . . to Marc Mangel and Qi Gong for being such a gracious hosts!
. . . to the Burroughs Wellcome Fund for financial support.
◮
◮
. . . to Warren Ewens, and to all the members of the Plotkin lab,
for many stimulating discussions.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
These results invite a comparison to John Gillespie’s results on
fecundity variance polymorphism.
◮
The model is similar to the Wright-Fisher model, where each
individual’s offspring number is a type-dependent i.i.d. random
variable with mean 1 + µNi and variance σi2 .
◮
Each generation is composed of N individuals sampled
uniformly from the offspring of the previous generation.
◮
When σ12 − σ22 = O (1), the proportion of type 1 admits a
diffusion approximation with forward equation
∂f
d p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f
=−
∂t
dp
1 d2 +
p(1 − p) σ22 p + σ12 (1 − p) f .
2
2 dp
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
These results invite a comparison to John Gillespie’s results on
fecundity variance polymorphism.
◮
The model is similar to the Wright-Fisher model, where each
individual’s offspring number is a type-dependent i.i.d. random
variable with mean 1 + µNi and variance σi2 .
◮
Each generation is composed of N individuals sampled
uniformly from the offspring of the previous generation.
◮
When σ12 − σ22 = O (1), the proportion of type 1 admits a
diffusion approximation with forward equation
d ∂f
p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f
=−
∂t
dp
1 d2 +
p(1 − p) σ22 p + σ12 (1 − p) f .
2
2 dp
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
These results invite a comparison to John Gillespie’s results on
fecundity variance polymorphism.
◮
The model is similar to the Wright-Fisher model, where each
individual’s offspring number is a type-dependent i.i.d. random
variable with mean 1 + µNi and variance σi2 .
◮
Each generation is composed of N individuals sampled
uniformly from the offspring of the previous generation.
◮
When σ12 − σ22 = O (1), the proportion of type 1 admits a
diffusion approximation with forward equation
∂f
d p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f
=−
∂t
dp
1 d2 +
p(1 − p) σ22 p + σ12 (1 − p) f .
2
2 dp
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
These results invite a comparison to John Gillespie’s results on
fecundity variance polymorphism.
◮
The model is similar to the Wright-Fisher model, where each
individual’s offspring number is a type-dependent i.i.d. random
variable with mean 1 + µNi and variance σi2 .
◮
Each generation is composed of N individuals sampled
uniformly from the offspring of the previous generation.
◮
When σ12 − σ22 = O (1), the proportion of type 1 admits a
diffusion approximation with forward equation
∂f
d p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f
=−
∂t
dp
1 d2 +
p(1 − p) σ22 p + σ12 (1 − p) f .
2
2 dp
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
◮
Despite superficial similarities, these results are qualitatively
different.
To see this, consider the lifetime reproductive output of an
individual of type i born at time t = 0, which is
O ∼ Poisson(βi L)
◮
◮
where L is a non-stationary exponential process with rate
!
PK
N
j=1 Xj (t)
.
δi 1 +
N
The Law of Large Numbers tells us that to O N1 ,
δi
,
L ∼ Exponential
α
We then have, to O N1 ,
Var(O) = E [Var(O)|L] + Var (E [O|L]) =
α2 β 2
αβi
+ 2 i = 2.
δi
δi
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
◮
Despite superficial similarities, these results are qualitatively
different.
To see this, consider the lifetime reproductive output of an
individual of type i born at time t = 0, which is
O ∼ Poisson(βi L)
◮
◮
where L is a non-stationary exponential process with rate
!
PK
N
j=1 Xj (t)
.
δi 1 +
N
The Law of Large Numbers tells us that to O N1 ,
δi
,
L ∼ Exponential
α
We then have, to O N1 ,
Var(O) = E [Var(O)|L] + Var (E [O|L]) =
α2 β 2
αβi
+ 2 i = 2.
δi
δi
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE
◮
◮
Despite superficial similarities, these results are qualitatively
different.
To see this, consider the lifetime reproductive output of an
individual of type i born at time t = 0, which is
O ∼ Poisson(βi L)
◮
◮
where L is a non-stationary exponential process with rate
!
PK
N
j=1 Xj (t)
.
δi 1 +
N
The Law of Large Numbers tells us that to O N1 ,
δi
,
L ∼ Exponential
α
We then have, to O N1 ,
Var(O) = E [Var(O)|L] + Var (E [O|L]) =
α2 β 2
αβi
+ 2 i = 2.
δi
δi
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
Applications
F IXATION P ROBABILITY FOR THE M UTANT WITH
B OTTLENECK , K = 2
0.5
0.4
0.3
0.1
0.2
Fixation probability
0.6
If we impose bottlenecks by sampling a small number of individuals
at regular intervals to start a new population (β1 = 2, β2 = 0.4),
5
10
20
50
100
200
500
1000 2000
Bottleneck frequency (generations)
Fixation probabilities vary substantially with the interval length.
Outline
Foundational Models
Generalized Birth-Death-Mutation Processes
C OMPETITION A SSAYS
Measuring adaptation
1:1 mixture
Transfer sample to
solid media
Evolved population
Competition
Ancestral population
(Buckling et al., Nature 457, 2009, 824 – 829)
Estimate frequency of
each population
Applications