2014 05 lausanne

Social evolution in structured populations
Florence Débarre
University of Exeter
florence.debarre@normalesup.org
@flodebarre
Lausanne – May 2014
F. Débarre Social evolution in structured populations 1 / 26

Some theoretical perspectives on
Social evolution in structured populations
Florence Débarre
University of Exeter
florence.debarre@normalesup.org
@flodebarre
Lausanne – May 2014

Acknowledgements

Acknowledgements
Collaborators:
Michael DoebeliChristoph Hauert

Acknowledgements
Collaborators:
Special thanks:
Mike WhitlockSally Otto
S´ebastien Lion
Peter Taylor
Minus van Baalen
Fran¸cois Rousset
Wes Maciejewski

Acknowledgements
Collaborators:
Special thanks:
Mike WhitlockSally Otto
S´ebastien Lion
Peter Taylor
Minus van Baalen
Fran¸cois Rousset
Wes Maciejewski
Funding:
2011–2012 2012–2014 2013 – . . .

The puzzle of altruism

cMJGrimson&RLBlanton

cWikimedia

cWikimedia
cFD

cWikimedia
cFD
cpicturesforcoloring.com

. . . is already qualitatively solved

Assortment!

Assortment!
Altruists interact more with altruists
than defectors do.

Assortment!
than defectors do.
Population viscosity

Assortment!
than defectors do.
Conditional behaviour

. . . is already qualitatively solved. Well, almost.
Assortment!
than defectors do.
Conditional behaviour

Diﬀerent theoretical frameworks

Group Selection Inclusive Fitness
Game Theory

Game Theory
Hamilton’s rule
r b > c

Game Theory
cSMBCcomics

Game Theory
cLionetal,2011,TREE

Game Theory
A =
Donnors
Recipients a b
c d

Game Theory
A =
Donnors
Recipients
b − c + d − c
b 0

Game Theory
A =
Donnors
Recipients
b − c + d − c
b 0
d = (a + d) − (b + c)

Game Theory
A =
Donnors
Recipients a b
c d
d = (a + d) − (b + c)

Game Theory
A =
Donnors
Recipients
b − c − c
b 0

Game Theory

cUderzo&Goscinny
Game Theory

Conﬂicting theories. . .
cBiodiversitymuseum,UBC

Theory and conﬂict

Assumptions
+

Method
Assumptions
+

Method
Assumptions
+
Result

Method
Inclusive ﬁtnessAssumptions
+
Result

Method
Game TheoryAssumptions
+
Result

Method
Assumptions
+
Result
Results may be diﬀerent
Mathematical / simulation errors
Assumptions are actually not the same

Method
Assumptions
+
Result
Results may be different
Mathematical / simulation errors
Assumptions are actually not the same
Results may look different
Different viewpoints of the same result
Semantics

Assumptions and semantics

Assumptions and semantics
What is meant by . . .
Covariance
Relatedness
Weak selection
Evolutionary success

Weak selection
Wild & Traulsen, 2007, JTB

Weak selection
Small change in ﬁtness W , due to a
small change in x

Weak selection
small change in x
“Kin selection” weak selection
Small distance in phenotype space

Weak selection
small change in x
δ-weak selection

Weak selection
small change in x
δ-weak selection
“Game theory” weak selection
Small contribution from the game
w-weak selection

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)
Small distance in phenotype space; y = x + δ

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)
A =
B(x, x) − C(x) B(x, x) − C(x)
B(x, x) − C(x) B(x, x) − C(x)
+ δ
B(1)
(x, x) + B(2)
(x, x) − C (x) B(1)
(x, x) − C (x)
B(2)
(x, x) 0
+ O(δ2
).

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
Small contribution from the game; ω

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
A =ω B(x, x) − C(x) B(x, x) − C(x)
B(x, x) − C(x) B(x, x) − C(x)
+ ω B(y, y) − B(x, x) − (C(y) − C(x)) B(y, x) − B(x, x) − (C(y) − C(x))
B(x, y) − B(x, x) 0 .

Weak selections
A =
Donnors
Recipients
B(y, y) − C(y) B(y, x) − C(y)
B(x, y) − C(x) B(x, x) − C(x)
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
A =Constant(x) + ω
b − c + d −c
b 0
.

Weak selections – Other implications
Linearity
Pairwise interactions
come for
FREE

t1 t2
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.

0
∆p(t) > 01.
2.

0
?
∆p(t) > 01.
The ﬁxation probability of social behaviour is
greater than the ﬁxation probability of neutral
behaviour.
ρS > 1/N2.

0
?
∆p(t) > 01.
behaviour.
ρS > 1/N2.
greater than the ﬁxation probability of non-
social behaviour.
ρS > ρNS3.

0
?
∆p(t) > 01.
behaviour.
ρS > 1/N2.
greater than the ﬁxation probability of non-
social behaviour.
ρS > ρNS3.
⇔⇔

cUderzo&Goscinny
Game Theory

cUderzo&Goscinny

cUderzo&Goscinny
I used methods
from the diﬀerent frameworks

Evolutionary graph theory
Population of size N (ﬁxed)
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB

Dispersal graph D
dij : dispersal from i to j 12 3
4
5 67
8 9
10
11

Dispersal graph D
dij : dispersal from i to j
Weighted
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Oriented
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Interaction graph E
eij : beneﬁts given by i to j.
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Interaction graph E
Small eﬀects (ω 1),
→ weak selection.
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Interaction graph E
→ weak selection.
Eﬀects add up
δw + = δw +δw
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Interaction graph E
→ weak selection.
Eﬀects add up
→ Pairwise interactions.
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Interaction graph E
→ weak selection.
Eﬀects add up
Focus on
symmetric and transitive D graphs.
12 3
4
5 67
8 9
10
11

Dispersal graph D
Weighted
Interaction graph E
→ weak selection.
Eﬀects add up
Focus on
symmetric and transitive D graphs.
dij = dji
12 3
4
5 67
8 9
10
11

Symmetric and transitive dispersal graphs
Lattices
Other structures

Symmetric and transitive dispersal graphs
Lattices Island model
Other structures

Updating the population
Constant population size (N), so
between two time steps,
=# #

=# #
=N N
...
...
=k k
...
...
=1 1

=# #
=N N
...
...
=k k
...
...
=1 1
Wright-Fisher
Moran process
Wright-Fisher

Life-cycle: Moran process t t + dt
time

time
Death-Birth (DB)

time
Death-Birth (DB)
First step

time
Death-Birth (DB)
First step
Second step

time
Death-Birth (DB) Birth-Death (BD)
First step
Second step

Life-cycle: Moran process
First step
Second step
Outcome

First step
Second step
Outcome
Ohtsuki et al. (2006), Nature

First step
Second step
Outcome
? ?
Ohtsuki et al. (2006), Nature
Nakamaru & Iwasa (2006), JTB
Taylor (2010), JEB

Eﬀects on the ﬁrst step
Survival in DB
Fecundity in BD.
A[1] = a[1] b[1]
c[1] d[1]

Effects on the first step
Survival in DB
Fecundity in BD.
A[1] = a[1] b[1]
c[1] d[1]
Effects on the second step
Fecundity in DB
Survival in BD.
A[2] = a[2] b[2]
c[2] d[2]

Technical details

Technical details
Notation: pi = P ( i ) = 1 − P ( i )

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
D matrix of the dispersal graph
E matrix of the interaction graph

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
Weak selection (1st order)

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
Expected change in the frequency of
∆p(t) =
ω
N2
Tr ET
· P − Tr ET
· P · D (c[1] − d[1])
− Tr (P) − Tr (P · D) (d[1] − b[1])
+ Tr ET · P − Tr ET · Π · D (a[1] − b[1] − c[1] + d[1])
+ Tr ET · P − Tr ET · P · D · D (c[2] − d[2])
− Tr (P) − Tr (P · D · D) (d[2] − b[2])
+ Tr ET · P − Tr ET · Π · D · D (a[2] − b[2] − c[2] + d[2]) .

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
∆p(t) =
ω
N2
Tr ET
· P − Tr ET
· P · D b[1]
− Tr (P) − Tr (P · D) c[1]
+ Tr ET · P − Tr ET · Π · D d[1]
+ Tr ET · P − Tr ET · P · D · D b[2]
− Tr (P) − Tr (P · D · D) c[2]
+ Tr ET · P − Tr ET · Π · D · D d[2] .

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
∆p(t) =
ω
N2
direct eﬀects
Tr ET
· P − Tr ET
· P · D b[1]
− Tr (P) − Tr (P · D) c[1]
+ Tr ET · P − Tr ET · Π · D d[1]
+ Tr ET · P − Tr ET · P · D · D b[2]
− Tr (P) − Tr (P · D · D) c[2]
+ Tr ET · P − Tr ET · Π · D · D d[2] .

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
∆p(t) =
ω
N2
direct eﬀects
Tr ET
· P −
competition
= secondary eﬀects
Tr ET
· P · D b[1]
− Tr (P) − Tr (P · D) c[1]
+ Tr ET · P − Tr ET · Π · D d[1]
+ Tr ET · P − Tr ET · P · D · D b[2]
− Tr (P) − Tr (P · D · D) c[2]
+ Tr ET · P − Tr ET · Π · D · D d[2] .

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
First step
D
Second step
D · D
Competitive radius

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
Separation of time scales:
∆ ∆

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
∆ ∆
∆p(t) = (sσ + sτ p(t))
s(p(t))
E[varS(p(t))]

Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
N population size
∆ ∆
∆p(t) = (sσ + sτ p(t))
s(p(t))
E[varS(p(t))]
ρ =
1
N
+
N − 1
2 N
sσ + sτ
N + 1
3N

Fixation probability
Test with (sτ = 0).
ρ =
1
N
+
N − 1
2 N
sσ

N ρ − 1 =
N − 1
2
sσ
15 2510 20
−0.05
0.00
0.05
0.10
0.15
0.20
Population size N
Scaledfixationprobability
NρS−1
Neutral

N ρ − 1 =
N − 1
2
sσ
15 2510 20
−0.05
0.00
0.05
0.10
0.15
0.20
Population size N
Scaledfixationprobability
NρS−1
Groups,
public good
Lattice,
two−player
Neutral

Equivalent payoﬀ matrix
Condition for evolutionary success: ρ > ρ

Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
;

A∞ =
a b
c d
; a + b > c + d

A∞ =
b + d − c −c
b 0
; d − 2 c > 0

A∞ =
a b
c d
; a + b > c + d
Structured population

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]
[1] eﬀects on 1st step

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]
[2] eﬀects on 2nd step

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]
σ[1]
= 1 −
2
N
,
N population size

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]
σ[1]
= 1 −
2
N
, σ[2]
=
1 + dself + ed − 4/N
1 + dself − ed
,
N population size
dself
ed = i j eij dji /N

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]
σ[1]
= 1 −
2
N
, σ[2]
=
1 + dself + ed − 4/N
1 + dself − ed
,
ξ = 1 + dself − ed.
N population size
dself
ed = i j eij dji /N

A∞ =
a b
c d
; a + b > c + d
˜A =
σ[1]
a[1]
b[1]
c[1]
σ[1]
d[1]
+ ξ
σ[2]
a[2]
b[2]
c[2]
σ[2]
d[2]
σ[1]
= 1 , σ[2]
=
1 + dself + ed
1 + dself − ed
,
ξ = 1 + dself − ed.
∞ population size
dself
ed = i j eij dji /N

Survival vs. fecundity
Death-Birth updating
Beneﬁts
B
Costs
C

[1]
Survival
[2]
Fecundity
Beneﬁts
B
Costs
C

0 1
λb
[1]
Survival
[2]
Fecundity
Beneﬁts
B
0 1
λc
Costs
C

0 1
λb
[1]
Survival
[2]
Fecundity
Beneﬁts
B
0 1
λc
Costs
C
B[1]
= (1 − λB) B B[2]
= λB B
C[1]
= (1 − λC ) C C[2]
= λC C

Survival vs. fecundity: Prisoner’s Dilemma
Payoﬀs: Prisoner’s dilemma
Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0

Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0
Population structure
Large population size N → ∞
ed = ed
dself > 0 dself=0

q
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
[1] [2]
[1]
[2]
λC, Costs
λB,Benefits
q11
q22
Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0
ed = ed
dself > 0 dself=0

Survival vs. fecundity: Snowdrift
q
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
[1] [2]
[1]
[2]
λC, Costs
λB,Benefits
q11
q22
Payoﬀs: Snowdrift game
Survival
BS − CS /2 BS − CS
BS 0
Fecundity
BF − CF /2 BF − CF
BF 0
ed = ed
dself > 0 dself=0

Take Home Messages

Take Home Messages
Evolution of social behaviour

Take Home Messages
Death-Birth vs. Birth-Death

Take Home Messages
Death-Birth vs. Birth-Death Eﬀect on ﬁrst or second step

Take Home Messages
Beneﬁts: on the second step
Costs: depends

Take Home Messages
Costs: depends
Inclusive ﬁtness vs. Game theory

Take Home Messages
Costs: depends
Inclusive ﬁtness vs. Game theory
Semantic issues
Diﬀering assumptions (weak selection), with consequences

Thanks for your attention!

Semantics: Covariance
t1

t1 t2

t1 t2 t3

t1 t2 t3 t4

Replicate
1
Replicate
2
t1 t2 t3 t4

Replicate
1
Replicate
2
t1 t2 t3 t4
Average and Expectation

Replicate
1
Replicate
2
t1 t2 t3 t4
Ave(Xt)

Replicate
1
Replicate
2
t1 t2 t3 t4
Ave(Xt) = E[Xi,t]

Replicate
1
Replicate
2
t1 t2 t3 t4
Ave(Xt) = E[Xi,t] ; E[Ave(Xt)]
Covariance and Covariance

Replicate
1
Replicate
2
t1 t2 t3 t4
CovS(Wt, Xt) = Ave(Wt, Xt) − Ave(Wt)Ave(Xt)

Replicate
1
Replicate
2
t1 t2 t3 t4
CovS(Wt, Xt) = Ave(Wt, Xt) − Ave(Wt)Ave(Xt) =
Cov[Wi,t, Xi,t] = E[Wi,tXi,t] − E[Wi,t]E[Xi,t]

Replicate
1
Replicate
2
t1 t2 t3 t4
CovS(Wt, Xt) = Ave(Wt, Xt) − Ave(Wt)Ave(Xt) =
Cov[Wi,t, Xi,t] = E[Wi,tXi,t] − E[Wi,t]E[Xi,t]
E[CovS(Wt, Xt)]

Relatedness

Relatedness
No, but I would to save
two brothers or eight cousins.

Relatedness
A measure of identity
Often expressed as a regression of partner’s genotype (or
phenotype) on focal individual’s genotype (or phenotype)
Frank (2013), JEB

Relatedness
r =
CovS(g, g )
CovS(g, g)
Frank (2013), JEB
Gardner et al. (2011), JEB

Relatedness
r =
E[CovS(g, g )]
E[CovS(g, g)]
Frank (2013), JEB

Relatedness
r =
E[CovS(g, g )]
E[CovS(g, g)]
r =
E[x, x ]
E[x, x]
Frank (2013), JEB
Taylor (2013), JTB

Relatedness
r =
E[CovS(g, g )]
E[CovS(g, g)]
r =
E[x, x ]
E[x, x]
r =
Gij − G
1 − G
, where Gi,j = P[Xi = Xj ].
Frank (2013), JEB
Taylor (2013), JTB
Taylor et al. (2007), JTB

2014 05 lausanne

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2014 05 lausanne