1. Social evolution in structured populations
Florence D´ebarre
University of Exeter
florence.debarre@normalesup.org
@flodebarre
Lausanne – May 2014
F. D´ebarre Social evolution in structured populations 1 / 26
2. Some theoretical perspectives on
Social evolution in structured populations
Florence D´ebarre
University of Exeter
florence.debarre@normalesup.org
@flodebarre
Lausanne – May 2014
F. D´ebarre Social evolution in structured populations 1 / 26
6. Acknowledgements
Collaborators:
Michael DoebeliChristoph Hauert
Special thanks:
Mike WhitlockSally Otto
S´ebastien Lion
Peter Taylor
Minus van Baalen
Fran¸cois Rousset
Wes Maciejewski
Funding:
2011–2012 2012–2014 2013 – . . .
F. D´ebarre Social evolution in structured populations 2 / 26
7. The puzzle of altruism
F. D´ebarre Social evolution in structured populations 3 / 26
8. The puzzle of altruism
cMJGrimson&RLBlanton
F. D´ebarre Social evolution in structured populations 3 / 26
9. The puzzle of altruism
cMJGrimson&RLBlanton
cWikimedia
F. D´ebarre Social evolution in structured populations 3 / 26
10. The puzzle of altruism
cMJGrimson&RLBlanton
cWikimedia
cFD
F. D´ebarre Social evolution in structured populations 3 / 26
11. The puzzle of altruism
cMJGrimson&RLBlanton
cWikimedia
cFD
cpicturesforcoloring.com
F. D´ebarre Social evolution in structured populations 3 / 26
12. . . . is already qualitatively solved
F. D´ebarre Social evolution in structured populations 4 / 26
13. . . . is already qualitatively solved
Assortment!
F. D´ebarre Social evolution in structured populations 4 / 26
14. . . . is already qualitatively solved
Assortment!
Altruists interact more with altruists
than defectors do.
F. D´ebarre Social evolution in structured populations 4 / 26
15. . . . is already qualitatively solved
Assortment!
Altruists interact more with altruists
than defectors do.
Population viscosity
F. D´ebarre Social evolution in structured populations 4 / 26
16. . . . is already qualitatively solved
Assortment!
Altruists interact more with altruists
than defectors do.
Population viscosity
Conditional behaviour
F. D´ebarre Social evolution in structured populations 4 / 26
17. . . . is already qualitatively solved. Well, almost.
Assortment!
Altruists interact more with altruists
than defectors do.
Population viscosity
Conditional behaviour
F. D´ebarre Social evolution in structured populations 4 / 26
21. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
Hamilton’s rule
r b > c
F. D´ebarre Social evolution in structured populations 5 / 26
22. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
cSMBCcomics
F. D´ebarre Social evolution in structured populations 5 / 26
25. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
cLionetal,2011,TREE
F. D´ebarre Social evolution in structured populations 5 / 26
28. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
A =
Donnors
Recipients a b
c d
F. D´ebarre Social evolution in structured populations 5 / 26
29. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
A =
Donnors
Recipients
b − c + d − c
b 0
F. D´ebarre Social evolution in structured populations 5 / 26
30. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
A =
Donnors
Recipients
b − c + d − c
b 0
d = (a + d) − (b + c)
F. D´ebarre Social evolution in structured populations 5 / 26
31. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
A =
Donnors
Recipients a b
c d
d = (a + d) − (b + c)
F. D´ebarre Social evolution in structured populations 5 / 26
32. Different theoretical frameworks
Group Selection Inclusive Fitness
Game Theory
A =
Donnors
Recipients
b − c − c
b 0
F. D´ebarre Social evolution in structured populations 5 / 26
49. Theory and conflict
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
F. D´ebarre Social evolution in structured populations 8 / 26
51. Assumptions and semantics
What is meant by . . .
Covariance
Relatedness
Weak selection
Evolutionary success
F. D´ebarre Social evolution in structured populations 9 / 26
52. Weak selection
Wild & Traulsen, 2007, JTB
F. D´ebarre Social evolution in structured populations 10 / 26
53. Weak selection
Wild & Traulsen, 2007, JTB
F. D´ebarre Social evolution in structured populations 10 / 26
54. Weak selection
Small change in fitness W , due to a
small change in x
Wild & Traulsen, 2007, JTB
F. D´ebarre Social evolution in structured populations 10 / 26
55. Weak selection
Small change in fitness W , due to a
small change in x
“Kin selection” weak selection
Small distance in phenotype space
Wild & Traulsen, 2007, JTB
F. D´ebarre Social evolution in structured populations 10 / 26
56. Weak selection
Small change in fitness W , due to a
small change in x
“Kin selection” weak selection
Small distance in phenotype space
δ-weak selection
Wild & Traulsen, 2007, JTB
F. D´ebarre Social evolution in structured populations 10 / 26
57. Weak selection
Small change in fitness W , due to a
small change in x
“Kin selection” weak selection
Small distance in phenotype space
δ-weak selection
“Game theory” weak selection
Small contribution from the game
w-weak selection
Wild & Traulsen, 2007, JTB
F. D´ebarre Social evolution in structured populations 10 / 26
59. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
F. D´ebarre Social evolution in structured populations 11 / 26
60. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
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
).
F. D´ebarre Social evolution in structured populations 11 / 26
61. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
F. D´ebarre Social evolution in structured populations 11 / 26
62. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
F. D´ebarre Social evolution in structured populations 11 / 26
63. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
“Game theory” weak selection
Small contribution from the game; ω
F. D´ebarre Social evolution in structured populations 11 / 26
64. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
“Game theory” weak selection
Small contribution from the game; ω
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 .
F. D´ebarre Social evolution in structured populations 11 / 26
65. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
“Game theory” weak selection
Small contribution from the game; ω
A =Constant(x) + ω
b − c + d −c
b 0
.
F. D´ebarre Social evolution in structured populations 11 / 26
66. 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)
“Kin selection” weak selection
Small distance in phenotype space; y = x + δ
A =Constant(x) + δ
b − c −c
b 0
+ O(δ2
).
“Game theory” weak selection
Small contribution from the game; ω
A =Constant(x) + ω
b − c + d −c
b 0
.
F. D´ebarre Social evolution in structured populations 11 / 26
67. Weak selections – Other implications
“Kin selection” weak selection
Linearity
Pairwise interactions
come for
FREE
F. D´ebarre Social evolution in structured populations 12 / 26
69. Evolutionary success
t1 t2
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.
F. D´ebarre Social evolution in structured populations 13 / 26
70. Evolutionary success
0
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.
2.
F. D´ebarre Social evolution in structured populations 13 / 26
71. Evolutionary success
0
?
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.
The fixation probability of social behaviour is
greater than the fixation probability of neutral
behaviour.
ρS > 1/N2.
F. D´ebarre Social evolution in structured populations 13 / 26
72. Evolutionary success
0
?
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.
The fixation probability of social behaviour is
greater than the fixation probability of neutral
behaviour.
ρS > 1/N2.
The fixation probability of social behaviour is
greater than the fixation probability of non-
social behaviour.
ρS > ρNS3.
F. D´ebarre Social evolution in structured populations 13 / 26
73. Evolutionary success
0
?
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.
The fixation probability of social behaviour is
greater than the fixation probability of neutral
behaviour.
ρS > 1/N2.
The fixation probability of social behaviour is
greater than the fixation probability of non-
social behaviour.
ρS > ρNS3.
⇔⇔
F. D´ebarre Social evolution in structured populations 13 / 26
74. Evolutionary success
0
?
The expected frequency of social individuals
increases from one time step to the next.
∆p(t) > 01.
The fixation probability of social behaviour is
greater than the fixation probability of neutral
behaviour.
ρS > 1/N2.
The fixation probability of social behaviour is
greater than the fixation probability of non-
social behaviour.
ρS > ρNS3.
F. D´ebarre Social evolution in structured populations 13 / 26
79. 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
F. D´ebarre Social evolution in structured populations 15 / 26
80. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j 12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
81. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
82. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Oriented
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
83. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
84. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
85. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
86. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
Small effects (ω 1),
→ weak selection.
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
87. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
Small effects (ω 1),
→ weak selection.
Effects add up
δw + = δw +δw
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
88. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
Small effects (ω 1),
→ weak selection.
Effects add up
→ Pairwise interactions.
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
89. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
Small effects (ω 1),
→ weak selection.
Effects add up
→ Pairwise interactions.
Focus on
symmetric and transitive D graphs.
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
90. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
Small effects (ω 1),
→ weak selection.
Effects add up
→ Pairwise interactions.
Focus on
symmetric and transitive D graphs.
dij = dji
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
91. Evolutionary graph theory
Population of size N (fixed)
Dispersal graph D
dij : dispersal from i to j
Weighted
Interaction graph E
eij : benefits given by i to j.
Small effects (ω 1),
→ weak selection.
Effects add up
→ Pairwise interactions.
Focus on
symmetric and transitive D graphs.
12 3
4
5 67
8 9
10
11
Lieberman et al (2005), Nature
Taylor et al (2007), JTB
F. D´ebarre Social evolution in structured populations 15 / 26
92. Symmetric and transitive dispersal graphs
Lattices
Other structures
F. D´ebarre Social evolution in structured populations 16 / 26
93. Symmetric and transitive dispersal graphs
Lattices Island model
Other structures
F. D´ebarre Social evolution in structured populations 16 / 26
94. Symmetric and transitive dispersal graphs
Lattices Island model
Other structures
F. D´ebarre Social evolution in structured populations 16 / 26
95. Symmetric and transitive dispersal graphs
Lattices Island model
Other structures
F. D´ebarre Social evolution in structured populations 16 / 26
96. Updating the population
Constant population size (N), so
between two time steps,
=# #
F. D´ebarre Social evolution in structured populations 17 / 26
97. Updating the population
Constant population size (N), so
between two time steps,
=# #
=N N
...
...
=k k
...
...
=1 1
F. D´ebarre Social evolution in structured populations 17 / 26
98. Updating the population
Constant population size (N), so
between two time steps,
=# #
=N N
...
...
=k k
...
...
=1 1
Wright-Fisher
Moran process
Wright-Fisher
F. D´ebarre Social evolution in structured populations 17 / 26
99. Updating the population
Constant population size (N), so
between two time steps,
=# #
=N N
...
...
=k k
...
...
=1 1
Wright-Fisher
Moran process
Wright-Fisher
F. D´ebarre Social evolution in structured populations 17 / 26
100. Life-cycle: Moran process t t + dt
time
F. D´ebarre Social evolution in structured populations 18 / 26
101. Life-cycle: Moran process t t + dt
time
F. D´ebarre Social evolution in structured populations 18 / 26
102. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
F. D´ebarre Social evolution in structured populations 18 / 26
103. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
First step
F. D´ebarre Social evolution in structured populations 18 / 26
104. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
First step
F. D´ebarre Social evolution in structured populations 18 / 26
105. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
First step
F. D´ebarre Social evolution in structured populations 18 / 26
106. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
107. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
108. Life-cycle: Moran process t t + dt
time
Death-Birth (DB)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
109. Life-cycle: Moran process t t + dt
time
Death-Birth (DB) Birth-Death (BD)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
110. Life-cycle: Moran process t t + dt
time
Death-Birth (DB) Birth-Death (BD)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
111. Life-cycle: Moran process t t + dt
time
Death-Birth (DB) Birth-Death (BD)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
112. Life-cycle: Moran process t t + dt
time
Death-Birth (DB) Birth-Death (BD)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
113. Life-cycle: Moran process t t + dt
time
Death-Birth (DB) Birth-Death (BD)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
114. Life-cycle: Moran process t t + dt
time
Death-Birth (DB) Birth-Death (BD)
First step
Second step
F. D´ebarre Social evolution in structured populations 18 / 26
115. Life-cycle: Moran process
Death-Birth (DB) Birth-Death (BD)
First step
Second step
Outcome
F. D´ebarre Social evolution in structured populations 18 / 26
116. Life-cycle: Moran process
Death-Birth (DB) Birth-Death (BD)
First step
Second step
Outcome
Ohtsuki et al. (2006), Nature
F. D´ebarre Social evolution in structured populations 18 / 26
117. Life-cycle: Moran process
Death-Birth (DB) Birth-Death (BD)
First step
Second step
Outcome
Ohtsuki et al. (2006), Nature
F. D´ebarre Social evolution in structured populations 18 / 26
118. Life-cycle: Moran process
Death-Birth (DB) Birth-Death (BD)
First step
Second step
Outcome
? ?
Ohtsuki et al. (2006), Nature
Nakamaru & Iwasa (2006), JTB
Taylor (2010), JEB
F. D´ebarre Social evolution in structured populations 18 / 26
119. Pairwise interactions
Effects on the first step
Survival in DB
Fecundity in BD.
A[1] = a[1] b[1]
c[1] d[1]
F. D´ebarre Social evolution in structured populations 19 / 26
120. Pairwise interactions
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]
F. D´ebarre Social evolution in structured populations 19 / 26
123. Technical details
Notation: pi = P ( i ) = 1 − P ( i )
Pij = P i j
Πijk = P i j k
F. D´ebarre Social evolution in structured populations 20 / 26
124. 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
F. D´ebarre Social evolution in structured populations 20 / 26
125. 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
Weak selection (1st order)
F. D´ebarre Social evolution in structured populations 20 / 26
126. 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
Weak selection (1st order)
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]) .
F. D´ebarre Social evolution in structured populations 20 / 26
127. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
128. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
129. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
130. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
131. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
132. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
133. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
134. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
135. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
136. 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
Weak selection (1st order)
Expected change in the frequency of
∆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] .
F. D´ebarre Social evolution in structured populations 20 / 26
137. 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
Weak selection (1st order)
First step
D
Second step
D · D
Competitive radius
F. D´ebarre Social evolution in structured populations 20 / 26
138. 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
Weak selection (1st order)
First step
D
Second step
D · D
Competitive radius
F. D´ebarre Social evolution in structured populations 20 / 26
139. 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
Weak selection (1st order)
Separation of time scales:
∆ ∆
F. D´ebarre Social evolution in structured populations 20 / 26
140. 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
Weak selection (1st order)
Separation of time scales:
∆ ∆
∆p(t) = (sσ + sτ p(t))
s(p(t))
E[varS(p(t))]
F. D´ebarre Social evolution in structured populations 20 / 26
141. 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
Weak selection (1st order)
Separation of time scales:
∆ ∆
∆p(t) = (sσ + sτ p(t))
s(p(t))
E[varS(p(t))]
F. D´ebarre Social evolution in structured populations 20 / 26
142. 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
Weak selection (1st order)
Separation of time scales:
∆ ∆
∆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
F. D´ebarre Social evolution in structured populations 20 / 26
143. Fixation probability
Test with (sτ = 0).
ρ =
1
N
+
N − 1
2 N
sσ
F. D´ebarre Social evolution in structured populations 21 / 26
144. Fixation probability
Test with (sτ = 0).
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
F. D´ebarre Social evolution in structured populations 21 / 26
145. Fixation probability
Test with (sτ = 0).
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
F. D´ebarre Social evolution in structured populations 21 / 26
146. Fixation probability
Test with (sτ = 0).
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
F. D´ebarre Social evolution in structured populations 21 / 26
148. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
;
F. D´ebarre Social evolution in structured populations 22 / 26
149. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
F. D´ebarre Social evolution in structured populations 22 / 26
150. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
b + d − c −c
b 0
; d − 2 c > 0
F. D´ebarre Social evolution in structured populations 22 / 26
151. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
F. D´ebarre Social evolution in structured populations 22 / 26
152. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
F. D´ebarre Social evolution in structured populations 22 / 26
153. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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]
F. D´ebarre Social evolution in structured populations 22 / 26
154. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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
F. D´ebarre Social evolution in structured populations 22 / 26
155. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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
[2] effects on 2nd step
F. D´ebarre Social evolution in structured populations 22 / 26
156. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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
,
[1] effects on 1st step
[2] effects on 2nd step
N population size
F. D´ebarre Social evolution in structured populations 22 / 26
157. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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
,
[1] effects on 1st step
[2] effects on 2nd step
N population size
dself
ed = i j eij dji /N
F. D´ebarre Social evolution in structured populations 22 / 26
158. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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
,
[1] effects on 1st step
[2] effects on 2nd step
N population size
dself
ed = i j eij dji /N
F. D´ebarre Social evolution in structured populations 22 / 26
159. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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.
[1] effects on 1st step
[2] effects on 2nd step
N population size
dself
ed = i j eij dji /N
F. D´ebarre Social evolution in structured populations 22 / 26
160. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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.
[1] effects on 1st step
[2] effects on 2nd step
∞ population size
dself
ed = i j eij dji /N
F. D´ebarre Social evolution in structured populations 22 / 26
161. Equivalent payoff matrix
Condition for evolutionary success: ρ > ρ
Well-mixed, N → ∞ (“Ideal” population)
A∞ =
a b
c d
; a + b > c + d
Structured population
˜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.
[1] effects on 1st step
[2] effects on 2nd step
∞ population size
dself
ed = i j eij dji /N
F. D´ebarre Social evolution in structured populations 22 / 26
163. Survival vs. fecundity
Death-Birth updating
[1]
Survival
[2]
Fecundity
Benefits
B
Costs
C
F. D´ebarre Social evolution in structured populations 23 / 26
164. Survival vs. fecundity
Death-Birth updating
0 1
λb
[1]
Survival
[2]
Fecundity
Benefits
B
0 1
λc
Costs
C
F. D´ebarre Social evolution in structured populations 23 / 26
165. Survival vs. fecundity
Death-Birth updating
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
F. D´ebarre Social evolution in structured populations 23 / 26
166. Survival vs. fecundity: Prisoner’s Dilemma
Death-Birth updating
Payoffs: Prisoner’s dilemma
Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0
F. D´ebarre Social evolution in structured populations 24 / 26
167. Survival vs. fecundity: Prisoner’s Dilemma
Death-Birth updating
Payoffs: Prisoner’s dilemma
Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0
Population structure
Large population size N → ∞
ed = ed
dself > 0 dself=0
F. D´ebarre Social evolution in structured populations 24 / 26
168. Survival vs. fecundity: Prisoner’s Dilemma
Death-Birth updating
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: Prisoner’s dilemma
Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0
Population structure
Large population size N → ∞
ed = ed
dself > 0 dself=0
F. D´ebarre Social evolution in structured populations 24 / 26
169. Survival vs. fecundity: Prisoner’s Dilemma
Death-Birth updating
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: Prisoner’s dilemma
Survival
BS − CS −CS
BS 0
Fecundity
BF − CF −CF
BF 0
Population structure
Large population size N → ∞
ed = ed
dself > 0 dself=0
F. D´ebarre Social evolution in structured populations 24 / 26
170. Survival vs. fecundity: Snowdrift
Death-Birth updating
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
Population structure
Large population size N → ∞
ed = ed
dself > 0 dself=0
F. D´ebarre Social evolution in structured populations 24 / 26
171. Survival vs. fecundity: Snowdrift
Death-Birth updating
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
Population structure
Large population size N → ∞
ed = ed
dself > 0 dself=0
F. D´ebarre Social evolution in structured populations 24 / 26
173. Take Home Messages
Evolution of social behaviour
F. D´ebarre Social evolution in structured populations 25 / 26
174. Take Home Messages
Evolution of social behaviour
Death-Birth vs. Birth-Death
F. D´ebarre Social evolution in structured populations 25 / 26
175. Take Home Messages
Evolution of social behaviour
Death-Birth vs. Birth-Death Effect on first or second step
F. D´ebarre Social evolution in structured populations 25 / 26
176. Take Home Messages
Evolution of social behaviour
Death-Birth vs. Birth-Death Effect on first or second step
Benefits: on the second step
Costs: depends
F. D´ebarre Social evolution in structured populations 25 / 26
177. Take Home Messages
Evolution of social behaviour
Death-Birth vs. Birth-Death Effect on first or second step
Benefits: on the second step
Costs: depends
Inclusive fitness vs. Game theory
F. D´ebarre Social evolution in structured populations 25 / 26
178. Take Home Messages
Evolution of social behaviour
Death-Birth vs. Birth-Death Effect on first or second step
Benefits: on the second step
Costs: depends
Inclusive fitness vs. Game theory
Semantic issues
Differing assumptions (weak selection), with consequences
F. D´ebarre Social evolution in structured populations 25 / 26
179. Take Home Messages
Evolution of social behaviour
Death-Birth vs. Birth-Death Effect on first or second step
Benefits: on the second step
Costs: depends
Inclusive fitness vs. Game theory
Semantic issues
Differing assumptions (weak selection), with consequences
F. D´ebarre Social evolution in structured populations 25 / 26
180. Thanks for your attention!
F. D´ebarre Social evolution in structured populations 26 / 26
196. Relatedness
No, but I would to save
two brothers or eight cousins.
F. D´ebarre Social evolution in structured populations 29 / 26
197. Relatedness
No, but I would to save
two brothers or eight cousins.
F. D´ebarre Social evolution in structured populations 29 / 26
198. 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
F. D´ebarre Social evolution in structured populations 29 / 26
199. Relatedness
A measure of identity
Often expressed as a regression of partner’s genotype (or
phenotype) on focal individual’s genotype (or phenotype)
r =
CovS(g, g )
CovS(g, g)
Frank (2013), JEB
Gardner et al. (2011), JEB
F. D´ebarre Social evolution in structured populations 29 / 26
200. Relatedness
A measure of identity
Often expressed as a regression of partner’s genotype (or
phenotype) on focal individual’s genotype (or phenotype)
r =
E[CovS(g, g )]
E[CovS(g, g)]
Frank (2013), JEB
Gardner et al. (2011), JEB
F. D´ebarre Social evolution in structured populations 29 / 26
201. Relatedness
A measure of identity
Often expressed as a regression of partner’s genotype (or
phenotype) on focal individual’s genotype (or phenotype)
r =
E[CovS(g, g )]
E[CovS(g, g)]
r =
E[x, x ]
E[x, x]
Frank (2013), JEB
Gardner et al. (2011), JEB
Taylor (2013), JTB
F. D´ebarre Social evolution in structured populations 29 / 26
202. Relatedness
A measure of identity
Often expressed as a regression of partner’s genotype (or
phenotype) on focal individual’s genotype (or phenotype)
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
Gardner et al. (2011), JEB
Taylor (2013), JTB
Taylor et al. (2007), JTB
F. D´ebarre Social evolution in structured populations 29 / 26