Supplementary materials:
Figure S1 Other typical snapshots of the downward evolution in the same iteration as
in Fig. 2. See also Video S1.
1
(b) Mobility rate in directional
and random migration
Dir C
Dir D
Ran C
Ran D
0.08
0.10
Mobility rate
0.12
0.9
0.8
0.7
Fraction of cooperation
0.14
1.0
0.16
(a) Fraction of Cooperation
0.6
0.04
0.06
Directional
Random
Fixed
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Benefit / Cost, (b/c)
Benefit / Cost, (b/c)
(c) Mobility rate in
fixed migration
(d) Entropy of C and D
in directional migration
4.5
2.0
1.0
1.0
Entropy
0.6
1.0
0.4
Mobility rate
1.5
0.8
Fix C
Fix D
0.5
0.0
0.2
Dir C
Dir D
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.0
Benefit / Cost, (b/c)
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Benefit / Cost, (b/c)
Figure S2 General analysis as a function of the benefit/cost ratio. (a) Fraction of
cooperation from the 1000th to 2000th generation (averaged over 1000 simulation runs,
excluding the runs resulting in extinction) as a function of the benefit-to-cost ratio of
cooperation. (b) Mobility rates from the 1000th to 2000th generations (averaged over
individuals and 1000 simulation runs, excluding the runs resulting in extinction) as a
function of the benefit-to-cost ratio of cooperation in random and directional migration.
(c) Mobility rate for fixed migration. (d) Average entropy of cooperators and defectors
in directional migration.
2
1.0
1.0
1.0
0.2
0.5
1.0
2.0
5.0
10.0
0.9
0.8
Fraction of cooperation
0.6
0.6
0.1
0.7
0.9
0.8
Fraction of cooperation
0.8
0.7
Directional
Random
0.7
0.9
Directional
Random
0.6
Fraction of cooperation
Directional
Random
0.001
Initial density (N/(M x M))
0.005
0.020
0.050
0.001
Sigma
(a) Initial density(N/(M × M ))
0.005
0.020
0.050
Mu
(b) σ
(c) µ
60
40
20
Frequency
80
100
Figure S3 Sensitivity analysis. Fraction of cooperation (over 1000 simulation runs) as
a function of the (a) initial density (N/(M × M )), (b) σ, and (c) µ. The result of fixed
migration is excluded because the extinction rate was almost 100% (See Fig. 8). (a)
N = 1000 is the initial number of individuals. The square lattice is composed of M × M ,
and only M takes values of 10, 30, 50, and 100. Thus, N/(M × M ) denotes the initial
density. (b) σ is the SD of the normal distribution in the mutation of pm and d. (c) µ is
the mutation rate. At µ = 0.1, all three models had died out (See Fig. 8).
0
Extinction
Divergence
0.34
0.36
0.38
0.40
0.42
0.44
Death probability
Figure S4 Sensitivity to death probability (pd ) in directional migration. We conduct
100 independent simulation runs. If the global population reaches 100,000, it is defined
as divergence. At pd ≤ 0.33(≥ 0.43), all runs show divergence (extinction). Thus,
0.34 ≤ pd ≤ 0.42 is the realistic situation in which the finite global population is observed.
3
0.1
0.2
0.5
2.0
5.0
10.0
(a) Initial density(N/(M × M ))
2.0
0.5
0.001
Directional
Random
0.0
Directional
Random
0.0
1.0
Initial density (N/(M x M))
1.0
Entropy
1.5
2.0
0.5
1.0
Entropy
1.5
2.0
1.5
1.0
Entropy
0.5
0.0
Directional
Random
0.005
0.020
Sigma
(b) σ
0.050
0.001
0.005
0.020
0.050
Mu
(c) µ
Figure S5 Entropy (over 1000 simulation runs) as a function of the (a) initial density
(N/(M × M )), (b) σ, and (c) µ with error bars (SD).
4
10000 20000 30000 40000 50000
4e+05
(a)
2e+05
0
0e+00
1e+05
# individuals
3e+05
Cooperators
Defectors
0
20
40
60
80
100
Generation
(b)
Figure S6 (a) Typical snapshots of cooperation divergence when individuals are allocated in the center with only one defector (µ = 0). In this case, one defector does not
expand and cooperators diverge quickly. After the 100th generation, the population is
over 400,000. (b) The population dynamics of the cooperation divergence.
5
10000 20000 30000 40000 50000
2e+05
4e+05
Cooperators
Defectors
0
0e+00
# individuals
6e+05
(a)
0
20
40
60
80
100
Generation
(b)
Figure S7 (a) Typical snapshots of defective invasion from the inner territory when
individuals are allocated in the center with only one defector (µ = 0). In this case, one
defector invades cooperators, but they only survive at the border with cooperators. After
the 100th generation, there are over 600,000 cooperators (left y-axis) and around 20,000
defectors (right y-axis). (b) The population dynamics of defective invasion.
6