Supporting Information
Wakano et al. 10.1073/pnas.0812644106
SI Text

F共w兲 ⫽ 1 ⫹ 共r ⫺ 1兲wN⫺1 ⫺Model Analysis. Consider the following selection–diffusion dy-

namics for the density of cooperators, u(x, y, t), and defectors,v(x, y, t), as a function of the spatial coordinates x, y, and time t:⭸tu ⫽ DCⵜ2u ⫹ u关w共 fC ⫹ b兲 ⫺ d兴[S1a]

⭸tv ⫽ DDⵜ v ⫹ v关w共 fD ⫹ b兲 ⫺ d兴

[S1b]

2

The per capita birth rate of cooperators and defectors, w( fC ⫹b) and w( fD ⫹ b), is determined by the baseline birthrate, b, theirperformance in public-goods interactions, fC and fD, respectively,
and is negatively correlated with the population density u ⫹ vthrough w ⫽ 1 ⫺ u ⫺ v. Thus, competition results in decreasing
birth rates for increasing population densities and vice versa. The
per capita death rate, d, is equal and constant for both types.
Spatial migration of cooperators and defectors is determined by
the diffusion constants, DC and DD, and the diffusion operator
ⵜ2 ⫽ ⭸x2 ⫹ ⭸2y , where ⭸i denotes the shorthand notation for thepartial derivative with respect to i. The spatial domain ⍀ is L ⫻L, where L denotes the linear extension of space, unless otherwise specified. We only consider Neumann boundary condition(i.e., zero-flux boundaries).
Derivation of payoffs. Public-goods games are played in randomly
formed groups with a maximum size of N. Participants are
sampled by interpreting the densities u (v) as probabilities fordrawing a cooperator (defector) and w for failing to find aparticipant, respectively. Thus, the population density, u ⫹ v,
determines the effective interaction group size S with an average
size of (u ⫹ v)N. An individual finds itself in a group of size Swith probability

冉

冊

N⫺1
共1 ⫺ w兲S⫺1wN⫺S.
S⫺1In this group, the individual faces m cooperators and S ⫺ 1 ⫺m defectors with probability冉 冊冉 冊 冉 冊
u m
u⫹v

S⫺1
m

S⫺1⫺m
v
u⫹v

and the payoff for defectors and cooperators is given byr
PD共S兲 ⫽
S

冘冉

S⫺1

m⫽0

冊 冉 冊冉 冊

u
S⫺1
m
m
u⫹v

m

PC共S兲 ⫽ PD共S兲 ⫹

v
u⫹v

S⫺1⫺m

r
⫺ 1.
S

[S2a]

[S2b]

For convenience, the costs of cooperation are set to c ⫽ 1.
Averaging over all group sizes S yields

冘冉
N

fi ⫽

S⫽2

fD ⫽ r冊冉u
1 ⫺ wN
1⫺1⫺w
N共1 ⫺ w兲冊

fC ⫽ fD ⫺ F共w兲Wakano et al. www.pnas.org/cgi/content/short/0812644106[S3c]

The difference between the payoffs of cooperators and defectors, F(w), is always positive for r ⬍ 2 such that defectors prevail.
For r ⬎ 2, F(w) ⫽ 0 has a single and unique root ŵ 僆 [0,1) (1).
For w ⬎ ŵ, F(w) ⬍ 0 holds, and hence, cooperators thrive. A
more detailed derivation and analysis is provided in refs. 2
and 3).
Nonspatial dynamics. The dynamics in the absence of space (see Eq.
1 in the main text), which corresponds to Eq. S1 without the
diffusion terms DCⵜ2u and DDⵜ2v) reduces to a set of ordinary
differential equations (ODE) that can be analyzed in detail (3).
This provides the basis for the analysis of the spatial system ofpartial differential equations (PDE) in Eq. S1. Here, we focus on
the scenario with baseline birthrates b ⱖ 1 and the moreinteresting case with death rates d ⬍ b, such that defectors cannot
survive in the absence of cooperators (⭸tv ⬍ 0 for u ⫽ 0). In thiscase, the ODE admits 2 fixed points that are of particularinterest: O and Q. The state O denotes the extinction of the
population (u ⫽ 0, v ⫽ 0) and is always stable. The existence andstability of Q, where cooperators and defectors can coexist,
depends on the multiplication factor of the public good, r, but thethresholds of r are analytically inaccessible because ŵ cannot be
determined for arbitrary N. However, it can be shown that Q is
unique, if it exists (3). A lower bound for the existence of Q isgiven by r ⬎ 2 because otherwise ŵ does not exist. At rHopf, thesystem undergoes a subcritical Hopf bifurcation (4) that gives
rise to unstable limit cycles for r close to but above rHopf. Notethat with b ⱖ 1, supercritical Hopf bifurcations accompanied bystable limit cycles are impossible (3). For r ⬍ rHopf ⫽ 2.3658, Q
is unstable and O is globally stable (numerical values areprovided for the parameters used in all figures, N ⫽ 8, b ⫽ 1, d ⫽1.2). For rHopf ⬍ r ⬍ r2 ⫽ 6.53, Q is stable, and the basin of
attraction increases with r but remains limited. For unfavorable
initial configurations, either because of low population densities
that result in a shortage of interaction partners [as described by
the Allee effect (5)] or because of high relative abundance of
defectors that diminish the public good, the population cannot
recover and goes extinct, i.e., converges to O. Finally, for highmultiplication factors, r ⬎ r2, defectors disappear and cooperators thrive unchallenged.
Spatial Dynamics
Selection and diffusion in spatial ecological public-goods games
can trigger spontaneous pattern formation through diffusioninduced instability (Turing instability) or diffusion-induced coexistence. The heterogeneous density distributions of cooperators and defectors can be static, quasistatic with intermittent
bursts of rapid changes or dynamic patterns of spatiotemporalchaos. The different dynamical regimes are summarized in aschematic phase plane diagram, Fig. S1.
Diffusion-Induced Instability. If the coexistence equilibrium QN⫺1共1 ⫺ w兲S⫺1wN⫺SPi共S兲
S⫺1with i ⫽ C or D and reduces tor 1 ⫺ wN.
N 1⫺w

exists and is stable in the absence of spatial extension (r ⬎ rHopf),
then diffusion can destabilize Q because the ecological publicgoods game represents an activator–inhibitor system (6, 7) in thevicinity of Q. The Jacobian matrix J at Q is given by

[S3a]
[S3b]

冉

冊

J11 J21J⫽⫽
J12 J22冢冣dd⫹ uw⭸u fC ⫺ u ⫹ uw⭸v fC
w
w
.
d
d
⫺ v ⫹ vw⭸u fD ⫺ v ⫹ vw⭸v fD
w
w

⫺u

[S4]

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The fitness of cooperators and defectors is determined by theirperformance in public-goods interactions and adjusted by the
population density. Increasing population densities (and hence
decreasing w) reduces the fitness, which is reflected in the negative
first term in each component of J. The second term indicates theeffect of population densities on the payoffs. It is easy to see that
increasing the defector density reduces the payoffs of both cooperators and defectors, i.e., ⭸vfC ⬍ 0 and ⭸vfD ⬍ 0 (see Eq. S3. Thus,
defectors act as inhibitors. Similarly, increasing the cooperator
density increases the payoffs of both types, i.e., ⭸ufC ⬎ 0 and ⭸ufD
⬎ 0, which indicates that cooperators act as activators whenever the
population densities are sufficiently low. An activator–inhibitor
system requires J1i ⬎ 0 and J2i ⬍ 0 at Q.
In this case, diffusion induced instability (or Turing instability)
occurs if inhibitors (defectors) diffuse faster than activators(cooperators). Various critical parameters can be calculated that
determine the spatial pattern formation (6–8). In particular, we
can derive the minimum ratio of the diffusion coefficients ofinhibitors and activators, DD/DC, as well as the characteristic
length scale, l, of the emerging patterns (see Pattern formation
process below). Any spatial perturbation in the square domain ⍀is decomposed into spatial modes
cos

n␲x
m␲y
cos
.
L
L

[S5]

with spatial eigenvalues k2 ⫽ ␲2(n2 ⫹ m2)/L2, where n,m are
integers. The linearlized temporal dynamics of each spatial modeis given by

冉冊 冋

冉

d uk
DC 0
⫽ J ⫺ k2
0 DD
dt vk

冊册冉 冊

uk
,
vk

[S6]

冉 冊

uk
denotes deviations from Q. The dynamics is notvknecessarily stable even if it is stable against homogeneous
perturbations (i.e., n ⫽ m ⫽ 0 or k ⫽ 0). The necessary andsufficient condition for instability is determined by J, DC, DD and
k. In particular, if we neglect the condition that n,m should be
integers, the minimum ratio of diffusion coefficient is given by
where

DD
⫺ 2J21⫹J 12 ⫹ J 11J 22 ⫹ 2 冑 ⫺ J 12 J 21det J
⬎
.
DC
J 211

[S7]

The minimum ratio must be ⬎1. Numerically we obtain Jij to
confirm that the system satisfies the conditions for activator–
inhibitor dynamics and derive the threshold value for DD/DC thatis required for the spontaneous emergence of spatial patterns(see Fig. S1).
Pattern formation process. The pattern formation process can beunderstood and analyzed in greater detail in a 1-dimensional spatial
setting. The most unstable mode, k*, (with the largest eigenvalue␭k*) determines the characteristic length scale, l, of the emerging
patterns with l ⫽ 2␲/k*. k* is estimated by d␭k/dk ⫽ 0 where

␭k ⫽

冉
冑

1
J ⫹ J22 ⫺ k2共DC ⫹ DD兲
2 11

⫹

冊

共J11 ⫹ J22 ⫺ k2共DC ⫹ DD兲兲2.
⫺ 4共det J ⫺ k 2共D CJ 22 ⫹ D DJ 11兲 ⫹ k 4D CD D兲

[S8]The analytic estimation of l provides an excellent approximationof the characteristic scale for the observed patterns both in 1
dimension (see Fig. S2) and in 2-dimensional space (see Fig. 2 A
in the main text).
Starting from an initially homogeneous distribution withdensities near Q and a small disturbance in the center, then the
spreading of the disturbance nicely illustrates the pattern forWakano et al. www.pnas.org/cgi/content/short/0812644106

mation process (see Fig. S2). First the perturbation grows, andthe densities of cooperators and defectors increases. Due to thehigher diffusion rate of defectors, cooperators suffer in theadjacent regions on either side, which decreases the densities ofboth strategies. The low defector densities then enable cooperators still further away from the initial perturbation to thrive,
which triggers an increase in both strategies. This process repeats
until the boundaries are reached. The distance between subsequent density peaks (or valleys) depends on the diffusion
coefficients, DC, DD, and indicates the characteristic lengthscale, l.
Diffusion-induced coexistence. Diffusion-induced coexistence isclosely related to diffusion-induced instability with the only
difference being that the coexistence equilibrium Q is unstablein the absence of space (r ⬍ rHopf). Consequently, any deviationfrom Q is amplified over time in an oscillatory manner until the
population goes extinct and reaches the globally stable equilibrium O. In the vicinity of r ⫽ rHopf, the system again exhibits anactivator–inhibitor dynamics, and the analysis of the Jacobian J
at Q remains the same. Indeed, cooperators behave as activators
as long as r ⬍ rTuring ⫽ 3.31. However, the homogeneous mode(k ⫽ 0) is already unstable, and a linear analysis is less powerfulfor diffusion-induced instability. Only recently, diffusioninduced coexistence attracted increasing attention (9), but arigorous analysis is lacking. Diffusion-induced coexistence is not
restricted to static patterns but allows much richer dynamics,
which include spatiotemporal chaos (see Fig. S1).
The detailed pattern formation process for diffusion-induced
coexistence is again nicely illustrated when considering a 1-dimensional system with a homogeneous initial distribution withdensities of cooperators and defectors near Q and a small
disturbance in the center. The densities in homogeneous regions
oscillate and approach extinction, O, but the disturbance spreads
and establishes stable heterogeneous strategy distributions (seeFig. S3). The basic spreading of the disturbance is the same asfor diffusion-induced instability (see Fig. S2), because in bothcases, the propagation is driven by the activator-inhibitor dynamics. In addition, the instability of Q manifests itself near thefront of the propagating disturbance in that defectors destabilize
peaks of high cooperator densities and repeatedly split them into
2 peaks until the boundaries are reached, and a stable heterogeneous strategy distribution takes shape.
Robustness of Results. To support our claims that spatial patternformation promotes cooperation, we verified the robustness of
our results for different initial configurations as well as othernumerical methods and domain shapes.
Numerics. For the numerical integration of Eq. S1, the spatial
domain is discretized by a square grid with an equal spacing ofdx in both dimensions, and the time evolution is determined byusing the Crank–Nicholson method (10) in time increments dt.
Results in the chaotic regime are particularly sensitive to numerical methods. To verify the robustness, the numerical integration shown in Fig. 1 of the main text was carried out for all
combinations of dx ⫽ 0.25, 0.4, 1.0, 2, 4 and dt ⫽ 0.001, 0.01, 0.1,
1. No qualitative differences were found for any combination. In
addition, we checked the results using a different method basedon finite elements [Hecht F, Pironneau O, Le Hyaric A, Ohtsuka
K (2007) Freefem⫹⫹: Finite element method to solve partial
differential equations. www.freefem.org] with different boundary shapes (see Fig. 4). Again, no qualitative differences were
found.
Initial configuration. Numerical integration of PDEs is often susceptible to changes in the initial configuration. To verify therobustness of the pattern formation, we carried out simulations
using initial configurations with a homogeneous disk of equal
densities of cooperators and defectors located in the center ofvacant space (see Fig. S5) instead of heterogeneous distributions
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of random cooperator and defector densities (see Fig. 4 in maintext). The qualitative features of the dynamics and the characteristic patterns remain unchanged. Only the boundaries of thechaotic regime are slightly different, but this is not surprisingbecause this regime is defined by its susceptibility with respect tosmall configurational changes.
Patterns. In the absence of space, the basin of attraction of thecoexistence equilibrium Q is determined by r. For r ⱕ rHopf, Q is
unstable, and the population invariably disappears (the extinction
state O with w ⫽ 1 is globally stable). For r ⬎ rHopf, Q is stable, andthe basin of attraction increases with r, but O remains locally stable
for any r. The dynamics of the nonspatial ODE is shown in Fig. S6a.
Spatial extension and diffusion enlarge the basin of attraction of Q
by forming heterogeneous strategy distributions. Because measuring the basin of attraction is challenging, we consider r just below
rHopf such that O is globally stable in the absence of space.
Nevertheless, in the spatial system, cooperators and defectors
coexist (see Fig. S6b) for a broad range of initial population
densities. Thus, spatial extension sustains the population by maintaining cooperation through pattern formation.

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public goods games. J Theor Biol 218:187–194.
2. Hauert C, Holmes M, Doebeli M (2006) Evolutionary games and population dynamics:
maintenance of cooperation in public goods games. Proc R Soc London Ser B 273:2565–
2570.
3. Hauert C, Wakano JY, Doebeli M (2008) Ecological public goods games: cooperation
and bifurcation. Theor Pop Biol 73:257–263.
4. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory (Springer, New York).
5. Stephens PA, Sutherland WJ (1999) Consequences of the Allee effect for behavior,ecology and conservation. Trends Ecol Evol 14:401– 405.
6. Murray J (2003) Mathematical Biology II: Spatial Models and Biomedial Applications(Springer, Berlin).

7. Segel LA, Jackson JL (1972) Dissipative structure: An explanation and an ecologicalexample. J Theor Biol 37:545–559.
8. Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc London Ser
B 237:37–72.
9. Baurmann M, Gross T, Feudel U (2007) Instabilities in spatially extended predator–preysystems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations.
J Theor Biol 245:220 –229.
10. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1988) Numerical Recipes in C
(Cambridge Univ Press, Cambridge, UK).

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10000

ratio of diffusion constants DD / DC

heterogeneous coexistence

rHopf

1000
diffusioninducedcoexistence100

diffusion induced instability

10
chaos

homogeneous coexistence1
extinction

0.1
2

2.2

2.4

2.6

2.8

3

3.2

3.4

multiplication factor r
Fig. S1. Schematic phase plane diagram of the different dynamical regimes in terms of the ratio of the diffusion of defectors to cooperators, DD/DC and themultiplication factor r in spatial ecological public goods games. The vertical red line indicates rHopf, which separates the region of extinction (r ⬍ rHopf) and possible
stable coexistence (r ⬎ rHopf) in the absence of space. If cooperator diffusion exceeds defector diffusion, DD/DC ⬍ 1 the dynamics is barely affected by space, except
for a small chaotic region near rHopf. The dashed lines indicate approximate boundaries based on numerical results. These thresholds are analytically inaccessible
and for the chaotic region even numerically challenging because the sensitivity of the system with respect to initial configurations increases when approaching
the boundary of the parameter range where the population persists. The dynamics becomes much richer for DD/DC ⬎ 1. For r ⬍ rHopf, the chaotic regime increases
substantially and is replaced by diffusion-induced coexistence for high DD/DC. For r ⬎ rHopf the homogeneous spatial distributions are replaced by heterogeneous
distributions triggered by diffusion induced instability (Turing patterns) for high DD/DC. The solid line indicates the analytical threshold ratio of the diffusion
constants required for Turing patterns (see Eq. S7.) For very large r ⬎ rTuring ⫽ 3.32 (shown by the vertical dashed line) all patterns disappear. All thresholds refer
to N ⫽ 8, d ⫽ 1.2, b ⫽ 1, DC ⫽ 1 and are based on numerical integration using L ⫽ 283, dx ⫽ 1.4, dt ⫽ 0.1.

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a

b

c

b
c

0.125

d

density

0.1
e

0.075

0.05

0.125

0.125

0.1

0.1

0.075

0.075

densitydensity

f

0.05

0.05
0

d

0.25

0.5

0.75

1

space L

0

e

0.25

0.5

space L

0.75

1

0

f

0.25

0.5

0.75

1

space L

Fig. S2. Pattern formation process in ecological public goods games through diffusion-induced instability in 1-dimensional spatial systems. Initially the strategy
distribution is homogeneous with densities near the stable coexistence equilibrium Q and a perturbation in the center (increase of the cooperator density by0.1). (a) Space–time plot of the pattern formation process where the abscissa denotes space, and the ordinate the time (from top to bottom). The brightness ofthe colors indicates the density of cooperators (green) and defectors (red) or both (yellow). The horizontal lines in a, labeled b–f, indicate the times of thecross-sections of cooperator (green) and defector (red) densities in t ⫽ 10 (b), t ⫽ 24 (c), t ⫽ 40 (d), t ⫽ 200 (e), and t ⫽ 398 (f). In b–f, the dashed horizontal lines
indicate the equilibrium frequencies of cooperators, û, and defectors, v̂ at Q. The dashed vertical lines in f indicate the characteristic length scale, l, according
to Eq. S8. The game parameters are N ⫽ 8, b ⫽ 1, d ⫽ 1.2, r ⫽ 2.5, which yield û ⫽ 0.078 and v̂ ⫽ 0.11. Diffusion coefficients are DC ⫽ 1 and DD ⫽ 10, spatial extension
is L ⫽ 100 and numerical integration uses dx ⫽ 0.5 and time increments of dt ⫽ 0.1.

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c

0.2

e

0.1

densityfdensity

b
b
c
d

0

0.2

0.2

0.1

0.1

0

density

a

0
0

d

0.25

0.5

0.75

1

space L

0

e

0.25

0.5

space L

0.75

1

0

f

0.25

0.5

0.75

1

space L

Fig. S3. Pattern formation process in ecological public goods games through diffusion-induced coexistence in 1-dimensional spatial systems. As in Fig. S2, theinitial strategy distribution is homogeneous with densities near the unstable equilibrium Q and a perturbance in the center (increase of the cooperator density
by 0.1). (a) Space–time plot of the pattern formation process where the ordinate denotes spatial extension and the abscissa the time (from top to bottom). The
brightness of the colors indicates the density of cooperators (green) and defectors (red), both (yellow), or none (black). The horizontal lines in a, labeled b–f,
indicate the times of the cross-sections of cooperator (green) and defector (red) densities t ⫽ 30 (b), t ⫽ 60 (c), t ⫽ 99 (d), t ⫽ 300 (e), and t ⫽ 597 (f). In b-f, thedashed horizontal lines indicate the frequencies of cooperators, û, and defectors, v̂ at the unstable Q. The game parameters are N ⫽ 8, b ⫽ 1, d ⫽ 1.2, r ⫽ 2.3,
which yields û ⫽ 0.058, v̂ ⫽ 0.066. Diffusion coefficients are DC ⫽ 1 and DD ⫽ 10, spatial extension is L ⫽ 200 and numerical integration uses dx ⫽ 1 and timeincrements of dt ⫽ 0.1.

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abcFig. S4. Chaotic pattern formation in spatial ecological public goods. The setup is identical to Fig. 1 in the main text with the only differences that the spatial
domain is circular instead of rectangular and the numerical integration of Eq. S1 is based on the finite element method FreeFEM⫹⫹ [Hecht F, Pironneau O, Le
Hyaric A, Ohtsuka K (2007) Freefem⫹⫹: Finite element method to solve partial differential equations. www.freefem.org]. The qualitative features of theemerging spatial patterns remain unchanged. (a) Initial configuration. (b) Snapshot at time t ⫽ 2,000. (c) Snapshot at t ⫽ 7,000. The parameters of the ecologicalpublic goods are N ⫽ 8, d ⫽ 1.2, b ⫽ 1, r ⫽ 2.34, DC ⫽ 1, DD ⫽ 2, and the initial configuration is a vacant disk with radius R ⫽ 200 (no flux boundary conditions)and a disk with radius R/10 in the center, where cooperators (green) and defectors (red) coexist at equal densities (udisk ⫽ vdisk ⫽ 0.1 and u ⫽ v ⫽ 0 elsewhere).

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ratio of diffusion constants DD / DC

100

10

4

1
2.14

2.16

2.2

2.3

2.36

rHopf 2.4

2.5

3

4

multiplication factor r
Fig. S5. Diversity of spatial distributions in terms of the ratio of the diffusion of defectors to cooperators, DD/DC, and the multiplication factor, r, in spatial
ecological public goods games. This figure is identical to Fig. 4 in the main text with the only exception being that instead of a randomly mixed initialconfiguration, the initial state is given by a vacant L ⫻ L square and a homogeneous disk with radius L/10 in the center, where cooperators and defectors coexist
at equal densities (udisk ⫽ vdisk ⫽ 0.1 and u ⫽ v ⫽ 0 elsewhere; see Fig. 1 in the main text). The symmetry of the initial configuration leads to more symmetricalstatic patterns for diffusion induced instability (green frame) and diffusion induced coexistence (red frame). Homogeneous coexistence (yellow frame) remains
unaffected. In all cases the qualitative features of the emerging strategy distribution remain unaffected by the initial configuration. Only the boundaries (butnot the patterns) of the chaotic region (blue frame) are slightly shifted, but this is not surprising because chaotic dynamics is highly susceptible to perturbations.
The parameters are N ⫽ 8, d ⫽ 1.2, b ⫽ 1, Du ⫽ 1, L ⫽ 283, dx ⫽ 1.4, dt ⫽ 0.1, rHopf ⫽ 2.3658.

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u

b

0.0

5

0.5

7

a

w 0.050.57

v

Fig. S6. Robustness of pattern formation in spatial ecological public goods games. (a) At or below the Hopf bifurcation, rHopf, extinction O is globally stablein the absence of space. (b) For the same parameters cooperators and defectors can coexist in a spatial setting through the spontaneous formation ofheterogeneous strategy distributions. Diffusion stabilizes the unstable coexistence equilibrium Q. Qualitatively the same patterns emerge for different initial
population densities, as indicated by the red area in a. Only for very low (u ⫹ v ⬍ 0.05) and for high (u ⫹ v ⬎ 0.57) population densities, coexistence is unreachable,
and the population disappears just as in the absence of space. The density thresholds in a are numerically derived for N ⫽ 8, r ⫽ 2.36, (rHopf ⫽ 2.3658), b ⫽ 1, d ⫽1.2, DC ⫽ 1, DD ⫽ 10. The initial density, p, is fixed and we randomly set the cooperator density u(x,y,0) according to a uniform distribution in [0,p]. The defector
density is given by v(x,y,0) ⫽ p ⫺ u(x,y,0). The time evolution for various p determines the red area.

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Movie S1. Spatiotemporal chaos. Onset of chaotic dynamics when starting from a symmetrical initial configuration (see Fig. 1 in main text). The deterministic
dynamics should, in principle, preserve the symmetry, but it breaks down after some time because of limitations of the numerical integration combined withthe exponential amplification of arbitrarily small deviations in chaotic dynamics. Parameters: N ⫽ 8, r ⫽ 2.34, b ⫽ 1, d ⫽ 1.2, DC ⫽ 1, DD ⫽ 2, L ⫽ 400, dx ⫽ 0.8,
dt ⫽ 0.01 and the movie ends at time T ⫽ 4,000. The movies are highly compressed using the open standard H.264 and require QuickTime 7 or higher for playback.

Movie S1 (MOV)

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Movie S2. Intermittent activity. In the region of intermittent activity, the formation of quasistatic patterns alternates with rapid changes triggering local (orglobal) rearrangement and redistribution of cooperators and defectors. Parameters: N ⫽ 8, r ⫽ 2.34, b ⫽ 1, d ⫽ 1.2, DC ⫽ 1, DD ⫽ 4, L ⫽ 256, dx ⫽ 0.8, dt ⫽ 0.01
and the movie ends at time T ⫽ 5,000. The movies are highly compressed using the open standard H.264 and require QuickTime 7 or higher for playback.

Movie S2 (MOV)

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Movie S3. Diffusion induced instability (Turing patterns). Cooperators and defectors form an activator–inhibitor system that destabilizes the homogeneous
coexistence equilibrium Q. The antagonistic forces of cooperators (activators) and defectors (inhibitors) give rise to the formation of stable patterns. Parameters:
N ⫽ 8, r ⫽ 2.5, b ⫽ 1, d ⫽ 1.2, DC ⫽ 1, DD ⫽ 100, L ⫽ 256, dx ⫽ 0.8, dt ⫽ 0.01 and the movie ends at time T ⫽ 500. The movies are highly compressed using theopen standard H.264 and require QuickTime 7 or higher for playback.

Movie S3 (MOV)

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Movie S4. Diffusion induced coexistence. In the absence of space, the population goes extinct. The emerging spatial patterns are responsible for the survivalof the population and permit stable coexistence of cooperators and defectors. Parameters: N ⫽ 8, r ⫽ 2.2, b ⫽ 1, d ⫽ 1.2, DC ⫽ 1, DD ⫽ 100, L ⫽ 256, dx ⫽ 0.8,
dt ⫽ 0.01 and the movie ends at time T ⫽ 2,000. The movies are highly compressed using the open standard H.264 and require QuickTime 7 or higher for playback.

Movie S4 (MOV)

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