Spatial dynamics of ecological public goods
Joe Yuichiro Wakanoa, Martin A. Nowakb, and Christoph Hauertc,1aMeiji Institute for Advanced Study of Mathematical Sciences, 1-1-1 Higashi Mita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan; bProgram for
Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Department of Mathematics, Harvard University, One Brattle Square,
Cambridge, MA 02138; and cDepartment of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2

cooperation ! evolutionary game theory ! pattern formation !
population dynamics

S

pontaneous and complex pattern formation represents acommon principle in physical, chemical, and biological systems ranging from hydrodynamical phenomena such as the
ripples in the sand, to interacting chemical fronts (1), and thecoloration of plants and animals such as the spots of the jaguar(2, 3), or vegetation patterns in arid ecosystems (4). Mathematical models of pattern formation use either cellular automata (5)or differential equations (6). Here, we show that similar types ofpatterns emerge in evolutionary settings of social dilemmas (7,8). Social dilemmas are characterized by a conflict of interest
between individuals and the group. Such conflicts arise in
humans (9) and microorganisms alike (10–12). In antibiotic
resistance, for example, bacteria secrete an enzyme that prevents
cell wall degradation (11, 13, 14). Synthesizing the enzyme iscostly to the bacterium, and releasing the enzyme creates apublic good by protecting not only the bacterium itself but also
the surrounding bacteria. Thus, enzyme production represents
an act of cooperation that is prone to exploitation by mutant
strains, which synthesize fewer or no enzymes. Such situations
are captured by public-goods games (7, 15).
In a typical public-goods experiment, N individuals have theopportunity to cooperate and invest a fixed amount, c, into acommon pool or to defect and invest nothing. The total investment is multiplied by a factor, r ! 1, and distributed equally
among all participants—irrespective of whether they have invested or not. Thus, every invested unit returns rc/N units to thewww.pnas.org"cgi"doi"10.1073"pnas.0812644106

investor (as well as to all other participants). Consequently, forr " N, rational players withhold their investments—but if allparticipants reason in this manner, the group foregoes thebenefits of the public good, and no one receives anything. In
contrast, had everybody cooperated, they would have been
better off with a return of (r # 1)c. Conversely, if r ! N rational
players invest in the public good (7) because, even with only asingle investor, the return from the public goods exceeds theinvestment (rc/N ! c).
In populations containing a fraction, u, of cooperators and afraction, v $ 1 # u, of defectors, the evolutionary dynamics canbe described by the replicator equation (16): u̇ $ u(1 # u)(gC #gD). The fitness of cooperators and defectors, gC, gD, is determined by their performance in public-goods interactions ingroups of size N that are randomly formed according to binomial
sampling. The average number of cooperators among the N # 1
interaction partners is u(N # 1). Therefore, the average payoffof defectors is gD $ u(N # 1)rc/N, and of cooperators it is gC $gD % rc/N # c. Thus, rc/N # c measures the effective cost ofcooperation. For r " N, defectors win. For r ! N, cooperators
win. However, the replicator dynamics neglects the fact thatcooperator populations have a higher productivity than defector
populations. This should be reflected in the natural assumptionthat cooperators are capable of maintaining higher population
densities than defectors.
Model and Results
Ecological public goods allow the population density u % v tofluctuate (17). This extension establishes a link between ecological and evolutionary dynamics. Variable population densities
not only introduce feedbacks between reproductive success andcarrying capacity but also affect the effective group size, S, inpublic-goods interactions. With u % v ! 1, the densities u, v canbe interpreted in terms of probabilities when attempting to form
interaction groups with N participants [see supporting information (SI) Text]. In particular, each binomial sampling trial fails
with probability w $ 1 # u # v, such that the average interactiongroup size is reduced to S $ (u % v)N. For r " N, this results in
an interesting feedback, which enables cooperators and defectors to coexist: If population densities are high, interactiongroups are large (S ! r) and defectors increase, but this reduces
the return from the public good and results in a decline of the
population density. Decreasing population densities lead tosmaller interaction groups until eventually S " r holds, andcooperators thrive. This restores high population densities, andthe cycle continues. Similar feedback mechanisms have been
discussed in the complementary context of competitive interactions (18).
Author contributions: J.Y.W., M.A.N., and C.H. designed research; J.Y.W. and C.H. performed research; J.Y.W. contributed new reagents/analytic tools; J.Y.W. analyzed data; andJ.Y.W., M.A.N., and C.H. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed. E-mail: hauert@math.ubc.ca.

This article contains supporting information online at www.pnas.org/cgi/content/full/
0812644106/DCSupplemental.

PNAS Early Edition ! 1 of 5

APPLIED
MATHEMATICSThe production, consumption, and exploitation of common resources ranging from extracellular products in microorganisms to
global issues of climate change refer to public goods interactions.
Individuals can cooperate and sustain common resources at somecost or defect and exploit the resources without contributing. This
generates a conflict of interest, which characterizes social dilemmas: Individual selection favors defectors, but for the community,
it is best if everybody cooperates. Traditional models of public
goods do not take into account that benefits of the common
resource enable cooperators to maintain higher population densities. This leads to a natural feedback between population dynamics and interaction group sizes as captured by ‘‘ecologicalpublic goods.’’ Here, we show that the spatial evolutionary dynamics of ecological public goods in ‘‘selection-diffusion’’ systems
promotes cooperation based on different types of pattern formation processes. In spatial settings, individuals can migrate (diffuse)
to populate new territories. Slow diffusion of cooperators fosters
aggregation in highly productive patches (activation), whereas fastdiffusion enables defectors to readily locate and exploit thesepatches (inhibition). These antagonistic forces promote coexistence of cooperators and defectors in static or dynamic patterns,
including spatial chaos of ever-changing configurations. The local
environment of cooperators and defectors is shaped by the production or consumption of common resources. Hence, diffusioninduced self-organization into spatial patterns not only enhances
cooperation but also provides simple mechanisms for the spontaneous generation of habitat diversity, which denotes a crucial
determinant of the viability of ecological systems.

EVOLUTION

Edited by Richard E. Lenski, Michigan State University, East Lansing, MI, and approved March 16, 2009 (received for review December 11, 2008)

A

B

C

D

E

F

G

H

Fig. 1. Chaotic pattern formation in spatial ecological public goods. A sequence of snapshots A–H demonstrates the spatial density distribution of cooperators(green) and defectors (red) over time (see Movie S1). The symmetry of the initial configuration A should be preserved in a deterministic system, but after sometime it breaks down and disappears because of limitations of the numerical integration of Eq. 2. The exponential amplification of arbitrarily small disturbancescharacterizes chaotic systems. The initial configuration is a vacant L + L square (L $ 400) with no flux boundaries and a homogeneous disk with radius L/10 in
the center, where cooperators and defectors coexist at equal density (udisk $ vdisk $ 0.1 and u $ v $ 0 elsewhere). The parameters of the ecological public goods
are N $ 8, d $ 1.2, b $ 1, r $ 2.34, c $ 1. The multiplication factor r lies slightly below the Hopf bifurcation rHopf $ 2.3658, such that the fixed point Q is unstable,
and in the absence of space, the population disappears. Diffusion of defectors is twice that of cooperators (DC $ 1, DD $ 2). The color brightness indicates thedensity of cooperators (green) and defectors (red). The snapshots are taken at times t $ 0 (A), 1,200 (B), 1,800 (C), 2,000 (D), 2,200 (E), 2,600 (F), 2,800 (G), 4,000(H). The numerical integration uses a spatial grid with dx $ 0.8 and step size dt $ 0.01.

Unstructured Populations. The performance of cooperators anddefectors in ecological public goods, fC, fD (see SI Text), depends
not only on the composition of the interaction groups but also on
their size. Moreover, increasing population density exerts competitive pressure on the rates of reproduction, and the evolutionary dynamics become:u̇ " u&w'fC # b( $ d)[1a]v̇ " v&w'fD # b( $ d).

[1b]

Cooperators and defectors have an equal and constant per capita
death rate, d, and a per capita birth rate of w( fC % b) and w( fD %b), which decreases for increasing population densities. The
baseline birthrate, b, ensures fC % b % 0 because fC can become
negative in the limit v 3 1, which is not meaningful (17).
Hereafter, we assume that the death rate exceeds the baseline
birth rate (d ! b), such that defectors cannot survive in theabsence of cooperators. However, cooperators and defectors cancoexist at an equilibrium, Q, or exhibit rich population dynamics
depending on r (17). In particular, Q changes stability through a
Hopf bifurcation at rHopf. For r ! rHopf coexistence in Q is stable
but the basin of attraction limited. For unfavorable initialconfigurations, the returns from the public good are insufficient
to offset the death rate, d, and the population goes extinct: At low
densities, too few public-goods interactions occur, or, if defectors
abound, exploitation irrecoverably diminishes the public resource. Generally, for r " rHopf, the equilibrium Q is unstable,
and the population cannot survive (17).
Spatial Dynamics. To consider effects of spatial extension on the

dynamics of ecological public goods, we introduce diffusive
migration in 2 dimensions (19):u̇ " DC*2u # u&w'fC # b( $ d)2v̇ " DD* v # v&w'fD # b( $ d).

[2a]
[2b]

The functions u, v denote the density of cooperators anddefectors at location (x,y) and time t. As before, w $ 1 # u # v
determines the negative feedback between population densityand birth rates. The diffusion constants DC and DD specify the
2 of 5 ! www.pnas.org"cgi"doi"10.1073"pnas.0812644106

migration rates for cooperators and defectors and *2 denotes the
diffusion operator. This continuous spatial extension reveals afascinating world of dynamical pattern formation in ecologicalpublic goods [see Fig. 1, Fig. S1, Movie S1, Movie S2, Movie S3,
and Movie S4, and interactive online tutorials Hauert C (2009)
Virtuallabs: Interactive tutorials on evolutionary game theory,www.univie.ac.at/virtuallabs].
Spatial patterns unfold if defectors diffuse (migrate) fasterthan cooperators, DD % DC. Consequently, the dominant effectof spatial dynamics is not cooperators outrunning defectors but,instead, the defectors’ relentless search of productive patches.
Slow migration facilitates aggregation of cooperators, whereas
fast migration supports defectors to readily locate cooperator
patches, but it also impedes their ability to exploit one particular
patch.
The equilibrium Q of Eq. 1 translates into a trivial solution of
Eq. 2 in the form of a spatially homogeneous strategy distribution with densities according to Q. However, if Q is stable, thisdoes not necessarily imply the stability of the corresponding
homogeneous state. In the vicinity of Q, Eq. 2 takes on the formof an activator–inhibitor system. Any deviation from the equilibrium is amplified by cooperators (activators) but suppressedby defectors (inhibitors). For DD ! DC, these antagonistic forces
may give rise to the formation of complex patterns (Turing
instability) (3, 6) (see Fig. 2A). Any small local disturbance
propagates through the system and induces stable heterogeneous
strategy distributions (see SI Text, Fig. S2, and Movie S3). Local
disturbances may give rise to rearrangements of the patterns but
then quickly relax into another qualitatively indistinguishable
distribution of cooperators and defectors.
Conversely, if Q is unstable, then spatial extension and migration often prevents extinction and stabilizes coexistence ofcooperators and defectors either in static spots and stripes
similar to Turing patterns (see Fig. 2B) or in chaotic dynamics
of ever-changing patterns (see Fig. 1 and Movie S1, Movie S2,and Movie S4). In general, a homogeneous population near Q
exhibits periodic density oscillations of increasing amplitude thateventually result in extinction. However, any small local disturbance can propagate through space and trigger stationary,heterogeneous strategy distributions (see Fig. 2B). The patternformation is again driven by the opposing forces of cooperators
Wakano et al.

(activators) and defectors (inhibitors). However, the activator–
inhibitor system develops in the vicinity of an unstable fixed
point (see SI Text, Fig. S3, and Movie S4), which could be termed‘‘diffusion-induced coexistence’’ in contrast to the classical‘‘diffusion-induced instability’’ of Turing patterns. Also note
that, whereas Turing patterns rely on substantial differences inthe diffusion constants of activators and inhibitors (see SI Text
and Fig. S1), this does not apply to diffusion-induced coexistence, where dynamic patterns emerge even for DD $ DC (seeFig. 3).

ratio of diffusion constants DD / DC

100

10

4

1
2.14

2.16

2.2

2.3

2.36

r 2.4

Hopf

2.5

3

4

multiplication factor r

Fig. 3. 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 ecologicalpublic-goods games. The brightness of the colors indicates the density of cooperators (green) and defectors (red). In the absence of space, the population survives
for r ! rHopf. If cooperator diffusion exceeds defector diffusion, DD/DC ! 1, the dynamics is barely affected by space, except for a small chaotic region (blue frame)near rHopf. The dynamics becomes much richer for DD/DC ! 1. For r " rHopf, the chaotic regime increases (blue frame) and is replaced by diffusion induced
coexistence patterns for high DD/DC (red frame). For r ! rHopf, the homogeneous spatial distributions (orange frame) are replaced by diffusion-induced instability(Turing patterns; green frame) for high DD/DC. For very large r, all patterns disappear. The parameters are N $ 8, c $ 1, d $ 1.2, b $ 1, DC $ 1, L $ 283, dx $ 1.4,
dt $ 0.1 (rHopf $ 2.3658) and an initial configuration with random cooperator and defector densities in [0, 0.1]. For a detailed phase plane diagram and otherinitial configurations see SI Text and Figs. S1 and S5.
Wakano et al.

PNAS Early Edition ! 3 of 5

EVOLUTIONAPPLIED
MATHEMATICS

Fig. 2. Typical stationary patterns for diffusion-induced instability (Turing
patterns) (A) and diffusion induced coexistence (B) on an L + L square (no flux
boundaries). The density variation of cooperators (green) and defectors (red)is shown along a cross-section (solid blue line). (A) In the absence of space,cooperators and defectors coexist for suitable initial configurations (r $ 2.4 !
rHopf $ 2.3658). Diffusion destabilizes the spatially homogeneous state andinduces stable and static heterogeneous strategy distributions, where individuals spontaneously aggregate in spots or striped patterns. (B) In the absence of space, the population goes extinct (r $ 2.24 " rHopf). Diffusion
stabilizes persistence of the population and coexistence of cooperators anddefectors by inducing heterogeneous strategy distributions. In both scenarios,
the parameters of the ecological public goods are n $ 8, c $ 1, d $ 1.2, b $ 1,
DC $ 1, DD $ 10, with an initial configuration where densities are randomly
drawn in [0, 0.1]. Numerical integration is performed on a spatial grid with L $283, dx $ 1.4, and a step size of dt $ 0.1. The brightness of the colors indicatesthe strategy densities (Upper) and the dashed horizontal lines mark thedensities at Q (Lower).

Discussion
In ecology, Turing patterns occur in spatial predator–preymodels (20). Particularly rich dynamics develop in the vicinity ofTuring–Hopf bifurcations (3, 21), as in the present case. Turing
instabilities also occur in replicator–diffusion systems, but require at least 3 different strategic types (22). Two types are
sufficient, however, if population densities can vary across space(23). In ecological public goods, cooperators and defectors
suffice to produce rich spatiotemporal dynamics not only in theform of Turing patterns but also through diffusion-induced
coexistence or deterministic chaos. Spatial dimensions generally
support cooperation (24), and here, this holds in 3 different ways.
First, the ratio of cooperators versus defectors is altered in favorof cooperators, which leads to an increase of the overall population density (see Fig. 4). Second, the robustness with respect
to variations in the initial configuration is improved such thatcoexistence is much more easily achieved in spatial settings (seeSI Text and Figs. S4–S6). Finally, and most importantly, spatial
pattern formation based on diffusion-induced coexistence substantially extends the parameter range for the persistence ofcooperators to settings where, otherwise, defectors would drive
the population to extinction (see Figs. 3 and 4 for r " rHopf).
Spatial games are traditionally based on 2 distinct types of
discrete models that differ significantly from our approach. One
type considers single individuals located in each site of a latticeor general network (25–28). Interactions are limited to a local
neighborhood, and individuals are sedentary. Note that evensmall migration rates would severely challenge cooperation inthese models. The other type of model considers metapopulations that consist of patches that are arranged on a lattice.
Patches are linked to neighboring patches through migrating
individuals (29–33). In the case of predator–prey interactions,this corresponds to a system of coupled oscillators that can
exhibit highly complex dynamics (29–31). Our model represents
a continuum limit of this second type of models, applied to the
problem of cooperation.
The formulation in terms of continuous space based on partial
differential equations (PDE) allows for a deeper understandingof the relevant mechanisms that drive the spontaneous generation of spatial heterogeneity and temporal fluctuations, which
are ultimately responsible for supporting cooperation. Addition-

A 0.24
density u+v

0.2

0.16

0.12

0.08

B 1.4 2.1

2.2

2.3

1.2

2.5

diffusion induced
instability

1

ratio u/v

2.4

0.8

2.6

2.7

no spatial effectdiffusion induced
coexistence

chaos

0.6
0.4
0.2
2.1

2.2

2.3

2.4

multiplication factor r

2.5

2.6

2.7

Fig. 4. Average global population density (A) and ratio of cooperators to
defectors (B) as a function of the multiplication factor r in ecological publicgoods interactions. The nonspatial stable equilibrium (solid lines; requires r !
rHopf) is shown together with numerical results for the stationary spatial
distributions (dots). The chaotic regime depicts the average and standard
deviation of the time series from t $ 5,000 to 10,000 with dt $ 0.1, dx $ 1.4 andL $ 283. For small r, the population goes extinct, but for increasing r, thepopulation persists and exhibits chaotic dynamics that change into quasistaticand static patterns emerging through diffusion-induced coexistence (r "rHopf). For r ! rHopf, static patterns are triggered by diffusion-induced instability (Turing patterns) and relax into spatially homogeneous coexistence forhigh r. Diffusion supports cooperation by significantly increasing the persistence region of the population and in the chaotic regime, cooperator densities
even exceed those of defectors. Snapshots illustrate typical patterns emerging
in the different dynamical regimes. The brightness of the colors indicates thedensity of cooperators (green) and defectors (red). The parameters for the
ecological public goods game are N $ 8, c $ 1, d $ 1.2, b $ 1, DC $ 1, DD $ 10
such that rHopf $ 2.3658.

ally, PDEs allow one to derive macroscopic quantities such as therelevant scale as well as the threshold for the onset of patterns
emerging through Turing instabilities—features that are only
empirically accessible in lattice models.
In our model, the local variability of the population density isa crucial feature combined with differences in migration rates ofcooperators and defectors. The feedback between population
density and interaction group size introduces a coupling of
ecological and evolutionary dynamics, which enables cooperators to survive in well-mixed models (17). Fixing population
densities at constant levels eliminates cooperation in both wellmixed and continuous spatial settings. In contrast to latticemodels, continuous spatial extension alone is not enough to avert
the extinction of cooperators, because diffusion prevents spatial
segregation of cooperators and defectors, and hence, cooperators
are unable to prevent exploitation by forming clusters. It is not
merely the spatial dimensions but the emerging spatial heteroge1. Lee KJ, McCormick WD, Ouyang Q, Swinney HL (1993) Patterns formation by interacting chemical fronts. Science 261:192–194.
2. Gell-Mann, M (1994) The Quark and the Jaguar: Adventures in the Simple and theComplex (Freeman, New York).
3. Pearson JE (1993) Complex patterns in a simple system. Science 261:189 –192.
4. Rietkerk M, Dekker SC, de Ruiter PC, van de Koppel J (2004) Self-organized patchinessand catastrophic shifts in ecosystems. Science 305:1926 –1929.
5. Wolfram S, ed (1986) Theory and Applications of Cellular Automata (World Scientific,
Singapore).
6. Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc London Ser
B 237:37–72.

4 of 5 ! www.pnas.org"cgi"doi"10.1073"pnas.0812644106

neity that enables cooperators to persist at higher densities, or evenmore importantly, to survive under conditions that would inevitably
result in the demise of the population otherwise.
Increased migration rates of defectors are motivated by thefact that defectors deplete common resources (or are unable
to sustain them), and this causes them to move elsewhere.
Conversely, migration rates of cooperators should be lower toenable them to take advantage of the locally sustained common resource.
In the context of antibiotic resistance, our results suggest thatin spatial settings, susceptible free-riding strains (defectors) withhigh motility may actually increase the chances of survival as wellas the density of the enzyme-producing, resistant strains (cooperators) and thereby aggravate the challenges for medical treatments. An interesting twist is introduced in microbial populations where migration rates or cell motility itself is linked to
public-goods interactions. Examples include collective lubricant
production in Paenibacillus to move across hard surfaces (34),the creation of an extracellular fibril matrix in Myxococcus toallow for cooperative swarming (12), or the formation of biofilms
in Pseudomonas to gain competitive advantages in accessing
limiting resources such as oxygen (10, 35). Particularly intriguing
pattern formations are observed in Escherichia coli (36, 37). The
chemotactic response to excreted aspartate amplifies spatial
heterogeneity and enables cells to efficiently exploit available
nutrients at the cost of producing the chemical signal. In thewake of the swarming cells, aggregates of sessile cells form. Cellaggregation may alleviate oxidative stress. Hence, in response to
2 common resources, nutrients and oxygen, E. coli strains
diversify into 2 types with different motility (37).
Spatial heterogeneity supports coexistence (29, 30). In ecology, the most prominent example of pattern formation refers to
the emergence of spatial vegetation patterns, which are usuallyassociated with the risk of impending desertification (4, 38)
because they indicate the ecosystems’ bistability (39). Actually,
the patterns stabilize the vegetated state, but environmental
changes (absence of precipitation, overgrazing) may induce
catastrophic shifts toward the desert state. In ecological public
goods, similarly sharp transitions to extinction or homogeneousstates are triggered by extreme external changes affecting theyield of the public resource.
Individuals consuming common resources, such as in Hardin’sTragedy of the commons (8), or producing common resources
may alter and shape their environment in an enduring manner.
This is particularly evident in microbial systems involving extracellular products such as in antibiotic resistance (11, 13, 14),
biofilms (10), or swarming (12) and represent crucial determinants of microbial ecology (40). Spatial ecological public goods
model concurrent spontaneous habitat diversification and species coexistence and hence suggest a mechanism to promotebiodiversity (30, 41).
ACKNOWLEDGMENTS. J.Y.W. acknowledges support from Japanese Science
Foundation Grant JSPS.KAKENHI 18-09976. M.A.N. and C.H. acknowledge
support from the John Templeton Foundation and the National Science
Foundation/National Institutes of Health (NIH) joint program in mathematicalbiology (NIH Grant R01GM078986). C.H. acknowledges support from the
Natural Sciences and Engineering Research Council of Canada.
7. Hauert C, Michor F, Nowak M, Doebeli M (2006) Synergy and discounting of cooperation in social dilemmas. J Theor Biol 239:195–202.
8. Hardin G (1968) The tragedy of the commons. Science 162:1243–1248.
9. Milinski M, Semmann D, Krambeck HJ, Marotzke M (2006) Stabilizing the earth’s
climate is not a losing game: Supporting evidence from public goods experiments. Proc
Natl Acad Sci USA 103:3994 –3998.
10. Rainey PB, Rainey K (2003) Evolution of cooperation and conflict in experimental
bacterial populations. Nature 425:72–74.
11. Neu HC (1992) The crisis in antibiotic resistance. Science 257:1064 –1073.
12. Velicer GJ, Yu YTN (2003) Evolution of novel cooperative swarming in the bacterium
Myxococcus xanthus. Nature 425:75–78.

Wakano et al.

EVOLUTION

27. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446:97–216.
28. Taylor PD, Day T, Wild G (2007) Evolution of cooperation in a finite homogeneousgraph. Nature 447:469 – 472.
29. Hassell MP, Comins HN, May RM (1991) Spatial structure and chaos in insect population
dynamics. Nature 353:255–258.
30. Hassell MP, Comins HN, May RM (1994) Species coexistence and self-organizing spatial
dynamics. Nature 370:290 –292.
31. Jansen VAA, Sigmund K (1998) Shaken not stirred: On permanence in ecologicalcommunities. Theor Pop Biol 54:195–201.
32. Alizon S, Taylor P (2008) Empty sites can promote altruistic behavior. Evolution (Lawrence, Kans) 62:1335–1344.
33. Sella G, Lachmann M (2000) On the dynamic persistence of cooperation: How lower
individual fitness induces higher survivability. J Theor Biol 206:465– 485.
34. Ben-Jacob E, Levine H (2006) Self-engineering capabilities of bacteria. J R Soc Interface
3:197–214.
35. Xavier JB, Foster KR (2007) Cooperation and conflict in microbial biofilms. Proc Natl
Acad Sci USA 104:876 – 881.
36. Budrene EO, Berg HC (1991) Complex patterns formed by motile cells of Escherichia
coli. Nature 349:630 – 633.
37. Budrene EO, Berg HC (1995) Dynamics of formation of symmetrical patterns bychemotactic bacteria. Nature 376:49 –53.
38. Scanlon TM, Caylor KK, Levin SA, Rodriguez-Iturbe I (2007) Positive feedbacks promote
power-law clustering of Kalahari vegetation. Nature 449:209 –212.
39. von Hardenberg J, Meron E, Shachak M, Zarmi Y (2001) Diversity of vegetation patterns
and desertification. Phys Rev Lett 87:198101.
40. Prosser JI, et al. (2007) The role of ecological theory in microbial ecology. Nat Rev
Microbiol 5:384 –392.
41. Tilman D, Reich PB, Knops JMH (2006) Biodiversity and ecosystem stability in a decadelong grassland experiment. Nature 441:629 – 632.

APPLIED
MATHEMATICS

13. Lenski RE, Hattingh SE (1986) Coexistence of two competitors on one resource andone inhibitor: A chemostat model based on bacteria and antibiotics. J Theor Biol
122:83–93.
14. Dugatkin LA, Perlin M, Lucas JS, Atlas R (2005) Group-beneficial traits, frequencydependent selection and genotypic diversity: An antibiotic resistance paradigm. Proc
R Soc London Ser B 272:79 – 83.
15. Kagel JH, Roth AE, eds (1995) The Handbook of Experimental Economics (Princeton
Univ Press, Princeton).
16. Hofbauer J, Sigmund K (1998) Evolutionary Games and Population Dynamics (Cambridge Univ Press, Cambridge, UK).
17. Hauert C, Wakano JY, Doebeli M (2008) Ecological public goods games: Cooperation
and bifurcation. Theor Pop Biol 73:257–263.
18. Rankin DJ (2007) Resolving the tragedy of the commons: The feedback between
intraspecific conflict and population density. J Evol Biol 20:173–180.
19. Wakano JY (2006) A mathematical analysis on public goods games in the continuous
space. Math Bio Sci 201:72– 89.
20. Mimura M, Murray J (1978) On a diffusive prey-predator model which exhibits patchiness. J Theor Biol 75:249 –262.
21. 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.
22. Vickers GT, Hutson VCL, Budd CJ (1993) Spatial patterns in population conflicts. J Math
Biol 31:411– 430.
23. Cressman R, Vickers GT (1997) Spatial and density effects in evolutionary game theory.
J Theor Biol 184:359 –369.
24. Doebeli M, Hauert C (2005) Models of cooperation based on the prisoner’s dilemma
and the snowdrift game. Ecol Lett 8:748 –766.
25. Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359:826 – 829.
26. Ohtsuki H, Hauert C, Lieberman E, Nowak MA (2006) A simple rule for the evolution ofcooperation on graphs and social networks. Nature 441:502–505.

Wakano et al.

PNAS Early Edition ! 5 of 5