We found that a network-organized metapopulation of cooperators, defectors, and destructive agents playing the public goods game with mutations can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics switches between these two steady states through a first order transition. Depending on the parameters determining the dynamical and topological properties, chimera states with different numbers of coherent and incoherent clusters are observed. Our results present the first systematic study of chimera states and their characterization in the context of evolutionary game theory. This provides a valuable insight into the details of their occurrence, extending the relevance of such states to natural and social systems.

1.
J.
Hofbauer
and
K.
Sigmund
,
Evolutionary Games and Population Dynamics
(
Cambridge University Press
,
Cambridge
,
1998
).
2.
K.
Sigmund
,
The Calculus of Selfishness
(
Princeton University Press
,
Princeton
,
2010
).
4.
C.
Hauert
,
S.
De Monte
,
J.
Hofbauer
, and
K.
Sigmund
,
Science
296
(
5570
),
1129
(
2002
).
5.
C.
Hauert
,
S. D.
Monte
,
J.
Hofbauer
, and
K.
Sigmund
,
J. Theor. Biol.
218
,
187
(
2002
).
6.
A.
Arenas
,
J.
Camacho
,
J. A.
Cuesta
, and
R. J.
Requejo
,
J. Theor. Biol.
279
(
1
),
113
(
2011
).
7.
R. J.
Requejo
,
J.
Camacho
,
J. A.
Cuesta
, and
A.
Arenas
,
Phys. Rev. E
86
,
026105
(
2012
).
8.
B.
Kerr
,
M. A.
Riley
,
M. W.
Feldman
, and
B. J.
Bohannan
,
Nature
418
,
171
(
2002
).
9.
D.
Semmann
,
H.-J.
Krambeck
, and
M.
Milinski
,
Nature
425
,
390
(
2003
).
11.
D. F. P.
Toupo
and
S. H.
Strogatz
,
Phys. Rev. E
91
,
052907
(
2015
).
12.
G. T.
Vickers
,
V. C. L.
Huston
, and
C. J.
Budd
,
J. Math. Biol.
31
,
411
430
(
1993
).
13.
J. Y.
Wakano
,
M. A.
Nowak
, and
C.
Hauert
,
Proc. Natl. Acad. Sci. U.S.A.
106
(
19
),
7910
(
2009
).
14.
J. Y.
Wakano
and
C.
Hauert
,
J. Theor. Biol.
268
(
1
),
30
(
2011
).
15.
G.
Szabó
and
I.
Borsos
,
Phys. Rep.
624
,
1
60
(
2016
).
16.
A.
Arenas
,
A.
Díaz-Guilera
,
J.
Kurths
,
Y.
Moreno
, and
C.
Zhou
,
Phys. Rep.
469
,
93
153
(
2008
).
17.
M. J.
Pannagio
and
D.
Abrams
,
Nonlinearity
28
,
R67
(
2015
).
18.
Y.
Kuramoto
and
D.
Battogtokh
,
Nonlinear Phenomena in Complex Systems
5
,
380
(
2002
).
19.
D. M.
Abrams
and
S. H.
Strogatz
,
Phys. Rev. Lett.
93
,
174102
(
2004
).
20.
L.
Schmidt
,
K.
Schönleber
,
K.
Krischer
, and
V.
García-Morales
,
Chaos
24
,
013102
(
2014
).
21.
G. C.
Sethia
and
A.
Sen
,
Phys. Rev. Lett.
112
,
144101
(
2014
).
22.
A.
Yeldesbay
,
A.
Pikovsky
, and
M.
Rosenblum
,
Phys. Rev. Lett.
112
,
144103
(
2014
).
23.
F.
Böhm
,
A.
Zakharova
,
K.
Lüdge
, and
E.
Schöll
,
Phys. Rev. E
91
,
040901
(
2015
).
24.
C. R.
Laing
,
Phys. Rev. E
92
,
050904
(
2015
).
25.
J.
Hizanidis
,
N.
Lazarides
, and
G. P.
Tsironis
,
Phys. Rev. E
94
,
032219
(
2016
).
26.
M. G.
Clerc
,
S.
Coulibaly
,
M. A.
Ferré
,
M. A.
García-Nũstes
, and
R. G.
Rojas
,
Phys. Rev. E
93
,
052204
(
2016
).
27.
J.
Hizanidis
,
N. E.
Kouvaris
, and
C. G.
Antonopoulos
,
Cybernetics and Physics
4
,
17
(
2015
).
28.
J.
Hizanidis
,
N. E.
Kouvaris
,
G.
Zamora-López
,
A.
Díaz-Guilera
, and
C. G.
Antonopoulos
,
Sci. Rep.
6
,
19845
(
2016
).
29.
M. S.
Santos
,
J. D.
Szezech
, Jr.
,
F. S.
Borges
,
K. C.
Iarosz
,
I. L.
Caldas
,
A. M.
Batista
,
R. L.
Viana
, and
J.
Kurths
, e-print arXiv:1609.01534.
30.
C. R.
Laing
and
C. C.
Chow
,
Neural Comput.
13
,
1473
(
2001
).
31.
H.
Sakaguchi
,
Phys. Rev. E
73
,
031907
(
2006
).
32.
N. C.
Rattenborg
,
C. J.
Amlaner
, and
S. L.
Lima
,
Neurosci. Biobehav. Rev.
24
,
817
(
2000
).
33.
R. G.
Andrzejak
,
C.
Rummel
,
F.
Mormann
, and
K.
Schindler
,
Sci. Rep.
6
,
23000
(
2016
).
34.
M.
Wolfrum
and
O. E.
Omel'chenko
,
Phys. Rev. E
84
,
015201
(
2011
).
35.
J.
Sieber
,
O. E.
Omel'chenko
, and
M.
Wolfrum
,
Phys. Rev. Lett.
112
,
054102
(
2014
).
36.
C.
Bick
and
E. A.
Martens
,
New J. Phys.
17
,
033030
(
2015
).
37.
T.
Isele
,
J.
Hizanidis
,
A.
Provata
, and
P.
Hövel
,
Phys. Rev. E
93
,
022217
(
2016
).
38.
I.
Omelchenko
,
O. E.
Omel'chenko
,
A.
Zakharova
,
M.
Wolfrum
, and
E.
Schöll
,
Phys. Rev. Lett.
116
,
114101
(
2016
).
39.
A. M.
Hagerstrom
,
T. E.
Murphy
,
R.
Roy
,
P.
Hovel
,
I.
Omelchenko
, and
E.
Scholl
,
Nat. Phys.
8
(
9
),
658
(
2012
).
40.
E. A.
Martens
,
S.
Thutupalli
,
A.
Fourrière
, and
O.
Hallatschek
,
Proc. Natl. Acad. Sci. U.S.A.
110
(
26
),
10563
(
2013
).
41.
M. R.
Tinsley
,
S.
Nkomo
, and
K.
Showalter
,
Nat. Phys.
8
(
9
),
662
(
2012
).
42.
S.
Nkomo
,
M. R.
Tinsley
, and
K.
Showalter
,
Phys. Rev. Lett.
110
,
244102
(
2013
).
43.
M.
Wickramasinghe
and
I. Z.
Kiss
,
PLoS One
8
,
e80586
(
2013
).
44.
R. J.
Requejo
and
A.
Díaz-Guilera
,
Phys. Rev. E
94
,
022301
(
2016
).
45.
P. D.
Taylor
and
L. B.
Jonker
,
Math. Biosci.
40
,
145
(
1978
).
46.
J.
Hofbauer
,
P.
Schuster
, and
K.
Sigmund
,
J. Theor. Biol.
81
,
609
(
1979
).
47.
I.
Omelchenko
,
O. E.
Omel'chenko
,
P.
Hövel
, and
E.
Schöll
,
Phys. Rev. Lett.
110
,
224101
(
2013
).
48.
J.
Hizanidis
,
E.
Panagakou
,
I.
Omelchenko
,
E.
Schöll
,
P.
Hövel
, and
A.
Provata
,
Phys. Rev. E
92
,
012915
(
2015
).
49.
F. P.
Kemeth
,
S. W.
Haugland
,
L.
Schmidt
,
I. G.
Kevrekidis
, and
K.
Krischer
,
Chaos
26
,
094815
(
2016
).
50.
Y.
Maistrenko
,
A.
Vasylenko
,
O.
Sudakov
,
R.
Levchenko
, and
V. L.
Maistrenko
,
Int. J. Bifurcation Chaos
24
,
1440014
(
2014
).
51.
A.
Vüllings
,
J.
Hizanidis
,
I.
Omelchenko
, and
P.
Hövel
,
New J. Phys.
16
,
123039
(
2014
).
52.
G. C.
Sethia
,
A.
Sen
, and
F. M.
Atay
,
Phys. Rev. Lett.
100
,
144102
(
2008
).
53.
S. S.
Wiltermuth
and
C.
Heath
,
Psychol. Sci.
20
,
1
5
(
2009
).

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