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On arbitrarily long periodic orbits of evolutionary games on graphs
1. | Center for Dynamics & Institute for Analysis, Dept. of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany |
2. | Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 30614, Pilsen, Czech Republic |
A periodic behavior is a well observed phenomena in biological and economical systems. We show that evolutionary games on graphs with imitation dynamics can display periodic behavior for an arbitrary choice of game theoretical parameters describing social-dilemma games. We construct graphs and corresponding initial conditions whose trajectories are periodic with an arbitrary minimal period length. We also examine a periodic behavior of evolutionary games on graphs with the underlying graph being an acyclic (tree) graph. Astonishingly, even this acyclic structure allows for arbitrary long periodic behavior.
References:
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G. Abramson and M. Kuperman, Social games in a social network, Physical Review E, 63 (2001), 030901.
doi: 10.1103/PhysRevE.63.030901. |
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B. Allen and M. A. Nowak,
Games on graphs, EMS Surv. Math. Sci., 1 (2014), 113-151.
doi: 10.4171/EMSS/3. |
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Evolutionary prisoner's dilemma in random graphs, Physica D: Nonlinear Phenomena, 208 (2005), 257-265.
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J. Epperlein, S. Siegmund and P. Stehlík,
Evolutionary games on graphs and discrete dynamical systems, Journal of Difference Equations and Applications, 21 (2015), 72-95.
doi: 10.1080/10236198.2014.988618. |
[7] |
J. Epperlein, S. Siegmund, P. Stehlík and V. Švígler,
Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics, Discrete and Continuous Dynamical Systems- Series B, 21 (2016), 803-813.
doi: 10.3934/dcdsb.2016.21.803. |
[8] |
J. Epperlein and V. Švígler, Periodic orbits of an evolutionary game on a tree, https://figshare.com/articles/6-periodic_orbit_of_an_evolutionary_game_on_a_tree/5110981, June 2017, DOI: 10.6084/m9.figshare.5110981. |
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J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9781139173179. |
[10] |
B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoners' dilemma game in small-world networks, Physical Review E, 66 (2002), 021907.
doi: 10.1103/PhysRevE.66.021907. |
[11] |
C. Marr and M.-T. Hütt,
Outer-totalistic cellular automata on graphs, Physics Letters A, 373 (2009), 546-549.
doi: 10.1016/j.physleta.2008.12.013. |
[12] |
N. Masuda and K. Aihara,
Spatial prisoner's dilemma optimally played in small-world networks, Physics Letters A, 313 (2003), 55-61.
doi: 10.1016/S0375-9601(03)00693-5. |
[13] |
M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Belknap Press of Harvard University Press, 2006. |
[14] |
M. A. Nowak and R. M. May,
Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829.
doi: 10.1038/359826a0. |
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M. Tomochi,
Defectors' niches: Prisoner's dilemma game on disordered networks, Social Networks, 26 (2004), 309-321.
doi: 10.1016/j.socnet.2004.08.003. |
show all references
References:
[1] |
G. Abramson and M. Kuperman, Social games in a social network, Physical Review E, 63 (2001), 030901.
doi: 10.1103/PhysRevE.63.030901. |
[2] |
B. Allen and M. A. Nowak,
Games on graphs, EMS Surv. Math. Sci., 1 (2014), 113-151.
doi: 10.4171/EMSS/3. |
[3] |
M. Broom and J. Rychtář, Game-Theoretical Models in Biology 1st edition, CRC Press, Taylor & Francis Group, 2013. |
[4] |
J. T. Cox, R. Durrett and E. A. Perkins, Voter Model Perturbations and Reaction Diffusion Equations, vol. 349 of Astérisque, Société Mathématique de France, 2013. |
[5] |
O. Durán and R. Mulet,
Evolutionary prisoner's dilemma in random graphs, Physica D: Nonlinear Phenomena, 208 (2005), 257-265.
doi: 10.1016/j.physd.2005.07.005. |
[6] |
J. Epperlein, S. Siegmund and P. Stehlík,
Evolutionary games on graphs and discrete dynamical systems, Journal of Difference Equations and Applications, 21 (2015), 72-95.
doi: 10.1080/10236198.2014.988618. |
[7] |
J. Epperlein, S. Siegmund, P. Stehlík and V. Švígler,
Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics, Discrete and Continuous Dynamical Systems- Series B, 21 (2016), 803-813.
doi: 10.3934/dcdsb.2016.21.803. |
[8] |
J. Epperlein and V. Švígler, Periodic orbits of an evolutionary game on a tree, https://figshare.com/articles/6-periodic_orbit_of_an_evolutionary_game_on_a_tree/5110981, June 2017, DOI: 10.6084/m9.figshare.5110981. |
[9] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9781139173179. |
[10] |
B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoners' dilemma game in small-world networks, Physical Review E, 66 (2002), 021907.
doi: 10.1103/PhysRevE.66.021907. |
[11] |
C. Marr and M.-T. Hütt,
Outer-totalistic cellular automata on graphs, Physics Letters A, 373 (2009), 546-549.
doi: 10.1016/j.physleta.2008.12.013. |
[12] |
N. Masuda and K. Aihara,
Spatial prisoner's dilemma optimally played in small-world networks, Physics Letters A, 313 (2003), 55-61.
doi: 10.1016/S0375-9601(03)00693-5. |
[13] |
M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Belknap Press of Harvard University Press, 2006. |
[14] |
M. A. Nowak and R. M. May,
Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829.
doi: 10.1038/359826a0. |
[15] |
M. Tomochi,
Defectors' niches: Prisoner's dilemma game on disordered networks, Social Networks, 26 (2004), 309-321.
doi: 10.1016/j.socnet.2004.08.003. |














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