# American Institute of Mathematical Sciences

July  2018, 23(5): 1895-1915. doi: 10.3934/dcdsb.2018187

## 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

* Corresponding author

Received  April 2017 Revised  August 2017 Published  July 2018 Early access  May 2018

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.

Citation: Jeremias Epperlein, Vladimír Švígler. On arbitrarily long periodic orbits of evolutionary games on graphs. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1895-1915. doi: 10.3934/dcdsb.2018187
##### References:
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##### 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.
Regions of admissible parameters $\mathcal{P}$ with normalization $a = 1, d = 0$
Example of the graph $\mathcal{G}$ with parameters $p = 5$, $q = 3$, $r = 2$ and $s = 4$. Cooperators are depicted by full circles
Example of the graph $\mathcal{G}$ with parameters $p = 5, o = 4, s = 6, r = 1, q = 2$ with strategy vector $X(4)$. Cooperators are depicted by full circles. Note, that this graph exhibits periodic behavior as described in Section 3.2 for $(a, b, c, d) = (1, 0.45, 1.24, 0)$
Regions of parameters $o, q$ satisfying the inequalities (15)and (16). The regions are depicted for $(a, b, c, d) = (1, -0.45, 1.35, 0)$ and $p = 10$
Example of the graph constructed in the proof of Theorem 4.1 with an initial condition. The cooperators are depicted by filled black circles, defectors by white ones. The parameters are $r = 3, q = 6$
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
Illustration of the local situation in Lemma 4.2. Generally, nothing can be stated about the behavior of the cooperating neighbor on the left
Illustration of the local situation in Lemma 4.3
The function $f$ governing the shrinking and expansion of cooperation among the special vertices for $q = 8$
Development of the number of cooperators for the evolutionary game on the tree $\mathcal{G}$ in Section 4 with $r = 3$ and game theoretic parameters $(a, b, c, d) = (1, 0.7, 2, 0)$. On the left the tree has depth $q = 6$, on the right $q = 9$
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