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# On arbitrarily long periodic orbits of evolutionary games on graphs

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• 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.

Mathematics Subject Classification: Primary: 91A22, 05C57, 91A43; Secondary: 37N40.

 Citation: • • Figure 1.  Regions of admissible parameters $\mathcal{P}$ with normalization $a = 1, d = 0$

Figure 2.  Example of the graph $\mathcal{G}$ with parameters $p = 5$, $q = 3$, $r = 2$ and $s = 4$. Cooperators are depicted by full circles

Figure 3.  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 $(a, b, c, d) = (1, 0.45, 1.24, 0)$

Figure 4.  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 $p = 10$

Figure 5.  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 $r = 3, q = 6$

Figure 10.  The example from Section 4.2

Figure 11.  The example from Section 4.2

Figure 12.  The example from Section 4.2

Figure 13.  The example from Section 4.2

Figure 14.  The example from Section 4.2

Figure 15.  The example from Section 4.2

Figure 6.  Illustration of the local situation in Lemma 4.2. Generally, nothing can be stated about the behavior of the cooperating neighbor on the left

Figure 7.  Illustration of the local situation in Lemma 4.3

Figure 8.  The function $f$ governing the shrinking and expansion of cooperation among the special vertices for $q = 8$

Figure 9.  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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