Topological chaos of the spatial prisoner׳s dilemma game on regular networks

• Dynamics of frequency of cooperation is sensitive to the initial conditions.
• Frequency of cooperation could not be adopted by simply averaging.
• Spatial prisoner׳s dilemma game is chaos.
• Procedure is applicable to other deterministic spatial models therein.

The spatial version of evolutionary prisoner׳s dilemma on infinitely large regular lattice with purely deterministic strategies and no memories among players is investigated in this paper. Based on the statistical inferences, it is pertinent to confirm that the frequency of cooperation for characterizing its macroscopic behaviors is very sensitive to the initial conditions, which is the most practically significant property of chaos. Its intrinsic complexity is then justified on firm ground from the theory of symbolic dynamics; that is, this game is topologically mixing and possesses positive topological entropy on its subsystems. It is demonstrated therefore that its frequency of cooperation could not be adopted by simply averaging over several steps after the game reaches the equilibrium state. Furthermore, the chaotically changing spatial patterns via empirical observations can be defined and justified in view of symbolic dynamics. It is worth mentioning that the procedure proposed in this work is also applicable to other deterministic spatial evolutionary games therein.