with the discrete forms of the replicator equation. Mathematically, two simple phenomenological versions---

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---are consistent with the Darwinian tenet of natural selection or any analogous evolutionary phenomena. Here, prime stands for the next time step. However, the discrete nature of the equations puts bounds on the payoff-matrix elements. Interestingly, for the simple case of two-player-two-strategy games, the type I replicator map is capable of showing period doubling bifurcation

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