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The replicator dynamics (RD) were introduced by Taylor and Jonker (1978) to model evolution of behaviour in intraspecific conflicts under random pairwise interaction in a large, ideally infinite population. It is used for modeling many biological processes including the evolution of animal behaviour, selection in population genetics, and prebiotic evolution.

This infographic presents the case of n = 3 behaviour patterns yielding a system of cubic differential equations. The complete list of possible phase portraits under the replicator dynamics is presented bellow, containing 49 qualitatively different cases up to flow reversal: 19 robust ones and 30 non-robust. [proof in Bomze (1983, 1985)].

Given this classification, we can see how the game theoretic solution concept of evolutionarily stable sets (ES sets) [introduced by Thomas (1985)] fairs in capturing the asymptotic behaviour under RD of this class of games.

Citations
Maynard Smith J (1974) The theory of games and the evolution of animal conflict. J Theor Biol 47:209-221
Taylor P, and Jonker L (1978) Evolutionarily stable strategies and game dynamics. Math Biosci 40:145 156
Thomas B (1985) On evolutionarily stable sets. J Math Biol 22:105-115
Zeeman EC (1981) Dynamics of the evolution of animal conflicts. J Theor Biol 89:249-270.

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