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 leading to chaos and it also gives a hint on how to generalize the concept of the evolutionary stable state to accommodate the periodic solutions of the map. Generalizations A generalization of the replicator equation which incorporates mutation is given by the replicator-mutator equation, which takes the following form in the continuous version: where the matrix gives the transition probabilities for the mutation of type to type , is the fitness of the and is the mean fitness of the population. This equation is a simultaneous generalization of the replicator equation and the quasispecies equation, and is used in the mathematical analysis of language. The discrete version of the replicator-mutator equation may have two simple types in line with the two replicator maps written above: and respectively. The replicator equation or the replicator-mutator equation can be extended to include the effect of delay that either corresponds to the delayed information about the population state or in realizing the effect of interaction among players. The replicator equation can also easily be generalized to asymmetric games. A recent generalization that incorporates population structure is used in evolutionary graph theory. References Further reading Cressman, R. (2003). Evolutionary Dynamics and Extensive Form Games The MIT Press. Taylor, P.D.; Jonker, L. (1978). "Evolutionary Stable Strategies and Game Dynamics". Mathematical Biosciences, 40: 145-156. Sandholm, William H. (2010). Population Games and Evolutionary Dynamics. Economic Learning and Social Evolution, The MIT Press. External links Izquierdo, S.S. & Izquierdo L.R. (2011). Online software: The Replicator-Mutator Dynamics with Three Strategies Game theory Differential equations Evolutionary game theory Evolutionary dynamics Mathematical and theoretical biology Mathematical economics