Replicator Equation

The replicator equation is a differential equation used in evolutionary game theory to describe how the proportion of each strategy or type within a population changes over time, based on the rule that a type whose payoff is above the population's current average share grows in frequency while a type below the average shrinks. It was formally introduced in 1978 by the mathematicians Peter Taylor and Leo Jonker, who used it to define the closely related notion of an evolutionarily stable strategy in dynamic terms, building on the foundational static concept of evolutionary stability that the biologist John Maynard Smith and the geneticist George Price had introduced a few years earlier. The replicator equation can be shown to be mathematically equivalent, after a suitable change of variables, to the Lotka-Volterra equations originally developed to describe predator and prey population dynamics, a connection that links evolutionary game theory directly to classical mathematical ecology. It remains the standard dynamic framework for studying how cooperation, competition, and mixed populations of strategies evolve over time in models across biology, economics, and the social sciences.