Replicator equation

In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. Unlike the quasispecies equation, the replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies.

The most general continuous form is given by the differential equation

where ${\displaystyle x_{i}}$ is the proportion of type ${\displaystyle i}$ in the population, ${\displaystyle x=(x_{1},\ldots ,x_{n})}$ is the vector of the distribution of types in the population, ${\displaystyle f_{i}(x)}$ is the fitness of type ${\displaystyle i}$ (which is dependent on the population), and ${\displaystyle \phi (x)}$ is the average population fitness (given by the weighted average of the fitness of the ${\displaystyle n}$ types in the population). Since the elements of the population vector ${\displaystyle x}$ sum to unity by definition, the equation is defined on the n-dimensional simplex.

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