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. Equation The most general continuous form of the replicator equation is given by the differential equation:where is the proportion of type in the population, is the vector of the distribution of types in the population, is the fitness of type (which is dependent on the population), and is the average population fitness (given by the weighted average of the fitness of the types in the population). Since the elements of the population vector sum to unity by definition, the equation is defined on the n-dimensional simplex. The replicator equation assumes a uniform population distribution; that is, it does not incorporate population structure into the fitness. The fitness landscape does incorporate the population distribution of types, in contrast to other similar equations, such as the quasispecies equation. In application, populations are generally finite, making the discrete version more realistic. The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing. Note that the continuous form can be obtained from the discrete form by a limiting process. To simplify analysis, fitness is often assumed to depend linearly upon the population distribution, which allows the replicator equation to be written in the form: where the payoff matrix holds all the fitness information for the population: the expected payoff can be written as and the mean fitness of the population as a whole can be written as . It can be shown that the change in the ratio of two proportions with respect to time is:In other words, the change in the ratio is driven entirely by the difference in fitness between types. Derivation of deterministic and stochastic replicator dynamics Suppose that the number of individuals of type is and that the total number of individuals is .