Define the proportion of each type to be . Assume that the change in each type is governed by geometric Brownian motion:where is the fitness associated with type . The average fitness of the types . The Wiener processes are assumed to be uncorrelated. For , Itô's lemma then gives us:The partial derivatives are then:where is the Kronecker delta function. These relationships imply that:Each of the components in this equation may be calculated as:Then the stochastic replicator dynamics equation for each type is given by:Assuming that the terms are identically zero, the deterministic replicator dynamics equation is recovered. Analysis The analysis differs in the continuous and discrete cases: in the former, methods from differential equations are utilized, whereas in the latter the methods tend to be stochastic. Since the replicator equation is non-linear, an exact solution is difficult to obtain (even in simple versions of the continuous form) so the equation is usually analyzed in terms of stability. The replicator equation (in its continuous and discrete forms) satisfies the folk theorem of evolutionary game theory which characterizes the stability of equilibria of the equation. The solution of the equation is often given by the set of evolutionarily stable states of the population. In general nondegenerate cases, there can be at most one interior evolutionary stable state (ESS), though there can be many equilibria on the boundary of the simplex. All the faces of the simplex are forward-invariant which corresponds to the lack of innovation in the replicator equation: once a strategy becomes extinct there is no way to revive it. Phase portrait solutions for the continuous linear-fitness replicator equation have been classified in the two and three dimensional cases. Classification is more difficult in higher dimensions because the number of distinct portraits increases rapidly. Relationships to other equations The continuous replicator equation on types is equivalent to the Generalized Lotka–Volterra equation in dimensions. The transformation is made by the change of variables: where is the Lotka–Volterra variable. The continuous replicator dynamic is also equivalent to the Price equation. Discrete replicator equation When one considers an unstructured infinite population with non-overlapping generations, one should work with the discrete forms of the replicator equation. Mathematically, two simple phenomenological versions--- ---are consistent with the Darwinian tenet of natural selection or any analogous evolutionary phenomena. Here, prime stands for the next time step. However, the discrete nature of the equations puts bounds on the