Equational Forms
The most general continuous form 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 .
Read more about this topic: Replicator Equation
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