Replicator equation
Encyclopedia
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 landscape
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.
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 
.
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 state
s 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.
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
(see Page & Nowak's (2002) paper Unifying Evolutionary Dynamics).
where the matrix
gives the transition probabilities for the mutation of type 
to type 
. This equation is a simultaneous generalization of the replicator equation and the quasispecies equation, and is used in the mathematical analysis of language.
The replicator equation can easily be generalized to asymmetric games. A recent generalization that incorporates population structure is used in evolutionary graph theory
.
Evolutionary game theory
Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...
. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness landscape
Fitness landscape
In evolutionary biology, fitness landscapes or adaptive landscapes are used to visualize the relationship between genotypes and reproductive success. It is assumed that every genotype has a well-defined replication rate . This fitness is the "height" of the landscape...
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
Selection
In the context of evolution, certain traits or alleles of genes segregating within a population may be subject to selection. Under selection, individuals with advantageous or "adaptive" traits tend to be more successful than their peers reproductively—meaning they contribute more offspring to the...
. Unlike the quasispecies equation, the replicator equation does not incorporate mutation
Mutation
In molecular biology and genetics, mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. They can be defined as sudden and spontaneous changes in the cell. Mutations are caused by radiation, viruses, transposons and mutagenic...
and so is not able to innovate new types or pure strategies.
Equational forms
The most general continuous form is given by the differential equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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 simplexSimplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
.
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 .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 theoremFolk theorem (game theory)
In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players' minimax conditions are satisfied. The minimax condition states that a player will minimize the maximum possible loss which he...
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 state
Evolutionarily stable state
"A population is said to be in an evolutionarily stable state if its genetic composition is restored by selection after a disturbance, provided the disturbance is not too large...
s 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 variableswhere
is the Lotka–Volterra variable.The continuous replicator dynamic is also equivalent to the Price equation
Price equation
The Price equation is a covariance equation which is a mathematical description of evolution and natural selection. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton's work on kin selection...
(see Page & Nowak's (2002) paper Unifying Evolutionary Dynamics).
Generalizations
A generalization of the replicator equation which incorporates mutation is given by the replicator-mutator equation, which takes the following form in the continuous version:where the matrix
gives the transition probabilities for the mutation of type to type . This equation is a simultaneous generalization of the replicator equation and the quasispecies equation, and is used in the mathematical analysis of language.The replicator equation can easily be generalized to asymmetric games. A recent generalization that incorporates population structure is used in evolutionary graph theory
Evolutionary graph theory
Evolutionary graph theory is an area of research lying at the intersection of graph theory, probability theory, and mathematical biology. Evolutionary graph theory is an approach to studying how topology affects evolution of a population...
.