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In mathematics, the replicator equation is a deterministic
monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...
non-linear and non-innovative game dynamic used in
evolutionary game theory Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John M ...
. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the
fitness function {{no footnotes, date=May 2015 A fitness function is a particular type of objective function that is used to summarise, as a single figure of merit, how close a given design solution is to achieving the set aims. Fitness functions are used in geneti ...
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 may refer to: Science * Selection (biology), also called natural selection, selection in evolution ** Sex selection, in genetics ** Mate selection, in mating ** Sexual selection in humans, in human sexuality ** Human mating strateg ...
. Unlike the quasispecies equation, the replicator equation does not incorporate
mutation In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA replication, DNA or viral repl ...
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 In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
: : \dot = x_i f_i(x) - \phi(x) \quad \phi(x) = \sum_^ where x_i is the proportion of type i in the population, x=(x_1, \ldots, x_n) is the vector of the distribution of types in the population, f_i(x) is the fitness of type i (which is dependent on the population), and \phi(x) is the average population fitness (given by the weighted average of the fitness of the n types in the population). Since the elements of the population vector x 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 In electronics, a limiter is a circuit that allows signals below a specified input power or level to pass unaffected while attenuating (lowering) the peaks of stronger signals that exceed this threshold. Limiting is a type of dynamic range comp ...
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: :\dot=x_i\left(\left(Ax\right)_i-x^TAx\right) where the
payoff matrix In game theory, normal form is a description of a ''game''. Unlike extensive form, normal-form representations are not graphical ''per se'', but rather represent the game by way of a matrix. While this approach can be of greater use in identifyin ...
A holds all the fitness information for the population: the expected payoff can be written as \left(Ax\right)_i and the mean fitness of the population as a whole can be written as x^TAx. It can be shown that the change in the ratio of two proportions x_/x_ with respect to time is:\left( \right) = \left f_(x) - f_(x) \right/math>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 i is N_ and that the total number of individuals is N. Define the proportion of each type to be x_ = N_/N. Assume that the change in each type is governed by
geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It i ...
:dN_ = f_N_dt + \sigma_N_dW_where f_ is the fitness associated with type i. The average fitness of the types \phi = x^f. The
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
es are assumed to be uncorrelated. For x_(N_,...,N_), Itô's lemma then gives us:\begin dx_(N_,...,N_) &= dN_ + dN_dN_ \\ &= dN_ + (dN_)^ \endThe partial derivatives are then:\begin &= \delta_ - \\ &= -\delta_ + \endwhere \delta_ is the Kronecker delta function. These relationships imply that:dx_ = -x_\sum_ - + x_\sum_Each of the components in this equation may be calculated as:\begin &= f_x_dt + \sigma_x_dW_ \\ -x_\sum_ &= -x_\left(\phi dt + \sum_\sigma_x_dW_ \right) \\ - &= -\sigma_^x_^dt \\ x_\sum_ &= x_\left( \sum_\sigma_^x_^\right )dt \endThen the stochastic replicator dynamics equation for each type is given by:dx_ = x_\left(f_ -\phi-\sigma_^x_ + \sum_\sigma_^x_^ \right)dt + x_\left(\sigma_dW_-\sum_\sigma_x_dW_ \right )Assuming that the \sigma_ 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 state A population can be described as being in an evolutionarily stable state when that population's "genetic composition is restored by selection after a disturbance, provided the disturbance is not too large" (Maynard Smith, 1982).Maynard Smith, J.. (1 ...
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 n types is equivalent to the Generalized Lotka–Volterra equation in n-1 dimensions. The transformation is made by the change of variables: :x_i = \frac \quad i=1, \ldots,n-1 :x_n = \frac, where y_i is the Lotka–Volterra variable. The continuous replicator dynamic is also equivalent to the
Price equation In the theory of evolution and natural selection, the Price equation (also known as Price's equation or Price's theorem) describes how a trait or allele changes in frequency over time. The equation uses a covariance between a trait and fitness, ...
.


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--- : x'_ = x_i + x_i \left left(Ax\right)_i-x^TAx\right\,(\rm type~I), : x'_ = x_i\left frac\right, (\rm type~II), ---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 payoff-matrix elements. Interestingly, for the simple case of two-player-two-strategy games, the type I replicator map is capable of showing period doubling bifurcation leading to
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and it also gives a hint on how to generalize the concept of the
evolutionary stable state A population can be described as being in an evolutionarily stable state when that population's "genetic composition is restored by selection after a disturbance, provided the disturbance is not too large" (Maynard Smith, 1982).Maynard Smith, J.. (1 ...
to accommodate the periodic solutions of the map.


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: : \dot = \sum_^ - \phi(x)x_i, where the matrix Q gives the
transition probabilities A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
for the mutation of type j to type i , f_i is the fitness of the i^ and \phi is the mean fitness of the population. This equation is a simultaneous generalization of the replicator equation and the quasispecies equation, and is used in the mathematical analysis of language. The discrete version of the replicator-mutator equation may have two simple types in line with the two replicator maps written above: : x'_ = x_i + \sum_^ - \phi(x)x_i, and : x'_ = x_i + \frac, respectively. The replicator equation or the replicator-mutator equation can be extended to include the effect of delay that either corresponds to the delayed information about the population state or in realizing the effect of interaction among players. The replicator equation can also easily be generalized to asymmetric games. A recent generalization that incorporates population structure is used in evolutionary graph theory.


References


Further reading

* Cressman, R. (2003).
Evolutionary Dynamics and Extensive Form Games
' The MIT Press. * Taylor, P.D.; Jonker, L. (1978). "Evolutionary Stable Strategies and Game Dynamics". ''Mathematical Biosciences'', 40: 145-156. *Sandholm, William H. (2010). ''Population Games and Evolutionary Dynamics''. Economic Learning and Social Evolution, The MIT Press.


External links

* Izquierdo, S.S. & Izquierdo L.R. (2011). Online software
The Replicator-Mutator Dynamics with Three Strategies
{{game theory Game theory Differential equations Evolutionary game theory Evolutionary dynamics Mathematical and theoretical biology Mathematical economics