Replicator equations

Replicator equations are non-linear differential equations used to describe the dynamics of a population in which successful individuals reproduce faster than less successful individuals. They are among the basic equations of evolutionary game theory and are used in theoretical biology as well as in evolutionary psychology to B. to explain behavior patterns in animals or humans as a result of selection .

The concept of replicator equations was introduced in 1978 to model the dynamics that lead to an evolutionarily stable state. These equilibrium states of the replicator equations are similar, but not identical, to the concept of evolutionarily stable strategy (ESS) . The underlying model is a population of infinitely many individuals who are divided into different types. How quickly individuals of a type reproduce depends on the fitness of the type. Most of the time this fitness is not constant, but results from the interaction with other individuals. It is assumed that every individual interacts with every other individual (mean-field approximation). This approximation can be justified by the fact that interaction and reproduction usually take place on different time scales, that is, that each individual interacts with many others before it reproduces.

Equations

In a relatively general form, continuous replicator equations are of the form

{\ displaystyle {\ dot {x_ {i}}} = x_ {i} [f_ {i} (x) - \ phi (x)], \ quad \ phi (x) = \ sum _ {i = 1} ^ {n} {x_ {i} f_ {i} (x)}}

with the proportion of a replicator species of the type in the total population, distribution vector, fitness of replicator type and average fitness. ${\ displaystyle x_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle x = (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle f_ {i} (x)}$ ${\ displaystyle i}$ ${\ displaystyle \ phi (x)}$

An assumption that is often made to simplify the model is that fitness is linearly dependent on the composition of the replicator population:

{\ displaystyle {\ dot {x_ {i}}} = x_ {i} \ left (\ left (Ax \ right) _ {i} -x ^ {T} Ax \ right),}

The payoff matrix contains the fitness information for the population: the payoff to be expected can be written as and the mean fitness of the total population as . ${\ displaystyle A}$ ${\ displaystyle \ left (Ax \ right) _ {i}}$ ${\ displaystyle x ^ {T} Ax}$

Generalizations

A generalization of the replicator equations that takes mutations into account are replicator-mutator equations:

{\ displaystyle {\ dot {x_ {i}}} = \ sum _ {j = 1} ^ {n} {x_ {j} f_ {j} (x) Q_ {ji}} - \ phi (x) x_ {i},}

Here, the matrix are the mutational transition probabilities of Replikatortypen according to. ${\ displaystyle Q}$ ${\ displaystyle j}$ ${\ displaystyle i}$

literature

IM Bomze: Lotka-Volterra equations and replicator dynamics: A two dimensional classification. In: Biol. Cybern. Volume 48, 1983, pp. 201-211.
IM Bomze: Lotka-Volterra equations and replicator dynamics: New issues in classification. In: Biol. Cybern. Volume 72, 1995, pp. 447-453.
R. Cressman: Evolutionary Dynamics and Extensive Form Games The MIT Press, 2003.
J. Hofbauer, K. Sigmund : Evolutionary game dynamics. In: Bull. Am. Math. Soc. Volume 40, 2003, pp. 479-519.
I. Hussein: An Individual-Based Evolutionary Dynamics Model for Networked Social Behaviors. In: Proceedings of the American Control Conference, St. Louis, MO. To appear, 2009.
E. Lieberman , C. Hauert, M. Nowak : Evolutionary dynamics on graphs. In: Nature. Volume 433, No. 7023, 2005, pp. 312-316.
M. Nowak, K. Page: Unifying Evolutionary Dynamics. In: Journal of Theoretical Biology. Volume 219, 2002, pp. 93-98.
M. Nowak: Evolutionary Dynamics: Exploring the Equations of Life Belknap Press, 2006.

proof

↑ P. Taylor, L. Jonker: Evolutionary stable strategies and game dynamics . In: Mathematical Biosciences . tape 40 , no. 1-2 , 1978, pp. 145-156 .

[1] P. Taylor, L. Jonker: Evolutionary stable strategies and game dynamics . In: Mathematical Biosciences . tape 40 , no. 1-2 , 1978, pp. 145-156 .