# Replicator Dynamics

# Replicator Dynamics

The replicator dynamics are part of evolutionary game theory and are especially prominent in models of cultural evolution. Evolutionary game theory uses principles of interactive behavior to explain the emergence of behavioral regularities in organisms forming a population. The results of organisms’ interactions affect their fitness, measured as their ability to reproduce. If one organism is fitter than another, then it is more likely to reproduce than the other. An organism’s offspring inherit its traits. The offspring may differ from the parent in fitness, however, because the offspring’s fitness depends on their success in interactions with their contemporaries. As the population changes, the traits that confer fitness may change, too.

The replicator dynamics explain changes in fitness that arise from changes in a population’s composition. Applications divide a population according to organisms’ behavioral traits, in particular strategies for interacting with others. For instance, in a dispute over food, one strategy may be to fight. Another strategy may be to retreat. A strategy’s consequences are assessed in terms of fitness, or average number of offspring.

A mathematical equation characterizes the replicator dynamics. It applies to a population given simplifying assumptions. They specify four conditions: (1) The population is infinite. Assuming an infinite population makes the relative frequency of an outcome match its probability. (2) An individual in the population has the same probability of interacting with any other member of the population. Pairs of interacting individuals form as if individuals were matched at random. (3) Strategies breed true. That is, if an organism has offspring, its offspring adopt the same strategy it follows. An organism’s fitness equals the average number of offspring following its strategy. (4) Reproduction is asexual. An organism has a single parent and so inherits the parent’s strategy. This condition puts aside the possibility that an organism’s strategy differs from a parent’s strategy because it has two parents with differing strategies. The proportion of individuals following a strategy changes only if some organisms with that strategy are more or less fit than their parents and so reproduce more or less frequently than their parents did.

Brian Skyrms’s (1996, pp. 51-53) notation is used here to present the replicator equation. First, assume that a population evolves in steps from one generation to the next. Using the proportion of individuals following a strategy in one generation, and the strategy’s consequences for their fitness, one may compute the proportion of individuals following the strategy in the next generation. Let *U* (*A* ) be the average fitness of a strategy *A*, and let *U* be the average fitness of all strategies. Then the proportion of the population using *A* in the next generation equals the proportion of the population using *A* in the current generation times the ratio *U* (*A* )/ *U*. That is, if *p* (*A* ) is the proportion using strategy *A* in the current generation and *p'* (*A* ) is the proportion using *A* in the next generation, then *p'* (*A* ) = *p* (*A* ) *U* (*A* )/ *U*. If *A* has greater than average fitness, its proportion increases. A little algebra yields the following equation specifying the change from one generation to the next in the proportion of the population using strategy *A*.

*p'* (*A* ) – *p* (*A* ) = *p* (*A* )[ *U* (*A* ) – *U* ]/ *U*

Next, suppose that evolution is continuous with respect to time. Then the population evolves according to this differential equation.

*dp* (*A* )/ *dt* = *p* (*A* )[ *U* (*A* ) – *U* ]/ *U*

The equation gives the rate of change in the proportion of the population using strategy *A*. Given that the average fitness of the population is positive, the following simpler differential equation describes the structural features of the population’s evolution.

*dp* (*A* )/ *dt* = *p* (*A* )[ *U* (*A* ) – *U* ]

This equation characterizes the replicator dynamics. Peter Taylor and Leo Jonker (1978) were the first theorists to formulate the equation (using different notation). Peter Schuster and Karl Sigmund (1983) called it the replicator equation and the pattern of change it describes the replicator dynamics.

Social scientists use the replicator dynamics to construct models of human behavior, for example, models of the emergence of cultural norms. When applying the dynamics to cultural evolution, imitation may replace reproduction as the mechanism responsible for replication of strategies. Because human populations do not satisfy exactly the idealizations of the replicator dynamics, the models the dynamics yield are simplifications. However, in some cases the dynamics may approximate the course of evolution.

Applications of the replicator dynamics look for the emergences of stable behavior. Consider, for example, food sharing in a hunter-gatherer society. This strategy may enhance fitness, propagate within a population, and drive out rival strategies. It may be evolutionarily stable in the sense that once a population adopts it, even if a few organisms with different strategies arrive, those new strategies do not propagate within the population. The stable strategy resists invasion by mutants. A strategy that is evolutionarily stable with respect to the replicator dynamics corresponds to a stable Nash equilibrium in a game representing individuals’ interactions.

**SEE ALSO** *Game Theory; Nash Equilibrium; Strategy*

## BIBLIOGRAPHY

Hofbauer, Joseph, and Karl Sigmund. 1998. *Evolutionary Games and Population Dynamics.* Cambridge, U.K.: Cambridge University Press.

Schuster, Peter, and Karl Sigmund. 1983. Replicator Dynamics. *Journal of Theoretical Biology* 100: 533-538.

Skyrms, Brian. 1996. *Evolution of the Social Contract*. Cambridge, U.K.: Cambridge University Press.

Taylor, Peter, and Leo Jonker. 1978. Evolutionarily Stable Strategies and Game Dynamics. *Mathematical Biosciences* 40: 145-156.

*Paul Weirich*

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