Evolutionarily stable state
Encyclopedia
"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. Such a population can be genetically monomorphic or polymorphic." --Maynard Smith (1982).
Contrast this with the definition of an evolutionarily stable strategy
, which is a game theory
concept.
"An ESS or evolutionarily stable strategy is a strategy such that, if all the members of a population adopt it, no mutant strategy can invade." --Maynard Smith (1982).
, given in its linear payoff form by
The state
is said to be evolutionarily stable if for all 
in some neighborhood of 
This is similar but distinct from the concept of an evolutionarily stable strategy
.
The concept of evolutionary stability is equivalent to the concept of strong stability for normal form game
s, due to Cressman.
Contrast this with the definition of an evolutionarily stable strategy
Evolutionarily stable strategy
In game theory and behavioural ecology, an evolutionarily stable strategy , which is sometimes also called an evolutionary stable strategy, is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium...
, which is a game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
concept.
"An ESS or evolutionarily stable strategy is a strategy such that, if all the members of a population adopt it, no mutant strategy can invade." --Maynard Smith (1982).
Mathematical Formulation
Evolutionarily stable states are frequently used to identify solutions to the replicator equationReplicator equation
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...
, given in its linear payoff form by
The state
is said to be evolutionarily stable if for all in some neighborhood of This is similar but distinct from the concept of an evolutionarily stable strategy
Evolutionarily stable strategy
In game theory and behavioural ecology, an evolutionarily stable strategy , which is sometimes also called an evolutionary stable strategy, is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium...
.
Properties
It can be shown that for the replicator equation if a state is evolutionarily stable then it is an asymptotically stable rest point, so the evolutionarily stable states are often taken as solutions of the replicator equation. The converse is true if the game matrix is symmetric.The concept of evolutionary stability is equivalent to the concept of strong stability for normal form game
Normal form game
In game theory, normal form is a way of describing 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 identifying strictly dominated strategies and Nash equilibria, some...
s, due to Cressman.