Evolutionarily stable state

While related to the concept of an evolutionarily stable strategy (ESS), evolutionarily stable states are not identical and the two terms cannot be used interchangeably.

An ESS is a strategy that, if adopted by all individuals within a population, cannot be invaded by alternative or mutant strategies. This strategy becomes fixed in the population because alternatives provide no fitness benefit that would be selected for. In comparison, an evolutionarily stable state describes a population that returns as a whole to its previous composition even after being disturbed. In short: the ESS refers to the strategy itself, uninterrupted and supported through natural selection, while the evolutionarily stable state refers more broadly to a population-wide balance of one or more strategies that may be subjected to temporary change.Thomas, B. (1984). Evolutionary stability: States and strategies. Theoretical Population Biology, 26(1), 49-67. https://doi.org/10.1016/0040-5809(84)90023-6

The term ESS was first used by John Maynard Smith in an essay from the 1972 book On Evolution.Maynard Smith, J. (1972). Game Theory and the Evolution of Fighting. On Evolution. Edinburgh University Press. {{ISBN|0-85224-223-9}}. Maynard Smith developed the ESS drawing in part from game theory and Hamilton's work on the evolution of sex ratio.Maynard Smith, J., Price, G. R. (1973). The logic of animal conflict. Nature 246 (5427), 15-18. https://doi.org/10.1038/246015a0Maynard Smith, J. (1974). The theory of games and the evolution of animal conflicts. J Theor Biol. 47(1). 209-221.https://doi.org/10.1016/0022-5193(74)90110-6 The ESS was later expanded upon in his book Evolution and the Theory of Games in 1982, which also discussed the evolutionarily stable state.

There has been variation in how the term is used and exploration of under what conditions an evolutionarily stable state might exist. In 1984, Benhard Thomas compared "discrete" models in which all individuals use only one strategy to "continuous" models in which individuals employ mixed strategies. While Maynard Smith had originally defined an ESS as being a single "uninvadable strategy," Thomas generalized this to include a set of multiple strategies employed by individuals. In other words, a collection of simultaneously present strategies could be considered uninvadable as a group. Thomas noted that evolutionary stability can exist in either model, allowing for an evolutionarily stable state to exist even when multiple strategies are used within the population.

The strategy employed by individuals (or ESS) is thought to depend on fitness: the better the strategy is at supporting fitness, the more likely the strategy is to be used. When it comes to an evolutionarily stable state, all of the strategies used within the population must have equal fitness.Taylor, P. D, Jonker, L. B. (1978). Evolutionarily stable states and Game Dynamics. Mathematical Biosciences 40, 145-156. https://doi.org/10.1016/0025-5564(78)90077-9 While the equilibrium may be disturbed by external factors, the population is considered to be in an evolutionarily stable state if it returns to the equilibrium state after the disturbance.

One of the base mathematical models for identifying an evolutionarily stable state was outlined by Taylor & Jonker in 1978. Their base equilibrium model for ES states stipulates that

A state p is called an ESS (evolutionary stable state) if for every state q ≠ p, if we let p̅ =(1-ε)p + εq (the perturbed state), then F(q|p) < F(p|p̅) for sufficiently small ε>0.

It has been suggested by Ross Cressman that criteria for evolutionary stability include strong stability, as it would describe evolution of both frequency and density (whereas Maynard Smith's model focused on frequency).Cressman, R. (1990). Strong stability and density-dependent evolutionarily stable strategies. Journal of Theoretical Biology, 145(3), 319-330. https://doi.org/10.1016/S0022-5193(05)80112-2 Cressman further demonstrated that in habitat selection games modeling only a single species, the ideal free distribution (IFD) is itself an evolutionarily stable state containing mixed strategies.Cressman, R., & Křivan, V. (2010). The ideal free distribution as an evolutionarily stable state in density‐dependent population games. Oikos, 119(8), 1231-1242. https://doi.org/10.1111/j.1600-0706.2010.17845.x

Evolutionary game theory as a whole provides a theoretical framework examining interactions of organisms in a system where individuals have repeated interactions within a population that persists on an evolutionarily relevant timescale.Cowden, C. C. (2012) [https://www.nature.com/scitable/knowledge/library/game-theory-evolutionary-stable-strategies-and-the-25953132/#:~:text=Glossary,different%20strategy%20through%20natural%20selection. Game Theory, Evolutionary Stable Strategies and the Evolution of Biological Interactions]. Nature Education Knowledge 3(10):6. This framework can be used to better understand the evolution of interaction strategies and stable states, though many different specific models have been used under this framework. The Nash Equilibrium (NE) and folk theorem are closely related to the evolutionarily stable state. There are various potential refinements proposed to account for different theory games and behavioral models.Li, J., Kendall, G., and John, R. (2015). [https://ieeexplore.ieee.org/document/7296643 Computing Nash Equilibria and Evolutionarily Stable States of Evolutionary Games.] IEEE Transactions on Evolutionary Computation, 20(3), 460-469.

For the purpose of predicting evolutionary outcomes, the replicator equation is also a frequently utilized tool. Cressman, R. (2003) Evolutionary Dynamics and Extensive Form Games. The MIT Press. {{ISBN|9780262033053}}Cressman, R., & Tao, Y. (2014). The replicator equation and other game dynamics. Proceedings of the National Academy of Sciences, 111(Supplement 3), 10810-10817. https://doi.org/10.1073/pnas.1400823111 Evolutionarily stable states are often taken as solutions to the replicator equation, here in linear payoff form:

:$\backslash dot\{x\_i\}=x\_i\backslash left(\backslash left(Ax\backslash right)\_i-x^TAx\backslash right),$

The state $\backslash hat\{x\}$ is said to be evolutionarily stable if for all $x\; \backslash neq\; \backslash hat\{x\}$ in some neighborhood of $\backslash hat\{x\}$.

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{{Game theory}}

Category:Evolutionary game theory