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Correction: Wolpert, D.H. The Free Energy Requirements of Biological Organisms; Implications for Evolution. Entropy 2016, 18, 138
Open AccessArticle

Stationary Stability for Evolutionary Dynamics in Finite Populations

Independent, Playa del Rey, CA 90293, USA
Department of Mathematics & Statistics, San José State University, One Washington Square, San Jose, CA 95192, USA
Author to whom correspondence should be addressed.
Academic Editor: Raúl Alcaraz Martínez
Entropy 2016, 18(9), 316;
Received: 3 April 2016 / Revised: 24 July 2016 / Accepted: 16 August 2016 / Published: 25 August 2016
(This article belongs to the Special Issue Information and Entropy in Biological Systems)
We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the Moran process with mutation and generalizations, as well as a generalized notion of evolutionary stability that includes mutation called an incentive stable state (ISS) candidate. For sufficiently large populations, extrema of the stationary distribution are ISS candidates and we give a family of Lyapunov quantities that are locally minimized at the stationary extrema and at ISS candidates. In various examples, including for the Moran and Wright–Fisher processes, we show that the local maxima of the stationary distribution capture the traditionally-defined evolutionarily stable states. The classical stability theory of the replicator dynamic is recovered in the large population limit. Finally we include descriptions of possible extensions to populations of variable size and populations evolving on graphs. View Full-Text
Keywords: evolutionary stability; finite populations; information entropy; stationary distributions evolutionary stability; finite populations; information entropy; stationary distributions
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Harper, M.; Fryer, D. Stationary Stability for Evolutionary Dynamics in Finite Populations. Entropy 2016, 18, 316.

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  • Externally hosted supplementary file 1
    Description: There are some supplementary images posted at the above link -- they are simply extra examples and not necessary for the main manuscript. The exact same page is available at and this may be a more permanent location (I don't control the UCLA website) so I updated the URL listed in the manuscript. Please note that (now) Figure 1 was intended to be a graphical abstract. I'm not sure why it was moved to the middle of the paper. I have moved it to the second page.
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