The Time Invariance and Nondecreasing Expectation of an Evolutionary Path Characteristic under Weak Selection.

Bibliographic Details
Title:	The Time Invariance and Nondecreasing Expectation of an Evolutionary Path Characteristic under Weak Selection.
Authors:	Yu YY; School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China., Hui C; Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa; National Institute for Theoretical and Computational Sciences, African Institute for Mathematical Sciences, Stellenbosch 7602, South Africa; and International Initiative for Theoretical Ecology, London N1 2EE, United Kingdom., Feng TJ; Key Laboratory of Animal Ecology and Conservation Biology, Center for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing 100101, China., Li C; School of Ecology and Environment, Northwestern Polytechnical University, Xi'an 710072, China., Wang C; School of Ecology and Environment, Northwestern Polytechnical University, Xi'an 710072, China., Tao Y; Key Laboratory of Animal Ecology and Conservation Biology, Center for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing 100101, China.; School of Ecology and Environment, Northwestern Polytechnical University, Xi'an 710072, China., Wang RW; College of Life Sciences, Zhejiang University, Hangzhou 310058, China.
Source:	The American naturalist [Am Nat] 2026 Jun; Vol. 207 (6), pp. 860-869. Date of Electronic Publication: 2026 Mar 31.
Publication Type:	Journal Article
Language:	English
Journal Info:	Publisher: University of Chicago Press Country of Publication: United States NLM ID: 2984688R Publication Model: Print-Electronic Cited Medium: Internet ISSN: 1537-5323 (Electronic) Linking ISSN: 00030147 NLM ISO Abbreviation: Am Nat Subsets: MEDLINE
Imprint Name(s):	Publication: Chicago, IL : University of Chicago Press Original Publication: Salem, Mass. : Essex Institute
MeSH Terms:	Selection, Genetic* , Biological Evolution* , Genetic Fitness*, Genetic Drift ; Phenotype ; Stochastic Processes ; Models, Genetic
Abstract:	AbstractFisher's fundamental theorem states that the rate of change in mean fitness due to natural selection equals the additive genetic variance in fitness, suggesting that selection generally drives populations toward higher average fitness. Yet in real populations, this increase can be altered or reversed by mutation, frequency-dependent selection, or environmental fluctuations, underscoring the need for complementary measures of adaptation. We analyze stochastic game dynamics of phenotypic frequency in large finite populations under weak frequency-dependent selection and genetic drift. Using the Fokker-Planck equation for the probability density of phenotypic frequency and a path integral formulation, we characterize the feature of possible evolutionary paths over finite timescales. Within this framework, we define the "evolutionary path characteristic" as the likelihood ratio between the probability densities of paths reaching a given endpoint under selection plus drift versus under neutrality. This ratio is time invariant and captures the cumulative effect of directional selection relative to drift. Importantly, its expected value, which equals 1 plus the χ² divergence between evolutionary paths under selection and drift versus those under drift alone, is nondecreasing. In the presence of fitness variance, the effect of directional selection on phenotypic frequency accumulates over time, resulting in a divergence from paths shaped solely by drift. This framework complements Fisher's theorem by providing a robust measure of accumulating adaptation in stochastic evolutionary dynamics.
Contributed Indexing:	Keywords: Fisher’s fundamental theorem; finite population; invariance; path integral; stochastic evolutionary game; weak selection
Entry Date(s):	Date Created: 20260526 Date Completed: 20260717 Latest Revision: 20260717
Update Code:	20260717
DOI:	10.1086/739954
PMID:	42190682
Database:	MEDLINE

Description
ISSN:	1537-5323
DOI:	10.1086/739954