{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T07:41:16Z","timestamp":1775029276018,"version":"3.50.1"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,30]],"date-time":"2021-10-30T00:00:00Z","timestamp":1635552000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Suppose we have n different types of self-replicating entity, with the population Pi of the ith type changing at a rate equal to Pi times the fitness fi of that type. Suppose the fitness fi is any continuous function of all the populations P1,\u2026,Pn. Let pi be the fraction of replicators that are of the ith type. Then p=(p1,\u2026,pn) is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher\u2019s fundamental theorem of natural selection. We compare it to Fisher\u2019s original result as interpreted by Price, Ewens and Edwards.<\/jats:p>","DOI":"10.3390\/e23111436","type":"journal-article","created":{"date-parts":[[2021,11,1]],"date-time":"2021-11-01T22:21:08Z","timestamp":1635805268000},"page":"1436","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["The Fundamental Theorem of Natural Selection"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0609-9836","authenticated-orcid":false,"given":"John C.","family":"Baez","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Riverside, CA 92521, USA"},{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Fisher, R.A. (1930). The Genetical Theory of Natural Selection, Clarendon Press.","DOI":"10.5962\/bhl.title.27468"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1111\/j.1469-1809.1972.tb00764.x","article-title":"Fisher\u2019s \u201cfundamental theorem\u201d made clear","volume":"36","author":"Price","year":"1972","journal-title":"Ann. Hum. Genet."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0040-5809(89)90028-2","article-title":"An interpretation and proof of the Fundamental Theorem of Natural Selection","volume":"36","author":"Ewens","year":"1989","journal-title":"Theor. Popul. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1111\/j.1469-185X.1994.tb01247.x","article-title":"The fundamental theorem of natural selection","volume":"69","author":"Edwards","year":"1994","journal-title":"Biol. Rev."},{"key":"ref_5","first-page":"309","article-title":"On the mathematical foundations of theoretical statistics","volume":"222","author":"Fisher","year":"1922","journal-title":"Philos. Trans. A Math. Phys. Eng. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Akin, E. (1979). The Geometry of Population Genetics, Springer.","DOI":"10.1007\/978-3-642-93128-4"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lessard, S. (1990). The differential geometry of population genetics and evolutionary games. Mathematical and Statistical Developments of Evolutionary Theory, Springer.","DOI":"10.1007\/978-94-009-0513-9"},{"key":"ref_8","first-page":"211","article-title":"A new mathematical framework for the study of linkage and selection","volume":"17","author":"Shahshahani","year":"1979","journal-title":"Mem. Am. Math. Soc."},{"key":"ref_9","unstructured":"Harper, M. (2009). Information geometry and evolutionary game theory. arXiv."},{"key":"ref_10","unstructured":"Harper, M. (2009). The replicator equation as an inference dynamic. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Amari, S. (2016). Information Geometry and Its Applications, Springer.","DOI":"10.1007\/978-4-431-55978-8"},{"key":"ref_12","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley-Interscience. [2nd ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Baez, J.C., and Pollard, B.S. (2016). Relative entropy in biological systems. Entropy, 18.","DOI":"10.3390\/e18020046"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Leinster, T. (2021). Entropy and Diversity: The Axiomatic Approach, Cambridge Press.","DOI":"10.1017\/9781108963558"},{"key":"ref_15","unstructured":"Baez, J.C. (2021, October 28). Information Geometry, Part 7. Available online: https:\/\/math.ucr.edu\/home\/baez\/information."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1436\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:23:22Z","timestamp":1760167402000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1436"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,30]]},"references-count":15,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["e23111436"],"URL":"https:\/\/doi.org\/10.3390\/e23111436","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,30]]}}}