The Fundamental Theorem of Natural Selection

The Fundamental Theorem of Natural Selection

Suppose we have <i>n</i> different types of self-replicating entity, with the population <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>i</mi></msub>...

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Autor principal: John C. Baez
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
Fisher information metric
natural selection
population biology
replicator equation
Lotka–Volterra equation
Science
Q
Astrophysics
QB460-466
Physics
QC1-999
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spelling oai:doaj.org-article:7f2826959cfe469e956c73ccfab82aa12021-11-25T17:29:39ZThe Fundamental Theorem of Natural Selection10.3390/e231114361099-4300https://doaj.org/article/7f2826959cfe469e956c73ccfab82aa12021-10-01T00:00:00Zhttps://www.mdpi.com/1099-4300/23/11/1436https://doaj.org/toc/1099-4300Suppose we have <i>n</i> different types of self-replicating entity, with the population <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>i</mi></msub></semantics></math></inline-formula> of the <i>i</i>th type changing at a rate equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>i</mi></msub></semantics></math></inline-formula> times the fitness <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>i</mi></msub></semantics></math></inline-formula> of that type. Suppose the fitness <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>i</mi></msub></semantics></math></inline-formula> is any continuous function of all the populations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>P</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula> be the fraction of replicators that are of the <i>i</i>th type. Then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>p</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> 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’s fundamental theorem of natural selection. We compare it to Fisher’s original result as interpreted by Price, Ewens and Edwards.John C. BaezMDPI AGarticleFisher information metricnatural selectionpopulation biologyreplicator equationLotka–Volterra equationScienceQAstrophysicsQB460-466PhysicsQC1-999ENEntropy, Vol 23, Iss 1436, p 1436 (2021)
institution DOAJ
collection DOAJ
language EN
topic Fisher information metric
natural selection
population biology
replicator equation
Lotka–Volterra equation
Science
Q
Astrophysics
QB460-466
Physics
QC1-999
spellingShingle Fisher information metric
natural selection
population biology
replicator equation
Lotka–Volterra equation
Science
Q
Astrophysics
QB460-466
Physics
QC1-999
John C. Baez
The Fundamental Theorem of Natural Selection
description Suppose we have <i>n</i> different types of self-replicating entity, with the population <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>i</mi></msub></semantics></math></inline-formula> of the <i>i</i>th type changing at a rate equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>i</mi></msub></semantics></math></inline-formula> times the fitness <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>i</mi></msub></semantics></math></inline-formula> of that type. Suppose the fitness <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>i</mi></msub></semantics></math></inline-formula> is any continuous function of all the populations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>P</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula> be the fraction of replicators that are of the <i>i</i>th type. Then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>p</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> 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’s fundamental theorem of natural selection. We compare it to Fisher’s original result as interpreted by Price, Ewens and Edwards.
format article
author John C. Baez
author_facet John C. Baez
author_sort John C. Baez
title The Fundamental Theorem of Natural Selection
title_short The Fundamental Theorem of Natural Selection
title_full The Fundamental Theorem of Natural Selection
title_fullStr The Fundamental Theorem of Natural Selection
title_full_unstemmed The Fundamental Theorem of Natural Selection
title_sort fundamental theorem of natural selection
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/7f2826959cfe469e956c73ccfab82aa1
work_keys_str_mv AT johncbaez thefundamentaltheoremofnaturalselection
AT johncbaez fundamentaltheoremofnaturalselection
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