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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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/7f2826959cfe469e956c73ccfab82aa1 |
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| Sumario: | 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. |
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