Journal of Mathematical Biology
June 2018, Volume 76, Issue 7, pp 1589–1622 | Cite as
The fundamental theorem of natural selection with mutations
Authors and affiliations
William F. Basener 1
John C. Sanford 2
1.Rochester Institute of Technology Rochester USA
2.Horticulture Section NYSAES Geneva USA
Open AccessArticle
First Online: 07 November 2017
Abstract
The mutation–selection process is the most fundamental mechanism of evolution. In 1935, R. A. Fisher proved his fundamental theorem of natural selection, providing a model in which the rate of change of mean fitness is equal to the genetic variance of a species. Fisher did not include mutations in his model, but believed that mutations would provide a continual supply of variance resulting in perpetual increase in mean fitness, thus providing a foundation for neo-Darwinian theory. In this paper we re-examine Fisher’s Theorem, showing that because it disregards mutations, and because it is invalid beyond one instant in time, it has limited biological relevance. We build a differential equations model from Fisher’s first principles with mutations added, and prove a revised theorem showing the rate of change in mean fitness is equal to genetic variance plus a mutational effects term. We refer to our revised theorem as the fundamental theorem of natural selection with mutations. Our expanded theorem, and our associated analyses (analytic computation, numerical simulation, and visualization), provide a clearer understanding of the mutation–selection process, and allow application of biologically realistic parameters such as mutational effects. The expanded theorem has biological implications significantly different from what Fisher had envisioned.
Keywords
Population genetics Population dynamics Mutations Fitness Fisher Fundamental theorem of natural selection Natural selection Mutational meltdown
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