Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Friday, January 14, 2011

Fisher's Fundamental Theorem on Natural Selection

"The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time." (Fisher 1930)
What it means
In brief, simplified terms, it means that natural selection will in all organisms tend to increase fitness. Evolution is in this simplified sense an optimizing process. Fitness, defined as per capita growth rate, is what is being optimized.
In more precise terms the statement needs a fair amount of qualification, almost word by word, to be as general as claimed. Fisher was by no means clear about the qualifications - they are mostly due to later interpretations (Price 1972).
increase in fitness” - is the increase in the population mean additive genetic values of fitness. Further, it is the additive genetic values at the time of selection that counts.
”genetic variance” - the additive genetic variance, i.e. the variance in additive effects in the population.
Fitness can not increase forever, and Fisher was perfectly aware of it. However, natural selection will always tend to increase fitness while changes in the environment (such as an increased population density) can decrease fitness.
Edwards (1994) suggested a revised, modernized version of the theorem:
The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genic variance in fitness at that time.
For further details see Price (1972) and Grafen (2003).
Implications and importance
The theorem is a key link between the mechanics of Mendelian genetics and evolution through natural selection, and thus a keystone of the modern evolutionary synthesis.
It has been viewed as a ’license’ for naturalists to think of organisms as optimizing agents, and pointing out exactly what is being optimized (Grafen 2003).  (Note: The process of evolution by natural selection is by no means dependent on genetics as we know it - evolution can work with many types of heritability. In this sense, organic life on Earth is but an example)
The theorem was for a long time disregarded as only applicable to special, simplified cases, but was later resurrected to its general status (Price 1972). This long delay can in most part be explained by the obscureness of Fisher’s writing and his unwillingness to express the theorem in more formal mathematics.

Further reading
Edwards, A. W. F. (1994) The fundamental theorem of natural selection. Biol. Rev. 69: 443-474
Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Oxford, Oxford University Press.
Grafen, A. (2003). Fisher the evolutionary biologist. The Statistician 52(3): 319-329
Price, G. R. (1972) Fisher's "fundamental theorem" made clear. Ann. Hum. Genet., 36: 129-140