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


  1. An interesting, non-genetic version of the theorem is dealt with in the book by Vincent & Brown (2005) "Evolutionary game theory, natural selection and Darwinian dynamics" (CUP).

    Per L.

  2. For me, the Fundamental Theorem is mainly of interest as a conceptual understanding of the opposing effects of natural selection (improving organism's fitness) and the "deteriotion of the environment" caused by density-dependent depletion of resources when populations grow in size (there is an excellent paper by Frank and Slatkin in TREE from 1992 explaining this). Thus, the Fundamental Theorem does not provide us with a lot of empirical guidelines what to do and what to measure, only a good understanding (which is not bad and certainly important). To some extent one could argue that the same is true for Sewall Wright's conceptual model of the Adaptive Landscape, I guess.

    I do agree that natural selection is an ecological process that is independent of the underlying genetic details (i. e. it does not matter if we live in a DNA-world a RNA-world or in some other world). I also agree with the notion that natural selection is an optimization process that tends to make organisms better adapted (=achieve higher local population growth rate) to their local environments. I disagree, however, with the conclusion that organisms are "optimal" or always "well-adapted" to their local environments, and that we can therefore ignore genetics and only do phenotypic models and analyses. It does not logically follow that organisms must be optimal or well-adapted from the correct observation that natural selection is an optimization process. Non-adaptive forces (i. e. other evolutionary forces than natural selection) tend to oppose the optimizing effects of natural selection. These non-adaptive forces (genetic drift, gene flow, recombination) require some knowledge about genetics. Therefore, natural selection might very well increase an organisms temporary adaptation to a local environment, but this adaptation might immediately get destroyed by other forces. This is one insight from Fisher's Fundamental Theorem and from population genetics in general that I think are very important for ecologists to keep in mind, to avoid falling in to the Panglossian Paradigm trap.

  3. Thank you Erik for your comment.
    It is full of insight and nicely complements our brief introduction to a vast topic (the process and consequences of natural selection will re-appear on this blog, rest assure). I do however have some comments to your views.

    First of all, the Fundamental Theorem does provide empiricists with guidelines, although they might seem so obvious today we hardly think of them. It shows that the appropriate fitness measure is 'per capita growth rate' or 'the Malthusian parameter', nothing else. There are other possibilities, like for example lifetime reproductive success or equilibrium population size, that might seem suitable but really only work properly under certain circumstances.

    Another important insight from the theorem is that evolution is an extremely 'time-local' process - it is the current additive genetic fitness-values, given by the current genetic composition of the population and its current environment, that matters for natural selection, nothing else! In this way, the theorem implies both frequency and density dependence, albeit rather implicitly.

    Regarding other, non-adaptive, mechanisms for evolultionary change, it is true they exist and may play a major role in particular cases. However, Fisher's theorem helps explaining why we find so many well adapted species out there. Why adaptation is the rule rather than the exception. If you want to explain the exceptions, there are many alternatives, and not all of them are 'genetic' in nature. First of all, adaptation takes time and in a changing environment perfect optimization can not be expected. Further, 'genetic drift' is basically what in population ecology is called 'demographic stochasticity'. It would occur in any evolving system with finite populations. 'Gene flow' can be regarded as 'dispersal' and incorporated in the fitness concept (Brown and Pavlovic , 1992, Evol. Ecol. 6:360-382, have a nice discussion on this). Nevertheless, recombination, and dominance, remain as a quite particular genetic mechanisms.

  4. Although I agree that "genetic drift" and "gene flow" could potentially be translated to population ecological parameters such as "demographic stochasticity" and "dispersal", I am not 100 % convinced of a 1:1 mapping. For instance, individual dispersal is likely to differ from "genetic" dispersal, since all dispersers might not be able to reproduce (or even be selected against, through reproductive pre-mating isolation, for instance). Nevertheless, this is perhaps mainly a technical detail, I am not sure how important it is.

    Also, "mutation" is certainly a genetic phenomena, and I am not sure it can be translated to a population ecological parameter? Or can it? Mutation and mutation load plays a central role in Fisher's Fundamental Theorem since it generates new variation that is removed every generation due to selection. At equilibrium the theorem states (cited above):

    "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)"

    This IMPLIES that the amount of new genetic variation generated by novel (mainly deleterious) mutations (" mutational input") must EXACTLY counter balance the amount of genetic variation that is removed by selection (each generation). Hence, here we definitely have two forces (one adaptive and one non-adaptive that are operating in different directions, opposing each others effects). Selection is needed to keep the population "free" from deleterious mutations, but there will always be some genetic variation in fitness, due to mutation.

    I do also still have some issue with the notion that organisms are in general or always "well adapted". Although an optimization approach is a respectable scientific tradition, I think one could actually use Fisher's important insights and his Fundamental Theorem to argue exactly the opposite: Organisms might often be mal-adapted to their local environments and are unlikely to sit on their adaptive peaks, simply because of the fact that the non-adaptive forces in each generation will "push" them down, and selection has to start all over again (see my comments about mutation load above, in principle the same arguments apply to the role of recombination and gene flow).

  5. The one supreme observation one makes of Nature is that organisms are surprisingly fit of form and function. This does not preclude the fact that they are not exactly at an adaptive peak - the (evolutionary) environment is ever changing.
    Maladaptation is a much misused and confusing concept.
    Adaptation (the process) is travel towards any (local) peak in the adaptive landscape. This becomes a maladaption when this travel amounts to climbing a sinking landscape (due to frequency dependence).
    If we say that a population with a trait distribution that is off the optimal one is "maladapted", then most populations would be, but they are not.

  6. Agree, the term "maladaption" might be confusing, even misleading, but I could not find a better word, though. To be fair, the term "adaption" has has also a long and controversial history (see e. g. Rose & Lauders book "Adaptation" from 1996), and we might never find THE correct definition of neither adaptation nor maladaption. The way I used "maladaption" above was a matter of degree, not in an absolute sense. I view organisms as (mostly) residing close to their adaptive peaks (or hover close around the peaks), but frequency-dependent selection, evolutionary lags, mutation pressure and changing environments are constantly pushing organisms downhill, against the force of selection.

    I would like to distinguish selection (a process) from adaption (a state), mainly for conceptual purposes, although both are of course closely related.

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