gregor

Yes, it is a magnificent title, but it is a little out of place, because the premise that evolution selects genes is doubtful – not because Fisher’s calculations are wrong.

Sorry, I got my quotation wrong. “Variance” should be “genetic variance”. So a better formulation of the theorem should be: The rate of increase of mean fitness of any organism at any time is equal to its genetic variance at that time.
In the Fisher case, fitness is a mathematical measure of the ability of a gene to contribute to survival. Or according to Maynard Smith: ”Fitness is a property, not of an individual, but of a class of individuals – for example homozygous for allele A at a particular locus. Thus the phrase ’expected number of offspring’ means the average number, not the number produced by some one individual."
This definition is certainly useful in breeding programs. But unfortunately, a theory based on this is completely useless as a basis of a model of an evolution selecting individuals, because the selection of individuals rules the selection of genes. For more details see Maynard Smith in my references (I forgot to tell that this is my reference to the theorem).
http://www.evolution-in-a-nutshell.se/references.htm
From a mathematical expression Fisher calculates the mean value of fitness over the set of all genes of a large population. The result is – what is called - the fundamental theorem. This way of thinking has inspired biologists since 1930, leading to the opinion that egoism is a law of nature. An example is Dawkins: The selfish gene. See also
http://www.evolution-in-a-nutshell.se/egoism.htm

As long as the genetic variance (variability) is > 0, mean fitness will increase and therefore the process strives or converges towards a maximum in mean fitness. On the other hand – as Maynard Smith points out – evolution may reach a state of selective equilibrium, in which case there will be no increase in mean fitness, even though the genetic variance is > 0. This contradicts the theorem.

The theorem of Gaussian adaptation may be formulated in a similar manner. Say that the gradient of mean fitness is grad[ P(m) ] is a measure of increase in mean fitness P. Then the theorem of GA states:
grad[ P(m) ] = inverse-of(M)* P ( m* – m ), where M is the moment matrix of the Gaussian distribution, m and m* defined as before. I should perhaps not use the word fundamental, but the GA-theorem is fundamental in the sense that it is valid for all regions of acceptability of any structure.

In contrast to the fundamental theorem due to Fisher, this result seems more reliable, because in a state of selective equilibrium, we have m* = m and consequently no increase in P. But the phenotypic variance (disorder) – displayed by M - must not be equal to zero.

It is also possible to maximize the logarithm of the determinant of M (proportional to the disorder of the Gaussian) keeping P constant using Lagrange multipliers. The condition of optimality will be the same, m = m*, meaning that GA effectuates a simultaneous maximization of men fitness and disorder.
I have never observed any data. My interpretation is purely mathematical philosophical. The basis of this philosophy is according to the following six theorems.
http://www.evolution-in-a-nutshell.se/six-theorems.htm
I hope the theorems are correctly proved. But of course, their application to natural evolution can always be questioned. They constitute a second order approximation of what may happen to the mean fitness and the phenotypic disorder in large populations, nothing more. But the approximation may still be good. Evolution is viewed as a statistical optimisation algorithm. See for instance Kjellström, 1996, in references.


I dont understand what you mean by time-frame. In natural evolution there will be millions of years. In simulated evolution some hours, days or months perhaps. It has nothing to do with Lamarck.