1. Fisher's Fundamental Theorem on Natural Selection states that the rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time. 2. The theorem has been revised and clarified over time, but generally means that natural selection will tend to increase the fitness of organisms through changes in gene frequencies. 3. Fisher framed his argument in terms of parental expenditure, suggesting that natural selection will act to equalize the total parental expenditure between male and female offspring to maximize reproductive value.

1. 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."
Explanation
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.
”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 cannot 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 organismat any time ascribable to natural selection acting through changesi n
gene frequenciesis exactly equal to its genic variance in fitness at that time.
Basic explanation
W.D. Hamilton gave the following basic explanation in his 1967 paper on "Extraordinary sex ratios" given the condition that
males and females cost equal amounts to produce:
1. Suppose male births are less common than female.
2. A newborn male then has bettermating prospects than a newborn female, and therefore can expect to have more
offspring.
3. Therefore parents genetically disposed to produce males tend to have more than average numbers of grandchildren born
to them.
4. Therefore the genes for male-producing tendencies spread,and male births become more common.
5. As the 1:1 sex ratio is approached,the advantage associated with producing males dies away.
6. The same reasoning holds if females are substituted formales throughout.Therefore 1:1 is the equilibrium ratio.
In modern language,the 1:1 ratio is the evolutionarily stable strategy (ESS). In a population of individuals playing the ESS (i.e.
producing equal numbers of sons and daughters),any otherstrategy (e.g. 51% sons and 49% daughters)would have lower fitness ,
and hence would be selected against.
Parental expenditure Specifically, Fisher frames his argument in terms of parental expenditure. In organisms of all
kinds the young are launched upon their careers endowed with a certain amount of biological capital derived from their parents.
This varies enormously in amount in different species,but,in all, there has been, before the offspring is able to lead an
independent existence, a certain expenditure of nutriment in addition, almost universally, to some expenditure of time or activity,
which the parents are induced by their instincts to make for the advantage of their young.
Let us consider the reproductive value of these offspring at the moment when this parental expenditure on their behalf has just
ceased.If we consider the aggregate of an entire generation of such offspring it is clear that the total reproductive value of the
males in this group is exactly equal to the total value of all the females, because each sex must supply half the ancestry of all
future generations of the species.From this it follows that the sex ratio will so adjust itself, under the influence of Natu ral
Selection, that the total parental expenditure incurred in respect of children of each sex, shall be equal; for if this were not so and
the total expenditure incurred in producing males, for instance,were less than the total expenditure incurred in producing females,
then since the total reproductive value of the males is equal to that of the females, it would follow that those parents,the innate
tendencies of which caused them to produce males in excess, would, for the same expenditure, produce a greater amount of
reproductive value; and in consequence would be the progenitors of a larger fraction of future generations than would parents
having a congenital bias towards the production of females. Selection would thus raise the sex-ratio until the expenditure upon
males became equal to that upon females.
Development of the argument Fisher's principle is an early example of a model in which genes for greater production of
either sex become equalized in the population, because each sex supplies exactly half the genes of all future generations. Fisher's
principle is rooted in the concept of frequency-dependent selection,though Fisher's principle is not frequency-dependent selection.
Fisher's principle extends frequency dependence to explain how natural selection can act on genes that affect the frequency of an
individual's grandchildren without affecting the frequency of their children. Fisher predicted that parents will invest their
resources equally between each sex of offspring, because each sex supplies exactly half the genes of all future generations .As a
result, those genes that cause parents to invest unequally in the sexes will tend to be selected against. Fisher was aware th at in
humans, more boys are born, whilst boys are also more likely to die in infancy. As a consequence,he reasoned that because
parents tend to invest less in boys - because more boys die before the end of the period of parental care - there is a higher rate of
male births to equalize parental investment in each sex. Fisher's concept of parental expenditure (now termed parental
investment), developed particularly by Robert Trivers is now an important concept in ecology.