The fate of new beneficial mutations within a population was first addressed by Fisher, Haldane, and Wright in the 1920's and 1930's.
In his 1927 paper, Haldane proved the classic result that the probability of fixation of a new beneficial mutation (P) is approximately 2s in a large population of constant size.
Natural populations do not, however, remain constant in size, but experience expansions, contractions, or fluctuations over time.
With changing population size, what is the probability of fixation of an allele and how does this affect the rate of evolution of a species?
Haldane used a branching process to calculate P, noting that for an allele to be lost, all offspring copies of the allele must also be lost.
By assuming that the number of offspring per parent is, on average, one in a population of constant size, is 1+s for a new beneficial mutant, and is Poisson distributed,
When s is small,
In this case,
When r>0, beneficial mutations will fix at a faster rate.
When r<0, beneficial mutations will fix at a slower rate.
Why? In a growing population, every individual has more offspring on average, which protects the new mutation from loss while rare.
By solving the inhomogeneous branching process,
This reduces to 2(s+r) when the population is far from carrying capacity and to 2s at carrying capacity.
We therefore define a fixation effective population size (Ne) such that:
This equation may be used to determine the probability of fixation for
The probability of incorporation depends strongly on the current population size relative to future population sizes.
Growing populations incorporate more beneficial mutations and fewer deleterious mutations than predicted based on current population size.
Shrinking populations incorporate fewer beneficial mutations and more deleterious mutations than predicted based on current population size.
Evolutionary forces are likely to reinforce, rather than counteract, externally caused changes in population size, such as those currently wrought by humans.