This paper gives a proof that certain quantities are independent of the geographical structure of a population. The quantities are: (1) the fixation probability of a mutant; (2) the sum of the quantity x(1 − x), where x is the mutant frequency, while the mutant is segregating; and (3) the quantity x(1 − x) summed over the generations during which the gene frequency in the whole population assumes a specified value. The independence of geographical structure for the latter two quantities is not exact if there is selection, but is a close approximation.
The model is a geographically structured version of Moran's haploid overlapping generation model. The population consists of colonies connected genetically by migration. Each individual has the same negative exponential lifetime distribution. When an individual dies, it is immediately replaced by an individual born in the same colony with a probability proportional to the frequency and fitness of the type giving birth. In a diploid population the quantity x(1 − x) is proportional to the heterozygosity.
[1]
T. Maruyama.
An invariant property of a structured population.
,
1971,
Genetical research.
[2]
P. Moran.
The survival of a mutant gene under selection
,
1959,
Journal of the Australian Mathematical Society.
[3]
T. Maruyama,et al.
Evidence for the Neutral Hypothesis of Protein Polymorphism
,
1972,
Science.
[4]
Motoo Kimura,et al.
Some Problems of Stochastic Processes in Genetics
,
1957
.
[5]
T. Maruyama,et al.
On the fixation probability of mutant genes in a subdivided population.
,
1970,
Genetical research.
[6]
P. A. P. Moran,et al.
Random processes in genetics
,
1958,
Mathematical Proceedings of the Cambridge Philosophical Society.
[7]
T. Maruyama.
Some invariant properties of a geographically structured finite population: distribution of heterozygotes under irreversible mutation.
,
1972,
Genetical research.