Malecot's method of coancestry

In a finite size population, after some generations, all individuals will have a common ancestor : ${\displaystyle f\rightarrow 1}$ . Consider a non-sexual population of fixed size ${\displaystyle N}$ , and call ${\displaystyle f_{i}}$  the inbreeding coefficient of generation ${\displaystyle i}$ . Here, ${\displaystyle f}$  means the probability that two individuals picked at random will have a common ancestor. At each generation, each individual produces a large number ${\displaystyle k\gg 1}$  of descendants, from the pool of which ${\displaystyle N}$  individual will be chosen at random to form the new generation.

At generation ${\displaystyle n}$ , the probability that two individuals have a common ancestor is "they have a common parent" OR "they descend from two distinct individuals which have a common ancestor" :

${\displaystyle f_{n}={\frac {k-1}{kN}}+{\frac {k(N-1)}{kN}}f_{n-1}}$ 

What is the source of the above formula? Is it in a later paper than the 1948 Reference.

${\displaystyle \approx {\frac {1}{N}}+(1-{\frac {1}{N}})f_{n-1}.}$ 

This is a recurrence relation easily solved. Considering the worst case where at generation zero, no two individuals have a common ancestor,

${\displaystyle f_{0}=0}$ , we get

${\displaystyle f_{n}=1-(1-{\frac {1}{N}})^{n}.}$ 

The scale of the fixation time (average number of generation it takes to homogenize the population) is therefore

${\displaystyle {\bar {n}}=-1/\log(1-1/N)\approx N.}$ 

This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing ${\displaystyle N}$  to ${\displaystyle 2N}$  (the number of gametes).

• Coefficient of relationship
• Consanguinity
• Genetic distance

• Malécot, G. (1948). Les mathématiques de l'hérédité. Paris: Masson & Cie.