Derivation of the inbreeding effective population size N_e

    The inbreeding coefficient F in any closed, finite population will increase over time, as an increasing proportion of individuals will necessarily come to have related parents. The effective population size N_e is defined by observing how the observed inbreeding coefficient changes from one generation to the next, and defining N_e as the size of the ideal population with the same change as the real population under consideration. 

    In any finite population, the inbreeding coefficient F in the current generation t changes from the previous generation t-1 as a recursion equation:

${\displaystyle F_{t}={\frac {1}{N}}\left({\frac {1+F_{t-2}}{2}}\right)+\left(1-{\frac {1}{N}}\right)F_{t-1}.}$

    The equation has two terms, (1/N) and its complement (1 - 1/N).

    (1) For a draw & replace experiment in a population size N in the previous generation t-1, for any individual drawn at random, the expectation of drawing the same individual a second time is 1/N, and if so the expectation of drawing the same allele is 1/2. Therefore, the expectation of drawing the same allele twice so as to obtain an individual with two alleles that are identical by descent (I by D) is 1/2N. This calculation is weighted by the expectation F_t-2 that any pair of alleles were already I by D in the next previous generation.

    (2) The expectation of not drawing the same individual twice is  (1 - 1/N), weighted by the expectation F_t-1 that any random pair of alleles is already I by D in the previous generation. 

    To extend this recursion calculation from two generations to t generations, consider the complement of the inbreeding coefficient F as the Panmixia Index P = (1 − F), which is the expectation that any two individuals in a finite population size N do not share a common ancestor.  For an initial P₀, the expectation that any two alleles drawn in the subsequent generation are not I by D is (1 - 1/2N). The same is true in each subsequent generation, so that if population size N is constant, the joint probability that this is never so over all generations i = 1, 2, 3, ..., t is simply the product of all those "not" terms,

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Text material © 2020 by Steven M. Carr

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