Editing Quantitative genetics (section)

==== "Islands" random fertilization====
The breeding population consists of '''s''' small '''dispersed random fertilization''' gamodemes of sample size <math display="inline"> 2N_k </math> ( '''k''' = 1 ... ''s'' ) with " ''overlaps'' " of proportion <math display="inline"> m_k </math> in which '''non-dispersive random fertilization ''' occurs. The '' dispersive proportion '' is thus <math display="inline"> \left( 1 - m_k \right) </math>. The bulk population consists of ''weighted averages'' of sample sizes, allele and genotype frequencies and progeny means, as was done for genetic drift in an earlier section. However, each ''gamete sample size'' is reduced to allow for the ''overlaps'', thus finding a <math display="inline"> 2 N_k </math> effective for <math display="inline"> \left( 1 - m_k \right) </math>.
[[File:RF Inbreeding C c.jpg|thumb|300px|right|"Islands" random fertilization]]

For brevity, the argument is followed further with the subscripts omitted. Recall that <math display="inline"> \tfrac {1}{2N} </math> is <math display="inline"> \Delta f </math> in general. [Here, and following, the ''2N'' refers to the ''previously defined'' sample size, not to any "islands adjusted" version.]

After simplification,<ref name="Wright 1951"/> <math display="block"> ^{\mathsf{islands}} \Delta f = \frac { \left( 1-m \right)^2}{2N - m^2 \left( 2N - 1 \right) } </math> 
Notice that when ''m = 0'' this reduces to the previous ''Δ f''. The reciprocal of this furnishes an estimate of the " <math display="inline"> 2 N_k </math> ''effective for <math display="inline"> \left( 1 - m_k \right) </math>'' ", mentioned above.

This Δf is also substituted into the previous ''inbreeding coefficient'' to obtain <ref name="Wright 1951"/>
<math display="block"> {^{\mathsf{islands}} f_t} = \ {^{\mathsf{islands}} \Delta f_t} + \left( 1 - \ {^{\mathsf{islands}} \Delta f_t} \right) \ {^{\mathsf{islands}} f_{t-1}} </math> where ''t'' is the index over generations, as before.

The effective ''overlap proportion'' can be obtained also,<ref name="Wright 1951"/> as 
<math display="block"> m_t = 1 - \left[ \frac {2N \ {^{\mathsf{islands}} \Delta f_t }}{\left( 2N-1 \right) \ {^{\mathsf{islands}} \Delta f_t + 1 }} \right] ^{ \tfrac{1}{2}} </math>

The graphs to the right show the ''inbreeding'' for a gamodeme size of ''2N = 50'' for ''ordinary dispersed random fertilization '' '''(RF)''' ''(m=0)'', and for ''four overlap levels ( m = 0.0625, 0.125, 0.25, 0.5 )'' of '''islands'''  ''random fertilization''. There has indeed been reduction in the inbreeding resulting from the ''non-dispersed random fertilization'' in the overlaps. It is particularly notable as '''m → 0.50'''. Sewall Wright suggested that this value should be the limit for the use of this approach.<ref name="Wright 1951"/>