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Quantitative genetics
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=====Dispersion via ''f''===== The method for examining the inbreeding coefficient is similar to that used for ''σ <sup>2</sup><sub>p,q</sub>''. The same weights as before are used respectively for ''de novo f'' ( '''Δ f''' ) [recall this is '''1/(2N)''' ] and ''carry-over f''. Therefore, <math display="inline"> f_2 = \left( 1 \right) \Delta f + \left( 1 - \Delta f \right) f_1 </math> , which is similar to '''Equation (1)''' in the previous sub-section. [[File:RF Inbreeding.jpg|thumb|300px|left|Inbreeding resulting from genetic drift in random fertilization.]] In general, after rearrangement,<ref name="Crow & Kimura"/> <math display="block"> \begin{align} f_t & = \Delta f + \left( 1 - \Delta f \right) f_{t-1} \\ & = \Delta f \left( 1 - f_{t-1} \right) + f_{t-1} \end{align} </math> The graphs to the left show levels of inbreeding over twenty generations arising from genetic drift for various ''actual gamodeme'' sizes (2N). Still further rearrangements of this general equation reveal some interesting relationships. '''(A)''' After some simplification,<ref name="Crow & Kimura"/> <math display="inline"> \left( f_t - f_{t-1} \right) = \Delta f \left( 1 - f_{t-1} \right) = \delta f_t </math>. The left-hand side is the difference between the current and previous levels of inbreeding: the ''change in inbreeding'' ('''δf<sub>t</sub>'''). Notice, that this ''change in inbreeding'' ('''δf<sub>t</sub>''') is equal to the ''de novo inbreeding'' ('''Δf''') only for the first cycle—when f<sub>t-1</sub> is ''zero''. '''(B)''' An item of note is the '''(1-f<sub>t-1</sub>)''', which is an "index of ''non-inbreeding''". It is known as the ''panmictic index''.<ref name= "Crow & Kimura"/><ref name="Falconer 1996"/> <math display="inline"> P_{t-1} = \left( 1 - f_{t-1} \right) </math>. '''(C)''' Further useful relationships emerge involving the ''panmictic index''.<ref name="Crow & Kimura"/><ref name="Falconer 1996"/> <math display="block"> \begin{align} \Delta f & = \frac {\delta f_t} {P_{t-1}} \\ & = 1 - \frac {P_t} {P_{t-1}} \end{align} </math>. '''(D)''' A key link emerges between ''σ <sup>2</sup><sub>p,q</sub>'' and ''f''. Firstly...<ref name="Crow & Kimura"/> <math display="block"> \begin{align} f_t & = 1 - \left( 1 -1 \Delta f \right) ^t \left( 1 - f_0 \right) \end{align} </math> Secondly, presuming that '''f<sub>0</sub>''' = '''0''', the right-hand side of this equation reduces to the section within the brackets of '''Equation (2)''' at the end of the last sub-section. That is, if initially there is no inbreeding, <math display="inline"> \sigma^2_t = p_g q_g f_t </math> '''!''' Furthermore, if this then is rearranged, <math display="inline"> f_t = \tfrac {\sigma^2_t} {p_g q_g} </math>. That is, when initial inbreeding is zero, the two principal viewpoints of ''binomial gamete sampling'' (genetic drift) are directly inter-convertible.
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