Editing Effective population size (section)

===Inbreeding effective size===

Alternatively, the effective population size may be defined by noting how the average [[inbreeding coefficient]] changes from one generation to the next, and then defining ''N''<sub>''e''</sub> as the size of the idealized population that has the same change in average inbreeding coefficient as the population under consideration.  The presentation follows Kempthorne (1957).<ref>{{cite book |author=Kempthorne O |year=1957 |title=An Introduction to Genetic Statistics |publisher=Iowa State University Press}}</ref>

For the idealized population, the inbreeding coefficients follow the recurrence equation

:<math>F_t = \frac{1}{N}\left(\frac{1+F_{t-2}}{2}\right)+\left(1-\frac{1}{N}\right)F_{t-1}.</math>

Using Panmictic Index (1&nbsp;&minus;&nbsp;''F'') instead of inbreeding coefficient, we get the approximate recurrence equation

:<math>1-F_t = P_t = P_0\left(1-\frac{1}{2N}\right)^t. </math>

The difference per generation is

:<math>\frac{P_{t+1}}{P_t} = 1-\frac{1}{2N}. </math>

The inbreeding effective size can be found by solving

:<math>\frac{P_{t+1}}{P_t} = 1-\frac{1}{2N_e^{(F)}}. </math>

This is

:<math>N_e^{(F)} = \frac{1}{2\left(1-\frac{P_{t+1}}{P_t}\right)} </math>.

==== Theory of overlapping generations and age-structured populations ====

When organisms live longer than one breeding season, effective population sizes have to take into account the [[life table]]s for the species.

===== Haploid =====
Assume a haploid population with discrete age structure.  An example might be an organism that can survive several discrete breeding seasons.  Further, define the following age structure characteristics:

: <math>v_i = </math> [[Fisher's reproductive value]] for age <math>i</math>,

: <math>\ell_i = </math> The chance an individual will survive to age <math>i</math>, and

: <math>N_0 = </math> The number of newborn individuals per breeding season.

The [[generation time]] is calculated as

: <math>T = \sum_{i=0}^\infty \ell_i v_i = </math> average age of a reproducing individual

Then, the inbreeding effective population size is<ref>{{cite journal |author=Felsenstein J |year=1971 |title=Inbreeding and variance effective numbers in populations with overlapping generations | journal= [[Genetics (journal)|Genetics]]|volume= 68|issue=4 |pages=581–597|doi=10.1093/genetics/68.4.581 |pmid=5166069 |pmc=1212678 }}</ref>

:<math>N_e^{(F)} = \frac{N_0T}{1 + \sum_i\ell_{i+1}^2v_{i+1}^2(\frac{1}{\ell_{i+1}}-\frac{1}{\ell_i})}.</math>

===== Diploid =====
Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure.  This was first given by Johnson,<ref>{{cite journal |author=Johnson DL |year=1977 |title=Inbreeding in populations with overlapping generations |journal=[[Genetics (journal)|Genetics]] |volume=87 |issue=3 |pages=581–591|doi=10.1093/genetics/87.3.581 |pmid=17248780 |pmc=1213763 }}</ref> but the notation more closely resembles Emigh and Pollak.<ref>{{cite journal |doi=10.1016/0040-5809(79)90028-5 |vauthors=Emigh TH, Pollak E |year=1979 |title=Fixation probabilities and effective population numbers in diploid populations with overlapping generations |journal=Theoretical Population Biology |volume=15 |issue=1 |pages=86–107|bibcode=1979TPBio..15...86E }}</ref>

Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as ''N''<sub>0</sub><sup>''ƒ''</sup> and ''N''<sub>0</sub><sup>''m''</sup> for the number of newborn females and males, respectively (notice lower case ''ƒ'' for females, compared to upper case ''F'' for inbreeding).

The inbreeding effective number is

:<math>
\begin{align}
\frac{1}{N_e^{(F)}} = \frac{1}{4T}\left\{\frac{1}{N_0^f}+\frac{1}{N_0^m} + \sum_i\left(\ell_{i+1}^f\right)^2\left(v_{i+1}^f\right)^2\left(\frac{1}{\ell_{i+1}^f}-\frac{1}{\ell_i^f}\right)\right. \,\,\,\,\,\,\,\, & \\
 \left. {} + \sum_i\left(\ell_{i+1}^m\right)^2\left(v_{i+1}^m\right)^2\left(\frac{1}{\ell_{i+1}^m}-\frac{1}{\ell_i^m}\right) \right\}. &
\end{align}
</math>