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Effective population size
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=== Variance effective size === In the [[Idealized population|Wright-Fisher idealized population model]], the [[conditional variance]] of the allele frequency <math>p'</math>, given the [[allele frequency]] <math>p</math> in the previous generation, is :<math>\operatorname{var}(p' \mid p)= {p(1-p) \over 2N}.</math> Let <math>\widehat{\operatorname{var}}(p'\mid p)</math> denote the same, typically larger, variance in the actual population under consideration. The variance effective population size <math>N_e^{(v)}</math> is defined as the size of an idealized population with the same variance. This is found by substituting <math>\widehat{\operatorname{var}}(p'\mid p)</math> for <math>\operatorname{var}(p'\mid p)</math> and solving for <math>N</math> which gives :<math>N_e^{(v)} = {p(1-p) \over 2 \widehat{\operatorname{var}}(p)}.</math> In the following examples, one or more of the assumptions of a strictly idealised population are relaxed, while other assumptions are retained. The variance effective population size of the more relaxed population model is then calculated with respect to the strict model. ==== Variations in population size ==== Population size varies over time. Suppose there are ''t'' non-overlapping [[generation]]s, then effective population size is given by the [[harmonic mean]] of the population sizes:<ref>{{Cite journal|last=Karlin|first=Samuel|date=1968-09-01|title=Rates of Approach to Homozygosity for Finite Stochastic Models with Variable Population Size|journal=The American Naturalist|volume=102|issue=927|pages=443β455|doi=10.1086/282557|bibcode=1968ANat..102..443K |s2cid=83824294|issn=0003-0147}}</ref> :<math>{1 \over N_e} = {1 \over t} \sum_{i=1}^t {1 \over N_i}</math> For example, say the population size was ''N'' = 10, 100, 50, 80, 20, 500 for six generations (''t'' = 6). Then the effective population size is the [[harmonic mean]] of these, giving: :{| |- |<math>{1 \over N_e}</math> |<math>= {\begin{matrix} \frac{1}{10} \end{matrix} + \begin{matrix} \frac{1}{100} \end{matrix} + \begin{matrix} \frac{1}{50} \end{matrix} + \begin{matrix} \frac{1}{80} \end{matrix} + \begin{matrix} \frac{1}{20} \end{matrix} + \begin{matrix} \frac{1}{500} \end{matrix} \over 6} </math> |- | |<math>= {0.1945 \over 6}</math> |- | |<math>= 0.032416667</math> |- |<math>N_e</math> |<math>= 30.8</math> |} Note this is less than the [[arithmetic mean]] of the population size, which in this example is 126.7. The harmonic mean tends to be dominated by the smallest [[population bottleneck|bottleneck]] that the population goes through. ==== Dioeciousness ==== If a population is [[dioecious]], i.e. there is no [[self-fertilisation]] then :<math>N_e = N + \begin{matrix} \frac{1}{2} \end{matrix}</math> or more generally, :<math>N_e = N + \begin{matrix} \frac{D}{2} \end{matrix}</math> where ''D'' represents dioeciousness and may take the value 0 (for not dioecious) or 1 for dioecious. When ''N'' is large, ''N''<sub>''e''</sub> approximately equals ''N'', so this is usually trivial and often ignored: :<math>N_e = N + \begin{matrix} \frac{1}{2} \approx N \end{matrix}</math> ==== Variance in reproductive success ==== If population size is to remain constant, each individual must contribute on average two [[gamete]]s to the next generation. An idealized population assumes that this follows a [[Poisson distribution]] so that the [[variance]] of the number of gametes contributed, ''k'' is equal to the [[mean]] number contributed, i.e. 2: :<math>\operatorname{var}(k) = \bar{k} = 2.</math> However, in natural populations the variance is often larger than this. The vast majority of individuals may have no offspring, and the next generation stems only from a small number of individuals, so :<math>\operatorname{var}(k) > 2.</math> The effective population size is then smaller, and given by: :<math>N_e^{(v)} = {4 N - 2D \over 2 + \operatorname{var}(k)}</math> Note that if the variance of ''k'' is less than 2, ''N''<sub>''e''</sub> is greater than ''N''. In the extreme case of a population experiencing no variation in family size, in a laboratory population in which the number of offspring is artificially controlled, ''V''<sub>''k''</sub> = 0 and ''N''<sub>''e''</sub> = 2''N''. ==== Non-Fisherian sex-ratios ==== When the [[sex ratio]] of a population varies from the [[Ronald Fisher|Fisherian]] 1:1 ratio, effective population size is given by: :<math>N_e^{(v)} = N_e^{(F)} = {4 N_m N_f \over N_m + N_f}</math> Where ''N''<sub>''m''</sub> is the number of males and ''N''<sub>''f''</sub> the number of females. For example, with 80 males and 20 females (an absolute population size of 100): :{| |- |<math>N_e</math> |<math>= {4 \times 80 \times 20 \over 80 + 20}</math> |- | |<math>={6400 \over 100}</math> |- | |<math>= 64</math> |} Again, this results in ''N''<sub>''e''</sub> being less than ''N''.
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