Editing Effective population size (section)

==== 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=443–455|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.