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# Effective population size

The effective population size (Ne) is defined as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies under random genetic drift or the same amount of inbreeding as the population under consideration" (Sewall Wright). It is a basic parameter in many models in population genetics. The effective population size is usually smaller than the absolute population size (N). The concept of effective population size was first derived by the American geneticist Sewall Wright, who wrote two landmark papers on it (Wright 1931, 1938) See also small population size.

 Contents

## Definitions

Effective populaiton size may be defined in two ways, variance effective size and inbreeding effective size. These are closely linked, and derived from F-statistics.

### Variance effective size

In an idealized population, the variance in allele frequency (p) is given by:

$\operatorname{var}(p)= {p(1-p) \over 2N}$

then this gives:

$N_e^{(v)} = {p(1-p) \over 2 \widehat{\operatorname{var}}(p)}$

### Inbreeding effective size

Alternatively, the effective population size may be defined by how much the inbreeding coefficient changes from one generation to the next, and then define Ne by the same amount of inbreeding there would be in the idealized population.

$N_e^{(f)} = {1 - {\hat f}_t \over 2({\hat f}_{t + 1} - {\hat f}_t)}$

${\hat f}_t$is usually taken to be 0, because it is its size relative to ${\hat f}_{t+1}$ that is important, meaning that the above equation simplifies to:

$N_e^{(f)} = {1 \over 2 {\hat f}_{t + 1}}$

## Examples

### Variations in population size

Population size varies over time. Suppose there are t non-overlapping generations, then effective population size is given by the harmonic mean of the population sizes:

${1 \over N_e} = {1 \over t} \sum_{i=1}^t {1 \over N_i}~~~~~~~~~~~~~~~(1)$

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:

 ${1 \over N_e}$ $= {\begin{matrix} \frac{1}{10} \end{matrix} + \begin{matrix} \frac{1}{100} \end{matrix} + \begin{matrix} \frac{1}{50} \end{matrix} + \begin{matrix} \frac{1}{80} \end{matrix} + \begin{matrix} \frac{1}{20} \end{matrix} + \begin{matrix} \frac{1}{500} \end{matrix} \over 6}$ $= {0.9145 \over 6}$ = 0.032416667 Ne = 30.8

Note this is less than the arithmetic mean of the population size, which in this example is 126.7

Of particular concern is the effect of a population bottleneck.

### Dioeciousness

If a population is dioecious, i.e. there is no self-fertlisation then

$N_e = N + \begin{matrix} \frac{1}{2} \end{matrix}~~~~~~~~~~~~~~~(2)$

or more generally,

$N_e = N + \begin{matrix} \frac{D}{2} \end{matrix}$

where D represents dioeciousness and may take the value 0 (for not dioecious) or 1 for dioecious.

When N is large; Ne approximately equals N, so this is usually trivial and often ignored:

$N_e = N + \begin{matrix} \frac{1}{2} \approx N \end{matrix}$

### Non-Fisherian 1:1 sex-ratios

When the sex ratio of a population varies from the Fisherian 1:1 ratio, effective population size is given by:

$N_e = {4 N_m N_f \over N_m + N_f}~~~~~~~~~~~~~~~(3)$

Where Nm is the number of males and Nf the number of females. For example, with 80 males and 20 females (and thus an absolute population size of 100):

 Ne $= {4 \times 80 \times 20 \over 80 + 20}$ $={6400 \over 100}$ = 64

Again, this results in Ne being less than N.

### Unequal contributions to the next generation

If population size is to remain constant, each individual must contribute on average two gametes 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:

$\operatorname{var}(k) = \bar{k} = 2$

However, in natural populations the variance is larger than this, i.e.

$\operatorname{var}(k) > 2$

The effective population size is then given by:

$N_e = {4 N - 2D \over 2 + \operatorname{var}(k)}~~~~~~~~~~~~~~~(3)$

Note that if the variance of k is less than 2, Ne is less than N. Heritable variation in fecundity, usually pushes Ne lower.

### Overlapping generations and age-structured populations

If generations overlap (i.e. they are not discrete) then:

Ne = TNa

where T is the mean time to reproduction and Na is the number of individuals born per year that survive to adulthood.