# Effective population size

In population genetics, the concept of effective population size Ne was introduced by the American geneticist Sewall Wright, who wrote two landmark papers on it (Wright 1931, 1938). He defined it 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". It is a basic parameter in many models in population genetics. The effective population size is usually smaller than the absolute population size (N). See also small population size.

## Definitions

Effective population 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:

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

### Inbreeding effective size

Alternatively, the effective population size may be defined by noting how the inbreeding coefficient changes from one generation to the next, and then defining Ne as the size of the idealized population that has the same change in inbreeding. The presentation follows Kempthorne (1957).

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

$F_t = \frac{1}{N}\left(\frac{1+F_{t-2}}{2}\right)+\left(1-\frac{1}{N}\right)F_{t-1}.$

Using Panmictic Index (1-F) instead of inbreeding coefficient, we get the approximate reccurrence equation

$1-F_t = P_t = P_0\left(1-\frac{1}{2N}\right)^t.$

The difference per generation is

$\frac{P_{t+1}}{P_t} = 1-\frac{1}{2N}.$

The inbreeding effective size can be found by solving

$\frac{P_{t+1}}{P_t} = 1-\frac{1}{2N_e^{(F)}}.$

This is

$N_e^{(F)} = \frac{1}{\frac{1}{2}\left(1-\frac{P_{t+1}}{P_t}\right)}$

although researchers rarely use this equation directly.

## 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:

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

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:

 ${1 \over N_e}$ $= {\begin{matrix} \frac{1}{10} \end{matrix} + \begin{matrix} \frac{1}{100} \end{matrix} + \begin{matrix} \frac{1}{50} \end{matrix} + \begin{matrix} \frac{1}{80} \end{matrix} + \begin{matrix} \frac{1}{20} \end{matrix} + \begin{matrix} \frac{1}{500} \end{matrix} \over 6}$ $= {0.1945 \over 6}$ $= 0.032416667$ $N_e$ $= 30.8$

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-fertilisation then

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

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 sex-ratios

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

$N_e^{(v)} = N_e^{(F)} = {4 N_m N_f \over N_m + N_f}$

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

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

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:

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

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

### Overlapping generations and age-structured populations

When organisms live longer than one breeding season, effective population sizes have to take into account the life tables 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:

$v_i =$ Fisher's reproductive value for age $i$,
$\ell_i =$ The chance an individual will survive to age $i$, and
$N_0 =$ The number of newborn individuals per breeding season.

The generation time is calculated as

$T = \sum_{i=0}^\infty \ell_i v_i =$ average age of a reproducing individual

Then, the inbreeding effective population size is (Felsenstein 1971)

$N_e^{(F)} = \frac{N_0T}{1 + \sum_i\ell_{i+1}^2v_{i+1}^2(\frac{1}{\ell_{i+1}}-\frac{1}{\ell_i})}.$

#### Diploid

Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure. This was first given by Johnson (1977), but the notation more closely resembles Emigh and Pollak (1979).

Assume the same basic parameters for the life table as given for the Haploid case, but distinguishing between male and female, such as $N_0^f$ and $N_0^m$ for the number of newborn females and males, respectively (notice lower case f for females,compared to upper case F for inbreeding).

The inbreeding effective number is calculated from

$\frac{1}{N_e^{(F)}} = \frac{1}{4T}\left\{\frac{1}{N_0^f}+\frac{1}{N_0^m} + \sum_i\left(\ell_{i+1}^f\right)^2\left(v_{i+1}^f\right)^2\left(\frac{1}{\ell_{i+1}^f}-\frac{1}{\ell_i^f}\right) + \sum_i\left(\ell_{i+1}^m\right)^2\left(v_{i+1}^m\right)^2\left(\frac{1}{\ell_{i+1}^m}-\frac{1}{\ell_i^m}\right) \right\}.$

## References

• Emigh, T. H. and E. Pollak (1979). Fixation probabilities and effective population numbers in diploid populations with overlapping generations. Theoretical Population Biology 15: 86-107.
• Felsenstein, J. (1971). "Inbreeding and variance effective numbers in populations with overlapping generations". Genetics 68: 581–597.
• Johnson, D. L. (1977). Inbreeding in populations with overlapping generations. Genetics 87:581-591.
• Kempthorne, O. (1957, [1969]). An Introduction to Genetic Statistics. 1969 Printing is Iowa State University Press.
• Wright, S. (1931). Evolution in Mendelian populations. Genetics 16: 97-159 Offsite pdf file
• Wright, S. (1938). Size of population and breeding structure in relation to evolution. Science 87:430-431