In population genetics, the concept of effective population size N_e 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 the Wright-Fisher idealized population model, the conditional variance of the allele frequency ${\displaystyle p'}$, given the allele frequency ${\displaystyle p}$ in the previous generation, is

${\displaystyle \operatorname {var} (p'\mid p)={p(1-p) \over 2N}.}$

Let ${\displaystyle {\widehat {\operatorname {var} ))(p'|p)}$ denote the same, typically larger, variance in the actual population under consideration. The variance effective population size ${\displaystyle N_{e}^{(v)))$ is defined as the size of an idealized population with the same variance. This is found by equating ${\displaystyle {\widehat {\operatorname {var} ))(p'|p)}$ with ${\displaystyle {var}(p'|p)}$ and solving for ${\displaystyle N}$ which gives

${\displaystyle N_{e}^{(v)}={p(1-p) \over 2{\widehat {\operatorname {var} ))(p)}.}$

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 N_e 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

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

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

${\displaystyle 1-F_{t}=P_{t}=P_{0}\left(1-{\frac {1}{2N))\right)^{t}.}$

The difference per generation is

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

The inbreeding effective size can be found by solving

${\displaystyle {\frac {P_{t+1)){P_{t))}=1-{\frac {1}{2N_{e}^{(F)))}.}$

This is

${\displaystyle N_{e}^{(F)}={\frac {1}{2\left(1-{\frac {P_{t+1)){P_{t))}\right)))}$

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:

${\displaystyle {1 \over N_{e))={1 \over t}\sum _{i=1}^{t}{1 \over N_{i))}$

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:

${\displaystyle {1 \over N_{e))}$	${\displaystyle =((\begin{matrix}{\frac {1}{10))\end{matrix))+{\begin{matrix}{\frac {1}{100))\end{matrix))+{\begin{matrix}{\frac {1}{50))\end{matrix))+{\begin{matrix}{\frac {1}{80))\end{matrix))+{\begin{matrix}{\frac {1}{20))\end{matrix))+{\begin{matrix}{\frac {1}{500))\end{matrix)) \over 6))$
	${\displaystyle ={0.1945 \over 6))$
	${\displaystyle =0.032416667}$
${\displaystyle N_{e))$	${\displaystyle =30.8}$

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.

Another way of calculating it

Equation for calculating Ne for populations with variation in family size: N_e = (4N)/(V_k + 2) Where V_k is the variance in population size. Large variance in family size is bad because large variations in family size lead to inbreeding.

Dioeciousness

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

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {1}{2))\end{matrix))}$

or more generally,

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {D}{2))\end{matrix))}$

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

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

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {1}{2))\approx N\end{matrix))}$

Non-Fisherian sex-ratios

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

${\displaystyle N_{e}^{(v)}=N_{e}^{(F)}={4N_{m}N_{f} \over N_{m}+N_{f))}$

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

${\displaystyle N_{e))$	${\displaystyle ={4\times 80\times 20 \over 80+20))$
	${\displaystyle ={6400 \over 100))$
	${\displaystyle =64}$

Again, this results in N_e 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:

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

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

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

The effective population size is then given by:

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

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

${\displaystyle v_{i}=}$ Fisher's reproductive value for age ${\displaystyle i}$,

${\displaystyle \ell _{i}=}$ The chance an individual will survive to age ${\displaystyle i}$, and

${\displaystyle N_{0}=}$ The number of newborn individuals per breeding season.

The generation time is calculated as

${\displaystyle T=\sum _{i=0}^{\infty }\ell _{i}v_{i}=}$ average age of a reproducing individual

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

${\displaystyle N_{e}^{(F)}={\frac {N_{0}T}{1+\sum _{i}\ell _{i+1}^{2}v_{i+1}^{2}({\frac {1}{\ell _{i+1))}-{\frac {1}{\ell _{i))}))).}$

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 ${\displaystyle N_{0}^{f))$ and ${\displaystyle N_{0}^{m))$ 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

${\displaystyle {\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\}.}$