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:^[10]

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

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. The harmonic mean tends to be dominated by the smallest bottleneck that the population goes through.

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:

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

Variance in reproductive success

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 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

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

The effective population size is then smaller, and 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. 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_k = 0 and N_e = 2N.

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.

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^[12]

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

Diploid

Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure. This was first given by Johnson,^[13] but the notation more closely resembles Emigh and Pollak.^[14]

Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as N₀^ƒ and N₀^m for the number of newborn females and males, respectively (notice lower case ƒ for females, compared to upper case F for inbreeding).

The inbreeding effective number is

${\displaystyle {\begin{aligned}{\frac {1}{N_{e}^{(F)}}}={\frac {1}{4T}}\left\{{\frac {1}{N_{0}^{f}}}+{\frac {1}{N_{0}^{m}}}+\sum _{i}\left(\ell _{i+1}^{f}\right)^{2}\left(v_{i+1}^{f}\right)^{2}\left({\frac {1}{\ell _{i+1}^{f}}}-{\frac {1}{\ell _{i}^{f}}}\right)\right.\,\,\,\,\,\,\,\,&\\\left.{}+\sum _{i}\left(\ell _{i+1}^{m}\right)^{2}\left(v_{i+1}^{m}\right)^{2}\left({\frac {1}{\ell _{i+1}^{m}}}-{\frac {1}{\ell _{i}^{m}}}\right)\right\}.&\end{aligned}}}$