# Definition of “Effective population size”

Could you explain to me, what is the meaning of "Effective population size ($N_e$)"?

I would appreciate an example as well.

• This is potentially a good question, but too vague in its current form. You should say what context you mean so that a better explanation can be given. What formula did it occur in? – Superbest Sep 8 '14 at 2:38

I'll add an informal answer to complement @Remi.b's excellent answer. In a very simple sense, you can think of the effective population size as the number of reproducing (breeding) individuals in a population. Nature Education has a very good (and free) Scitable article on Genetic Drift and Effective Population Size. The article makes four points, which I've annotated below.

• The population has an equal number of males and females. All individuals are able to reproduce.

Populations of many species contain individuals that have not yet reached sexual maturity, have passed an age of reproductive capability, or might have a genetic condition that prevents reproduction. All of these individuals count towards the census population size (actual number of individuals present) but not count towards effective population size because they can't reproduce.

• All individuals in the population are equally likely to reproduce.

• Mating is random.

Not all indivduals (assuming individuals that meet assumption #1 above) are equally likely to reproduce. For example, the process of sexual selection means that some individuals are much more likely to reproduce than other individuals. Sexual selection is not random, and lowers effective population size.

• The number of breeding individuals is constant from one generation to the next.

Population sizes for some species can fluctuate over many generations. For example, freshwater mussels have declined rapidly in population size due to habitat loss and other factors. Nearly 72% of the species in the United State and Canada are endangered, threatened or of conservation concern (Williams et al. 1993). The population sizes for the majority of mussels species are but a fraction of what they once were. This is the case for many endangered species. This too lowers effective population size.

Here's a simple mathematical treatment to highlight points 1-3. Consider a population of zebras. Zebras like the Plains Zebra (Equus quagga) form harems where one male mates with several females. He prevents other males from mating with the females in his harem. Let's make the following assumptions: 1) the population has 250 males and 250 females, all capable of breeding and 2) each male that is able to form a harem gets five females. If so, then only 50 males will actually get to reproduce. The census size is 500 individuals but the effective population size, as shown in Remi.b's answer, is calculate by

$$N_e = \frac{4N_m N_f}{N_m+N_F}$$

where, $N_e$ is the effective population size, $N_m$ is the number of breeding males and $N_f$ is the number of breeding females. Given assumptions 1 and 2 above, $N_m$ is 50 and $N_f$ is 250 (50 males $\times$ 5 females each). Therefore,

$$N_e = \frac{4\times 50 \times 250}{50 + 250} = 166.67$$

Clearly, the effective population size is much smaller than the census population size.

Short answer: The effective population size of a population is the corresponding population size of a idealized (fisherian) population that would function in the same way with respect to genetic drift and inbreeding as the focal population under interest.

Definitions:

Heterogenity:
the probability that two randomly sampled alleles in the population are identical (identical by descent).

Census Population Size:
The census population size ($N$) is the number of individuals in the population.

Fisherian population

A fisherian population is an idealized population of size $N$, where all individuals mate at random, there is no selection, there is no population structure, there is no mutation and generations are non-overlapping. There probably are no perfectly fisherian populations in nature!

Genetic drift causes a decrease in heterogeneity. As the genetic drift is a function of the population size (see this post) this decrease of heterogeneity is a function of the population size. The heterogeneity at time $t$ is:

$$H_t = H_{0}\left( 1-\frac{1}{2N} \right)^t$$

, where $H_0$ is the heterogeneity at $t=0$. You can verify by yourself that as $t$ increase , $H_t$ decreases. Also, as $N$ decreases, $H_t$ decreases. In other words, the longer you observe, the more heterogeneity gets lost and the smaller is the population, the higher is the loss of heterogeneity.

Effective Population Size

Let Pop_A be a real population. The effective population size ($Ne$) of Pop_A is the size of an idealized fisherian population that would have the same loss of heterogeneity than Pop_A.

You can read the above sentence twice to make sure you understand it. It basically is the definition of the concept of effective population size.

In other words: Let $N$ be the population size of Pop_A. If Pop_A is a perfect fisherian population, then its loss of heterozygosity through time is given by:

$$H_t = H_{0}\left( 1-\frac{1}{2N} \right)^t$$

If Pop_A is not a perfect fisherian population, then it's decrease in heterozygosity is greater than the one given by the above formula. The decrease in heterozygosity of Pop_A corresponds to the decrease of heterozygosity of an idealize fisherian population of a different size. This size is called the effective population size $Ne$. So, we can rewrite the loss of heterogosity through time of Pop_A as:

$$H_t = H_{0}\left( 1-\frac{1}{2Ne} \right)^t$$

(Note $N$ has been replaced by $Ne$)

, where $Ne$ is the effective population size of Pop_A.

Usually $Ne < N$, meaning that a real population undergoes more genetic drift than does a fisherian population of the same census size. There are many reasons that may yield a real population to have a higher drift than it would be expected with the same census size but respecting the assumptions of the fisherian population. Below I discuss three reasons:

Examples

Unequal sex-ratio

Think as a first example of a population where the sex-ratio is extremely biased to the point of having one male for 99 females. In such population, half of the alleles of a given locus at the next generation will come directly from the father. In such situations, the genetic drift is much higher than what would be expected from a fisherian population of size 100. Therefore $Ne<N=100$. To calculate $Ne$ when the sex-ratio in unequal you can use this formula:

$$Ne=\frac{4N_mN_f}{N_m+N_f}$$

, where $N_m$ and $N_f$ are the census populatino size of males and females respectively. You can verify that when $N_f = N_m$, then $N_f + N_m = N = Ne$.

Variation in population size through time

If your population size varies throughout a given time (a year for example) and that at time $t=3$ (a given period of the year) the census population size is given by $N_i$, then the effective population size is:

$$Ne = \frac{1}{\frac{1}{t}\sum_{i=1}^t\frac{1}{N_i}}$$

This is particularly relevant for insects for example which population size varies a lot between seasons.

Selection

Think now of a population where there is a great variance in fitness (there is selection). In such population, maybe all offspring will be produced by only a tenth of the current population. As a result the effective population is ten times smaller than the census population size.

You can probably find all these informations from wikipedia: http://en.wikipedia.org/wiki/Effective_population_size

Let me know if the concept of the effective population makes more sense to you now.

• I hope you are ok with my edit suggestions. – fileunderwater Sep 8 '14 at 10:05