So, Ne = 1/0.04 = 25. Contrast this with the arithmetic mean of the census population size, which is 70. Only those alleles that actually pass through the genetic bottleneck of 10 individuals are represented in later generations, regardless of how large the census size is, so the mean census size is much too high to use to predict the behavior of allele frequencies since it will underestimate genetic drift. In this case, we expect allele frequencies in the population to behave similarly to allele frequencies in an ideal Wright-Fisher population with a constant size of 25 over three generations. Like finite population size, fluctuations in population size through time are a universal feature of biological populations. Populations obviously vary greatly in the degree of size fluctuations and the time scale of these fluctuations, but Ne < N caused by temporal fluctuations in N is a widespread phenomenon.

Founder effect The establishment of a population by one or a few individuals, resulting in small effective population size in a newly founded population. Genetic bottleneck A sharp but often transient reduction in the size of a population that increases allele frequency sampling error and has a disproportionate impact on the effective population size in later generations even if census sizes increase.

Although the term bottleneck is usually associated with sharp reductions in the overall population size, there are other aspects of biological populations that have the same impact by increasing the sampling error in allele frequency across generations. Mating patterns can have a major impact on effective population size when individuals of different sexes make unequal contributions to reproduction. This occurs in populations where individuals of one sex compete for mating access to individuals of the other sex. In such a situation the numbers of females and males that breed, or the breeding sex ratio, may not be equal (even if the population sex ratio is equal). The leads to increased genetic drift compared with the case of a breeding sex ratio of 1:1, since the pool of alleles passed to the next generation will be sampled from fewer individuals in one sex. Thus, the less frequent sex becomes an allelic bottleneck of sorts. The effective population size in such cases is:

where Nf is the number of females and Nm is the number of males breeding in the population and all other assumptions of Wright-Fisher populations are met. Equation 3.44 shows that the effective population size approaches four when the rarer sex approaches a single individual and that the effective size is maximized when there is a breeding sex ratio of 1 : 1.

Let's look at an case where Nm does not equal Nf to see the impact on the effective population size. Elephant seals (Mirounga leonina) are a classic example of highly unequal breeding sex ratios since the mating system is harem polygyny. In one study of breeding patterns on Sea Lion Island in the Falkland Islands, about 550 females and 75 males were observed on land where mating takes place (Fabiani et al. 2004). Using genetic markers to ascertain the parentage of pups, it was determined that only 28% of the males fathered offspring during the course of two breeding seasons. Therefore, the breeding sex ratio was about Nm = 21 and Nf = 550. The effective population size during each breeding season was:

or was equivalent to an ideal population of 40 females and 40 males where breeding sex ratio is 1 : 1. The strongly unequal breeding sex ratio for elephant seals results in an effective population size an order of magnitude less than the census size of 625 individuals.

A third factor that distinguishes the census and effective population sizes is the degree to which adult individuals in the population contribute to the next generation. One of the assumptions of Wright-Fisher populations is that all individuals contribute an equal number of gametes to the infinite gamete pool. To maintain a population that is not changing in size across generations, each individual must produce one surviving progeny, on average, to replace itself. In outcrossing species, when each pair of individuals produces an average of two progeny, then the population will be stable in size over time. In terms of the Wright-Fisher population, that means that each individual contributes an average of two gametes to the next generation from the infinite gamete pool.

There are many patterns of individual reproduction within an outcrossing population that can achieve a mean rate of reproduction that results in a stable population through time. It might be the case that all individuals produce exactly one progeny. Another possibility is that a few parents produce no offspring, whereas most parents produce two offspring, and a few parents produce four offspring that offset the reduction in the average caused by the parents with no progeny. In the extreme, one pair of parents could produce all N offspring and all other pairs fail to reproduce successfully. The variance in family size can be used to describe these different patterns of individual reproduction. As variance in family size increases, the alleles passed to the next generation come increasingly from those parents producing more offspring.

The effective population size due to variation in family size is:

where Nt-1 is the size of the parental population and k is the number of gametes that result in progeny, or family size for outcrossing organisms (Crow & Denniston 1988). The equation shows that for a stable population (Y = 2) when the variance in family size is equal to the average family size, then there is no "bottleneck" due to family size variation: the population size of parents is the effective population size. The Wright-Fisher model assumes that production of progeny has exactly this quality of the mean family size being equal to the variance in family size. In essence the Wright-Fisher model assumes that family sizes follow a Poisson distribution (the Poisson distribution is an approximation of the binomial distribution as the sample size grows very large and the chance of a given outcome becomes very small; see Denny & Gaines 2000 or the appendix in Rice 2004).

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