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is affected as the harmonic mean of population sizes in each generation, or 1/Ne = 1/t (11N, + 11N, + ... + 1/N,), where t = time in generations (Franklin 1980). Comparison of the following two populations illustrates the importance of fluctuations to Ne. In the first, there are 100 fish for each of 5 consecutive generations, for an arithmetic mean of 100; 1/Ne = 1/5 (1/100 + 1/100 + 1/100 + 1/100 + 1/100) = .01, and Ne = 100. In the second case, the arithmetic mean is also 100, but the population fluctuates each generation as follows: 100, 10, 300, 10, 80; 1/Ne = 1/5 (1/100 + 1/10 + 1/300 + 1/10 + 1/80) = .045, and Ne = 22. In this case, Ne for five generations is reduced by 78% through popu- lation crashes. The importance of Ne to population genetic structure is immediately realized in consideration of three closely related problems of small populations: bottlenecks, drift, and in- breeding. Genetic Bottlenecks. A bottleneck is a sudden and dra- matic decline in numbers (Nei et al. 1975). Bottlenecks ef- fectively sample (althou&h not necessarily randomly) a few individuals from a larger gene pool, resulting in a remnant population with less overall variation. The degree to which the new population is genetically depauperate will depend at least on genetic diversity of the source population, size of the bottleneck, and the degree of randomness of selection of individuals. Loss of variation during a bottleneck has two components (Nei et al. 1975). First, is reduction in variance of quantitative traits. The proportion of quantitative variation remaining after a single bottleneck is approximately 1 - 1 GN ,where N is the number of individuals surviving (Frankel and Soule 1981). A small number of individuals contains most of the (D _Z Z_ Q W fY W U Q tY Q o. POPULATION SIZE (Ne) Figure 1. Proportion of original genetic variance remaining in pop- ulations of various sizes after a single bottleneck. 16 genetic variation of a source population (Fig. 1): two indi- viduals contain 75% and 10 contain 95%. Therefore, unless a bottleneck is very severe or is prolonged over several gen- erations, it will not drastically reduce the amount of quan- titative variation. Second, and more critical, is loss of specific, and usually rare alleles in a bottleneck (Denniston 1978). Alleles at fre- quencies of, say, 5% or less,. contribute little to overall ge- netic variance, but may periodically be important to the population as a whole (Frankel and Sould 1981). Figure 2 illustrates the effect of bottlenecks of various sizes on rare alleles, assuming a locus with initially six alleles at frequen- cies of .90, .02, .02, .02, .02 and .02. With a bottleneck pop- ulation size of 10, there would be an average loss of more than three alleles at this locus. Considering the repetition of this loss over hundreds or even thousands of similarly variable loci, such losses can be considerable. Genetic Drift. Genetic drift is random change in gene fre- quency due to sampling error in small populations (Li 1976). It is, in effect, a prolonged bottleneck leading to repeated loss of variance until, in its ultimate form, all loci are fixed, with complete absence of genetic variance. Even in the pres- ence of moderate selection pressures, drift can be a potent force in small populations (Wright 1931, 1948). The depletion-/of genetic variation through drift is esti- mated by l1 - I L t, where t is the number of genera- 6 Ln W 5 J W J J 4 Q O 3 r'r W 2 co Z POPULATION SIZE (Ne) Figure 2. Expected number of alleles present after a single bottle- neck in populations of various sizes, derived from a source pop- ulation with six alleles at frequencies of .90, .02, .02, .02, .02 and .02. Expected number of alleles remaining at a locus is calculated as E(n) = m - E (1 - p,)-,, where m = original number of alleles and p; is the frequency of the ith allele (from Denniston 1978) Fisheries, Vol. 11, No. 1 1 10 100 1 10 100