Abstract
Pacific salmon are semelparous but have overlapping year classes, which presents special challenges for the application of standard population genetics theory to these species. This article examines the relationship between the effective number of breeders per year (N_{b}) and singlegeneration and multigeneration effective population size (N_{e}) in salmon populations that fluctuate in size. A simple analytical model is developed that allows calculation of N_{e} on the basis of the number of spawners in individual years and their reproductive contribution (productivity) to the next generation. Application of the model to a 36year time series of data for a threatened population of Snake River chinook salmon suggests that variation in population dynamic processes across years reduced the multigeneration N_{e} by ∼40–60%, and reductions may have been substantially greater within some generations. These reductions are comparable in magnitude to, and in addition to, reductions in N_{b} within a year due to unequal sex ratio and nonrandom variation in reproductive success. Computer simulations suggest that the effects of variable population dynamics on N_{e} observed in this dataset are not unexpected for species with a salmon life history, as random variation in productivity can lead to similar results.
IT has long been known that the genetic behavior of a population depends not on the number of individuals it contains (N) but rather on its effective population size (N_{e}; Wright 1931 and later). In an “ideal” population (random mating, binomial variance in reproductive success among individuals) N_{e} is equal to N, but in real populations N_{e} will generally be <N because of unequal sex ratio (resulting in unequal reproductive success of the average male compared to the average female) and larger than binomial variance in reproductive success among individuals of the same sex. Wright (1938) also showed long ago that if population size varies over time, a population behaves genetically as if it had a constant N_{e} approximately equal to the harmonic mean of the singlegeneration N_{e} values.
This theory of effective population size was originally developed to model genetic processes in organisms with discrete generations. Since most species do not fit the assumptions of the discrete generation model, various authors have evaluated robustness of this theory for species with more complex life histories. In general, these studies have found that discretegeneration models for effective population size also provide a good description of processes of genetic change in organisms with overlapping generations, provided that demographic parameters of the population are stable (Felsenstein 1971; Hill 1972).
Demographic parameters are not stable, however, in populations that change in size. Fluctuating population size is an important consideration for evolutionary biologists because variability in N is one of the most important factors that determine extinction risk. Furthermore, because the rate and/or magnitude of most genetic processes are inversely related to N_{e}, the genetic effects of small population size are nonlinear; for example, although it may make little difference in the short term whether a population has N_{e} of 10^{3} or 10^{4}, it can make a great deal of difference whether N_{e} is 10 or 10^{2}. It is important, therefore, to consider in more rigorous detail the concept of effective size for populations with complex life histories that are (or may become) relatively small and that also fluctuate in size.
Here I consider this topic for Pacific salmon (Oncorhynchus spp.). These species have an unusual life history that combines features of both discrete and overlapping generation models (Figure 1): Adults invariably die after spawning (so there is no overlap in the breeding population from one year to the next—a feature shared with discrete generation models), but most species and populations produce offspring that mature at a variety of ages (which means that breeding populations in different years are not connected by a firstorder Markov process—a feature shared with overlapping generation models). Waples (1990a,b) modeled genetic changes over time in Pacific salmon populations and examined the relationship between the effective number of breeders per year (N_{b}) and the effective size per generation (N_{e}). He showed that when population size is constant, N_{e} per generation is simply the sum of the yearly N_{b} values over a period of a generation: N_{e} = gN_{b}, where g is the generation length (average age at spawning).
Waples (1990b) also performed a limited number of simulations with variable population size and concluded that N_{e} for Pacific salmon is a function of the harmonic mean of the N_{b} values in individual years:
It turns out, however, that Equation 1 is valid only for a particular demographic assumption implicit in Waples' model—specifically, that each year's spawning population contributes equally to the next generation regardless of the number of spawners. If instead we assume that each year's spawning population contributes to the next generation in direct proportion to the number of spawners, then (as is shown below) the relationship
The harmonic mean is smaller than the arithmetic mean for any variable series, so N_{e} computed using Equation 1 will be less than the value obtained using Equation 2. The difference can be substantial, as illustrated by a simple time series of abundance data for a salmon population with mean generation length of 4 years: N_{1} = 100; N_{2} = 100; N_{3} = 10; N_{4} = 100. Assume for the moment that N_{b}_{i} = N_{i} each year. The arithmetic mean for this data series is N_{b} = 77.5 whereas the harmonic mean is only Ñ_{b} = 30.8, leading to Equation 1, N_{e} = gÑ_{b} = 4 × 30.8 = 123; and Equation 2, N_{e} = gN_{b} = 4 × 77.5 = 310. In this example, the computed N_{e} per generation differs by a factor of almost 3 depending on the demographic assumption used.
It is clear from this simple example that population dynamic processes can profoundly influence effective population size in species with Pacific salmon life histories. In particular, whereas N_{b} is a function only of demographic processes occurring within a single cohort, N_{e} per generation is also a function of the relative reproductive success (productivity) of different cohorts within a generation. In this article I examine this issue in more depth using both analytical and simulation approaches, with the objective being to determine which of these ways of computing N_{e} is more realistic for Pacific salmon.
METHODS
Definition of terms:

N_{i} is the number of spawners in year i

N_{T} is the total number of spawners in a generation (= ΣN_{i} over a generation)

g is the generation length (average age at spawning)

N_{b}_{i} is the effective number of breeders in year i

N_{e} is the effective population size in a generation

N_{e(}_{k}_{)} is the effective population size over k generations (harmonic mean of singlegeneration N_{e} values)

R_{i} is the recruits = spawners in the next generation produced by spawners in year i

R_{T} is the total spawners in the next generation produced by all the spawners in the current generation (= Σ R_{i} over a generation)

λ_{i} is the productivity of spawners in year i (R_{i}/N_{i})

σ is the standard deviation of λ

X_{i} is the proportional contribution of spawners in year i to the next generation (R_{i}/R_{T}).
Analytical model: The genetic consequences of the contrasting demographic assumptions implicit in Equations 1 and 2 can be evaluated quantitatively by use of a model developed by Ryman and Laikre (1991). Although their article focused on amplification of part of a population's gene pool through a captive breeding program, Ryman and Laikre also provided a more general formula for evaluating the genetic effects of differential reproductive success by different segments of a population,
Scenario 1: Spawners in each year contribute an equal number of progeny to the next generation, regardless of N_{b}_{i} or N_{i}. (All X_{i} = 1/g.)
Scenario 2: Spawners in each year contribute to the next generation in proportion to N_{b}_{i}. (X_{i} = N_{b}_{i}/N_{T}.)
Equation 3 was used with empirical data for salmon to compute N_{e} for comparison with predictions based on Equations 1 and 2.
Empirical data: Marsh Creek, in central Idaho, is a tributary of the Middle Fork Salmon River, which flows into the Salmon River and thence the Snake River. All native chinook salmon populations in the Snake River were listed as threatened under the U.S. Endangered Species Act in 1992 (Federal Register 57(78):14653–14662, 22 April 1992). A time series (1958–1993) of abundance data for Marsh Creek spring chinook salmon (Beamsderferet al. 1998 and unpublished data) was used to provide an example for analytical evaluation of single and multigeneration N_{e}. The total number of spawning adults was estimated each year on the basis of expansions from sampling a portion of the population. Estimates of the age composition of the spawners each year, on the basis of lengths measured on a sample of returning adults, allowed a partitioning of the spawners into individual cohorts and, therefore, a reconstruction of the proportional contribution of each year to the next generation (X_{i} from Equation 3).
In the Marsh Creek population, the mean frequencies of spawners ages 3, 4, and 5 over the time series of data were 0.04, 0.25, and 0.71, respectively (Beamsderferet al. 1998), leading to g ≈ 4.7 years. The 3yearold spawners are all males (called “jacks”), and the spawnerrecruit relationships were based only on age 4 and age 5 spawners (“adults”). Therefore, to apply Equation 3 to these data, the time series was divided into either 4year or 5year segments, starting backward from the most recent year (1993) for which complete adult return data were available. These segments correspond roughly to salmon generations. For each generation, an estimate of N_{e} was computed in three ways, using Equations 1, 2, 3.
Computer simulations: To evaluate more generally the effects of population dynamic processes on N_{e} in Pacific salmon, I modified the computer model used by Waples (1990b). Each simulation was characterized by an initial number of spawners per year (N_{0}) and an average age distribution (progeny of spawners in year i matured at age i + k with probability A_{k}; ΣA_{k} = 1; maximum age at spawning = A_{max} = 5). Thus, the demographic trajectory of the population followed maturity schedule B in Figure 1, which is more realistic for fluctuating populations than maturity schedule A (as considered by Waples 1990b). Each replicate was started by creating an initial population with A_{max} years of N_{0} adults having allele frequency P_{0} = 0.5.
The replacement rate, or productivity (λ), for each year was selected randomly from a lognormal distribution (Peterman 1981), with specified mean (λ) and standard deviation (σ_{λ}). The product N_{i}λ_{i} (rounded to the nearest integer) determined the number of individuals produced by that year that would mature in subsequent years. Random numbers were used to assign each of these N_{i}λ_{i} individuals and their associated genes to subsequent years' spawners. This process modeled a population that fluctuated in size but otherwise was “ideal” (N_{b}_{i} = N_{i}) within each year. To retain the fixed initial population size and allele frequency for one complete life cycle (A_{max} years), new adults maturing in years 2 to A_{max} were ignored. After allowing the system to “warm up” for 20 years to allow the random allele frequency changes among years to reach a dynamic equilibrium, population data were collected each year for up to 80 years. Because λ was chosen randomly from a series that often included both very high and very low values, N_{max} and N_{min} values were chosen to prevent the population from growing too large or going extinct.
At periodic intervals, allele frequencies in the current year were compared to those in the reference year (A_{max} + 20), and the difference was used to estimate F (Nei and Tajima 1981), the standardized variance of allele frequency change,
F has been widely used in the temporal method for estimating effective population size from allele frequency change because its expectation is well known and independent of initial allele frequency. Waples (1990b) and Tajima (1992) showed that for species with Pacific salmon life history, the following describes the relationship between F and N_{b}:
Each simulation was run 5000 times, and mean F̂ values were computed for a range of numbers of years of elapsed time. Because the interest here is on parametric genetic processes, population allele frequencies, rather than samples thereof, were used in computing F̂_{.} Loss of alleles during the simulation could downwardly bias F̂ and upwardly bias estimates of N_{e} (once an allele goes extinct it can no longer change in frequency), so the incidence of allelic extinction [defined as P_{i}(1 – P_{i}) = 0in A_{max} consecutive years] was monitored in the simulations.
RESULTS
Empirical data analysis: From the series of data for Marsh Creek chinook salmon on yearly spawner abundance and recruits (spawners in the next generation summed over all ages), it is possible to calculate recruitsperspawner ratios, or productivities (λ), for each year and the relative contribution of that year to the next generation (X_{i}; see Table 1). The data shown in Table 1 are arrayed in 4year generation blocks to facilitate analysis and discussion of the data; analysis based on 5year segments would take a similar form. With the data arranged in this way, it is straightforward to compute estimates of N_{e} using the three methods (Table 2).
N̂_{e} values computed using the harmonic mean method (Equation 1) were always lower than using the additive method (Equation 2), and the magnitude of the difference between the two estimates was largest in the generations with the greatest annual variability in spawner counts. For example, in generation 6 the spawner counts ranged 30fold among years, from 16 to 491. Whereas the arithmetic mean number of spawners per year in this generation was 176, the harmonic mean was only 47. As a result, N̂_{e} using the harmonic mean method (188) was less than onethird of the estimate from the additive method (705). In contrast, in generations 2–4 N_{i} was generally high with little annual variation, and N̂_{e's} based on the two methods were more similar.
Surprisingly, N̂_{e} based on Equation 3 was even lower than the estimate based on the harmonic mean method (Equation 1) in seven of the nine generations. This same result was found in six of the seven generations using a 5year generation length (Table 2). Inspection of data for individual generations illustrates why these effects occurred. In generation 6, the range in productivity of cohorts (nearly 100fold, from a low of λ= 0.14 in 1978 to λ> 10 in 1980) was even greater than the 30fold variation in abundance. Furthermore, these population dynamic processes occurred in such a way that the year with the largest spawner escapement (1978) had the lowest λ and the year with the lowest escapement (1980) had the highest λ. As a result, the 16 spawners in year 1980 contributed 2.5 times as many total adults to the next generation as did the 491 spawners in 1978. Put another way, the 16 adults in 1980 represented just over 2% of the total spawners in the parental generation, but they were responsible for 35% of the genes transmitted to the progeny generation. These factors greatly increased the variance in reproductive success among individuals in different years (but within the same generation), thus by definition reducing N_{e}.
A different picture is seen in generation 8 (years 1986–1989). In this case, the yearly spawner counts were more stable, ranging from 80 to 395. Although λ was below replacement for all years in the generation, the values were roughly comparable, ranging only from 0.2 to 0.69. Furthermore, the highest λ occurred in the year with the largest population size, and the year with the lowest N_{i} had relatively low reproductive success. As a consequence, the contribution of spawners in each year to the next generation (X_{i}) was much closer to the relative size of N_{i} than was the case in generation 6, and N̂_{e} calculated by Equation 3 (762) was larger than the value (650) calculated using the harmonic mean method.
The total number of adult spawners within a generation (N_{T} = ΣN_{i}) provides a benchmark for comparing N_{e} to N ratios within a generation. Because the effective size estimates shown in Table 2 assume “ideal” conditions within a year, they provide an indication of the reduction in N_{e} due entirely to the effects of annual variance in mean reproductive success. For the nine generations of data shown in Table 2, the estimates of N_{e}/N_{T} using N̂_{e} from Equation 3 ranged from 0.16 to 0.89—indicating that population dynamic processes among years can be a substantial factor in reducing effective population size in Pacific salmon. Estimates of N_{e}/N based on the 5year model were not as extreme, falling in the range 0.53–0.88 (Table 2). The difference in the estimates for the 4 and 5year generation models can be attributed primarily to the different way years 1978 and 1980 were allocated into generations. In the 4year model, these two years fell in the same generation, leading to the extreme contrast in productivity noted above and an estimated 84% reduction in N_{e}. In the 5year model, these years fell in different generations within which there was much less variance in productivity among years.
Over the entire dataset (seven to nine generations), a longterm effective size can be calculated for each of the three methods using the harmonic mean of the estimates for the individual generations. If the assumptions behind the additive model are met, then N_{e} = N_{T} within each generation, and the genetic behavior of the population over the 36year period will be a function of the harmonic mean of the singlegeneration N_{T} values. These multigeneration estimates of effective size using Equation 2 are N̂_{e(9)} =Ñ_{T(9)} = 964 for the 4year generation model and N̂_{e(7)} =Ñ_{T(7)} = 992 for the 5year generation model (Table 2), the small difference being due to 1 more year of data in the 4year model. These values provide a benchmark for comparing the longterm N̂_{e(}_{k}_{)} values computed using the demographic data (Equation 3). For the 4year model we can estimate the longterm ratio N̂_{e(9)/}Ñ_{T(9)} as 376/964 = 0.39; for the 5year model the estimate is 607/992 = 0.61. Because to this point it has been assumed that N_{b}_{i} = N_{i} each year, these reductions in the N_{e}/N ratio are due entirely to annual differences in productivity. Data shown in Table 2 suggest that for Marsh Creek chinook salmon, these population dynamic processes have reduced effective size over a 36year time period by ∼40–60%.
Although N_{b}_{i}/N_{i} < 1 was not formally considered in these analyses, all of the methods discussed here can easily accommodate annual estimates of N_{b}_{i} if they are available. In that case, the estimates of N_{e} can be scaled by the factor N_{b}_{i}/N_{i} in the individual years. For example, assuming N_{b}_{i}/N_{i} = 0.3 each year (consistent with empirical estimates for Snake River chinook salmon; Wapleset al. 1993; Waples 2002), the singlegeneration estimates of N_{e} in Table 2 would all be reduced by the factor 0.3, and singlegeneration N_{e}/N estimates for Marsh Creek chinook salmon would range from 0.05 to 0.27 (using the 4year model) and from 0.16 to 0.26 (using the 5year model).
Computer simulations: Figure 2 shows results of a simulation that mimicked parameters previously considered by Waples (1990a,b) in evaluating temporal change in Pacific salmon: A_{k} = 0.25, 0.5, 0.25 for ages k = 3, 4, and 5, respectively; N_{b}_{i} = N_{i} = 200 every year. Because N_{b}_{i} was constant,Ñ_{b} = N_{b} for every time period considered and the expectations for longterm N_{e(}_{k}_{)} and F under scenarios 1 and 2 are the same. Results shown in Figure 2 demonstrate that F̂ values calculated from the simulations over periods ranging from 10 to 80 years (2.5 to 20 generations with g = 4) agree well with expectations for salmon populations, on the basis of previous work by Waples (1990a,b) and Tajima (1992).
When λ was allowed to vary randomly among years, N_{i} and N_{b}_{i} fluctuated over time, and empirical results from the simulations allowed a comparison of the longterm genetic behavior of the fluctuating population with expectations using the additive and harmonic mean methods. In the simulation shown in Figure 3, conditions were the same as in Figure 2 except that λ for each year was chosen from a lognormal distribution with λ= 1.0 and σ_{λ} = 0.5. With this moderate level of variability (σ_{λ} = 2.3 for the Marsh Creek data), expectations under the two scenarios are quite different, and it is clear that F̂ from the simulations agrees much better with the harmonic mean method than with the additive method. In fact, for every time period the observed F̂ was even larger than expected under scenario 1, indicating that longterm N_{e(}_{k}_{)} in the modeled population was lower than expected using the harmonic mean method and much lower than expected using the additive method.
To evaluate sensitivity of this result to particular values of key variables (initial N, σ_{λ}, age structure, generation length) I conducted additional simulations encompassing a wide range of parameter sets (data not shown). The following general results were obtained:

Larger variance in λ leads to larger F̂ and smaller effective size. This result was found consistently across a wide range of age structures and other parameter values. Under most scenarios with σ_{λ} > 1, F̂ was larger (and N̂_{e} lower) than predicted using the harmonic mean method.

For a given σ_{λ}, F̂ was larger (and N̂_{e} lower) if the population had more than two age classes of spawners.

Reductions in N_{e} are greatest with an even age distribution and diminish if any single age class constitutes >70% of the spawners.
DISCUSSION
It might be assumed that an analysis that fully accounted for variable demographics would show that N_{e} for species with salmontype life history falls in the range bounded by the additive and harmonic mean methods. Results presented here show that this assumption is not true, at least in a general sense. The additive method does provide an upper bound for N_{e}; if the genetic contribution of individual cohorts to the next generation is exactly proportional to N_{b}_{i}, N_{e} cannot be increased further except by increasing N_{b}_{i}/N_{i} within years. However, it is clear that the harmonic mean method does not provide a lower limit to N_{e}, which in some cases can be much lower than predicted by Equation 1. In fact, it is easy to show using Equation 3 that N_{e} for a generation can be as small as N_{b}_{i} in a single year (i.e., if λ= 0 in all other years). Years in which X_{i} (R_{i}/R_{T}) is large relative to N_{b}_{i}/N_{T} are primarily responsible for these reductions in effective size. With random variation in productivity, this phenomenon occurs frequently due to stochastic processes alone, and because the effects on N_{e} are nonlinear they are not completely offset by years in which X_{i} and N_{b}_{i}/N_{T} are more similar.
The simulations evaluated how well F̂ and longterm N_{e(}_{k}_{)} were predicted by functions of the harmonic mean and arithmetic mean of the yearly N_{b}_{i} values. Mathematically, taking the harmonic mean of a series of N_{b}_{i} values (as in the simulations) is equivalent to first computing singlegeneration N_{e} values as the harmonic mean of the N_{b}_{i} within a generation and then taking the harmonic mean of the generational N_{e} values (as in the analytical model; see Table 2). This is not true for the additive method, since even if N_{e} per generation were an additive function of the N_{b}_{i} values within a generation, the longterm N_{e} must be a function of the harmonic mean N_{e} values per generation. Thus, we would expect that the additive method would overestimate N_{e(}_{k}_{)} unless effective size were constant across generations.
Collectively, the results presented here lead to the following conclusions: (1) Variability across years in population dynamic processes can substantially reduce effective population size in Pacific salmon; (2) this reduction is in addition to, and can be comparable in size to, reductions in the ratio N_{b}_{i}/N_{i} within individual years; (3) under most realistic conditions, the harmonic mean method is much better than the additive method in describing the relationship between N_{b} and N_{e} in Pacific salmon; (4) in Marsh Creek chinook salmon, temporal variance in productivity over a 36year period has reduced effective size by an estimated 40–60%.
The simulation results corroborate these conclusions and provide more general insight into the effects of population dynamic processes on N_{e} in Pacific salmon. This is important because the analytical models all have limitations. First, the additive and harmonic mean methods make fixed assumptions about annual productivity that are unrealistic for most real populations. Equation 3 makes no assumptions about the nature of the variation in productivity, but it must be applied to a specific time series of data. The simulations allowed me to consider random variation in productivity and a large number of time series of data.
Second, the analytical approach is somewhat artificial in that no sharp temporal boundaries exist to indicate where one generation ends and another begins in Pacific salmon (except in pink salmon, which have a fixed 2year life cycle). Estimates of N_{e} can differ depending on how the years are organized into generations (Table 2, results for the 4year vs. the 5year model). In contrast, the simulations do not deal with individual generations but instead provide information about the genetic behavior of a population over continuous periods of time.
Third, Equation 3, which was derived on the basis of the concept of identity by descent (Ryman and Laikre 1991), applies to the inbreeding effective size, which will differ from the variance effective size in populations that vary in size (as do those considered here). The simulations, which focus on allele frequency change over time, provide information about the variance effective size of a population and thus provide a check for the general relevance of Equation 3.
Finally, the demographic data for Marsh Creek chinook salmon include an unquantified magnitude of uncertainty (measurement error in counting the fish and aging the spawners and sampling error associated with estimating the total number of spawners and age structure on the basis of sampling only a portion of the population). The simulations allowed an evaluation of genetic behavior of the population under a known set of parameters. The strong agreement of the analytical and modeling results suggests that the analytical approach can provide useful insights in spite of some of its simplifying assumptions.
The simulation results show that reductions in N_{e} in Pacific salmon as large or larger than those found in Marsh Creek chinook salmon can occur through random variation in productivity among years. Collectively, the simulations show that N_{e} in Pacific salmon generally will be as low or lower than predicted using the harmonic mean method if σ_{λ} is high, if adults mature at three or more age classes, or if age distribution is even. The genetic consequences will be less severe if the variance in λ is low or if age at maturity is strongly unimodal.
Salmon are unusual, but not unique, in being semelparous yet having variable age structure; other species with these traits include opossum shrimps (Morgan 1980) and a variety of monocarpic plants (DeJonget al. 1987). In these species, annual variability in λ can be expected to affect N_{e} in a fashion similar to that for Pacific salmon. See Nunney (2002) for analysis of a closely related problem involving N_{e} in annual plants with seed banks. In general, however, the effects of fluctuating population size on N_{e} are expected to be less for iteroparous species because lifetime reproductive success will not be dominated so strongly by events that occur in only 1 year.
Acknowledgments
The impetus for this study came from a question posed by Mike Gilpin about the relationship between N_{b} and N_{e} in salmon. Eric Anderson, Mike Ford, Steven Kalinowski, and Chris Ray provided useful comments on an earlier draft. I thank Chi Do and Chris Jordan for providing an algorithm for sampling from a lognormal distribution and Pete Lawson for information on other species with life histories similar to salmon.
Footnotes

Communicating editor: F. Tajima
 Received May 23, 2001.
 Accepted March 15, 2002.
 Copyright © 2002 by the Genetics Society of America