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Robustness of Hill’s overlapping-generation method for calculating Ne to extreme patterns of reproductive success


For species with overlapping generations, the most widely used method to calculate effective population size (Ne) is Hill’s, the key parameter for which is lifetime variance in offspring number (\({V}_{k\bullet }\)). Hill’s model assumes a stable age structure and constant abundance, and sensitivity to those assumptions has been evaluated previously. Here I evaluate the robustness of Hill’s model to extreme patterns of reproductive success, whose effects have not been previously examined: (1) very strong reproductive skew; (2) strong temporal autocorrelations in individual reproductive success; and (3) strong covariance of individual reproduction and survival. Genetic drift (loss of heterozygosity and increase in allele frequency variance) was simulated in age-structured populations using methods that generated no autocorrelations or covariances (Model NoCor); or created strong positive (Model Positive) or strong negative (Model Negative) temporal autocorrelations in reproduction and covariances between reproduction and survival. Compared to Model NoCor, the other models led to greatly elevated or reduced \({V}_{k\bullet }\), and hence greatly reduced or elevated Ne, respectively. A new index is introduced (ρα,α+), which is the correlation between (1) the number of offspring produced by each individual at the age at maturity (α), and (2) the total number of offspring produced during the rest of their lifetimes. Mean ρα,α+ was ≈0 under Model NoCor, strongly positive under Model Positive, and strongly negative under Model Negative. Even under the most extreme reproductive scenarios in Models Positive and Negative, when \({V}_{k\bullet }\) was calculated from the realized population pedigree and used to calculate Ne in Hill’s model, the result accurately predicted the rate of genetic drift in simulated populations. These results held for scenarios where age-specific reproductive skew was random (variance ≈ mean) and highly overdispersed (variance up to 20 times the mean). Collectively, these results are good news for researchers as they demonstrate the robustness of Hill’s model even in extreme reproductive scenarios.

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Fig. 1: Mean values of the lifetime variance in reproductive success (Vk) for simulated data.
Fig. 2: Observed (colored symbols) and expected (black lines) rates of genetic drift in a simulated population under Model Positive, Scenario ModerateSkew (in which ϕ = 5 for all ages in males).
Fig. 3: Observed (colored symbols) and expected (black lines) rates of loss of heterozygosity in simulated populations for three models that lead to positive correlations (Model Positive), negative correlations (Model Negative), and independence of individual reproductive success over time (Model NoCor).
Fig. 4: Observed (colored lines) and expected (solid black line) decline in observed heterozygosity for 10 different sets of 50 diallelic loci tracked on a single, 500-year pedigree.

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Data availability

All results presented here were generated by simulations. R code to conduct the simulations is available in Supplementary Information.


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The author is grateful to Bill Hill for many insightful discussions over the years, relating to effective population size as well as other topics. I thank Steinar Engen and Bernt-Erik Saether for useful discussions. Per Erik Jorde provided comments that substantially improved the manuscript.

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Correspondence to Robin S. Waples.

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Waples, R.S. Robustness of Hill’s overlapping-generation method for calculating Ne to extreme patterns of reproductive success. Heredity 131, 170–177 (2023).

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