Population genetic consequences of the Allee effect and the role of offspring-number variation

Abstract

A strong demographic Allee effect in which the expected population growth rate is negative below a certain critical population size can cause high extinction probabilities in small introduced populations. But many species are repeatedly introduced to the same location and eventually one population may overcome the Allee effect by chance. With the help of stochastic models, we investigate how much genetic diversity such successful populations harbor on average and how this depends on offspring-number variation, an important source of stochastic variability in population size. We find that with increasing variability, the Allee effect increasingly promotes genetic diversity in successful populations. Successful Allee-effect populations with highly variable population dynamics escape rapidly from the region of small population sizes and do not linger around the critical population size. Therefore, they are exposed to relatively little genetic drift. It is also conceivable, however, that an Allee effect itself leads to an increase in offspring-number variation. In this case, successful populations with an Allee effect can exhibit less genetic diversity despite growing faster at small population sizes. Unlike in many classical population genetics models, the role of offspring-number variation for the population genetic consequences of the Allee effect cannot be accounted for by an effective-population-size correction. Thus, our results highlight the importance of detailed biological knowledge, in this case on the probability distribution of family sizes, when predicting the evolutionary potential of newly founded populations or when using genetic data to reconstruct their demographic history.

Keywords: family size; founder effect; genetic diversity; introduced species; stochastic modeling.

Figure 1

The expected number of surviving offspring per individual (see Equation 1) as a function of the current population size without an Allee effect (no AE, shaded line) or with an Allee effect (AE, solid line) and critical size a = 50 (indicated by a dotted vertical line). k₁ = 1000; r = 0.1.

Figure 2

Consequences of the Allee effect for the population genetics and dynamics of successful populations under the binomial (first column: A, E, I), Poisson (second column: B, F, J), Poisson–Poisson (third column: C, G, K), and Poisson-geometric model (fourth column: D, H, L). Top row: proportion of genetic variation maintained with an Allee effect (AE, solid lines) or without (no AE, shaded lines). Middle row: percentage change in genetic diversity in Allee-effect populations compared to those without an Allee effect (see Equation 6). Dashed lines in the top two rows represent populations whose size trajectories were simulated from the respective offspring-number model, but where the genealogies were simulated assuming the Poisson model. Dotted lines show the results for the respective effective-size rescaled Poisson model. Bottom row: average number of generations spent by successful populations at each of the population sizes from 1 to z − 1 before reaching population size z, either with an Allee effect (solid lines) or without (shaded lines). The founder population size for the plots in the bottom row was 15. Note that in the case of the rescaled Poisson model, the values on the x-axis correspond to the founder population sizes before rescaling. Dotted vertical lines indicate the critical size of Allee-effect populations in the original model. Every point in A–L represents the average over 20,000 simulations. For the proportion of variation maintained, the maximum standard error of the mean was 0.0019. Parameters in the original model: k₁ = 1000, k₀ = 10,000, z = 100, r = 0.1.

Figure 3

Population-genetic consequences of the Allee effect under the mate-finding model where offspring-number variation differs between populations with and without an Allee effect. (A) Proportion of genetic variation maintained by populations without an Allee effect (a = 0, offspring model: Poisson), with a component Allee effect (a = 0, offspring model: mate finding with m = 0.005), or with a demographic Allee effect (a = 50, offspring model: mate finding with m = 0.005). Each point represents the average over 20,000 successful populations. The maximum standard error of the mean was 0.0017. (B) Corresponding values for the average number of generations that successful populations starting at size 15 have spent at each of the population sizes from 1 to z − 1 before reaching the target population size z. k₁ = 1000, k₀ = 10,000, z = 100, r = 0.1.

Figure 4

Overview over the various mechanisms by which the Allee effect influences the amount of genetic variation in successful introduced populations. Arrows represent positive effects while T-shaped symbols represent inhibitory effects. Variation in offspring number enhances the speed-up of population growth caused by the Allee effect, but prevents the Allee effect from causing a strong slow-down in population growth above the critical size. Variation in offspring number also magnifies the genetic consequences of both a speed-up and a slow-down in population growth. Dashed lines represent additional processes acting in the mate-finding model: The Allee effect increases offspring-number variation and therefore decreases genetic variation.