Approximate Bayesian estimation of extinction rate in the Finnish Daphnia magna metapopulation

Abstract

Approximate Bayesian computation (ABC) is useful for parameterizing complex models in population genetics. In this study, ABC was applied to simultaneously estimate parameter values for a model of metapopulation coalescence and test two alternatives to a strict metapopulation model in the well-studied network of Daphnia magna populations in Finland. The models shared four free parameters: the subpopulation genetic diversity (θS), the rate of gene flow among patches (4Nm), the founding population size (N0) and the metapopulation extinction rate (e) but differed in the distribution of extinction rates across habitat patches in the system. The three models had either a constant extinction rate in all populations (strict metapopulation), one population that was protected from local extinction (i.e. a persistent source), or habitat-specific extinction rates drawn from a distribution with specified mean and variance. Our model selection analysis favoured the model including a persistent source population over the two alternative models. Of the closest 750,000 data sets in Euclidean space, 78% were simulated under the persistent source model (estimated posterior probability = 0.769). This fraction increased to more than 85% when only the closest 150,000 data sets were considered (estimated posterior probability = 0.774). Approximate Bayesian computation was then used to estimate parameter values that might produce the observed set of summary statistics. Our analysis provided posterior distributions for e that included the point estimate obtained from previous data from the Finnish D. magna metapopulation. Our results support the use of ABC and population genetic data for testing the strict metapopulation model and parameterizing complex models of demography.

Fig. 1

Graphical representation of the simulated coalescent model, with four populations instead of the 14 considered in our analysis. Subpopulation effective sizes (width of population bars), pairwise migration rates (solid and dotted arrows), and founding population sizes (width of small boxes in the history of each population) were constrained to be equal across the system. The model illustrated corresponds to the strict metapopulation model or the variable e model, as all patches were recolonized in the recent past.

Fig. 2

Prior (dotted lines) and posterior distributions (solid lines) estimated for the four parameters of the persistent source model. Estimates are from an ABC analysis considering only the data sets simulated with a persistent source population and using the neural net methodology with a tolerance of 25% (250 000 data sets).

Fig. 3

Euclidean distances between standardized summary statistics from simulated and observed data. Distances for simulations conducted under a strict metapopulation model (dashed line; n = 1,000,000), a persistent source model (dotted line; n = 1,000,000), a model that allowed among population variation in extinction rate (dot-dashed line; n = 1,000,000), and pseudo-observed data sets (PODS) simulated under the persistent source model with parameter values drawn from posterior distributions (heavy solid line; n = 1000) are shown.