Experimental studies on genetic structure carried on in the last 10 years have been used as a benchmark to compare the results produced by our mechanistic, i.e. process-based, model of eel demography, and population genetics under alternative and competing assumptions on large-scale geographic differentiation and small-scale temporal differentiation. The demographic component describing the eel life cycle during the pre-reproductive continental phase was built over the extensive modeling work carried out in the last 20 years by Vøllestad and Jonsson (1988) in Norway, De Leo and Gatto (1995) in Italy, Dekker (2000) in the Netherlands, Bevacqua et al. (2007) in France, etc. As for the migration of eel larvae from the Sargasso Sea to the continental waters, we relied on the results of Kettle and Haines (2006) and Bonhommeau et al. (2009) to estimate larval survival, while we used information on the estimated abundance of silver eels leaving the continental waters (Dekker 2000) and eel fertility (Boëtius and Boëtius 1980) to derive a realistic range of oceanic survival of spawners and their reproductive success. With the exception of an over-simplified version of eel whole life cycle by Åström and Dekker (2007), which did not account for the spatial structure of the stock in the continental phase, no models have been published yet in the literature describing neither the full life cycle nor the genetic structure of the European eel.
Population dynamic features were simulated by a life-stage- and age-structured model from a continental perspective, as described hereafter and in Fig. 1. In order to explore the hypothesis proposed in earlier studies (Daemen et al. 2001; Wirth and Bernatchez 2001, Wirth and Bernatchez 2003 and Maes and Volckaert 2002) on the large-scale geographic structure of eel stock and to enable the joint assessment of spatial and temporal variation while keeping the population structure straightforward, we simplified the model for analytic reasons assuming that:
Figure 1. Life cycle of the European eel and model parameters: h (parameter controlling glass eel survival, Appendix A); δ, γij, ρ (parameters controlling sexual differentiation and maturation, Appendix A); M and F (natural and fishing mortality on the continent, Appendix A); Lij (body length, Appendix A); σL and σS (larval and silver eel survival rates, Appendix B); ASI and LSI (adult and larval segregation indices, eqns 1, 2); NB (total number of breeders); NG (number of reproductive groups); and NBGe (effective number of breeders per group, eqn 3).
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the eel stock in the continental phase is divided into three major subpopulations (Fig. 2
) corresponding to Northern (N, north of 50°N), Atlantic (A, between 35°N and 50°N), and Mediterranean (M, south of 35°N) regions;
the SG is also divided into three putative distinct areas (N-SG, A-SG, and M-SG) representing the primary spawning area of the corresponding hypothetical continental subpopulations.
the three subpopulations can be connected, either weakly (discrete populations) or strongly (panmixia), depending on how many individuals from one subpopulation end up in a different subpopulation during their journey either from the Sargasso Sea to continental waters in the larval phase or from the continental waters to the Sargasso Sea as reproductive adults, as described hereafter.
Figure 2. Schematic representation of the population structure hypotheses tested in our model. Following larval dispersal, the groups of larvae are delivered to the subpopulation corresponding to their spawning area or to one of the other two subpopulations according to ψk,j (top scheme). Similarly, following adult migration, silver eels can end in the spawning ground corresponding to their subpopulation or in one of the other two spawning grounds according to ϕk,j (bottom scheme). N, Northern; A, Atlantic; M, Mediterranean.
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Let ψk,j be the probability of progeny produced in spawning area k (k = 1 for N-SG, 2 for A-SG, and 3 for M-SG) to be delivered to continental region j (j = 1 for N, 2 for A and 3 for M):
where j, k = 1, 2, 3 and 0 ≤ LSI ≤ 1 and LSI represents a larval segregation index (0 corresponds to complete random mixing and 1 to complete segregation).
Similarly, the fraction ϕj,k of silver eels migrating from the continental region j to the spawning area k is computed as
where j, k = 1, 2, 3 and 0 ≤ ASI ≤ 1 and ASI represents an adult segregation index (Fig. 2). As a consequence, ASI = LSI = 1 represents the extreme case of three completely distinct and independent populations; ASI = LSI = 0 the opposite case of perfectly overlapping SGs or events (i.e. of a single homogenous population); any combination of ASI and LSI between 0 and 1 implies some gene flow among subpopulations. ASI > 0 can be also interpreted as temporal separation of breeders: for instance, in the extreme case of ASI = 1, we can either assume that there exist three different SGs or, alternatively, that there exist three subpopulations breeding potentially in the same spawning area but at different times during the reproductive season. This may happen if the migration to the Sargasso Sea is scarcely or not completely synchronized among eels coming from distant geographic regions, concordant to departure time variation (Anthony Acou, pers. comm.). In summary, the model can mimic any combination of spatial and temporal segregation that would lead to an overall level of adult segregation measured by the parameter ASI.
In order to account for small-scale structure of the breeding stock, as suggested by Pujolar et al. (2009) and Maes et al. (2006), breeding in each spawning areas was assumed to occur in small spawning groups where reproduction effectively takes place. Reproduction was never observed in the wild; laboratory studies are contrasting, suggesting a batch spawning as well as a mass spawning behavior (Pedersen 2003; van Ginneken and Maes 2005). In the present model, we assumed spawning to take place in a single mass event among eels belonging to the same reproductive group (van Ginneken and Maes 2005).
For a given number of breeders NB, the size of each group (NBG, number of breeders per group) depends upon the number of groups NG in which breeders are randomly divided:
Offspring originating in each spawning group determines a glass eel recruitment wave, assigned to a specific continental region. The sex ratio of reproductive eels was taken into account by defining an effective number of breeders per group as:
where η is the fraction of females in each spawning event (Crow and Kimura 1970; see Appendix B for the estimation of η).
Each cohort in each subpopulation in the continental phase is characterized by its own genotype frequency distribution (GFD). For those silver eels surviving the migration back to the Sargasso Sea for reproduction, the genotype of each breeding individual is randomly drawn from the GFD of the cohort and subpopulation it belongs to. Number of eggs produced by each female in each reproductive group is estimated from its body size according to Boëtius and Boëtius (1980) and eggs are randomly fertilized by breeding males of the same reproductive group. Offspring genotype frequencies are calculated as the product of male and female gamete frequencies (Crow and Kimura 1970, p. 44). Mutation occurs before union of gametes according to a stepwise mutation model with mutation rate μ = 5 × 10−4 (Balloux and Lugon-Moulin 2002). Breeders die after reproduction (semelparity). Offspring produced in each reproductive group remain subdivided into larval waves characterized by their own GFD. When these larval waves approach the continental coasts, individuals that survived the migration from the Sargasso Sea metamorphose and represent new glass eel recruitment waves that are then recruited in one of three continental sub-populations according to eqn (1). This means that each wave comes from only a single spawning area. The GFD of glass eel waves are used to estimate genetic differentiation without taking into account the sampling error encountered in empirical studies (see ‘Indices of genetic differentiation’). The estimation of genetic differentiation indices occurs before merging the glass eel waves of the same cohort in each continental region so as to derive the GFD of the newly recruited cohort.
The process of drawing genotypes for each reproducing individual from the corresponding GFD of the cohort it belongs to as well as the assignment of larval groups to different continental regions according to eqn (1) add a stochastic component to the model, all the other parameters being constant. To account for these sources of variability, the indices of genetic differentiation were averaged on 100 simulation replicates for each parameter set (ASI, LSI, NG, etc.). We initially set genotype frequencies in Hardy–Weinberg proportions, based on allelic frequencies drawn from a uniform distribution. Simulations were run for 300 years, which was largely sufficient for genetic differentiation to converge to stable values (drift-migration equilibrium), except for extreme cases of segregation (i.e. ASI or LSI ≥ 0.99) for which convergence was slower, requiring running the model for 3000 years.
The model component describing eel demography in the continental phase explicitly accounts for key aspects of its life history, such as natural and fishing mortality (Dekker 2000), sex-specific body growth (as silver females are remarkably larger than silver males; Melià et al. 2006a), size- and sex-specific maturation rates (as females takes substantially longer than male to reach sexual maturity; Bevacqua et al. 2006), and size-dependent fertility of females (as larger females produce substantially more eggs than smaller females; Boëtius and Boëtius 1980). Moreover, the model accounts for geographic variations in life-history traits, as yellow eels in the Mediterranean area grow faster and take less to reach sexual maturity than eels from the central and northern Europe (Vøllestad 1992) but suffer higher mortality rates. Accordingly, a detailed demographic model was developed so as to mimic these geographic and between-sex differences in age and size at sexual maturity and to allow us to realistically simulate the gene flow occurring when spawners from different cohorts and different geographic areas mate in the Sargasso Sea. A detailed description of the full demographic model and its parameterization is reported in Appendix A. By setting larval survival equal to 1.5 × 10−3 as in Bonhommeau et al. (2009), survival of silver eels migrating to the Sargasso Sea and density-dependent mortality of glass eels have been derived under steady-state hypothesis (Appendix B), by assuming that silver eel abundance at equilibrium is about 2 × 107 and glass eel abundance 109 as estimated by Dekker (2000). All the other demographic parameters have been set according to published works.
Indices of genetic differentiation
Genetic differentiation was estimated using genotype and allele frequencies of glass eels measured through hierarchical F-statistics. The components of the overall differentiation between recruitment waves FST were partitioned into a within-subpopulation within-cohort FSC (temporal component) and a between-subpopulation within-cohort component FCT (geographic component). In empirical studies, estimates of genetic differentiation are derived from sample data; therefore, statistical techniques are used to correct for sampling bias [e.g. the θ parameter developed by Weir and Cockerham (1984) to estimate FST]. In our model, however, glass eel genotype frequencies are assumed to be known and can be used directly to define the true FST and related indices as ratio of heterozygosities. FSC is computed using all the recruitment waves at a continental region:
where HC is the expected within-subpopulation heterozygosity, computed as the weighted average of the expected heterozygosities in the three geographically distinct subpopulations; HS is the expected within-recruitment wave heterozygosity, computed as the weighted average of the expected heterozygosities in all the recruitment waves within a region. Between-subpopulation differentiation is computed as
where HT is the expected heterozygosity in the whole population. Finally, the overall index of genetic differentiation FST is:
These indices satisfy the well-known relationship (1 − FST) = (1 − FSC)(1 − FCT).
A first set of simulations was run by using 5–10–20 alleles, respectively, and a number of loci ranging from 1 to 10. As reported in Appendix D, a sensitivity analysis showed that the number of loci and alleles did not significantly affect model results for FST, FSC, and FCT. Hence, even though multiple loci would provide greater precision in a single model run, for reasons of computational tractability, genetic structure was modeled by using only one locus with 10 alleles.
The model was used to estimate values of FST, FCT and FSC under different assumptions about (i) the level of small-scale genetic differentiation, modulated through the number of reproductive groups (NG = 1500, 3000 and 6000); (ii) the number of breeders (NB = 0.42 × 106, 0.83 × 106 and 1.66 × 106); and (iii) the connectivity level among subpopulations (LSI and ASI = 0, 0.5, 0.99 and 1; the combination ASI = LSI = 1 was not considered because it corresponds to three independent populations). The F-statistics were computed by both using all the glass eel waves recruiting at each continental region and a finite number (i.e. 5 and 10) of glass eel waves to mimic the fact that only a subset of arrival waves is actually sampled in field studies.
Results were used to investigate the relationship between FSC and NBGe and whether a particular value of NBGe yields an FSC equal to that estimated from microsatellites in Maes et al. (2006), i.e. 0.0018 ± 0.0014 (hereafter this specific value is referred to as NBGe*). Similarly, simulation results were used to explore if the same level of genetic differentiation could be derived by assuming large-scale geographic segregation. Finally, the model was used to investigate how FSC, FCT, and FST estimates change when a small number of glass eel waves per continental region are randomly sampled. Median values of the indices of differentiation were compared by a Kruskal–Wallis nonparametric ANOVA (KW test).