The balance between stabilizing selection and migration of maladapted individuals has formerly been modeled using a variety of quantitative genetic models of increasing complexity, including models based on a constant expressed genetic variance and models based on normality. The infinitesimal model can accommodate nonnormality and a nonconstant genetic variance as a result of linkage disequilibrium. It can be seen as a parsimonious one-parameter model that approximates the underlying genetic details well when a large number of loci are involved. Here, the performance of this model is compared to several more realistic explicit multilocus models, with either two, several or a large number of alleles per locus with unequal effect sizes. Predictions for the deviation of the population mean from the optimum are highly similar across the different models, so that the non-Gaussian infinitesimal model forms a good approximation. It does, however, generally estimate a higher genetic variance than the multilocus models, with the difference decreasing with an increasing number of loci. The difference between multilocus models depends more strongly on the effective number of loci, accounting for relative contributions of loci to the variance, than on the number of alleles per locus.

Migration of individuals from one population to another is a common phenomenon, as is local adaptation. This combination results in migrants often having lower fitness in their new environment than resident individuals, so that migration and selection work as opposing forces on the genetic make-up of a population. Migration–selection balance has been thoroughly explored in population genetic single-locus models (Haldane 1930; Wright 1931), and to some extent for quantitative traits, coded for by a large number of loci. To allow analytical tractability, this has generally been done using the simplifying assumptions of normally distributed genotypes and a constant genetic variance (Via and Lande 1985; Hendry et al. 2001; Tufto 2001), or a normal distribution with a dynamically changing variance (Bulmer 1980, p. 181; Barton 1999, 2001; Tufto 2000). It has been less extensively studied in a situation with both high migration rates and strong stabilizing selection, when the approximation of the genotype distribution with a Gaussian breaks down.

This situation potentially occurs when a wild species interbreeds with its domesticated relative. The domestic population is usually much larger than the wild one, making high immigration rates possible, and selection acting against escaped or released individuals is generally strong. Such interbreeding occurs in various species of both plants and animals, ranging from sunflowers (Whitton et al. 1997) to wolves (Randi et al. 2000) and wildcats (Beaumont et al. 2001), and may pose a threat to the genetic integrity of a wild population, or even an entire species (Rhymer and Simberloff 1996). While such interbreeding is often unintentional, similar processes occur during supportive breeding and reintroduction programs, which promote breeding between wild individuals and individuals that have adapted to captivity. In both cases, the domestic or captive bred individuals are expected to have lower fitness than native wild individuals in the local natural environment, due to (genetic) differences created by adaptation to captivity (Frankham 2008) and/or artificial selection. As most fitness traits have a heritable component, the average fitness in an admixed population resulting from such interbreeding may be lower than in the original population, depending on the number of domestic/captive bred individuals and the relative fitness of them and their descendants. Lower average fitness in a population increases risk of extinction (Hutchings and Fraser 2008), by increasing the rate of selective mortality in density-dependent populations, or by decreasing population size in density-independent populations (Lande 1976b).

An example of interbreeding between a wild species and its domesticated relative can be found in the Atlantic salmon (Salmo salar) (McGinnity et al. 1997; Gross 1998; Naylor et al. 2005). In Norway, the number of farm salmon escaping yearly is several times the estimated size of the wild breeding population (Jonsson 1997; Gross 1998; Fleming et al. 2000). Their mortality before they reach the breeding grounds is high, but nevertheless the proportion of farm escapees among wild breeders is around 20%, and up to 80% in some rivers (Fiske et al. 2001). Farm salmon differ from wild salmon due to adaptation to the captive environment, which in salmonids has been shown to decrease breeding success in the wild already after one or two generations (Hansen 2002; Araki et al. 2007). Additionally, they have been selected for increased growth and several other traits for around 10 generations (Gjøen and Bentsen 1997; Gross 1998). This is much shorter than most other domesticated species, but farm salmon differ already considerably from their wild counterparts (Fleming and Einum 1997; Gjøen and Bentsen 1997; Gross 1998). As in many fish species, growth and body size are correlated with fitness traits such as survival and fecundity (Walsh et al. 2006; Sundt-Hansen et al. 2007). Therefore, the change in growth performance is likely to have moved the multitrait phenotype away from the fitness optimum in the wild, via indirect response to selection in other traits. Studies on the performance of farm and wild Atlantic salmon and their hybrids under natural conditions have been performed, but do not extend beyond the second generation (Fleming et al. 2000; McGinnity et al. 2003; Araki et al. 2008; Fraser et al. 2010), partly due to the relatively long generation time (3–10 years) of Atlantic salmon.

To make long-term predictions about, for example, the amount of introgression or the decrease in population fitness, modeling studies are useful. Historically, models needed to be analytically tractable to solve them without the aid of computers. Examples are the one- and two-locus models, which still prove useful under certain conditions (see, e.g., Bolnick et al. 2007). Some fitness traits may be predominantly determined by a single gene; for example, in salmon 83% of genetic variation in resistance against a certain viral disease is explained by a single gene though, it is (Moen et al. 2009).

Most fitness traits, however, are continuous (quantitative) rather than discrete. The number of genes typically coding for a quantitative trait is disputed, partly because it is not well understood what a “gene” is in this context, but is generally believed to be large (Lande 1981; Merilä and Sheldon 1999). Genetic details such as interactions between alleles at the same or different loci are commonly unknown. Generally, though, it is assumed a measuring scale can be found on which all genetic variance is close to additive (Lande 1981). This notion is supported by animal breeding practice and many (but not all) experiments, including experiments in which hybrid performance in salmon was explainable by additive effects only (McGinnity et al. 2003; Fraser et al. 2010).


To make inferences about quantitative genetic traits, different models have been developed, with different sets of assumptions to allow analytical tractability. A commonly used model is the infinitesimal model, which assumes an infinite number of unlinked and nonepistatic loci underlying the trait, each with an infinitesimal effect (Fisher 1918; Falconer and Mackay 1996; Lynch and Walsh 2008). From the central limit theorem, this leads to normal within-family distributions of additive genotypes (breeding values). Within each family, the mean equals the midparental value and the variance equals half the variance under linkage equilibrium (VG,LE, genic variance) (Turelli and Barton 1994; Dawson 1997). Under random mating and in absence of migration or selection (i.e., under Hardy–Weinberg and linkage equilibrium), the variance of the population breeding value distribution equals VG,LE (Turelli and Barton 1994). Also in the limiting case of weak migration and/or weak selection, the population variance is reasonably well approximated by VG,LE, which stays constant over time. In those cases, one needs to focus only on changes in the mean breeding value to describe the entire distribution (Tufto 2000). Needing to track only a single variable gives rather straight forward equations, used by, for example, Via and Lande (1985) and Hendry et al. (2001).

The infinitesimal model assumes that VG,LE stays constant, as even under strong selection on the phenotype, the strength of selection acting on each of the (infinitely) many loci is very small, so that the change in allele frequency at each locus is very small too and can be ignored (Lynch and Walsh 2008). Migration will, however, create positive linkage disequilibrium (LD), as alleles within immigrant gametes tend to occur in different combinations than within local gametes. This increases the population variance relative to the variance expected in a random mating population with the same allele frequencies (and allelic effects), the variance under linkage equilibrium VG,LE. We define the total amount of LD as the difference between the observed variance Vz and VG,LE; the sum of all covariance terms between and within loci. Similarly, both truncation and stabilizing selection tend to create some negative LD. When accounting for this changing variance, the breeding value distribution of the admixed population can still be adequately approximated by a Gaussian, even for surprisingly high migration rates and when immigrants are highly maladapted, or when selection is strong (Turelli and Barton 1994; Barton 1999; Tufto 2000).

Under some conditions, however, the mean and the variance in the next generation cannot be predicted accurately by the mean and variance in the current generation alone, because the distribution is bimodal or considerably skewed. In those cases, it may still be possible to use the infinitesimal model, assuming constant VG,LE and equal within-family variances, by keeping track of the whole population distribution (Turelli and Barton 1994) rather than assuming a Gaussian distribution.

In reality, however, the number of loci coding for a trait will be finite, so that allele frequencies and thereby VG,LE will change under influence of selection and migration. Additionally, the assumption of equal within-family variances may be violated when LD is strong (Dawson 1997); one could imagine more variation among offspring of local-by-immigrant matings than of local-by-local matings for example. Multilocus models can incorporate this, but also they involve assumptions to enable analytical solutions or decrease computational costs.


Two commonly used multilocus models are the continuum-of-alleles model and the exchangeable (equivalent) loci model. The first is (again) inspired by the central limit theorem, and assumes that each locus has an infinite number of possible alleles. Allelic effects are normally distributed, with the same variance at each locus (Kimura and Crow 1964; Lande 1976a). This is a more restrictive assumption than that breeding values are approximately Gaussian, and it has been argued that it is motivated by questionable assumptions about mutation–selection balance (Barton and Turelli 1989).

The exchangeable loci model assumes all loci are diallelic and have the same allelic effect, such that only the total number of “+” alleles needs to be traced (Barton 1992; Turelli and Barton 1994). A special case of this is the symmetric model, which in addition assumes that all allele frequencies are equal; a questionable assumption that leads to coupling of the dynamics of the variance to the dynamics of the population mean (Tufto 2000).

A more natural assumption may be exponentially distributed effect sizes across loci (Orr 1998). Allowing for different allelic effects among loci makes a model analytically intractable and more computationally intensive than the other two types of multilocus models. Now one needs to track the frequency of all possible haplotypes (2L for L diallelic loci), or all individual genotypes.

Here, the infinitesimal model for will be compared to these three types of multilocus models, as well as a model where the allelic effects are normally distributed as in the continuum-of-alleles model, but the number of possible alleles is limited. For the infinitesimal model, we keep track of the entire distribution of breeding values, thus allowing for LD and nonnormality. Both are expected under a situation of strong stabilizing selection and high rates of immigration of maladapted individuals. The main differences between the infinitesimal model and the multilocus models lie in two assumptions of the infinitesimal model: negligible changes in allele frequencies (constant VG,LE) and equal, constant within-family variances.

Whether infinitesimal models or multilocus models more closely resemble the genetic architecture underlying fitness traits is debatable. There are arguments in favor of both models; sustained response to selection after 100 or more generations favors the infinitesimal model, and despite its apparent lack of biological realism it has a proven record in animal breeding practice. On the other hand, quantitative trait loci mapping studies frequently identify loci of large effect. Their effect size may, however, be overestimated (Barton and Keightley 2002), as one such “locus” may be a cluster of several tightly linked loci. For the current simulation study, it is irrelevant whether a locus consists of such a cluster or a single point mutation; we assume the locus stays intact during the relatively limited time span of the simulations.

The lack of knowledge on the genetic details underlying traits of interest, especially in wild populations, also means it is unknown which of the multilocus models forms the most suitable approximation. Therefore, the focus of this article is on how much, and under what conditions, the more parsimonious infinitesimal model differs from the various multilocus models, with minor attention to the difference between the multilocus models.

The non-Gaussian infinitesimal model has been compared previously in the same setting to models assuming a normal distribution of breeding values, with and without taking LD into account, as well as to a symmetric multilocus model (Tufto 2000). It has also been compared to a wide range of different models in a similar setting by Turelli and Barton (1994). It has, to our knowledge, not been compared previously to the “exponential effects” diallelic multilocus model or the model with a limited number of alleles per locus, for which no analytical approximation exist.

The trait considered is a nonspecified heritable trait affecting fitness, which is under stabilizing selection in the recipient population. This population receives migrants from a donor population, in which the average trait value is some distance away from the optimum as a result of repeated truncation selection in that population. The models used here deal with a single trait rather than a set of genetically correlated fitness traits, an approximation that has been shown to hold well (Tufto 2010). Often only the limiting case of weak selection and low migration rates is considered (see, e.g., Barton and Turelli 1991; Lynch and Walsh 2008), but the comparison of the models considered here is especially relevant under strong selection and high rates of migration, when the breeding value distribution is expected to deviate from a Gaussian.

We shall see that the difference between the models is very small for the deviation of the population mean from the optimum. The difference in variance between the infinitesimal model and the multilocus models is mainly due to changes in VG,LE, which are up to 20% when the number of loci is limited. The difference between the models in total genetic variance is generally smaller than the difference in VG,LE, as the amount of LD is smaller for the multilocus models than for the infinitesimal model, especially when selection is strong. Despite these differences between the two model types in the predicted variance, as well as in the skew, they predict a highly similar response to selection and resulting population mean.


We consider the breeding value, the additive genotypic value, for a nonspecified trait affecting fitness. The fitness effect includes any effect of this trait on fecundity (breeding success) and/or on survival. The breeding values in the population are assumed to be initially normally distributed, but not necessarily so after selection and migration. Each generation, a fraction of the population is replaced by immigrants with a different mean breeding value, followed by stabilizing selection around a fixed optimum and random mating (Fig. 1).

Figure 1.

Distribution of the breeding values z in the recipient population (solid line) after 20% is replaced by immigrants from the donor population with z1=3 (dashed line) just after migration (left panel), in the selected parents (middle, s=0.1), and after reproduction (right). The gray shaded area indicates the distribution in the original population. The postreproduction distribution is the premigration distribution in the next time step.

A population of fixed size N is simulated, with no age or other structure and no sexes, and discrete generations. A population of finite rather than infinite size is used, as this allows us to trace individual genotypes rather than a potentially large number of haplotype frequencies (2L; e.g., about 1030 for L=100 loci). To ensure any differences between the models are not due to effects of a finite versus infinite population size, a finite population size model is used for the infinitesimal model as well. To minimize stochastic noise, a large population size is used (N=1000) and averages are taken over 100 replicate runs. Results of the finite population size infinitesimal model are nearly identical to an infinite population size version of this model, using the exact Fourier transform method as described in Tufto (2000), for the range of parameter values considered (unpublished results).

No true deterministic migration–selection equilibrium occurs in a finite population, as fluctuations around this level will keep occurring. Therefore, comparisons of the models are done after 50 generations; for most parameter value combinations considered changes in all statistics leveled off after about 10–15 generations.

Simulations are performed in R, and partly in C for faster computation time. Simulation code is available at doi:10.5061/dryad.s808f.


A population is simulated with breeding values for a fitness trait initially following a standard normal distribution, that is, mean breeding value inline image and variance under gametic phase equilibrium (genic variance) VG,LE=1, so that all genetic differences are expressed in genetic standard deviations. In the multilocus models, an individual’s breeding value is the sum of the allelic effects over all its loci, for details see Appendix. During the simulations, the total additive genetic variance Vz is measured as the variance of the breeding value distribution. Under the multilocus model, VG,LE is calculated from the allele frequencies and allelic effects, under the infinitesimal model it is fixed to 1.

The population experiences stabilizing selection around the optimum z0 that, without loss of generality, is set to 0. The population mean is thus initially at the optimum. Stabilizing selection is implemented by letting the relative probability to be sampled as parent be inline image, with parameter s representing the strength of selection acting on the genotypes. The parameter s relates to the strength of selection acting on the phenotype sP as 1/s= 1/sP+VE, where VE is the environmental variance. The selection coefficient sP is the inverse of the variance of the fitness function, inline image, as used by Lande (1976b). Selection may act on viability, fecundity or both, provided the fecundity of a pair is the product of their individual fecundities (Bodmer 1965). No assumptions about VE are made.

Each time step N pairs of individuals are sampled (with replacement) to produce N offspring whose genotypes are either sampled from a normal distribution around the mean of the parents, with a fixed variance equal to half VG,LE (infinitesimal model) (Fisher 1918; Turelli and Barton 1994), or a sample of parental alleles following Mendelian inheritance (multilocus models). There is no separate selection on offspring survival; the resulting offspring distribution can be thought of as the distribution of offspring that survive until reproductive age.

Migration takes place before the reproduction and selection steps. A constant fraction m of the population is replaced by Nm individuals from the donor population; individuals from both populations mate at random.

Donor (farm) population

The donor population has mean breeding value z1, and is assumed to have the same alleles as the recipient population, but with different frequencies. Therefore, the donor population is created via directional selection from the recipient population (Fig. 2), rather than by creating a second, independent population (also for infinitesimal model). The donor population is generated by taking a random sample of size N from the recipient population at t=0 and applying truncation selection on the phenotype P of the trait for several generations, followed by random mating, until the desired genetic difference inline image is obtained. Phenotypes are created by adding random terms E from a normal distribution with zero mean and variance VE to the breeding values z (P=z+E). The phenotypes are used solely for the simulation of truncation selection in the donor (farm) population.

Figure 2.

Schematic overview of the model used. The donor population is created by duplicating the original population at t=0 and applying truncation selection until the average breeding value z1 is obtained. Each time step, migration takes place from this donor (farm) to the recipient (wild, local) population, followed by random breeding and stabilizing selection.

Due to this set up, variance in the donor population is generally lower than in the base population, and some negative LD among immigrants is created (Table A1 in Appendix). In this way, it differs from analytical models, where sometimes optionally a lower or higher variance in immigrants can be incorporated (Tufto 2000), but generally linkage equilibrium among them is assumed.

Table A1.  Genetic variance Vz and variance under linkage equilibrium VG,LE (medians over 250 replicates) in the donor population for the various models with five or 20 effective loci Le and for three levels of genetic difference between donor and recipient population z1 (the total response to selection in the donor population).
Model Le z 1=1 z 1=3 z 1=5
V z V G,LE V z V G,LE V z V G,LE
Diallelic (eq)50.8560.9370.6130.6760.5180.568
  20 0.868 0.978 0.796 0.895 0.659 0.736
Diallelic (exp)50.8320.9230.5780.6420.2650.279
  20 0.866 0.977 0.792 0.901 0.696 0.769
Multiallelic (Pois)50.8350.9210.5780.6280.2510.272
  20 0.878 0.979 0.799 0.898 0.676 0.752
Multi-allelic (2N)50.8540.9560.7920.8840.6910.767
  20 0.872 0.988 0.852 0.968 0.817 0.924

Parameter ranges

The setup described allows three parameters to vary: the migration rate m, strength of stabilizing selection s, and genetic difference between the populations z1z0 (=z1 since z0=0), the latter two of which are scaled relative to the genetic variance VG,LE. The ranges used for those parameters are chosen so as to correspond to values observed in nature. For the migration rate, defined here as the proportion of nonnative individuals in the breeding population, the entire range of 0–100% is considered, to include potentially low rates in some species and the very high rates in some salmon rivers (up to 80%, Fiske et al. 2001), and to allow indirect comparison of studies with abruptly changing optima (m=1). Two levels of selection intensity are considered, mild (s=0.1) and relatively strong (s=0.5) (Kingsolver et al. 2001), corresponding, for example, to inline image, respectively, inline image for a heritability of 0.5. The genetic difference between the recipient and donor population ranges from 0.5 to 5 genetic standard deviations, so that the average relative fitness of immigrants is 94–31% for the weakest selection (s=0.1), and 78–1.3% for the strongest selection (s=0.5).


Three different multilocus models are compared, which have as main difference the number of possible alleles per locus: n=2, a random number (inline image Pois(5) + 2), or n=2N (infinite) (Table 1). For the latter two, the allelic effects are assumed to be normally distributed around 0 with a constant variance across loci. For the diallelic model, exponentially distributed effects across loci are compared to a situation where all loci have the same effect size (exchangeable loci). Allelic effects are scaled at initiation to ensure VG,LE=1. Allelic effects are constant during a run, no mutations are incorporated.

Table 1.  Characteristics of the multilocus models.
ModelNo. alleles per locusDistribution of allelic effectsNo. loci
Le=5 Le=20
Diallelic (eq)2Equal6–926–34
Diallelic (exp) 2 Exponential 10–53 92–236
Multiallelic (Pois)Poisson(5)+2Normal6–1029–40
Multiallelic (2N) 2N Normal 5 20

Initial allele frequencies are distributed either uniform or U-shaped, and are sampled from a Beta distribution for the diallelic models, and from a symmetric Dirichlet distribution for the multiallelic distributions. For brevity, only the results for the U-shaped distribution will be shown in the section Results, which is the distribution expected under mutation–selection equilibrium (Bulmer 1989). Only for the diallelic model did the allele frequency distribution affect the results, partly via the different scaling of allelic effects at initiation (example in Fig. S4).

Effective number of loci

Because allele frequencies p and allelic effects a are sampled independently at the start of each replicate simulation, the relative contribution of loci to the genetic variance inline image will differ between runs, with a smaller number of loci contributing most to the total genetic variance. We found that a measure of the effective number of loci Le (defined in the Appendix) rather than the actual number of loci better predicted model behavior. This measure equals the actual number of loci if those loci contribute equally to the total genetic variance. With unequal contributions, Le becomes smaller than the actual number. For example, if four loci contribute 1%, 1%, 49%, and 49% to the total variance, Le= 2.08. Therefore, we used replicate runs with an equal effective number of loci Le. The actual number of loci L consequently differs between replicate runs, between boundaries that are based on the empirical relationship between L and Le (Fig. S1), specific to each model and allele frequency distribution (Table 1; see Appendix for details).

This “variance effective number” is not necessarily the same as the effective (minimum) number of loci contributing to the difference between two lines or populations, estimated using crossing experiments (Lande 1981; Barton and Turelli 1989). They are probably in the same order of magnitude; therefore, (variance) effective number of loci of 5 and 20 are used, as the minimum number of genes contributing to differences between population has been estimated to be about 5–10, up to 20 (Lande 1981). Using fewer than five effective loci would allow for more in depth study of the effect of the number of loci underlying the trait, but limits the maximum possible breeding value (Fig. A1 in Appendix), and therefore hampers the creation of donor populations.

Figure A1.

Initial mean breeding value and maximum possible breeding value inline image (when fixated for the [most] positive allele at all loci) for the diallelic model with exponential distributed allelic effects (black circles) and the multiallelic model with normally distributed effects (gray triangles), both with initially U-shaped distribution of allele frequencies.


Statistical analysis on the difference between the infinitesimal model and the various multilocus models is performed using two-sided two-sample t-tests assuming unequal variances (Welch’s t-test), with sample sizes n1=n2=100 replicate runs. This is done at regular intervals of m for z1=3, and at regular intervals of z1 for m=0.2, both for s=0.1 and s=0.5. Those parameter combinations were initially chosen to reflect the migration rate and wild-farm genetic difference in Norwegian Atlantic salmon. It was found that the difference between the models is largest around m=0.2 and z1=3, and as such they provide a “worst case scenario” for the comparison of the models. P-values of those tests are binned into the standard categories P<0.001 (***), 0.001< P<0.01 (**), 0.01< P<0.05 (*), 0.05< P<0.1 (.), and P>0.1 (n.s.).


The differences between the various models will be shown as differences in the mean, variance, and skewness of the breeding value distribution, as well as in total LD, quantified by the difference between the observed and genic variance. First an example will be given how those variables change over time during admixture, followed by comparisons after 50 generations of admixture between the infinitesimal model and the multilocus models, and among the various multilocus models. For clarity, results of the diallelic model with equal effects are not included in the figures, as those were generally very similar to the diallelic model with exponentially distributed effects (see Fig. S5).

As can be seen in Figure 3, the deviation of the mean from the optimum, inline image, approaches an asymptotic value within a dozen or so generations. The predictions for inline image of the infinitesimal model and multilocus models differ only slightly for relatively weak selection (s=0.1, left), and are nearly indistinguishable for stronger selection (s=0.5, right). The difference between the models is smaller than the variation between replicate runs of the same model, indicated by the error bars.

Figure 3.

Example of change in mean, variance (observed and under linkage equilibrium) and skewness over time, and amount of variation between models and between replicate runs. Each generation, a proportion m=0.2 of the population is replaced by immigrants with mean breeding value z1=3. Stabilizing selection around the optimum z0=0 is either relatively weak (s=0.1, left) or strong (s=0.5, right). For each model, the average over 100 replicate runs is shown (see legend) and error bars indicate average inline image 1 SD, with order as in legend. (For 10–90% quantiles see Fig. S6).

In those examples, as under most conditions (see later), the observed variance Vz of the multilocus models is lower than that of the infinitesimal model (Fig. 3, second row). For weak selection, Vz is slightly higher than VG,LE, and much higher in the first generations of admixture, as migration generates positive LD. Under stronger selection, this effect of migration is more than balanced out by the negative LD created by selection, resulting in Vz being somewhat lower than VG,LE.

Skew stays very close to zero for the infinitesimal model, whereas it is more extreme and variable for the multilocus models. When intensity of selection is low, the peak of the breeding value distribution is shifted away from the optimum in the direction of the immigrants, to the right, leaving a long tail behind at the left side of the peak—a left-skewed (negatively skewed) distribution. Under stronger selection, the peak stays closer to the optimum, but there is a long tail in the direction of the immigrants—the distribution becomes positively or right-skewed.


The small difference between the models in inline image occurs not only in the two earlier described examples, but is a general trend when the values after 50 generations of admixture are plotted against migration rate (upper row in Fig. 4) or the immigrant mean (upper row in Fig. 5). Where the models differ, the multilocus models predict a higher inline image than the infinitesimal model for intermediate values of the immigrant breeding value z1 (Fig. 4, z1=3), and lower for more extreme values of z1 (Fig. 5). The differences are larger for mild selection (s=0.1) than for strong selection (s=0.5). Although the difference between the infinitesimal model and the least different multilocus model are sometimes highly significant (P<0.001, indicated with *** in the figures), they are never more than 11% for z1=3, or more than 18% for larger z1 (difference between averages over replicate runs).

Figure 4.

Model differences as function of migration rate m: mean breeding value inline image, genetic variance Vz, linkage disequilibrium LD, and skewness for the different models (see legend, multilocus models Le=5) after 50 generations of recurrent immigration with rate m of immigrants with mean breeding value z1=3 and relatively weak (s=0.1, left) or strong (s=0.5, right) stabilizing selection. Error bars indicate average inline image 1 SD (with order as in legend), asterisks indicate significance levels of Welch’s t-tests between the infinitesimal model and the least different multilocus model (based on 100 replicate runs).

Figure 5.

Model differences as function of differences between the populations z1: mean breeding value inline image, genetic variance Vz, and linkage disequilibrium LD for the three different models (see text, for both multilocus models Le=5) after 50 generations of recurrent immigration with rate m=0.2 of immigrants with mean breeding value z1=3 and relatively weak (s=0.1, left) or strong (s=0.5, right) stabilizing selection. Error bars indicate average inline image 1 SD, asterisks indicate significance levels of Welch’s t-tests between the infinitesimal model and the least different multilocus model (based on 100 replicate runs). Note different ranges on Y-axes for Vz and LD than in previous figure.

The major difference between the model types is in the estimate of the variance (second row in Figs. 4, 5), as expected from theory. For the infinitesimal model, all changes in genetic variance Vz are due to changes in LD, as the genic variance VG,LE is fixed at 1. For a given strength of selection and immigrant mean, the negative LD created by stabilizing selection dominates when migration rate is low, so that Vz < VG,LE. For intermediate migration rates, positive LD created by migration dominates, causing Vz > VG,LE. For still higher migration rates, the negative LD among immigrants dominates, so that again Vz < VG,LE. What migration rates are “intermediate” or “high” in this context depends on both the strength of selection (Fig. 4) and z1 (not shown).

For the multilocus models, any change in Vz is generally mainly due to changes in VG,LE, with LD being smaller (closer to zero) than for the infinitesimal model with the same parameter values (Figs. 4 and 5, third row), even though the variance Vz is further away from one (Figs. 4 and 5, second row). Exception is when selection is strong and migration rate rather high (m=0.40.6 for z1=3), where the infinitesimal model predicts a balance between those forces such that LD = 0, whereas the multilocus models predict positive LD (Fig. 4).

It must be noted that the variance is not only determined by the amount of LD generated by migration and selection, but also by the reduced variance in the donor population (Table A1 in Appendix), especially when m and z1 are large. This reduced variance in the donor population is due to both changed allele frequencies (in the multilocus models) and negative LD created by truncation selection. The reduction in both VG,LE and Vz in the donor population is larger for lower Le and higher z1. Its effect on the admixed population is generally largest for parameter values resulting in the highest levels of introgression, that is, for weak to moderate intensity of selection and high migration rate.

The deviation of the breeding value distribution from a Gaussian is here depicted by the skew. Under none of the parameter combinations explored was the distribution at t=50, averaged over 100 replicate runs, markedly bimodal (not shown). Individual runs of the diallelic model resulted often in ragged distributions (see Fig. S2), of which modality was hard to judge. Skew is rather small (<0.2) and positive (distribution “leaning” to the left) for the infinitesimal model, and does not consistently increase or decrease with any of the parameters (Figs. 4 and 5, fourth row). For the multilocus models, skew is generally larger but never more extreme than ()0.4.


The number of possible alleles at each locus (2, a Poisson random number or 2N) has only a minor effect on the mean breeding value and LD, but in some cases a considerable effect on the genetic variance and the skewness. These differences in genetic variance Vz are thus due mainly to differences in VG,LE. Those are partly caused by the method used to scale the allelic effects, so that the multiallelic models have on average alleles with smaller effect sizes, and partly due to the larger genetic drift when the total number of alleles is smaller.

The difference between the various multilocus models does not increase or decrease with any of the parameters in an obvious, consistent manner. In general, the difference is small when m or z1 is low, but not necessarily at its smallest. Where the models differ, the model with 2N alleles per locus is generally more similar to the infinitesimal model than the diallelic model. The model with a limited, Poisson distributed number of alleles is not always intermediate between the other two models. Considering genetic variance Vz, this model is the most divergent of the three when selection is weak (s=0.1) and inline image, and the least divergent when selection is stronger (s=0.5).

For the diallelic models, the allelic effects (equal or exponentially distributed) have a small effect on the various characteristics of the breeding value distribution, which is only significant for weak selection and high migration rates (inline image, Fig. S5). Where they differ, the model with equal effects differs more from the infinitesimal model than the model with exponentially distributed effects.

The variation between replicate runs (error bars in Figs. 4 and 5) is due to both genetic drift and variation at initiation. It is smallest for the infinitesimal model and model with 2N alleles, as for the latter model allele frequencies at initiation are always 1/2N and the distribution of allelic effects approaches a continuous distribution.

Number of loci

When the effective number of loci is increased from Le=5 to Le=20, the results of all multilocus models become more similar to the infinitesimal model. The effective number of loci has a considerable effect on the genetic variance Vz, especially when selection is strong (s=0.5) and migration is limited (m=0.05) (Fig. 6, left column); the difference in Vz between the same model with Le=5 versus Le=20 is larger than the difference between the different models with the same Le.

Figure 6.

Difference in results between models with Le=5 (black) and Le=20 (gray): Mean inline image, variance Vz of the breeding value distribution, and LD plotted against time for the various models (see legend) under relatively strong stabilizing selection (s=0.5) and a difference between populations of z1=3. Error bars indicate average inline image 1 SD based on 100 replicate runs.

The two components that make up Vz, VG,LE, and LD, act in opposite directions on this difference. The change in VG,LE is larger for the models with Le=5 than for those with Le=20, because allelic effects and therefore strength of selection acting on each allele decrease with an increasing number of loci (due to the scaling method used, ensuring VG,LE=1). The amount of LD, on the other hand, rises with an increasing Le, not only as a proportion of the total genetic variance, but also in absolute terms (Fig. 6, bottom left panel; compare right columns Figs. 4 and 7). As a result, the difference due to Le in Vz is smaller than the difference in VG,LE.

Figure 7.

As Figure. 4, with Le=20 for the multilocus models instead of Le=5.

The amount of LD is highly similar for the various models for a given Le, despite their considerable variation in the actual number of loci; it does not matter whether the trait is coded for by 10–50 diallelic loci or by 6–10 loci with on average seven alleles each (Table 1). This suggests that for LD, as for the variance Vz, Le is a more relevant parameter than L.


We compared the non-Gaussian infinitesimal model to various multilocus models, differing in the number of alleles per locus, under a range of migration rates and strong stabilizing selection. We found that all models predicted highly similar deviations of the population mean breeding value from the optimum, suggesting the more parsimonious infinitesimal model is a good approximation for predicting the mean.

When the interest is in the genetic variance, however, the infinitesimal model always predicts less change compared to the original situation (Vz=1), which generally implies it predicts a somewhat higher variance than the multilocus models. In the latter models, variance changes not only due to LD, as in the infinitesimal model, but also due to changes in allele frequencies. In addition, in multilocus models the within-family variance may differ between families and across time, whereas in the infinitesimal model it is constant (inline image).

As expected, the relative importance of LD and VG,LE in the multilocus models to the total additive genetic variance changed with the effective number of loci (Bulmer 1971). When the number of loci is large or infinite, changes in variance are mainly due to the generation of LD, and VG,LE stays (nearly) constant. When the number is reduced, allele frequency changes cause VG,LE to change more, and the predicted amount of LD is generally (but not always) smaller. Overall, when the number of loci was increased, predicted deviation from the optimum slightly decreased, observed variance and LD increased, and skew decreased.

The difference between models with different number of alleles per locus, but with the same effective number of loci, was generally smaller than between models of the same type with five versus 20 effective loci. This is in agreement with results by, for example, Chevalet (1994), who found that “simulation runs involving either several unlinked loci with many alleles taken from a normal distribution, or several clusters of tightly linked loci with only 2 alleles, lead to very similar responses to directional selection”. The multilocus model with an infinite (2N) number of alleles was often more similar to the infinitesimal model than the other two multilocus models, but still most similar to the other multilocus models.


In both the infinitesimal and the multilocus models considered here, the genetic variance may change as a result of LD. In the multilocus models, the genetic variance may also change as a result of changes in VG,LE. In many genetic models of selection and migration, changes in the genetic variance are assumed to be negligible, to allow or simplify analytical tractability. In many systems, the difference between local optima will be small and/or the migration rate very limited, so that changes in the variance are indeed small and these simpler approximations can be used. Systems in which gene flow is studied, however, will typically be chosen specifically because they have widely different optima, strong selection, and/or high rates of migration (e.g., Bolnick et al. 2007; Moore et al. 2007). As a result, changes in variance are likely to affect the response to selection, as may deviations from normality (including skew) (Hendry et al. 2001).

Yeaman and Guillaume (2009) therefore compared simulation results from a non-Gaussian multilocus model of mutation–selection–migration balance to that of the Gaussian approximation model (GAM) described by Hendry et al. (2001). Their comparison uses the genetic variance at mutation–migration–selection balance from the simulation plugged into the GAM (rather than the variance prior to migration and selection (e.g., Tufto 2000, “No linkage disequilibrium”). This plug-in method implicitly assumes the difference in variance before and after selection is negligible. The discrepancy between the two models is attributed to deviations from normality only (mainly skew), and a strong positive correlation between skew and the discrepancy between models was indeed found (Yeaman and Guillaume 2009, Fig. 3).

In our results, skew after 50 generations explained little of the differences between the non-Gaussian infinitesimal and multilocus models, although the importance of the degree of nonnormality on the relative performance of the models was not investigated in detail. Combined with the finding of Yeaman and Guillaume (2009), it might be concluded that the discrepancy between the Gaussian and non-Gaussian infinitesimal increases with skew. However, the multilocus models used differ (e.g., theirs included mutation, while ours did not), as does the range of migration rates considered. More importantly, because Yeaman and Guillaume (2009) estimated the equilibrium variance independently, it is not clear from their analysis that models of migration–selection balance relying on a constant variance, such as that of Hendry et al. (2001), will underestimate divergence.

Interestingly, Turelli and Barton (1994) found that greatest steady-state skew is produced with relatively weak truncation selection (but that it was generally small and of negligible biological importance), whereas Yeaman and Guillaume (2009) found that skew increased with increasing strength of stabilizing selection in their system with two-way migration. Here, with one-way migration, there is no clear correlation between skew and the strength of selection (Figs. 4 and 5). There does seem to be a trend of increasing skew with increasing deviation of immigrants from the optimum, as was found by Yeaman and Guillaume (2009).

Practical relevance

When comparing the results presented here to results from models with constant variance, the most striking difference is in predictions of the deviation of the population mean from its optimum for a range of immigrant means. When variance is constant, the relationship is linear, with a slope depending on the strength of selection (Fig. 8, upper panels). When incorporating changes in variance, the deviation of the mean levels off above a certain level of maladaptation of the immigrants, and when additionally allowing for nonnormality the deviation even decreases after this point. This agrees with a recent simulation study on supportive breeding of salmon, which included additional parameters such as density dependence, who found the largest drop in population fitness at intermediate difference between wild and hatchery reared salmon (M. Baskett and R. Waples, unpubl. ms.). A similar result was obtained from simulations of a fragmented metapopulation by Lopez et al. (2008), who found that genetic load can increase and then decrease with increasing difference between optima.

Figure 8.

Difference between infinitesimal models for infinite population size: deviation of mean breeding value from optimum (inline image), divergence between populations at equilibrium (inline image), and variance Vz at equilibrium when using the non-Gaussian infinitesimal model (solid line), the Gaussian approximation (dashed), and the approximation assuming a constant variance (dotted), for a wide range of initial difference between the populations z1 when m=0.2 and two strengths of stabilizing selection s. (Three-dimensional plots of inline image vs. m and z1 in Fig. S7) (equations for those models and detailed comparison in Tufto 2000).

The biological interpretation of this is that as maladaption of immigrants increases, a smaller fraction of them (and their descendants) successfully reproduces. When their maladaptation exceeds a certain threshold, this reduces their combined effect, despite their larger individual effects on the population mean breeding value. The location of this threshold is dependent on the strength of stabilizing selection. This implies, for example, that the expected decrease of fitness in the wild of farm salmon across generations, due to ongoing artificial selection, might make them less harmful to wild salmon populations. Might, as individuals that will not successfully reproduce may still compete for scarce resources with native individuals, lowering their chances of survival (Fleming and Einum 1997; Houde et al. 2010). In addition, immigrants and especially hybrids may outperform native individuals during some life stages or years, something not considered in many models, including the one used here.


The difference between results from the non-Gaussian infinitesimal model and multilocus models shown here are small relative to the uncertainty expected in parameter estimates from empirical studies. The same is true for the difference between the results of the Fourier transform and Gaussian approximation (with LD) for biologically relevant parameter ranges (Tufto 2000). It seems that in many practical cases, use of the analytical Gaussian approximation is adequate, and elaborate simulations can be restricted to theoretical studies. Disagreement on this exists, however, as some researchers have chosen to ignore LD, rather than to ignore changes in genic variance (VG,LE, as all infinitesimal models do), as they found the latter to make up a large part of the total genotypic variance and respond strongly to habitat heterogeneity and dispersal rate (Lopez et al. 2008). In any case, ignoring changes in both VG,LE and LD seems to give unreliable results under a wide range of conditions (see, e.g., Tufto 2000).

Increased complexity of the underlying methods does not necessarily require more input parameters or imply more complicated use. The non-Gaussian and fixed-variance Gaussian infinitesimal models require the same parameters (although some reparametrization may be needed, e.g., inline image in Hendry et al. 2001 equals 1/s+VG here). Thus, the non-Gaussian infinitesimal model provides an equally parsimonious alternative the fixed-variance Gaussian model, while at the same time better capturing the behavior of more realistic explicit multilocus models.

Associate Editor: N. Barton


We thank K. Hindar for discussions and comments, and R. Lande, I. Oleson and several anonymous referees for comments on earlier versions of the manuscript. This study has been funded by the Research Council of Norway (184007/S30).



Calculating breeding values

For the diallelic model, individual breeding values zi are calculated as


where matrix X contains the number of copies of the allele minus one (1, 0, or 1), with inline image. The allelic effects are scaled such that Vz= 1, and we define inline image so that E[zi]=z0=0. The breeding value then becomes


For the multiallelic models, the breeding value is calculated as


where Ai,l,k is the effect size of the allele at locus l on gamete k (inline image), and inline image is again the negative of the expectation,


where Kl is the number of different alleles at locus l, and al,k is the allelic effect of the kth locus.

For all models, the inline image and a in the donor population are equal to those in the recipient population.

Effective number of loci

To account for the fact that different allele frequencies and different allelic effects will result in different distribution of relative contributions of each locus to the genetic variance, the effective number of loci Le is used rather than the actual number L. The two are equal when all allele frequencies pl and allelic effects al are equal, but Le is lower otherwise.

Analogous to what is sometimes referred to as Simpson index used in ecology (Begon et al. 1990, p. 616), the effective number of loci is defined as


where vl is the contribution of locus l to the variance under linkage equilibrium, and the last step follows because here we have scaled so that inline image. For the diallelic model, the variance contributions are given by


and for the multiallelic models by


where Kl is the number of alleles at locus l, and the subscript l, k denotes the kth allele at this locus.

Determining L

At initiation, allelic effects a and allele frequencies p are randomly sampled from model-specific distributions (see Table 1). The number of loci L to obtain the desired effective number of loci inline image depends on these sampled values, and is determined iteratively:

  • 1inline image values for a and p are sampled;
  • 2 Le is calculated, using equations (A2)–(A4);
  • 3if inline image: add one locus, sample additional values of a and p, calculate new Le. Repeat (3) until either:
  • • inline image: Use these values of a, p and L;
  • • inline image: start anew from (1).
  • • inline image: start anew from (1).

The values inline image and inline image are based on the (model-specific) empirical relationship between L and Le, which is obtained as follows. For a range of values L between 2 and 200, thousand vectors a and p are sampled, and Le are calculated using equations (A2)–(A4). The 10% and 90% quantiles of Le for each value L are determined, which were found to be linearly dependent on L (Fig. S1). These quantiles were used to determine inline image and inline image. At inline image, inline image or smaller was obtained 90% of cases, and at inline imageinline image or larger in 90% of cases. Using values of L outside these boundaries was considered, but often characterized by extreme values of p and/or a, which generally made creation of the donor population (see below) impossible.

In the simulations, inline image and inline image were expressed as linear functions of inline image, obtained via linear regression of L on the 90% and 10% quantiles.


Creation of a donor population with a given mean breeding value z1 is limited by the maximum breeding value, which is obtained when all loci are fixed at the (most) positive allele. This maximum depends on both the number of alleles per locus and the effective number of loci (Fig. A1). To ensure that replicate runs in which a donor population with a given z1 could be obtained are not an a-typical subset, and that variance within the donor population was possible, Le=5 was chosen as the minimum included in the simulations.

Due to directional selection for several generations, variance decreases in the donor population, both due to fixation of alleles (decrease in VG,LE) and due to negative LD (Bulmer effect) (Table A1).