Introduction

When a species expands its range and colonizes a new habitat, it can encounter new environmental conditions and potentially interact with other species. Although new environments can impose novel selective pressure over the invading species, the contact with another species may lead to a reinforcement of the reproductive isolation, introgression of adaptive, deleterious, or neutral alleles, and even hybrid speciation (Barton, 2001). In absence of selection against the invaders, it has been shown that interbreeding between a local and an invading species can result in massive introgression of local genes into the invading gene pool (Currat et al., 2008). This asymmetric introgression is observed even in the face of a very low (<2%) rate of interbreeding (Currat and Excoffier, 2011). This is partially caused by the ‘surfing’ of introgressed alleles at the wave front, where the effective population size of the invading species is usually lower than that of the local population (Klopfstein et al., 2006; Currat et al., 2008).

Humans have had a complex demographic history including range expansions with admixture even between remotely related human populations (Jobling, 2012; Hellenthal et al., 2014; Homburger et al., 2015) as well as interbreeding with archaic hominids (Sankararaman et al., 2014). A few hundred years ago, Europeans settled in America and an intricate process of admixture took place, involving autochthonous groups, European settlers and African slaves (Ruiz-Linares et al., 2014; Bryc et al., 2015; Homburger et al., 2015). This resulted in an admixed population with varying degrees of genetic contribution from these continental groups; for instance, while the proportion of Amerindian alleles in certain regions of South America is substantial (Wang et al., 2008; Galanter et al., 2012; Ruiz-Linares et al., 2014; Salzano and Sans, 2014; Homburger et al., 2015; Salazar-Flores et al., 2015), there is little evidence of Amerindian genetic contribution in European Americans in North America (Halder et al., 2009; Lao et al., 2010; Lisabeth et al., 2011; Bryc et al., 2015). The latter observation is at odds with the expected high level of introgression of local genes into the invaders’ gene pool mentioned above (Currat et al., 2008) because, in spite of the known strong reproductive isolation between Europeans and Amerindians, interbreeding still might have occurred at low levels (Meinig, 1986). Thus, other mechanisms that prevent introgression of local genes may be required to explain the limited genetic contribution of Amerindians to North American populations of European descent.

A factor that could have suppressed introgression of Amerindian alleles is the pattern of colonial expansion involving long-distance dispersal (LDD) of Europeans. LDD during range expansions has been shown to lead to patchy allelic distributions (Nichols and Hewitt, 1994; Ibrahim et al., 1996), to boost genetic exchange between distant populations (Bialozyt et al., 2006; Ray and Excoffier, 2010), and to increase diversity along the expansion range, limiting allele surfing and the effect of drift, and therefore maximizing adaptive potential during range expansions (Berthouly-Salazar et al., 2013). In fact, during the expansion of European colonialists in North America, a peculiar form of LDD took place, where migrants arriving from Europe settled directly in the expansion front (Meinig, 1993). Strikingly, the effects of this specific kind of LDD over the genetic diversity of the colonizing species or population have not yet been evaluated. In this study, we simulated range expansions with and without LDD, using two different LDD models that reflect two distinct dispersal dynamics in order to compare their impact on levels of introgression of local alleles in the invading population throughout the colonized range.

Materials and methods

Simulation of range expansions with competition

We simulated two subsequent range expansions with a modified version of the software SPLATCHE2 (Ray et al., 2010). This software considers a two-step simulation process: (1) forward demographic simulations; and (2) backward coalescent genetic simulations. In each simulation, the first step begins with a first expansion wave (locals) starting at 3000 generations before present (t=0; Figure 1a) and colonizes an area of 100 × 100 demes in about 500 generations (Figure 1b). The square area at the lower left, defined as a zone of 50 × 50 demes, is only colonized during the second wave to represent a source of LDD. This second wave then invades the range colonized by the first-wave 1500 generations before present (t=1 500 generations) from a deme at the lower-left corner of the large area (Figure 1c). Note that, even though the first step consists in a forward demographic simulation, the simulated populations consist of haploid individuals. Thus, in this context, migration events correspond to the dispersal of genetic material (that is, gene copies at a given locus). The simulation of genetic processes such as interbreeding, admixture and gene flow is described below in more detail.

Figure 1
figure 1

Illustration of the colonization process under the front LDD model. (a) t=0, start of the first expansion in the upper square world. (b) 500 Generations later, the upper-right world has been fully colonized by the first wave, and a second expansion starts in the lower-left world. (c) 1000 generations later, the second wave is allowed to invade the upper-right world. (d) The study area is fully colonized and the genetic diversity of population samples is inferred by a coalescent approach.

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Both range expansions (of local and invading populations) are modeled according to a Poisson-distributed stochastic nearest-neighbor migration model, with N × m haploid emigrants (which can also be thought of as gene copies, as mentioned above) being sent at each generation to surrounding demes, where N is the current local population size and m is the migration rate, here set to 0.1. Once a deme is occupied by one or more migrants, logistic growth starts with intrinsic rate r=0.8 until the carrying capacity K is reached. In our simulations, we used K=50 for the first wave (locals) and K=500 for the second wave (invaders) and any individual colonizing an empty deme could establish a new population by itself in any scenario. Note that these parameters were arbitrarily chosen. Nonetheless, they allow the colonization of the simulated world to occur in the timeframe chosen and are relatively realistic and in keeping with previous modeling of range expansions (Currat and Excoffier, 2004; Currat and Excoffier, 2011).

When the invaders colonize an already occupied deme, they compete with locals for local resources following a standard Lotka–Volterra model (Currat et al., 2004; Currat and Excoffier, 2011), and they admix at a rate that is controlled by an interbreeding success rate γ as follows:

Where Ni and Nj are the haploid population size of the local and the invader populations respectively. Aji is then used to update local population densities as follows:

In this regard, a γ value of 1 implies random mating between populations, and a value of 0 implies reproductive isolation between populations (Currat and Excoffier, 2011). A more detailed explanation on this process is given in Currat and Excoffier, 2004.

This density-dependent competition eventually results in the extinction of the local population due to its lower carrying capacity and it restricts interbreeding to occur only on the invaders’ colonization front. In this regard, it is worth noting that the absolute values for carrying capacity K were also chosen arbitrarily. However, the important point here is that K should be lower for locals than for colonizers such that density-dependent competition leads to the disappearance of the local population, which, in the context of the settlement of the Americas by Europeans, models the decline in Amerindian population densities at the time of European colonization (Mulligan et al., 2004).

Modeling long-distance dispersal

LDD was implemented in two different ways. The first model, hereafter called ‘generalized LDD’, is described in more detail by Ray and Excoffier, 2010. Briefly, during the second range expansion (invaders), a given proportion δ of emigration events is chosen to be LDD events, and the remaining proportion (1-δ) are short-distance (nearest-neighbor) migration events as mentioned above. For each LDD event, a migration direction and a dispersal distance are chosen at random. The dispersal distance is drawn from a gamma distribution with shape parameter α=0.5 and scale parameter β=0.05, which leads to an average dispersal distance of α/β=10 demes for LDD events. We also arbitrarily imposed a maximum dispersal distance of 10 demes, such that the LDD kernel is a truncated Gamma distribution. Because it has been shown that the shape of the dispersal kernel can influence the colonization dynamics (Bohrer et al., 2005; Fayard et al., 2009), we performed additional simulations with an average LDD dispersal distance set to 5 or 20 demes.

In the second model, hereafter called ‘front LDD’, LDD events occur from an arbitrarily defined source population and exclusively target the wave front. In our simulations, the LDD source area is modeled as a zone of 50x50 demes adjacent to the 100x100 study area (Figure 1c), but the exact location of this source should not affect our results on rate of introgression. During the second range expansion in the study area (Figure 1c), demes on the wave front receive a certain proportion ɛ of haploid migrants (that is, gene copies) from the LDD source population. This model is thus directly inspired by the colonization of North America by European migrants who settled directly on the colonization wave front, as described by Meinig, 1993.

In the front LDD model, the wave front is defined as those demes having a population density N⩽0.5 K, which implies that all demes that have not reached 50% of their carrying capacity K will eventually receive LDD directly from the source population (Supplementary Figure S1). The parameter ɛ specifies at each generation t how much of the K—N(t) available room in the front-deme will be filled by LDD migrants, which are then drawn from a random deme of the LDD source population. Note that the proportion of front LDD migrants (ɛ) is very different from δ defined in the generalized LDD model; ɛ specifies how much of the empty space in the front-deme will be filled by LDD migrants drawn from the source population, whereas δ is the proportion of migratory events that are LDD events in the first model.

We considered four values for γ (0.02, 0.04, 0.06 and 0.08) and five values (0, 0.0001, 0.001, 0.01 and 0.1) for each LDD parameter (δ for the generalized LDD model and ɛ for the front LDD model). The combination of these parameters yields 20 different scenarios for each model of LDD, including the possibility of no LDD when δ or ɛ are equal to 0.

Estimates of introgression level

For each of these 40 tested scenarios, we performed 1000 independent forward simulations equivalent to the simulation of 1000 independent loci. At the end of each simulation, we drew population samples belonging to 25 demes distributed over a regular grid in the study area (Figure 1d). We used a backward coalescent approach (corresponding to the second step of the simulation process) to infer each gene copy’s origin (first or second wave) and recorded it for estimating introgression proportions (that is, the proportion of gene copies in the invader population coming from the local population) in the 25 sampled demes in the study area over the 1000 simulations. We also recorded the number of generations between the start of the invasion and the colonization of the whole map for each simulation (colonization time), as well as the number of generations during which locals and invaders coexisted in a given position of the simulated grid (cohabitation time), averaged over demes and simulations. Interbreeding events were quantified as the number of haploid individuals exchanged between locals and invaders in a given deme averaged over the whole map and over all simulations.

Results

The dynamics of the colonization process over the large (100 × 100 demes) square world is illustrated in Figure 2 for different LDD models. Although the front-LDD (Figure 2d) model yields a colonization pattern very similar to the case of no LDD (Figure 2a), where dispersal occurs as a single, contiguous wave, under the generalized LDD model (Figure 2b and d), LDD events lead to independent expansion waves in the habitat occupied by the local population. The number of these growing points is directly proportional to the proportion δ of LDD events (Figure 2b and c).

Figure 2
figure 2

Illustration of the dynamics of the colonization process under various models explored in this study. The left and right columns are showing areas occupied by the invaders at two successive arbitrary time points. The white part of the square world is occupied by locals when the invaders begin their colonization (from the lower left). Gray areas indicate regions occupied only by invaders, whereas black areas indicate regions where locals and invaders co-exist. (a) Dispersal without LDD. (b) Dispersal with generalized LDD (δ=0.0001). (c) Dispersal with generalized LDD (δ=0.1). (d) Dispersal with front LDD (ɛ=0.1).

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Levels of introgression increase with interbreeding rates γ but are negatively correlated with the proportion δ of long-distance events (Figure 3a and b). Indeed, when 10% of all migration events are generalized LDD, introgression levels are reduced by 27% and up to 61% depending on interbreeding rates (γ=0.08 to 0.02, respectively). However, generalized LDD is sufficient neither to completely suppress local introgression in the invading population (Figure 3b) nor to prevent gradients of introgression along the expansion axis (Figure 3a).

Figure 3
figure 3

Introgression levels under the generalized (a and b) and front (c and d) LDD models. (a and c) Shown are the proportion of introgressed genes (black section of the pie charts) in the invading population averaged over 1000 replicates for each of the 25 sampled demes. The average introgression level over all sampled demes is indicated under each map and interbreeding values (γ) and LDD parameters (the proportion δ of LDD events for the generalized LDD and ɛ for the front LDD model) are shown for each line and column respectively. The expansion of the invader started in the lower-left corner of the map. (b and d) Shown are the proportion of introgressed alleles as a function of interbreeding values (γ) and LDD parameters (δ) for the generalized LDD (b) and ɛ for the front LDD model (d).

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Contrastingly, we find that LDD events from a non-admixed population targeting directly the wave front—the front LDD model—can very strongly inhibit introgression (Figure 3c and d). Indeed, if 10% of individuals living on the wave front (ɛ=0.1) come directly from the source population, introgression is virtually no longer detectable at the end of the expansion (rightmost columns of both Figure 3c and d). This occurs even when interbreeding is relatively common between local and invading individuals (that is, γ=0.08), as depicted in the lower right corner of Figure 3c. Interestingly, the gradient of introgression that is still visible with generalized LDD (Figure 3a) is also suppressed when there is direct LDD to the wave front (Figure 3c).

Although introgression levels shown in Figure 3a and c are averaged over 1000 loci, individual loci display a much more spatially heterogeneous distribution of introgression as shown in Supplementary Figure S2. It is possible to see clear sectors of different introgression levels for colonization without LDD (Supplementary Figure S2a), and to some extent also in the generalized LDD model (Supplementary Figure S2b), but not with front LDD (Supplementary Figure S2c).

Although there is a negative correlation between δ and introgression in the generalized LDD model (Figure 3b), there is an increase in the number of interbreeding events and in the period of cohabitation between locals and invaders when there is a larger proportion of long-distance events (Table 1). On the other hand, front LDD causes both interbreeding and cohabitation time to decrease (Table 2). Colonization time decreases with both generalized and front LDD (Tables 1 and 2). In addition, very low levels of LDD lead to a larger variance in the dynamics of the colonization process, particularly in the generalized LDD model given the random nature of LDD events. Note that cohabitation time and interbreeding may be slightly underestimated for the front LDD model, since the source zone is modeled differently in the front and generalized LDD models. Indeed, the source zone is included in the computation of cohabitation times and interbreeding levels in the case of the front LDD model, but not for the generalized LDD model. Nevertheless, values within each table are comparable and trends for each case can be trusted.

Table 1 Average colonization time, cohabitation time and number of interbreeding events under the generalized LDD model
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Table 2 Average colonization time, cohabitation time, and number of interbreeding events under the front LDD model
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Finally, using different average dispersal distances of LDD events (5 or 20 demes; Supplementary Figure S3), we obtain results very similar to when the average LDD distance is 10 demes, thus suggesting that the effect of LDD on admixture does not depend much on the dispersal kernel.

Discussion

As previously described (Currat et al., 2008), introgression levels are expected to increase along the expansion axis due to recurrent interbreeding on the wave front, where surfing of introgressed alleles may occur as a consequence of the lower population density of the invaders relative to the locals. Interbreeding during range expansions, even at low frequency, can thus lead to high levels of introgression in the growing invading populations. However, as we show here, the presence of LDD events during the colonization lowers the introgression of local genes into the invaders gene pool, because genes migrating to the front from the core directly compete with local introgressed genes that are thus rapidly outnumbered. In addition, we show LDD can even completely suppress introgression when LDD events target exclusively the wave front (Figure 3). Indeed, the effect of front LDD is much more drastic than the effect of generalized LDD. For instance, with relatively high levels of interbreeding (γ=0.08) we have over 96% of introgression in absence of LDD, but if as little as 0.1% of the vacant space on the wave front is filled by migrants from the source population (ɛ=0.001), introgression levels drops to 43.2% (Figure 3c). Contrastingly, under the generalized LDD model, a similar introgression proportion is reached (42.6%) with much more extreme parameter settings, namely high proportion of LDD (δ=0.1) and a much lower interbreeding rate (γ=0.04). These results are consistent with the protective effect of intraspecific migrations against introgression (Currat et al., 2008; Petit and Excoffier, 2009). An important implication of this effect is that, for species that are able to disperse over long distances, such as some plants and birds, the absence of introgression after a range expansion may not necessarily be due to selection against hybrids (Barton and Bengtsson, 1986; Currat and Excoffier, 2004) or to selection against local alleles in the invader (Excoffier et al., 2009).

Strikingly, under the generalized LDD model, the number of interbreeding events increases with larger levels of LDD events (Table 1), in contrast to the introgression levels, which decrease with increasing LDD (Figure 3a). It implies that these interbreeding events are not as effective as in the absence of LDD. This seemingly paradoxical behavior occurs because demes colonized ahead of the wave front by LDD events take more time to reach their carrying capacity than demes on the wave front, as they do not receive migrants from the wake of the wave. This leads to a longer cohabitation time between locals and invaders (Table 1) and hence more time for introgression events to happen. On the other side, these introgression events are not as successful as in absence of LDD, because colonization by LDD events limits the dilution of the invader gene pool and prevents introgressed alleles to surf (Bialozyt et al., 2006; Ray and Excoffier, 2010; Berthouly-Salazar et al., 2013) and increase in frequency during the expansion. Since the number of interbreeding events per deme in case of front LDD is not much smaller than in the case of generalized LDD, it suggests that the much lower introgression level observed in the case of front LDD is not due to a lowering of the amount of interbreeding, but indeed to the prevention of the surfing of introgressed genes on the wave front.

Some loci show regions of high and low introgression (Supplementary Figure S2) that look similar to sectors of low allelic diversity created by gene surfing during range expansions (Currat et al., 2006; Hallatschek et al., 2007; Excoffier and Ray, 2008). Introgressed alleles, like any mutation, can thus potentially surf on the wave of advance and create sectors with high admixture levels. Nonetheless direct LDD to the wave front efficiently prevents the occurrence of these sectors of introgression.

We notice that the choice of demographic parameters for the simulations was done somehow arbitrarily, but the chosen parameter values are nonetheless relatively realistic. For instance, migration and growth rates, as well as carrying capacities are within the range of values explored previously for the analysis of modern and archaic humans evolutionary history (Currat and Excoffier 2004; Currat and Excoffier 2011). Changes in migration and growth rates would mainly affect the speed of the colonization but could also affect introgression levels. Indeed, as mentioned earlier, higher levels of gene flow between demes of the invading populations should prevent introgression (Petit and Excoffier, 2009) and higher growth rates should favor surfing (Klopfstein et al. 2006) and therefore have a positive effect on introgression. However, the main conclusion of our study, which is that LDD can drastically lower introgression, should be robust to alternative (but reasonable) values of the expansion model parameters.

On a related note, we identified no effect of changes in the shape of the dispersal kernel on the patterns of admixture (Supplementary Figure S3). This is surprising given that it has been shown that this parameter influences the colonization dynamics (Bohrer et al., 2005; Fayard et al., 2009). We notice, however, that the key feature of our simulations is to show that heavy-tailed distributions, as the ones we considered for the generalized LDD model, can indeed preserve initial genetic diversity during range expansions, as previously shown (Fayard et al., 2009; Goodsman et al., 2014; Alves et al., 2016), and that LDD directed to the wave front is more efficient in doing that, in agreement with an empirical study on starlings (Berthouly-Salazar et al., 2013).

Although our simulations may not be truly realistic by setting the source of LDD after the first wave, it is important to acknowledge that the exact location and exact settlement time of this source does not matter here as this source only acts as a reservoir of non-admixed migrants. Other factors like a high ratio of local to invader density or different interbreeding levels contribute more to final introgression levels (Currat et al., 2008).

The invasion of human populations into already occupied territories has led to a varying degree of admixture throughout the different parts of the world. For instance, the extent of Amerindian admixture in individuals of European descent is usually higher in South America than in North America (Wang et al., 2008; Halder et al., 2009; Lao et al., 2010; Lisabeth et al., 2011; Galanter et al. 2012; Ruiz-Linares et al., 2014; Salzano and Sans, 2014; Bryc et al., 2015; Homburger et al., 2015; Salazar-Flores et al., 2015). Our simulation results suggest that, among the various evolutionary, demographic and cultural mechanisms that may explain these varying levels of introgression, different modes of migration could also have an important role in such cases. In fact, the comparably efficient transportation system along rivers, roads, and, later, along railroads, facilitated migrations over large distances in the United States since early stages of colonialism, such that more than 30% of settlers were European-born in some areas (Meinig, 1993).

Earlier studies suggested that admixture levels in South America varied according to local Amerindian population density (Wang et al., 2008; Rubi-Castellanos et al., 2009), and pre-Colombian Amerindian population densities in North America were lower than those in Central or South America (Lange et al., 2006). Interbreeding in South America was mainly between European men and Native women resulting in a higher introgression of Amerindian alleles on the X-chromosome than the Y-chromosome (Bedoya et al., 2006; Bryc et al., 2010). Although mainly male colonialists arrived in South America, the situation was different in North America, where whole European families migrated together (Meinig 1986, 1993). Strong assortative mating in North America and lower population density could thus explain the absence of introgression. However, even with an interbreeding success rate of only 4% (γ=0.04), we would expect to see ~80% introgression after a range expansion without LDD (Figure 3). The absence of introgression of local alleles in North Americans thus suggests that this could be at least in part a result of the mode of colonization where many pioneers arrived directly to the front from Europe. We notice that our modeling of assortative mating may not capture all the complex aspects of admixture in humans. However, what we show is that even if there was a very strong disassortative mating or if hybrids were strongly rejected (which is captured by setting γ to 2%), one would still expect more than 30% of introgression in absence of LDD, and still more than 10% when generalized LDD is present (Figure 3). Allowing LDD to occur directly between the source and the wave front, as in our front LDD model, reduces this introgression rate dramatically: less than 1% introgression when only 1% of the individuals of the front come directly from the source with low levels of interbreeding (γ=2%). This implies that either extremely low levels of disassortative mating were tolerated (γ<<2%), or that some levels of direct gene flow from the source to the front indeed occurred. Since the latter is documented (Meinig, 1993), it is likely that current very low Amerindian introgression levels are partially due to the migratory behavior of Europeans during the colonization of North America.

It appears difficult to estimate the fraction of LDD events in a given species (Nathan et al., 2003), but our simulations of a wide range of δ values (from 0.01 up to 10%) may be suitable for explaining the observed genetic diversity of a variety of different organisms. Although generalized LDD may be realistic for many species, direct LDD from source to the front probably only occurs in humans or in highly mobile species like birds and some plants for which there may be seed dispersal by wind. In any case, it is remarkable that ɛ values lower than 1% already have a marked effect on introgression, which shows that the establishment of even very few non-admixed individuals at the wave front can thus have a large impact on final introgression levels.

Data archiving

This article does not report new empirical data or software. The input files as well as the executables necessary to perform the simulations are available through the Dryad electronic repository (http://datadryad.org/) under the following DOI: 10.5061/dryad.t281k.