Further adaptation at a single-locus barrier:

Assume that, initially, is the only polymorphic locus. The initial barrier strength is α and results entirely from local adaptation ( ). A new mutation occurs at a tightly linked locus, . This scenario is equivalent to adaptation at a single compound locus with alleles Ac and ac, and mutation generating new alleles AC with fitness and aC with fitness γ. At most, two alleles can be maintained on the island (Nagylaki and Lou 2001): the continental allele and the allele with the highest fitness on the island. Thus, any new adaptation on the island that produces a better allele than Ac will replace this allele (e.g., the AC allele for ). While any successful adaptation on the island increases the barrier strength ( ), adaptation on the continent can lead to a stronger or weaker barrier ( ), depending on whether γ is positive or negative. In particular, means that no polymorphism can be maintained. However, if there is a small but nonzero recombination probability among and , any adaptation on the continent will eventually also enter the island background. We then have two new alleles AC and aC replacing the old ones (Ac and ac) and the barrier strength remains unchanged as long as there is no epistasis among A and C.

We thus see that further adaptation at a single polymorphic locus will usually strengthen the genetic barrier, rather than weaken it. In particular, this holds for any further adaptation on the island. A three-locus architecture, , with tight linkage among all three loci leads to an analogous single-locus problem (after appropriate relabeling of parameters).

Note that the genetic barrier formed by a single locus relies exclusively on local adaptation: any isolation observed is due to the impossibility of coexisting in a common environment and not due to a genetic mechanism. This is different for barriers with multiple interacting loci, which is our focus in the remainder of the manuscript.

Further adaptation at a two-locus barrier:

Assume now that we start with a two-locus polymorphism at two incompatible loci and (a two-locus DMI) in loose linkage. The continental haplotype is aB, and Ab is the fittest haplotype on the island. A new mutation appears on the island at locus in tight linkage with . As discussed in the previous section, this generates a compound locus . The new mutation generates a third allele at this compound locus (e.g., AC), which we will call the A’ allele in the following. We denote the fitness advantage of the new allele A’ as and its epistatic interaction with the B allele at the locus as This leads to the dynamics of a triallellic locus (with alleles a, A, and A’) that interacts with a loosely linked biallelic locus in the genomic background (alleles b and B):(2)For tight linkage, we can assume that a fourth allele A” (e.g., A” = aC) will only originate by mutation or rare recombination after one of the alleles a, A, or A’ is lost. This leads again to the three-allele dynamics described by Equation (2). Results for the four-allele dynamics are given in section C2.4 in File S3. Further scenarios with adaptation at the locus or continental adaptation at the locus are discussed in section B7 in File S2.

The dynamic system given in Equation (2) allows up to nine equilibria, up to three of which can be simultaneously stable. In the section B2 in File S2, we show that alleles A and A’ can never coexist at a stable equilibrium [extending the single-locus result of Nagylaki and Lou (2001)]. Nevertheless, interaction of with an unlinked locus considerably adds to the complexity and can lead to qualitatively different results.

Whereas A and A’ cannot coexist, allele A’ can still invade the equilibrium formed by the DMI between loci and . In contrast to the single-locus case, the potential for A’ to invade no longer depends only on the fitness values, but also on the strength of migration (analytical expressions of the bounds are given in Equation B7 in File S2). In Figure 1, invasion of A’ is possible in all colored regions. Figure 1A shows invasion of an allele A’ with larger direct effect If negative epistasis is less severe for A’ than for A (), A’ will always invade (i.e., up to the maximal migration rate, of the original two-locus polymorphism). However, for strong negative epistasis of the new allele (), invasion of A’ is only possible for weak migration and a sufficiently low frequency of the competing B allele on the island. Figure 1B shows that the A’ allele can also invade if its direct effect is weaker (), provided that negative epistasis is also weaker (). This requires that migration is sufficiently strong, because the marginal fitness of A’ becomes larger than the marginal fitness of A only for a sufficiently large frequency of B alleles.

• Download figure
• Open in new tab
• Download powerpoint

Figure 1

The impact of the invasion of a new allele A’ at locus on an existing two-locus barrier (loci , in loose linkage). We show the strength of a genetic barrier against swamping for a new allele A’ as a function of its (scaled) epistatic coefficient. The strength of the original genetic barrier is indicated by the black line. In both examples, we have = Invasion of allele A’ can only happen in a finite interval for m (Equation (B7) in File S2 for explicit expressions), corresponding to the colored area. There are four possible outcomes to the successful invasion of the A’ allele, denoted by the background color: A’ replaces A and b remains present (in blue), A’ replaces A but allele B fixes (in cyan), the polymorphism at locus is lost (orange), and the continental haplotype fixes (red). If A’ successfully replaces A, the new two-locus barrier strength, is given by the blue line. The dashed cyan line shows the strength of the new single-locus barrier, whenever the locus is swamped and The yellow hatched area indicates that the genetic barrier at the locus is strengthened by the invasion of allele A’. Panel A is obtained for and panel B for

Successful invasion of A’ can have qualitatively different outcomes, indicated by the different colors in Figure 1. In many cases, an invading A’ allele displaces the old A allele and the system settles at a new equilibrium with an a/A’ polymorphism. The new equilibrium can either be a two-locus polymorphism (blue areas in Figure 1) or a single-locus polymorphism with the locus fixed for the B allele (cyan area). In both cases, the strength of the genetic barrier with respect to swamping can either increase (blue or cyan line above the black line) or decrease (blue or cyan line below the black line). Parameter ranges where invasion leads to a stronger genetic barrier, or are indicated by yellow hatches.

Strengthening of the two-locus barrier (blue area with yellow hatches in Figure 1) can be due to two mechanisms. First, selection against migrants can be stronger due to additional local adaptation () and therefore leads to a larger fitness deficit () for the continental haplotype on the island. This is the same mechanism as for the single-locus case. The genetic barrier is strengthened as long as epistasis, does not deviate too much from the epistasis generated by the previous allele, (Figure 1A, blue line above the black line). Indeed, if epistasis is too weak, the boost provided by the increased selection against migrants is negated by the weakening of selection against hybrids (since ). If epistasis is too strong, on the other hand, the marginal fitness of allele A’ is decreased due to the increased cost of hybrids. Allele A can invade such an equilibrium as soon as migration increases and A’ cannot strengthen the genetic barrier (see also section B6 in File S2).

The alternative mechanism corresponds to the reduction of selection against hybrids (Figure 1B). It works only if the continental B allele is deleterious on the island. Indeed, in this scenario, selection against hybrids does not contribute to the genetic barrier, as B is already maladaptive on the island. Nevertheless, epistasis still generates a cost for the island adaptation through the production of hybrids. Therefore, releasing the selective pressure on locus due to the hybrid cost () can strengthen the genetic barrier, even if this relief is associated with a reduction of the direct selective advantage of the island adaptation (). The reduction of the selection against migrants is here compensated by the much lower hybrid cost paid by allele A’ relative to allele A.

In contrast to the single-locus case, invasion of A’ does not imply that this allele is maintained in the population. Indeed, we find significant parameter regions, where the following scenario happens. First, allele A’ invades the island population (at its initial equilibrium with two-locus polymorphism), leading to the loss of the A allele. In the absence of allele A, allele B is no longer repressed and increases in frequency, making it impossible for allele A’ to maintain itself in the population. Consequently, the continental a allele swamps the island and the polymorphism at the locus is lost altogether (red and orange are as in Figure 1). Again, the polymorphism at the locus can either be maintained (orange area, Figure 1B) or destroyed (red, Figure 1A). Clearly, a necessary condition for such behavior is that the original two-locus polymorphism is not globally stable in the original a/A, b/B state space, but bistable together with an equilibrium with the a allele fixed. Numerical evidence strongly suggests that the fate of an invading A’ allele depends on the existence of a stable a/A’ polymorphism in the state space spanned by a/A’, b/B. If it does, the A’ allele will eventually establish (as discussed above); if it does not, the a allele will take over (we did not find a case where the A allele would return and displace A’ once the latter has been able to invade; see also section B3 in File S2 for a more detailed discussion and some proofs for specific cases).

A new allele A’ can thus function as a temporary state that enables switching among different equilibria of the original two-locus two-allele system. In the examples discussed above, temporary invasion of A’ will destroy a DMI polymorphism in this case. However, we also observe the opposite phenomenon: invasion of A’ may create an a/A-b/B polymorphism rather than destroying it. This is illustrated in Figure 2. Bank et al. (2012) described the origin of such a DMI as a result of secondary contact (or a similar starting condition). Here, we provide an alternative explanation that does not require an interruption of gene flow.

• Download figure
• Open in new tab
• Download powerpoint

Figure 2

Evolutionary trajectory with A’ as transient state. Panel A Frequency of derived alleles B (black), A (red), and A’ (blue) as a function of time. At t = 0, the population is almost monomorphic for the continental haplotype aB, with both the alleles A and A’ present at an extremely low frequency (). Colored blocks T1–T4 indicate when the population is close to an equilibrium, with the color matching the corresponding equilibrium. Panel B We represent the same trajectory in the space. Dots indicate equilibria, as indicated. Arrows indicate the evolutionary trajectory. Parameters used are: One can observe a similar behavior with locus starting polymorphic and allele B deleterious on the island (see Figure B2 in File S2).

We can compare the consequences of further adaptation on the island at an existing genetic barrier in the two cases discussed so far: a single polymorphic locus and a polymorphic locus that interacts with a second polymorphic locus . There are two notable differences. First, while further adaptation on the island always leads to a stronger barrier in the single-locus case, this is not the case for a two-locus barrier. Furthermore, invasion of a new allele no longer even guarantees establishment of this allele. On the contrary, we see that such an event can erase the existing barrier entirely. Second, the potential to strengthen the genetic barrier does not only depend on the fitness landscape, but also on the migration rate. Suppose that an allele A’ exists that leads to a stronger barrier than A, if it invades. Figure 1 shows that invasion may either require sufficiently weak (Figure 1A) or sufficiently strong migration (Figure 1B). The latter scenario leads to the interesting observation that stronger gene flow can sometimes trigger the evolution of stronger barriers to gene flow [in Figure 1B, the blue line is above the black line; invasion of the new mutant is only possible with relatively strong migration (colored area in the figure)].

We also observe a general trend to replace a polymorphism that is maintained by selection against hybrids by one that is maintained due to selection against immigrants. Indeed, whereas it is possible to strengthen the genetic barrier by weakening the strength of epistasis without affecting the amount of local adaptation, the opposite is impossible. Any increase in the strength of selection against hybrids needs to be associated to some increase in local adaptation.

Extension of a single-locus genetic barrier:

Assume that is the only polymorphic locus on the island ( therefore = ) and a new mutation C occurs on the island at a loosely linked locus . In the absence of epistasis, this mutation can invade and establish if, and only if, does not affect the barrier at all.

Figure 3B shows the effect of epistasis between C and B on the barrier strength. As expected, negative epistasis can strengthen the genetic barrier (blue area), while positive epistasis will almost always weaken it (orange and red areas). However, Figure 3 also shows that negative epistasis is not sufficient to strengthen the barrier. Indeed, a C allele may invade for sufficiently weak migration, but will be the first polymorphism swamped once migration increases (gray area). Obviously, in this case, the barrier strength at the polymorphic locus remains unaffected.

• Download figure
• Open in new tab
• Download powerpoint

Figure 3 C

strengthens the genetic barrier formed by a single polymorphic locus: comparison between a new mutation in tight linkage and one in loose linkage. The x-axis shows the strength of epistasis between B and C. The y-axis shows the selective advantage of new allele C. The background color indicates the consequence of the invasion of allele C on the genetic barrier at the locus. Gray: the genetic barrier remains unchanged; blue: the genetic barrier is strengthened; orange: the genetic barrier is weakened; and red: the polymorphism at locus is lost. In addition, on panel B, the solid black line gives the necessary condition for invasion of allele C on the island. Below this, bound invasion is always impossible. The black dashed line gives the sufficient condition for invasion. Above this bound, allele C can always invade, regardless of the migration rate (provided the polymorphism at the locus still exists). Analytical expressions for the two lines are given in Equation (C2) in File S3.

It is instructive to see how linkage affects the parameter range where further adaptation leads to a stronger barrier. For tight linkage (adaptation at the polymorphic locus itself), any allele with will invade the island and will strengthen the barrier (see Figure 3A) as long as epistasis does not cancel the selective disadvantage of allele B (). In contrast, strengthening the barrier by adaptation at a loosely linked locus is much more difficult. To reinforce the barrier at the loosely linked locus, the new C allele has to withstand both migration for and the hybrid cost generated by its interaction with allele B. The first condition alone implies as a necessary condition for a stronger barrier. Indeed, for a given the genetic barrier is strengthened as long as negative epistasis is not too strong. Stronger epistasis results in larger hybrid cost for C and therefore a larger direct effect (larger γ) is needed to compensate for it. For the barrier is strengthened for any negative epistasis, including a lethal incompatibility.

Finally, even if a new C allele would strengthen the barrier (blue area), it is not always able to invade. Invasion of allele C requires(3)From Equation (3), one can deduce that a necessary condition for invasion of allele C is and a sufficient one is For any γ value between these two limits, invasion will be possible only if migration is sufficiently small. Such a constraint does not exist for the tight linkage case, as migration does not affect the fate of the new allele (given that remains polymorphic).

So far, we have considered the interaction of a new island adaptation C with a continental adaptation B. Alternatively, there are two other possibilities: we can also study interactions between B and a new continental adaptation or interactions among two island adaptations. The results are similar (see Figures C2 and C3 in File S3 and our discussion in the Supplemental Material).

Extension of a two-locus genetic barrier:

To complete the analysis of this section, we now ask how an existing genetic barrier of two interacting loci in loose linkage is affected by adaptation at a third locus that is also in loose linkage with the previous ones. In particular, we focus on the question of how a continental allele (the B allele at the locus) can be prevented from swamping the island. Depending on the direct fitness effect β of this allele on the island, we find similarities or differences to the extension from one to two loci discussed above.

If the continental adaptation B is deleterious on the island (), direct selection against migrants (all carrying allele B) contributes to the genetic barrier, at that locus. As Figure 4A shows, transition from two to three loci is analogous to the step from one to two loci and also the qualitative results agree (see section C1.2 in File S3 for details). Indeed, the presence of a first island adaptation (the A allele) does not make it any easier for a second, loosely linked island adaptation (the C allele) to strengthen the genetic barrier. In particular, the C allele still needs to have a stronger direct effect (in magnitude) than B, Allele C also needs to interact negatively with allele B, Finally, since this interaction generates some hybrid cost, this cost must be compensated by some extra local adaptation (larger γ). For example, a new mutation, interacting with allele B, with a direct selective advantage γ slightly larger than might not be able to strengthen the genetic barrier even if it fulfills the first criteria. These conditions are analogous to the one-to-two-locus barrier transition (see Figure 3B, gray area above the line). However, C can be weaker than the A adaptation () and still lead to a stronger barrier (see Figure 4A).

• Download figure
• Open in new tab
• Download powerpoint

Figure 4

for two and three loci in loose linkage. The x-axis corresponds to the epistasis between alleles B and C. The y-axis measures migration rate. The different lines correspond to the resistance to swamping at locus under different scenarios. The initial single-locus and two-locus barriers, are given in black and gray. The impact of a new allele C on the single-locus and two-locus genetic barrier is represented by the blue and orange lines, respectively. Allele B is deleterious on the island for panel A and advantageous on the island for panel B. The thin vertical black line indicates the absence of epistasis. Panel A is obtained for () and panel B for ().

We now consider a continental allele B that is beneficial also on the island (Figure 4B). In the haploid model, a single-locus genetic barrier is impossible. A genetic barrier can be formed if a second polymorphic locus, , interacts with through negative epistasis, generating selection against hybrids. However, a stable genetic barrier only exists if the direct and the epistatic effect of the C allele are both strong, and (see section C1.1.2 in File S3 for details), represented by the blue line in Figure 4.

Consider now such a two-locus barrier between loci and . We want to investigate under which conditions a polymorphism at a loosely linked locus strengthens the barrier against swamping at the locus, (orange line on Figure 4). With an A allele already present, there is no lower bound for the negative epistasis of the new C allele: any value () can increase the barrier strength. The new allele still has to fulfill a condition on the direct effect: Otherwise, allele C is the first allele that is lost when gene flow increases. However, the condition is weaker than the one on the A allele; indeed is a sufficient condition. Allele C can even have the weakest direct effect (Figure 4A) and still contribute to the strengthening of the barrier, in contrast to the case discussed above. The two island adaptations share the cost of forming hybrids, making it possible to prevent a strongly advantageous continental allele to fix on the island, despite their own relatively weak selective advantage () (Figure 4A).

Not only the maintenance, but also the invasion, of the new polymorphism in loose linkage is strongly affected by the existence of a polymorphism at locus . In its absence, the new mutation has to overcome the migration cost and the full incompatibility due to B being already fixed In addition, epistasis has an ambiguous effect: it hinders invasion of the new allele while the formation of the two-locus genetic barrier requires relatively strong negative epistasis (). This ambiguous effect makes invasion and establishment of a genetic barrier in this setting extremely unlikely. However, once a two-locus genetic barrier exists, invasion of a new allele is much easier and always possible if migration is sufficiently small (the invasion criterion tends to γ as ). Invasion of a third mutation is therefore similar to the previous case (allele B is deleterious on the island).

As we have seen, the constraints on the C allele are not so severe, but the flip side is that also the effect on the barrier strength is quite weak: roughly 10% of the direct effect of the C allele for Figure 4A and 5% for Figure 4B. In comparison, when all loci are in tight linkage, 100% of the direct effect of the new mutation contributes to strengthening the genetic barrier.

In the supplement, we explore a slightly different scenario, where the new mutation C does not interact directly with allele B but with allele a. Our results show that indirect strengthening of the genetic barrier, by increasing the marginal fitness of the A allele, can be the most efficient scenario (Figure C6 in File S3).

To summarize, if B is deleterious on the island, the presence of previous island adaptations does not much affect the invasion criterion for a new barrier gene, nor its impact on the barrier strength. However, if B is advantageous on the island, previous island adaptations change the invasion criteria from extremely stringent to somewhat more flexible. In addition, a genetic barrier can exist over a larger parameter range thanks to this third polymorphism.

Barrier strength and linkage architecture:

Assume now that , , and are placed without restrictions on recombination distance. For a given set of selection parameters, which linkage architecture will form the strongest barrier?

For a two-locus genetic barrier (loci and ), this question has been addressed by Bank et al. (2012). The main finding there is that selection against migrants is strongest for tight linkage while selection against hybrids is maximal in loose linkage, when most incompatible hybrids are produced. With both factors acting, the strongest barrier still results from one of these extreme architectures: is maximized for tight linkage whenever selection against migrants is the main driving force. In particular, this is the case whenever B is deleterious on the island (Figure C20a in File S3). In contrast, selection against hybrids is the only viable factor if the continental type, aB, also has the highest fitness on the island. In this case, we obtain the maximal in loose linkage (Figure C20b in File S3). Assuming that a genetic barrier can be formed both in tight and loose linkage, the loose linkage architecture forms the strongest barrier if:(4)In particular, there is never a maximum for intermediate recombination rates.

The case of three loci is more complicated because conflicting options can exist, e.g., the strongest barrier for pairs and is obtained with the different loci in tight linkage, but the strongest barrier for the pair is generated with the two loci in loose linkage. Still, numerical analysis suggests that the strongest barrier is obtained at the extreme ends of the recombination scale, either for or for between pairs of loci (we were not able to prove this claim, but did not find any counterexamples in numerical checks; see in Figures C24 and C25 in File S3). In more detail, we find the following. First, assume that C appears on the island. As long as tight linkage among and provides the strongest two-locus barrier, in tight linkage with and formed the strongest barrier (Figure 5, A and B red area, proof in section C3.2.1 in File S3). Selection against migrants is the key mechanism. If the strongest two-locus barrier is shaped by loci and in loose linkage, we obtain the strongest three-locus barrier for an additional adaptation C that occurs in tight linkage with either or . The new mutation contributes to a stronger barrier by either strengthening selection against hybrids (blue area, γ small, strongly negative), or by strengthening selection against migrants by reducing the direct effect of the B allele (green area, close to 0). Figure 5 shows that the parameter space for having the strongest barrier in tight linkage is much larger than for loose linkage. However, genomic regions around any locus that are effectively in tight linkage are small and randomly placed loci will more likely behave as loosely linked. Therefore, optimal nonlocal barriers with loose linkage between two loci may be easier to evolve than local (island-type) barriers with tight linkage among all loci.

• Download figure
• Open in new tab
• Download powerpoint

Figure 5

Linkage architecture forming the strongest genetic barrier for three mutations. In each panel, the x-axis corresponds to the epistasis between C and its interacting allele, B for the first row and A for the second row. The y-axis corresponds to the selective advantage of allele C on the island. The different colors indicate the linkage architecture and location at which a C mutation should appear to maximize Having all loci in tight linkage can be interpreted as the existence of a single island of divergence, and two loci in tight linkage and a third one in loose linkage as two islands of divergence, see Discussion. The case of three loci in loose linkage is not represented as it never provides the strongest barrier. Analytical expressions for the barrier strengths are given in the Equations (C7)–(C14) in File S3. If the initial barrier remains the strongest when C appears on the continent, it indicates that the new mutation will always weaken the barrier. If C appears on the island, the barrier is unaffected by the presence of the new mutation. The strongest genetic barrier corresponds to: Initial two-locus barrier: , black and , gray. Three-locus barrier: , red; , purple; , blue; and , green. Two-locus barrier after fixation of C: black and white chequered pattern, and gray and white chequered pattern, .

If C appears on the continent, we observe similar results (cf. Figure 5, C and D). Having all loci in tight linkage forms the strongest barrier as long as the continental adaptations are deleterious on the island and do not generate positive epistasis (see section C3.2.2 in File S3 for proof). If having all loci in tight linkage does not generate the strongest barrier, then two loci in tight linkage and the last one in loose linkage offers the strongest genetic barrier, with in tight linkage with , increasing both selection against migrants and hybrids (green area), or in tight linkage with , to only strengthen selection against migrants (rare, blue area). Fixing C is another possible mechanism to strengthen the genetic barrier if C generates positive epistasis with A (checkered areas). In this last case, the genomic location of locus does not matter.

Three loci in loose linkage never seems to be the strongest linkage architecture in our model. Indeed, we did not find such an architecture despite an extensive numerical search (although we were not able to prove this). Results can be different in more complex models. After extending our model to include general three-locus epistasis, we were able to construct a case where the strongest barrier has all three loci in loose linkage (cf. Figure C23 in File S3). However, the scenario requires a very specific type of three-locus interaction (epistasis between B and C is only expressed in the absence of A) and careful fine-tuning of the selection parameters.

As a general rule, the strongest barrier is usually formed with two clusters of loci in tight linkage: one cluster formed by all loci with selection against the immigrating allele, and the other cluster with all loci where the immigrating allele is beneficial on the island. These two clusters of loci are in loose linkage with each other, maximizing the expression of the incompatibilities among clusters and thus selection against the immigrating continental alleles that are not themselves deleterious on the island.

Extension of a two-locus genetic barrier, diploid populations:

Here, we extend our analysis to diploid populations. More precisely, we are interested in the similarities and differences between the haploid and diploid models. The diploid model is quite complex due to the number of equations and parameters. As mentioned in the model section, we focus on two specific dominance schemes for the interactions: codominance and recessivity. Despite this simplification, only few cases (mostly when the diploid case reduces to the haploid case) allow for analytical results. In Figure 6, we therefore compare numerical results for the strength of migration barriers. Rather than discussing each single linkage architecture (colored lines in Figure 6), we focus on the qualitative similarities and differences that emerge from comparing the total pattern.

• Download figure
• Open in new tab
• Download powerpoint

Figure 6

Maximal migration rate, in haploid and diploid models. Colored lines represent the maximal migration rate to maintain a polymorphism at loci and in the presence of a C adaptation for various linkage architectures. The black and gray lines serve as reference for a two-locus genetic barrier (in the absence of C) for tight linkage and loose linkage, respectively. We observe that the pattern of colored lines in (panel A and panel B) (haploid and diploid recessive) is very similar. For the codominant diploid case (panel C), there are a few notable differences, as explained in the text. Parameters: and if the C allele appears on the island (solid lines) and if C appears on the continent (dashed lines). The x-axis shows the epistatic interaction between C and its interacting allele ( if C appears on the continent or if C appears on the island).

Comparing the migration barriers for the haploid case (Figure 6A) with recessive diploids (Figure 6B), we find broad qualitative agreement (if all loci are in tight linkage, for both cases are identical). In particular, adaptation at the locus will weaken or strengthen the genetic barrier for the same linkage architectures among the three loci, and in approximately the same parameter ranges. Also, having all loci in loose linkage never seems to generate the strongest barrier for a given set of parameters. Furthermore, for a given set of parameters, numerical simulations suggest that we will observe qualitatively the same optimal linkage architectures as in the haploid case when we increase .

Also, the comparison of the haploid case and codominant diploids (Figure 6A vs. Figure 6C) shows many similarities. For several architectures, the dynamics (and thus the migration barriers) are identical. Indeed, as long as all interacting loci are in loose linkage, the haploid and diploid codominant model share their dynamics. This result holds for three different linkage architectures: all loci in loose linkage (orange lines), as well as and in tight linkage and in loose linkage if C appears on the island (blue solid line), or and in tight linkage and in loose linkage, if C appears on the continent (green dashed line). However, there is one major difference: epistasis between loci in tight linkage can be expressed. This is most noticeable when all loci are in tight linkage (red lines). Epistasis can be expressed directly in the F1 generation without any recombination event and therefore the behavior of DMIs in tight linkage differs strongly from its haploid or recessive counterparts. Surprisingly, positive epistasis between B and C, with C appearing on the island, can strengthen the genetic barrier. One can relate this case to the two-locus three-alleles model, where we have seen that reducing the hybrid cost is a viable option to strengthen the genetic barrier. A similar mechanism applies here as well.