The outcome of selection in structured populations with spatially varying selection pressures depends on the interaction of two factors: the level of gene flow and the amount of heterogeneity among the demes. Here we investigate the effect of three different levels of spatial heterogeneity on the levels of genetic polymorphisms for different levels of gene flow, using a construction approach in which a population is constantly bombarded with new mutations. We further compare the relative importance of two kinds of balancing selection (heterozygote advantage and selection arising from spatial heterogeneity), the level of adaptation and the stability of the resulting polymorphic equilibria. The different levels of environmental heterogeneity and gene flow have a large influence on the final level of polymorphism. Both factors also influence the relative importance of the two kinds of balancing selection in the maintenance of variation. In particular, selection arising from spatial heterogeneity does not appear to be an important form of balancing selection for the most homogeneous scenario. The level of adaptation is highest for low levels of gene flow and, at those levels, remarkably similar for the different levels of spatial heterogeneity, whereas for higher levels of gene flow the level of adaptation is substantially reduced.

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THE possibility of the maintenance of genetic variation by some form of balancing selection has been a long-standing issue in theoretical population genetics (LEWONTIN 1974). One form of balancing selection, heterozygote advantage, can maintain genetic variation, but standard population-genetic theory shows that the conditions for stable maintenance are highly restrictive: The proportion of random fitness arrays leading to stable, fully polymorphic equilibria becomes vanishingly small for even a moderate number of alleles (LEWONTIN et al. 1978). Another potential solution, balancing selection arising from spatial heterogeneity, what we call here local selection, has also often been suggested as a reason for the high levels of genetic variation found in natural populations (KASSEN 2002). Theoretically, by combining both forms of balancing selection, the potential for viability selection to maintain variation increases, reducing the restrictions that exist for heterozygote advantage alone (LEVENE 1953; KARLIN 1982; NAGYLAKI and LOU 2006b, 2007). Nevertheless, investigation of the proportion of random fitness arrays leading to stable, fully polymorphic equilibria in a spatial context shows that the potential for both forms of balancing selection to maintain variation is still very restrictive for higher numbers of alleles (STAR et al. 2007a).

While such a historic fitness-space investigation is useful for characterizing the restrictive parts of fitness space that maintain variation, by using a different methodological approach, it has been shown that such parts are quite easily reached by a construction approach incorporating a dynamic process of mutation and selection (SPENCER and MARKS 1988; MARKS and SPENCER 1991). Moreover, using such a construction approach in a spatial context shows that both forms of balancing selection, heterozygote advantage and local selection, emerge from such a model and the relative importance of each depends on the level of gene flow. For high levels of gene flow, heterozygote advantage is more important, whereas for lower gene-flow rates, local selection predominates in maintaining variation (STAR et al. 2007b). Thus the restricted parts of fitness space that maintain variation easily evolve out of a simple evolutionary process of mutation and selection. A critical assumption of this study, however, was that the fitness values for the mutants were uncorrelated between the demes (STAR et al. 2007b).

More realistically, fitness values for the same genotypes in different environments are likely to be correlated in some way, depending on the degree of heterogeneity of the environment (KASSEN 2002). In a highly heterogeneous environment, physical constraints may induce negative correlations between the fitnesses of genotypes in different demes, whereas a more homogeneous environment may induce positive correlations. Intuitively, more heterogeneous environments might be expected to enhance the levels of polymorphism, especially for lower levels of gene flow (SMITH and HOEKSTRA 1980; KARLIN 1982). Some examples from natural populations indicate that the actual fitness values can differ substantially between different demes (HANSKI and SACCHERI 2006; BOLNICK and NOSIL 2007). Such cases, however, are likely to be extreme because more moderate differences in fitness will be harder to detect. Moreover, balancing selection arising from spatial heterogeneity with such smaller differences in fitness is relatively more influenced by gene flow, reducing the potential of this particular form of balancing selection to maintain genetic variation. Nevertheless, the degree by which these different levels of environmental heterogeneity and gene flow may interact to influence the selective maintenance of multiallelic genetic variation is unknown, even in a simple single-locus construction model.

Here we investigate the effect of either negatively or positively correlated mutational fitness values, which imply different levels of environmental heterogeneity, on the amount of polymorphism maintained in a spatial two-deme constant-viability selection model incorporating recurrent mutation. We are also interested in the relative importance of both forms of balancing selection for the different environments. Furthermore, the average levels of fitness that result from this model are examined to see if the varying heterogeneity influences the final level of adaptation that is achieved by the evolutionary process of mutation and selection. Finally, we investigate the effect of the different levels of environmental heterogeneity on the stability of the resulting equilibria. All above variables are investigated in interaction with different levels of gene flow since these levels critically influence the outcome of selection in heterogeneous environments (LENORMAND 2002).

The model used here is based on the well-known recurrence equations for constant-viability selection at a single locus, which were adapted for two demes. The random effects of drift are ignored, mating is random within each deme, and generations are discrete. The frequency of the ith allele (i = 1, 2,..., k), A_i, in the dth deme (d {1, 2}), after selection is given by

(1)

in which is the current frequency of A_i in the dth deme,

is the current marginal fitness of A_i in the dth deme,

is the fitness of the A_iA_j genotypes in the dth deme, and

is the current mean fitness of the dth deme.

Selection acts locally and this model is therefore one of soft selection. Gene flow follows selection, and a proportion (m) of the frequency vector p_i,d is divided over both demes, giving the new frequency of A_i in deme d,

(2)

where if d = 1 and vice versa. The model is initiated with a single allele with a frequency of 1.0 and a homozygote fitness of 0.5 in both demes. If k is the number of alleles in the total population in a particular generation, the next generation is initiated by adding a new mutant allele, A_k+1, to a random deme with an initial allele frequency of 10^–4. Equations 1 and 2 are then applied and any allele is considered extinct if .

The genotypic fitnesses, drawn from the uniform distribution on [0, 1], were generated for each new mutant allele , using the sum-of-uniforms method (CHEN 2005). This method generates two sets of uniformly distributed random variables (one for each deme) with an average correlation coefficient (₀) between the sets. The genotypic fitnesses for each different mutant are mutually stochastically independent.

The model was run for three different levels of initial correlation (₀ {–0.5, 0, 0.5}) and seven different levels of gene flow (m {0, 0.01, 0.05, 0.1, 0.2, 0.5, 1.0}) up to 10,000 generations with 1000 replicates for each combination of m and ₀. Note that ₀ = –0.5 corresponds to high levels of environmental heterogeneity, whereas ₀ = 0.5 means the two demes are environmentally similar, and ₀ = 0 is the value used in STAR et al. (2007b).

After 10,000 generations, the number of alleles present (n) and the fitness sets were recorded for further analysis. For each fitness set, balancing selection was evaluated in two ways. First, overall levels of heterozygote advantage were calculated. For fitness sets with more than two alleles, the term heterozygote advantage or heterosis becomes ambiguous (LEWONTIN et al. 1978; NAGYLAKI and LOU 2006a), as the term can indicate at least three different properties: total heterosis (i.e., all heterozygotes are fitter than all homozygotes), pairwise heterosis (i.e., heterozygotes are fitter than their corresponding homozygotes), or average heterosis (i.e., heterozygotes are on average fitter than homozygotes). In this study, we consider only average heterosis, and we calculate the level of heterozygote advantage in fitness sets after 10,000 generations as , where the averaging is over all fitnesses and both demes.

Second, Pearson's correlation coefficients () between the fitnesses of genotypes in the two demes were calculated as a measure of local selection. Levels of after 10,000 generations can differ from the initial mutational distribution ₀ if mutants with correlations from a particular subset of that distribution ₀ are consistently favored by selection. Also, the level of adaptation after 10,000 generations was investigated by calculating the average fitness () over both demes for each combination of m and ₀. Because each combination of m and ₀ results in a varying number of alleles (n) after 10,000 generations, levels of H, , and may be confounded by different values of n. To correct for different numbers of n, the variables of interest were analyzed using ANCOVA, with m and ₀ as factors and n as the covariate.

Fully polymorphic equilibria in a two-deme model with spatially varying selection pressures might be equilibria that are locally stable rather than globally stable. Allele frequencies converge to these fully polymorphic equilibria only from some initial values. Both the presence of locally stable equilibria and the size of these convergence regions (also known as domains of attraction) were investigated by reiterating each recorded fitness set with 250 random initial allele-frequency vectors. These initial allele-frequency vectors were generated using the "broken-stick method" (HOLST 1980) and each of the 250 evaluations was iterated until equilibrium, , or until any allele became extinct at . We define three types of fitness sets as in STAR et al. (2007b): All initial allele-frequency vectors iterated to the recorded polymorphic equilibrium for type I fitness sets; some, but not all, initial allele-frequency vectors did so for type II fitness sets; and no vectors iterated to the recorded polymorphic equilibrium for type III fitness sets. Thus, type I fitness sets can be considered the most stable, type II fitness sets moderately stable, and type III fitness sets the least stable. Type III fitness sets may occur because of one or more transient alleles that can be present in the recorded fitness sets since iteration of the model was stopped after an arbitrary time (MARKS and SPENCER 1991). For type II fitness sets the proportion of frequency vectors leading to a fully polymorphic equilibrium was also recorded as a measure of the size of domain of attraction. A type II fitness set with a larger domain can be considered more stable.

Numbers of alleles:

Gene flow (m) and initial correlation (₀) have a strong interaction effect on the mean level of polymorphism after 10,000 generations (Figure 1 and Table 1). Only for intermediate levels of m (0.01 < m < 1.0), the difference in levels of polymorphism is profound for the different ₀; for ₀ = –0.5, the level of polymorphism drops much more slowly with increasing levels of m compared to ₀ = 0 and ₀ = 0.5. Interestingly, both ₀ = –0.5 and ₀ = 0.5 maintained higher levels of polymorphism compared to uncorrelated fitness sets for higher levels of m (m 0.5).

View larger version (13K): In this window In a new window Download PPT slide   FIGURE 1.— The average number of alleles (n) maintained after 10,000 generations as a function of gene flow and three levels of initial correlation (0) of mutant genotypic fitness values between the demes. The error bars are 95% confidence intervals. Gene flow and level of 0 have a significant interaction effect on the average number of alleles maintained (two-factor ANOVA, F12, 20,979 = 270, P < 0.0001).

 

View this table: In this window In a new window   TABLE 1 The number of alleles leading to n alleles for seven levels of gene flow (m) and three levels of initial correlation (0) of mutant genotypic fitness values between the demes

 

Heterozygote advantage:

Levels of heterozygote advantage (H) are used here as a heuristic for the relative importance of heterozygote advantage as a form of balancing selection in our two-deme model and we analyzed only combinations of m, ₀, and n for which at least 10 replicates were found (Table 1).

The levels of gene flow, the level of initial correlation, and the number of alleles have an interaction effect on H after 10,000 generations (Figure 2). For low n (n 4) patterns of H are not straightforward, but, for larger n (n 5) patterns become clearer. For these higher n, values of H are mainly determined by both m and ₀ and are largely consistent for different n. Again, as with levels of polymorphism, the largest differences between combinations of m and ₀ are found for intermediate levels of m. These results show that heterozygote advantage as a form of balancing selection is the most important for ₀ = 0.5, moderately important for ₀ = 0, and the least important for ₀ = –0.5 for comparable levels of m and n. Furthermore, these patterns are not confounded due to the presence of many rare alleles as a similar pattern for heterozygote advantage was found when performing this analysis using only the fitnesses of common alleles (, data not shown).

View larger version (21K): In this window In a new window Download PPT slide   FIGURE 2.— Levels of heterozygote advantage after 10,000 generations as a function of gene flow (m) and three levels of initial correlation (0) of mutant genotypic fitness values between the demes. The values of 0 used are –0.5 (open symbols), 0 (shaded symbols), and 0.5 (solid symbols). Standard errors are small (all <0.02, 82% < 0.01) and omitted for clarity. Levels of m, 0, and n have a significant interaction effect on H (ANCOVA, F12, 20,735 = 72.5, P 0.0001, m and 0 as factor, n as covariate).

 

Local selection:

As a measure of local selection, Pearson's correlation coefficients () were calculated for each pair of fitness sets for which n 2 after 10,000 generations. The genotypic mutant fitness sets are generated in our two-deme model with an initial distribution of ₀ between the demes. We are therefore interested in relative changes compared to these initial distributions. Relatively stronger negative correlations indicate a tendency for selection to lead to patterns of local selection, whereas relatively stronger positive correlations indicate the opposite. As in the analysis of heterozygote advantage above, only combinations of m, ₀, and n were used for which at least 10 replicates were found. In contrast to the analysis of heterozygote advantage, some patterns of local selection became more profound when using the fitnesses of common alleles () and both results are shown in Figure 3.

View larger version (31K): In this window In a new window Download PPT slide   FIGURE 3.— Levels of Pearson's correlation () after 10,000 generations as a function of gene flow (m) and three levels of initial correlation (0) of mutant genotypic fitness values for all alleles (a–d) and the subset of common alleles (e–h, ) between the demes. The values of 0 used are –0.5 (open symbols), 0 (shaded symbols), and 0.5 (solid symbols). Standard errors are small (all <0.15, 93% < 0.02) and omitted for clarity. For all alleles (a–d), levels of m, 0, and n have a significant interaction effect on (ANCOVA, F12, 20,735 = 17.5, P 0.0001, m and 0 as factor, n as covariate). For common alleles (e–h) the same effect is slightly less significant (ANCOVA, F12, 20,735 = 9.74, P 0.0001).

 
Obviously, the final level of correlation () that emerges from the model is strongly influenced by the initial level of correlation (₀) between fitnesses of the newly arising mutants (Figure 3). Regardless of the level of ₀, correlations () increase with increasing m. When including rare alleles in the analysis (Figure 3, a–d), for fitness sets that result from ₀ = 0.5, most levels of m and n do not lead to more local selection (i.e., < 0.5). Nevertheless, when excluding rare alleles (Figure 3, e and f), some patterns of local selection (i.e., < 0.5) do appear for these environments for low levels of m. These results suggest that in more homogeneous environments, common alleles are more likely than rare alleles to be locally adapted. In contrast, in the other two more heterogeneous environments, ₀ = –0.5 and ₀ = 0, fitnesses are substantially more negatively correlated for low levels of m, regardless of the types of alleles used for the analysis, although the patterns seem slightly stronger when excluding rare alleles. Thus, local selection in these more heterogeneous environments is strong enough to be detected, irrespective of the confounding effect of rare alleles. Moreover, the fitness sets that result from ₀ = –0.5 maintain their relatively stronger negative correlation for higher levels of m, when compared to fitness sets from ₀ = 0. Therefore, selection in more heterogeneous environments leads to these patterns of increased local selection for higher levels of gene flow compared to a more homogeneous environment.

An initially surprising result is that is negative for simulations with m = 0 and ₀ = 0, where perhaps an average of = 0 would be expected. This result is easily explained; in the absence of migration both demes develop independently and acquire their own set of alleles. Successful invaders therefore specifically require high genotypic fitness values in one deme only, regardless of its fitness characteristics in the other demes. Because these high genotypic fitness values are selected for in two different sets of alleles (one set of alleles in each deme), slightly negatively correlated fitness sets will evolve after many generations of selection. Overall, the level of fitness sets that result from ₀ = 0 is mostly influenced by m compared to both other levels of initial correlation. Perhaps initially uncorrelated fitness sets have a potentially wider range of either negatively or positively correlated mutant fitness sets and since these more extreme ranges are favored for the different levels of m, this wider range more easily emerges.

Fitness:

The average levels of fitness () achieved by the simulated populations are substantially higher than the initial fitness (0.5) with which the model was seeded. Nevertheless, the levels of gene flow (m), the level of initial correlation (₀), and the number of alleles (n) have an interaction effect on (Figure 4). Ignoring levels of ₀, fitness decreases with increasing levels of both m and n. Nevertheless, the more horizontal slopes of the plots for higher levels of m (m > 0.2) show that the relative effect of m on decreases, especially for the more homogeneous environments (₀ = 0 or ₀ = 0.5). Interestingly, for an increasing number of alleles (n > 4), the effect of different levels of ₀ on is small, especially for lower levels of m (m < 0.2). For these levels of m, each deme more-or-less evolves in isolation and as such the level of adaptation is less influenced by migrants from the other deme. In contrast, for higher levels of m (m 0.2) and for lower n (n 7), the effect of the different environments is substantial; more heterogeneous environments consistently lead to lower levels of average fitness. If mutants have substantial levels of local adaptation and subsequently migrate to the deme in which they are less fit, these unfit mutants have an effect on the average fitness of that deme. This effect is obviously strongest for the mutants that are relatively more locally adapted (i.e., negatively correlated) and have the highest levels of gene flow (i.e., more mutants with lower fitnesses are migrating). Overall, the simulated environmental heterogeneity strongly interacts with gene flow to influence the final level of adaptation.

View larger version (19K): In this window In a new window Download PPT slide   FIGURE 4.— The average level of fitness () after 10,000 generations as a function of gene flow (m) and three levels of initial correlation (0) of mutant genotypic fitness values between the demes. The values of 0 used are –0.5 (open symbols), 0 (shaded symbols), and 0.5 (solid symbols). Standard errors are small (all <0.02, 95% < 0.01) and omitted for clarity. Levels of m, 0, and n have a significant interaction effect on the average level of fitness (ANCOVA, F12, 20,735 = 40.6, P < 0.0001, m and 0 as factor, n as covariate).

 
The different values to which evolves in systems with different values of ₀ may be due in part to the different average numbers of alleles in these systems. Since less polymorphic populations tend to have higher mean fitnesses, they are less likely to admit new mutants. Thus the higher number of alleles in the ₀ = –0.5 simulations reduces the average compared with runs in which ₀ = 0.0 and 0.5.

Stability of equilibria:

The stability of the fitness sets was investigated by recording the proportion of simulations leading to type I, II, or III fitness sets for each level of gene flow (m) and level of correlation (₀). Preliminary analysis showed no particular trends for n (data not shown) and domain size was investigated only for combinations of m and ₀ for which at least 10 type II fitness sets were found.

Gene flow (m) and level of correlation (₀) have a particularly strong effect on the types of fitness sets that maintain polymorphism; for high levels of gene flow most fitness sets are type I, whereas for low levels of gene flow most were type II (Table 2). Differences between the levels of ₀ are most pronounced for intermediate levels of m (0.05 m 0.5); for comparable levels of m, the proportion of type I fitness sets increases with the value ₀ at the expense of the proportion of type II and type III fitness sets. The increased proportion of type I fitness sets coincides with higher levels of heterozygote advantage. This form of balancing selection is more likely to have a globally stable equilibrium in the absence of other forms of balancing selection (LEWONTIN et al. 1978; KARLIN 1982). Thus fitness sets in which heterozygote advantage is the more important form of balancing selection (which are the fitness sets in more similar environments with higher levels of gene flow) tend to have more globally stable equilibria.

View this table: In this window In a new window   TABLE 2 The proportion of simulations leading to type I, II, or III fitness sets for seven levels of gene flow (m) and three levels of initial correlation (0) of mutant genotypic fitness values between the demes

 
Interestingly, for fitness sets with locally stable equilibria (i.e., type II fitness sets) the domain of attraction is larger for those generated with ₀ = –0.5 than for those generated with ₀ = 0 or ₀ = 0.5 (Figure 5). Thus, even though the latter fitness sets have a higher proportion of type I fitness sets, their domain of attraction for type II fitness sets is smaller in comparison. For fitness sets resulting from a stronger negative ₀, individual genotypic fitness differences between the two demes are, on average, larger. This larger difference makes it likely that the points of attraction of the equilibria for each allele-frequency vector within each deme are quite far apart from each other and convergence toward these equilibria will be strong for a large proportion of the allele-frequency vectors. Failure to converge requires the initial allele-frequency vectors to be fairly extreme (and these vectors are therefore rare), resulting in a large domain of attraction. As fitness differences are smaller between the demes with more positively correlated fitness sets, a failure to converge can occur more easily from less extreme initial allele-frequency vectors, resulting in a smaller domain of attraction for these fitness sets. In summary, homogeneous environments have more globally stable equilibria than the more heterogeneous environments. In contrast, heterogeneous environments have locally stable equilibria with larger domains of attraction than more homogenous environments.

View larger version (13K): In this window In a new window Download PPT slide   FIGURE 5.— Size of domain of attraction measured by the proportion of initial allele-frequency vectors leading to fully polymorphic equilibrium as a function of gene flow for type II fitness sets (see text for explanation) for three levels of initial correlation (0) of mutant genotypic fitness values. The error bars are standard error intervals. The stability data could not be analyzed using ANOVA, since the data are not normally distributed and remain heteroscedastic even after transformation. Ignoring the effect of gene flow, 0 has a significant (Kruskal–Wallis, ) effect on the size of domain of attraction.

 

Numbers of alleles:

Differences in the spatial heterogeneity simulated in our two-deme model by different values of ₀ (the correlation between the mutant fitnesses in the demes) greatly influence the levels of polymorphism found after 10,000 generations of mutation and selection. More specifically, these effects are most profound for intermediate levels of gene flow; both for low levels and for high levels of gene flow the differences in levels of polymorphism are minimal. Thus, the effects of heterogeneous environments are tightly linked to the level of gene flow between these environments (LENORMAND 2002; KAWECKI and EBERT 2004).

Interestingly, both the most heterogeneous (₀ = –0.5) and the most homogenous (₀ = 0.5) simulated environments maintain slightly higher levels of polymorphism compared to the more intermediate environment (₀ = 0) for high levels of gene flow. This result is easily explained: Two forms of balancing selection can maintain variation in our model, heterozygote advantage and local selection. By manipulating the initial level of fitness correlation between the demes, we influence the ease which with each particular form of balancing selection can evolve. For example, for a more homogeneous environment (₀ = 0.5) the probability of generating mutants with substantial heterozygote advantage in both demes is higher in comparison to a more heterogeneous environment (e.g., ₀ = 0). This increased probability helps such more homogeneous environments to reach higher levels of polymorphism in the presence of high levels of gene flow when heterozygote advantage becomes the most important form of balancing selection. Using a similar argument, for a more heterogeneous environment (₀ = –0.5) the probability of generating mutants with substantial locally adaptive fitnesses is higher in comparison to a more homogeneous environment. Moreover, local selection has to be sufficiently strong to help maintain variation in the presence of high levels of gene flow and our simulated uncorrelated environment may have a small probability of generating mutants with these substantial levels of local selection. In other words, the intermediately heterogeneous environment has the worst of both worlds: It has a lower probability of generating mutants with substantial heterozygote advantage compared to a more homogeneous environment and it has a lower probability of generating mutants with enough local selection compared to a more heterogeneous environment. This lower probability of generating either form of balancing selection results in lower levels of polymorphism for the intermediate heterogeneous environment when compared to both other levels of spatial heterogeneity for high levels of gene flow.

Two possible problems with our model, due to the drawing of mutant fitnesses from [0, 1], may affect the number of alleles at the end of our simulations. First, it is possible that an extremely fit mutant could arise that drives existing alleles to extinction and prevents any further mutants from successfully invading the population. Nevertheless, such a possibility is very unlikely in a finite number of generations. Moreover, the major effect would be a small downward bias in our estimate of the number of alleles that could be maintained in a two-deme system (say, if mutational fitnesses were drawn from a distribution without an upper bound on fitnesses). Second, the mean fitness is bounded above by 1 and so, as it increases over time, successful invasions become less frequent, a feature also in the simulations of SPENCER and MARKS (1988). For both these reasons, our results may slightly underestimate the ability of spatially structured viability systems to maintain polymorphism.

Heterozygote advantage and local selection:

The amount of environmental variation and level of gene flow have a strong interaction effect on the relative importance of the two forms of balancing selection. Heterozygote advantage is an important form of balancing selection, especially for the more homogeneous environments and higher levels of gene flow. Furthermore, heterozygote advantage is present in some reduced form for lower levels gene flow as well. Therefore, most fitness sets in our model do develop either some or substantial levels of this form of balancing selection. If the process by which our fitnesses are generated is at all realistic, many examples of heterozygote advantage should be detectable in natural populations (SPENCER and MARKS 1993). Thus the rarity of such examples casts doubt on the generality of heterozygote advantage as an important form of balancing selection maintaining genetic polymorphism. Perhaps the generation of fitness sets in which heterozygote advantage easily develops is unrealistic and mutations that do develop this form of balancing selection should be exceptionally rare in natural populations. Nevertheless, high levels of variation can potentially be maintained without any substantial amount of heterozygote advantage for the more heterogeneous environments for lower levels of gene flow.

For these lower levels of gene flow, local selection is increasingly important, especially for highly heterogeneous environments. Two separate studies have recently found strong spatial differences in fitness in natural populations (HANSKI and SACCHERI 2006; BOLNICK and NOSIL 2007), indicating that substantial levels of local selection can be detected in specific situations. In contrast, for the most homogeneous environments, selection in a two-deme model with even the lowest levels of gene flow does lead to fitness sets with only subtle patterns of local selection. If these more homogeneous environments with positively correlated mutations are regularly occurring in natural populations, our results suggest that substantial local selection has a low chance of evolving and, therefore, may have a lower potential to help maintain variation in these populations. Interestingly, selection does lead to elevated levels of variation in these environments at sufficiently low levels of gene flow. Thus, since heterozygote advantage is similarly low for these scenarios, this variation is maintained with a minimum of both forms of balancing selection as measured by our methods. These results suggest that calculating average levels of heterozygote advantage and Pearson's correlation coefficients using genotypic fitness values may have a lower power to detect the more subtle forms of balancing selection that operate in more homogeneous environments at low levels of gene flow.

Fitness and stability:

The processes of selection and mutation lead to substantially higher levels of fitness for the lowest levels of gene flow, regardless of the level of simulated environmental variation. These higher levels of fitness suggest that local adaptation occurs for all environments if the level of gene flow becomes sufficiently low. Gene flow disturbs local adaptation by introducing migrants that carry alleles that are not locally advantageous, and the presence of these migrants reduces the average levels of fitness of the different demes. This reduction in fitness due to gene flow between locally adapted environments incurs an indirect fitness cost on the populations in these environments (GARCIARAMOS and KIRKPATRICK 1997). Interestingly, the effect of gene flow on the average fitness is capped: The highest levels of gene flow do not result in lower levels of fitness compared to intermediate levels of gene flow. These results indicate that while more migrants move between the demes for the highest levels of gene flow, these migrants are not particularly maladapted compared to those in the simulations with intermediate levels of gene flow. Therefore selection in populations with these high levels of gene flow leads to relatively lower levels of maladaptation, partially negating the potential indirect cost that gene flow has on fitness (BILLIARD and LENORMAND 2005).

While these fitness costs due to gene flow are partly negated by the reduced maladaptation of the alleles, the costs are nevertheless substantial, especially for the more heterogeneous environments. It has been argued that such costly dispersal would not be evolutionary stable (HASTINGS 1983; GREENWOOD-LEE and TAYLOR 2001). Therefore, our results show that highly heterogeneous environments are likely to be more susceptible than homogeneous environments to the invasion of alleles that reduce the level of gene flow (WIENER and FELDMAN 1993), in particular for higher levels of gene flow. Interestingly, however, the different environments do not differ much in final level of fitness for the lower levels of gene flow. These results indicate that the impact that different environments may have on the final level of adaptation is relatively reduced. Furthermore, at these low levels of gene flow, the indirect fitness costs are not very substantial, increasing the probability for an evolutionary stable level of gene flow between the demes, in particular when other forms of selection (e.g., kin selection) are included (BILLIARD and LENORMAND 2005).

The resulting polymorphic equilibria are fairly stable, perhaps because in our model mutants have to invade from a low frequency and therefore are likely to have properties that make them resistant toward extinction (STAR et al. 2007b). While most of the highly heterogeneous fitness sets are classified as type II fitness sets (fitness sets with a locally stable equilibrium), the domain of attraction of these fitness sets is quite large. In other words, the locally stable equilibria of highly heterogeneous environments are stable for fairly large perturbations of allele frequencies. These equilibria are present regardless of substantial levels of balancing selection due to heterozygote advantage and thus fitness sets in a highly heterogeneous environment can develop enough local adaptation to successfully maintain variation.

In summary, the levels of environmental heterogeneity and gene flow interact to critically influence both the amount of genetic variation and the relative importance of both heterozygote advantage and local selection in spatially structured models. Highly heterogeneous environments maintain elevated highest levels of genetic polymorphisms for a wide range of gene flow, but at a severe cost of average fitness.

Helpful comments from Reinhard Bürger and the reviewers significantly improved this manuscript; we are grateful for their help. This work was supported by the Allan Wilson Centre for Molecular Ecology and Evolution and the University of Otago Postgraduate Scholarships.

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Communicating editor: M. W. FELDMAN

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