Empirical tests for the importance of population mixing in constraining adaptive divergence have not been well grounded in theory for quantitative traits in spatially discrete populations. We develop quantitative-genetic models to examine the equilibrium difference between two populations that are experiencing different selective regimes and exchanging individuals. These models demonstrate that adaptive divergence is negatively correlated with the rate of population mixing (m̂, most strongly so when m̂ is low), positively correlated with the difference in phenotypic optima between populations, and positively correlated with the amount of additive genetic variance (G, most strongly so when G is low). The approach to equilibrium is quite rapid (fewer than 50 generations for two populations to evolve 90% of the distance to equilibrium) when either heritability or mixing are not too low (h2 > 0.2 or m̂ > 0.05). The theory can be used to aid empirical tests that: (1) compare observed divergence to that predicted using estimates of population mixing, additive genetic variance/covariance, and selection; (2) test for a negative correlation between population mixing and adaptive divergence across multiple independent population pairs; and (3) experimentally manipulate the rate of mixing. Application of the first two of these approaches to data from two well-studied natural systems suggests that population mixing has constrained adaptive divergence for color patterns in Lake Erie water snakes (Nerodia sipedon), but not for trophic traits in sympatric pairs of benthic and limnetic stickleback (Gasterosteus aculeatus). The theoretical framework we outline should provide an improved basis for future empirical tests of the role of population mixing in adaptive divergence.
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