Population mixing and the adaptive divergence of quantitative traits in discrete populations: A theoretical framework for empirical tests

Empirical tests for the importance of population mixing in constraining ada ptive divergence have not been well grounded in theory for quantitative tra its in spatially discrete populations. We develop quantitative-genetic mode ls to examine the equilibrium difference between two populations that are e xperiencing different selective regimes and exchanging individuals. These m odels demonstrate that adaptive divergence is negatively correlated with th e rate of population mixing (m, most strongly so when m is low), positively correlated with the difference in phenotypic optima between populations, a nd positively correlated with the amount of additive genetic variance (G, m ost 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 distan ce to equilibrium) when either heritability or mixing are not too low (h(2) > 0.2 or m > 0.05). The theory can he used to aid empirical tests that: (1 ) compare observed divergence to that predicted using estimates of populati on mixing, additive genetic variance/covariance, and selection; (2) test fo r a negative correlation between population mixing and adaptive divergence across multiple independent population pairs: and (3) experirnentally manip ulate the rate of mixing. Application of the first two of these approaches to data from two well-studied natural systems suggests that population mixi ng has constrained adaptive divergence for color patterns in Lake Erie wate r snakes (Nerodia sipedon. but nor 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.