Quantifying the effects of migration and mutation on adaptation and demography in spatially heterogeneous environments

How do mutation and gene flow influence population persistence, niche expansion, and local adaptation in spatially heterogeneous environments? In this article, we analyse a demographic and evolutionary model of adaptation to an environment containing two habitats in equal frequencies, and we bridge the gap between different theoretical frameworks. Qualitatively, our model yields four qualitative types of outcomes: (i) global extinction of the population (ii) adaptation to one habitat only, but also adaptation to both habitats with (iii) specialized phenotypes, or (iv) with generalized phenotypes; and we determine the conditions under which each equilibrium is reached. We derive new analytical approximations for the local densities and the distributions of traits in each habitat under a migration--selection--mutation balance, compute the equilibrium values of the means, variances and asymmetries of the local distributions of phenotypes, and contrast the effects of migration and mutation ...