How do genetic correlations affect species range shifts in a changing environment?

Species may be able to respond to changing environments by a combination of adaptation and migration. We study how adaptation affects range shifts when it involves multiple quantitative traits evolving in response to local selection pressures and gene flow. All traits develop clines shifting in space, some of which may be in a direction opposite to univariate predictions, and the species tracks its environmental optimum with a constant lag. We provide analytical expressions for the local density and average trait values. A species can sustain faster environmental shifts, develop a wider range and greater local adaptation when spatial environmental variation is low (generating low migration load) and multitrait adaptive potential is high. These conditions are favoured when nonlinear (stabilising) selection is weak in the phenotypic direction of the change in optimum, and genetic variation is high in the phenotypic direction of the selection gradient.