Abstract
We study the long time behaviour of a non-local parabolic integro-differential equation modelling the evolutionary dynamics of a phenotypically-structured population in a changing environment. Such models can arise in variety of contexts from climate change to chemotherapy to the ageing body. The main novelty is that there are two locally optimal traits, each of which shifts at a possibly different linear velocity. We determine sufficient conditions to guarantee extinction or persistence of the population in terms of associated eigenvalue problems. When the population does not go extinct, we study the behaviour of long time solutions in the case of rare mutations: the long time solution concentrates as a sum of Dirac masses on a point set of "lagged optima" which are strictly behind the true shifting optima as the mutation rate goes to zero. If we further assume the shift velocities are different, we show the solution concentrates specifically on the positive lagged optimum with maximum lagged fitness. Our results imply that for populations undergoing competition in temporally changing environments, both the true optimal fitness and the required rate of adaptation for each of the diverging optimal traits determine the eventual dominance of one trait.