Adaptive Dynamics of Diverging Fitness Optima

We analyse a non-local parabolic integro-differential equation modelling the evolutionary dynamics of a phenotypically-structured population in a changing environment. Such models arise in a 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 it does not go extinct, we analyse the solution in the long time, small mutation limits. If the optimas have equal shift velocities, the solution concentrates on a point set of "lagged optima" which are strictly behind the true shifting optima. If the shift velocities are different, we determine that the solution in fact concentrates as a Dirac delta function on the positive lagged optimum with maximum lagged fitness, which depends on the true optimum and the rate of shift. 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 contribute to the eventual dominance of one trait.