  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.
