  We study the evolutionary dynamics of a phenotypically structured population
in a changing environment , where the environmental conditions vary with a
linear trend but in an oscillatory manner. Such phenomena can be described by
parabolic Lotka-Volterra type equations with non-local competition and a time
dependent growth rate. We first study the long time behavior of the solution to
this problem. Next, using an approach based on Hamilton-Jacobi equations we
study asymptotically such long time solutions when the effects of the mutations
are small. We prove that, as the effect of the mutations vanishes, the
phenotypic density of the population concentrates on a single trait which
varies linearly with time, while the size of the population oscillates
periodically. In contrast with the case of an environment without linear shift,
such dominant trait does not have the maximal growth rate in the averaged
environment and there is a cost on the growth rate due to the climate shift. We
also provide an asymptotic expansion for the average size of the population and
for the critical speed above which the population goes extinct, which is
closely related to the derivation of an asymptotic expansion for the Floquet
eigenvalue in terms of the diffusion rate. By mean of a biological example,
this expansion allows to show that the fluctuations on the environment may help
the population to follow the climatic shift in a better way.
