  Invasion phenomena for heterogeneous reaction-diffusion equations are
contemporary and challenging questions in applied mathematics. In this paper we
are interested in the question of spreading for a reaction-diffusion equation
when the subdomain where the reaction term is positive is shifting/contracting
at a given speed $c$. This problem arises in particular in the modelling of the
impact of climate change on population dynamics. By placing ourselves in the
appropriate moving frame, this leads us to consider a
reaction-diffusion-advection equation with a heterogeneous in space reaction
term, in dimension $N\geq1$. We investigate the behaviour of the solution $u$
depending on the value of the advection constant~$c$, which typically stands
for the velocity of climate change. We find that, when the initial datum is
compactly supported, there exists precisely three ranges for $c$ leading to
drastically different situations. In the lower speed range the solution always
spreads, while in the upper range it always vanishes. More surprisingly, we
find that that both spreading and vanishing may occur in an intermediate speed
range. The threshold between those two outcomes is always sharp, both with
respect to $c$ and to the initial condition. We also briefly consider the case
of an exponentially decreasing initial condition, where we relate the
decreasing rate of the initial condition with the range of values of~$c$ such
that spreading occurs.
