  Many species see their range shifted poleward in response to global warming
and need to keep pace in order to survive. To understand the effect of climate
change on species ranges and its consequences on population dynamics, we
consider a space-time heterogeneous reaction-diffusion equation in dimension 1,
whose unknown~$u (t,x)$ stands for a population density. More precisely, the
environment consists of two patches moving with a constant climate shift speed
$c \geq 0$: in the invading patch $\{ t >0 , \, x \in \R \, | \ x < ct \}$ the
growth rate is negative and, in the receding patch $\{ t >0 , \, x \in \R \, |
\ x \geq ct \}$ it is of the classical monostable type. Our framework includes
species subject to a weak Allee effect, meaning that there may be a positive
correlation between population size and its per capita growth rate. We study
the large-time behaviour of solutions in the moving frame and show that whether
the population spreads or goes extinct depends not only on the speed $c$ but
also, in some intermediate speed range, on the initial datum. This is in sharp
contrast with the so-called `hair-trigger effect' in the homogeneous monostable
equation, and suggests that the size of the population becomes a decisive
factor under the joint influence of climate change and a weak Allee effect.
Furthermore, our analysis exhibit sharp thresholds between spreading and
extinction: in particular, we prove the existence of a threshold shifting speed
which depends on the initial population, such that spreading occurs at lower
speeds and extinction occurs at faster speeds.
