Extinction and spreading of a species under the joint influence of climate change and a weak Allee effect: a two-patch model

Abstract

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.