Spreading under shifting climate by a free boundary model: Invasion of deteriorated environment

In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed [Formula: see text] and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds [Formula: see text] and a complete classification of the long-time dynamics of the model, which reveals significant differences between the cases [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. For example, when [Formula: see text], for a suitably parameterized family of initial functions [Formula: see text] increasing continuously in [Formula: see text], we show that there exists [Formula: see text] such that the species vanishes eventually when [Formula: see text], it spreads with asymptotic speed [Formula: see text] when [Formula: see text], it spreads with forced speed [Formula: see text] when [Formula: see text], and it spreads with speed [Formula: see text] when [Formula: see text]. Moreover, in the last case, while the spreading front propagates with asymptotic speed [Formula: see text], the profile of the population density function [Formula: see text] approaches a propagating pair consisting of a traveling wave with speed [Formula: see text] and a semi-wave with speed [Formula: see text].