Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary

We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin \cite{DL}, where a spreading-vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed $c_0>0$. Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed $c>0$. We prove that when $c\geq c_0$, the species always dies out in the long-run, but when $0<c<c_0$, the long-time behavior of the species is determined by a trichotomy described by (a) {\it vanishing}, (b) {\it borderline spreading}, or (c) {\it spreading}. If the initial population is writen in the form $u_0(x)=\sigma \phi(x)$ with $\phi$ fixed and $\sigma>0$ a parameter, then there exists $\sigma_0>0$ such that vanishing happens when $\sigma\in (0,\sigma_0)$, borderline spreading happens when $\sigma=\sigma_0$, and spreading happens when $\sigma>\sigma_0$.