  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 $\sigma>\sigma_0$.
