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 (SIAM J Math Anal 42:377–405, 2010), 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 c0>0\documentclass[12pt]{minimal}
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\begin{document}$$c_0>0$$\end{document}. 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\documentclass[12pt]{minimal}
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\begin{document}$$c>0$$\end{document}. We prove that when c≥c0\documentclass[12pt]{minimal}
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\begin{document}$$c\ge c_0$$\end{document}, the species always dies out in the long-run, but when 0<c<c0\documentclass[12pt]{minimal}
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\begin{document}$$0<c<c_0$$\end{document}, the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form u0(x)=σϕ(x)\documentclass[12pt]{minimal}
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\begin{document}$$u_0(x)=\sigma \phi (x)$$\end{document} with ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} fixed and σ>0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma >0$$\end{document} a parameter, then there exists σ0>0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _0>0$$\end{document} such that vanishing happens when σ∈(0,σ0)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \in (0,\sigma _0)$$\end{document}, borderline spreading happens when σ=σ0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma =\sigma _0$$\end{document}, and spreading happens when σ>σ0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma >\sigma _0$$\end{document}.
