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 (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. 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 ≥ c₀, the species always dies out in the long-run, but when 0 < c < c₀, 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) with ϕ fixed and σ > 0 a parameter, then there exists σ₀ > 0 such that vanishing happens when σ ∈ (0,σ₀), borderline spreading happens when σ = σ₀, and spreading happens when σ > σ₀.