Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

Abstract

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 $c_0>0$. 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 $c\ge c_0$, 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) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written 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 $\sigma >\sigma _0$.