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 \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 $c\ge c_0$, 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) {\it vanishing}, (b) {\it borderline spreading}, or (c) {\it spreading}. If the initial population is writen in the form $u_0(x)=\sigma \varphi (x)$ with $\varphi$ 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 $\sigma >{\sigma }_0$.

References in corpus (1)

• Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments,

Cited by in corpus (1)

• A free boundary problem for the Fisher-KPP equation with a given moving boundary