Long time behavior for solutions of the diffusive logistic equation with advection and free boundary

Abstract

We consider the influence of a shifting environment and an advection on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is shifting and without advection (\(\beta =0\)), Du et al. (Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary. arXiv:1508.06246, 2015) showed that the species always dies out when the shifting speed \(c_*\ge \mathcal {C}\), and the long-time behavior of the species is determined by trichotomy when the shifting speed \(c_*\in (0,\mathcal {C})\). Here we mainly consider the problems with advection and shifting speed \(c_*\in (0,\mathcal {C})\) (the case \(c_*\ge \mathcal {C}\) can be studied by similar methods in this paper). We prove that there exist \(\beta ^* 0\) such that the species always dies out in the long-run when \(\beta \le \beta ^*\), while for \(\beta \in (\beta ^*,\beta _*)\) or \(\beta =\beta _*\), the long-time behavior of the species is determined by the corresponding trichotomies respectively.