Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments

Abstract

The current paper is concerned with the persistence and spreading speeds of the following Keller-Segel chemoattraction system in shifting environments, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(r(x-ct)-bu),\quad x\in\R\cr 0=v_{xx}- νv+μu,\quad x\in\R, \end{cases} \end{equation} where $\chi$, $b$, $\nu$, and $\mu$ are positive constants, { $c\in \mathbb{R}$ }, $r(x)$ is Hölder continuous, bounded, $r^{\ast }={\sup }_{x\in \mathbb{R}}r(x)>0$, $r(\pm \infty ):={\lim }_{x\to \pm \infty }r(x)$ exist, and $r(x)$ satisfies either $r(-\infty )<0<r(\infty )$, or $r(\pm \infty )<0$. Assume $b>\chi \mu$ and $b\ge (1+\frac{1}{2}\frac{(\sqrt{r^{\ast }}-\sqrt{\nu }{)}_{+}}{(\sqrt{r^{\ast }}+\sqrt{\nu })})\chi \mu$. In the case that $r(-\infty )<0<r(\infty )$, it is shown that if the moving speed $c>c^{\ast }:=2\sqrt{r^{\ast }}$, then the species becomes extinct in the habitat. If the moving speed $-c^{\ast }\le c<c^{\ast }$, then the species will persist and spread along the shifting habitat at the asymptotic spreading speed $c^{\ast }$. If the moving speed $c<-c^{\ast }$, then the species will spread in the both directions at the asymptotic spreading speed $c^{\ast }$. In the case that $r(\pm \infty )<0$, it is shown that if $|c|>c^{\ast }$, then the species will become extinct in the habitat. If ${\lambda }_{\infty }$, defined to be the generalized principle eigenvalue of the operator $u\to u_{xx}+c{u}_x+r(x)u$, is negative and the degradation rate $\nu$ of the chemo-attractant is grater than or equal to some number ${\nu }^{\ast }$, then the species will also become extinct in the habitat. If ${\lambda }_{\infty }>0$, then the species will persist surrounding the good habitat.