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}-\chi(uv_x)_x +u(r(x-ct)-bu),\quad x\in\R\cr 0=v_{ xx}- \nu v+\mu u,\quad x\in\R, \end{ cases} \end{ equation} where $\chi$, $b$, $\nu$, and $\mu$ are positive constants, { $c\in\R$ }, $r(x)$ is H\"older continuous, bounded, $r^*=\sup_{ x\in\R}r(x)>0$, $r(\pm \infty):=\lim_{ x\to \pm\infty}r(x)$ exist, and $r(x)$ satisfies either $r(-\infty)<0<r(\infty)$, or $r(\pm\infty)<0$. Assume $b>\chi\mu$ and $b\ge \big(1+\frac{ 1}{ 2}\frac{ (\sqrt{ r^*}-\sqrt{ \nu})_+}{ (\sqrt{ r^*}+\sqrt{ \nu})}\big)\chi\mu$. In the case that $r(-\infty)<0<r(\infty)$, it is shown that if the moving speed $c>c^*:=2\sqrt{ r^*}$, then the species becomes extinct in the habitat. If the moving speed $ -c^*\leq c<c^*$, then the species will persist and spread along the shifting habitat at the asymptotic spreading speed $c^*$. If the moving speed $c<-c^*$, then the species will spread in the both directions at the asymptotic spreading speed $c^*$. In the case that $r(\pm\infty)<0$, it is shown that if $|c|>c^*$, then the species will become extinct in the habitat. If $\lambda_{ \infty}$, defined to be the generalized principle eigenvalue of the operator $u\to u_{ xx}+cu_{ x}+r(x)u$, is negative and the degradation rate $\nu$ of the chemo-attractant is grater than or equal to some number $\nu^*$, then the species will also become extinct in the habitat. If $\lambda_{ \infty}>0$, then the species will persist surrounding the good habitat.
Categories: math.AP