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

TL;DR

This paper investigates the persistence and spreading speeds of a Keller-Segel model in shifting environments, establishing conditions under which species survive, spread, or become extinct based on environmental shifts and parameters.

Contribution

It provides new criteria for species persistence and extinction in shifting habitats within a Keller-Segel model, considering variable environments and chemoattractant dynamics.

Findings

Species become extinct if shift speed exceeds critical speed c*

Species persist and spread at speed c* when shift speed is within bounds

Extinction occurs if environmental conditions or eigenvalues indicate unfavorable growth

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}-\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.