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,(0.1){ut=uxx−χ(uvx)x+u(r(x−ct)−bu),x∈R0=vxx−νv+μu,x∈R, where χ, b, ν, and μ are positive constants, c∈R, r(x) is Hölder continuous, bounded, r⁎=supx∈R⁡r(x)>0, r(±∞):=limx→±∞⁡r(x) exist, and r(x) satisfies either r(−∞)<0<r(∞), or r(±∞)<0. Assume b>χμ and b≥(1+12(r⁎−ν)+(r⁎+ν))χμ. In the case that r(−∞)<0<r(∞), it is shown that if the moving speed c>c⁎:=2r⁎, then the species becomes extinct in the habitat. If the moving speed −c⁎≤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(±∞)<0, it is shown that if |c|>c⁎, then the species will become extinct in the habitat. If λ∞, defined to be the generalized principle eigenvalue of the operator u→uxx+cux+r(x)u, is negative and the degradation rate ν of the chemo-attractant is grater than or equal to some number ν⁎, then the species will also become extinct in the habitat. If λ∞>0, then the species will persist surrounding the good habitat.