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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Shen, Wenxian¹  Xue, Shuwen¹ 

[1] Auburn Univ, Dept Math & Stat, Auburn, AL 36849 USA

关键词: Parabolic-elliptic chemotaxis system;    Spreading speeds;    Persistence;    Shifting environment;   

DOI  :  10.1016/j.jde.2020.04.040

来源: Elsevier

【 摘 要 】

The current paper is concerned with the persistence and spreading speeds of the following Keller-Segel chemoattraction system in shifting environments, {u(t) = u(xx) - chi(uv(x))(x) + u(r(x - ct) - bu), x is an element of R (0.1) 0 = v(xx) - nu v + mu u, x is an element of R, where chi, b, nu, and mu are positive constants, c is an element of R, r(x) is Holder continuous, bounded, r* = sup(x is an element of R)r(x) > 0, r(+/-infinity) := lim(x ->+/-infinity) r(x) exist, and r(x) satisfies either r(-infinity) < 0 < r(infinity), or r(+/-infinity) < 0. Assume b > chi mu and b >= (1 + 1/2(root r*-root nu)+(root r*+root v) chi mu. In the case that r(-infinity) < 0 < r(infinity), it is shown that if the moving speed c > c* := 2 root r*, 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(+/-infinity) < 0, it is shown that if vertical bar c vertical bar > c*, then the species will become extinct in the habitat. If lambda(infinity), defined to be the generalized principle eigenvalue of the operator u -> 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(infinity) > 0, then the species will persist surrounding the good habitat. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free