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

Abstract

The current paper is concerned with the forced waves of Keller-Segel chemoattraction systems in shifting environments of the form, u t = u xx − χ(uv x) x + u(r(x − ct) − bu), x ∈ R 0 = v xx − νv + µu, x ∈ R, (0.1) where χ, b, ν, and µ are positive constants, c ∈ R, the resource function r(x) is globally Hölder continuous, bounded, r * = sup x∈R r(x) > 0, r(±∞) := lim x→±∞ r(x) exist, and either r(−∞) < 0 < r(∞), or r(±∞) < 0. Assume that b > 2χµ. In the case that r(−∞) < 0 < r(∞), it is shown that (0.1) has a forced wave solution connecting (r * b , µ ν r * b) and (0, 0) with speed c provided that c > χµr * 2 √ ν(b−χµ) − 2 r * (b−2χµ) b−χµ. In the case that r(±∞) < 0, it is shown that (0.1) has a forced wave solution connecting (0, 0) and (0, 0) with speed c provided that χ is sufficiently small and λ ∞ > 0, where λ ∞ is the generalized principal eigenvalue of the operator u(•) → u xx (•) + cu x (•) + r(•)u(•) on R in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.

Recent developments on spatial propagation for diffusion equations in shifting environments

“…Shen and Xue [88,89] incorporated the shifted feature into the parabolic-elliptic Keller-Segel models as follows…”

“…Theorem 3.4. [88,89] The persistence and spreading speeds as well as the forced waves of model (28) read that (i) For the case that r(−∞) < 0 < r(∞), the species becomes extinct in the habitat if the moving speed c > ĉ * := 2 √ r * ; and will persist and spread along the shifting habitat at the asymptotic spreading speed ĉ * if −ĉ * ≤ c < ĉ * ; and will spread in the both directions at the asymptotic spreading speed ĉ * if c < −ĉ * . Further, system (28) admits a forced wave solution connecting b−χµ .…”

“…In the past years, two-species competition system with monostable structure [4,20,95,98,102,111,112,114], cooperation system of weak type [52,97,108] and chemotaxis models [88,89] in shifting environments have been widely considered and studied. In future research, one should turn more attention to the strong competition case (which leads to a bistable system) and general cooperation situation.…”