Chemotaxis models with signal-dependent sensitivity and a logistic-type source, III: Bifurcation and spike-like solutions
Abstract
This third paper in a series on a parabolic-elliptic chemotaxis model studies bifurcation of positive constant equilibria in one space dimension. We consider { ll ut=Δu−χ0∇·(um/((1+v)β)∇ v)+au−bu1+α, & x∈Ω, 0=Δv−μv+νuγ, & x∈Ω, (∂ u)/(∂ n)=(∂ v)/(∂ n)=0, & x∈∂Ω, . where Ω is a bounded interval, α,γ,m,μ,ν>0, χ0∈ℝ, and a,b,β≥0. Building on earlier results on global existence, boundedness, persistence, and stabilization, we take χ0 as the bifurcation parameter and examine the effects of β and the domain size on the bifurcation. Center manifold reduction yields a coefficient βn0 at a simple bifurcation mode n0 of a positive constant equilibrium. The pitchfork bifurcation is supercritical if βn0>0 and subcritical if βn0<0. Stronger desensitization raises the sensitivity required for loss of linear stability. Bifurcation may be supercritical for small β but is always subcritical for sufficiently large β. Thus, desensitization makes patterns harder to trigger and favors abrupt rather than continuous emergence. The bifurcation mode increases with domain size, allowing more complex spatial patterns. Under suitable conditions, we prove unbounded continuation of local bifurcation branches. We also show that large-sensitivity stationary sequences satisfying a peak-gap hypothesis develop spike-like profiles. Numerical continuation and time-evolution experiments illustrate the bifurcating branches and spike-like solutions and suggest unbounded continuation and increasingly sharp stationary profiles.
Related
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- Infinite time blow-up of solutions to the classical Keller-Segel model of chemotaxis in higher dimensions
- Boundary layer analysis for the 2D Chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part II. viscous vanishing limit
BibTeX
@misc{chen2026chemotaxis,
title = {{Chemotaxis models with signal-dependent sensitivity and a logistic-type source, III: Bifurcation and spike-like solutions}},
author = {Le Chen and Ian Ruau and Wenxian Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20976},
eprint = {2610.20976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20976},
}
RIS
TY - UNPB
TI - Chemotaxis models with signal-dependent sensitivity and a logistic-type source, III: Bifurcation and spike-like solutions
AU - Le Chen
AU - Ian Ruau
AU - Wenxian Shen
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.20976
UR - https://arxiv.org/abs/2610.20976
AB - This third paper in a series on a parabolic-elliptic chemotaxis model studies bifurcation of positive constant equilibria in one space dimension. We consider { ll u_t=Δu−χ_0∇·(u^m/((1+v)^β)∇ v)+au−bu^1+α, & x∈Ω, 0=Δv−μv+νu^γ, & x∈Ω, (∂ u)/(∂ n)=(∂ v)/(∂ n)=0, & x∈∂Ω, . where Ω is a bounded interval, α,γ,m,μ,ν>0, χ_0∈ℝ, and a,b,β≥0. Building on earlier results on global existence, boundedness, persistence, and stabilization, we take χ_0 as the bifurcation parameter and examine the effects of β and the domain size on the bifurcation. Center manifold reduction yields a coefficient β_n_0 at a simple bifurcation mode n_0 of a positive constant equilibrium. The pitchfork bifurcation is supercritical if β_n_0>0 and subcritical if β_n_0<0. Stronger desensitization raises the sensitivity required for loss of linear stability. Bifurcation may be supercritical for small β but is always subcritical for sufficiently large β. Thus, desensitization makes patterns harder to trigger and favors abrupt rather than continuous emergence. The bifurcation mode increases with domain size, allowing more complex spatial patterns. Under suitable conditions, we prove unbounded continuation of local bifurcation branches. We also show that large-sensitivity stationary sequences satisfying a peak-gap hypothesis develop spike-like profiles. Numerical continuation and time-evolution experiments illustrate the bifurcating branches and spike-like solutions and suggest unbounded continuation and increasingly sharp stationary profiles.
ER -