Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

2018/12/28 by Rachidi B. Salako, Wenxian Shen, Salako, Rachidi B. +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models

paper · pdf · doi:10.48550/arxiv.1901.00045

openalex publication_date 2018/12/28 · openalex created_date 2019/01/11 · openalex updated_date 2026/07/28

Abstract

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \begincases u_t=u_xx-χ(uv_x)_x +u(a-bu), x∈\R\cr 0=v_xx- λv+μu, x∈\R, \endcases where χ, a, b, λ, and μ are positive constants. Assume b>χμ. Then if in addition, (1+(1)/(2)((√(a)-√λ)₊)/((√(a)+√(\la))))χμ ≤ b holds, it is proved that c₀^*=2√ a is the spreading speed of the solutions of \eqrefabstract-eq1 with nonnegative continuous initial function u₀ with nonempty compact support, that is, \limsup_{|x|≥ ct, t→∞}u(t,x;u₀)=0 ∀ cgt;c₀^* and \liminf_{|x|≤ ct,t→∞} u(t,x;u₀)>0 ∀ 02χμ and λ≥ a holds, then c₀^*=2√ a is the minimal speed of the traveling wave solutions of \eqrefabstract-eq1 connecting (0,0) and ((a)/(b),\fracμλ(a)/(b)), that is, for any c≥ c₀^*, \eqrefabstract-eq1 has a traveling wave solution connecting (0,0) and ((a)/(b),\fracμλ(a)/(b)) with speed c, and \eqrefabstract-eq1 has no such traveling wave solutions with speed less than c₀^*. Note that c₀^*=2√ a is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, u_t=u_xx+u(a-bu), x∈\R.

Citations

• Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model
• Multidimensional nonlinear diffusion arising in population genetics
• Model for chemotaxis

Cited by

• Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on ℝ^N: I. Global Solvability and Boundedness

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