Abstract
We are concerned with stationary solutions of a Keller-SegelModel with density-suppressed motility and without cell proliferation. we establish the existence and the analytical approximation of non-constant stationary solutions by applying the phase plane analysis and bifurcation analysis. We show that the one-step solutions is stable and two or more-step solutions are always unstable. Then we further show that two or more-step solutions possess metastability. Our analytical results are corroborated by direct simulations of the underlying system.
Highlights
Stripe pattern formation was observed in the experiment of the engineered E. coli strains with the behavior of density suppressing motility in an isolated apparatus, which showed that spatio-temporal patterns could be driven by a “self-trapping” mechanism besides diffusion-driven and chemotaxis-driven instabilities [10]
Where the domain Ω ⊂ RN, N ≥ 1 is bounded and has a smooth boundary ∂Ω, ν is the outward unit normal vector on ∂Ω; u(x, t) and v(x, t) denote the densities of E. coli cells and chemical substance acyl-homoserine lactone (AHL), respectively; The chemical substance AHL is produced by E. coli cells with a rate η > 0, degraded with a rate β > 0, and diffused with a rate D
In one-dimensional space we obtain the conditions for the existence of non-constant steady states of (1.3) by using the phase plane analysis
Summary
Stripe pattern formation was observed in the experiment of the engineered E. coli strains with the behavior of density suppressing motility in an isolated apparatus (see [9]), which showed that spatio-temporal patterns could be driven by a “self-trapping” mechanism besides diffusion-driven and chemotaxis-driven instabilities [10]. The existence of global classical solutions and the stability of constant steady state for Ω ⊂ R2 were investigated in [2]. Keller-Segel Model, density-suppressed motility, metastability, nonconstant steady states. In [15], the authors studied the global existence of classical solutions, the stability of constant steady states and the existence of non-constant solutions in any dimensions for the motility function r(v) given by r(v) = c0/vp, p > 0, c0 > 0 and small enough. The aim of this paper is to establish the existence of non-constant steady states of (1.3)-(1.4) by using a different method from [15] and to derive conditions for their stability and metastability in one-dimensional space.
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