Stability of spiky solution of Keller–Segel's minimal chemotaxis model

Chen, Xinfu; Hao, Jianghao; Wang, Xuefeng; Wu, Yaqi; Zhang, Yajing

doi:10.1016/j.jde.2014.06.008

Cited by 25 publications

(16 citation statements)

References 17 publications

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“…In the latter asymptotic methods were used to construct a central spike solution, shown to possess metastability and drift exponentially slowly to the boundary. For further results on the form and stability of patterned solutions to PKS models, see for example Wang (2013); Chen et al (2014); Zhang et al (2017). The introduction to Wang (2013) is highlighted for containing a review of earlier results.…”

Section: Self-organisation and Patterningmentioning

confidence: 99%

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Painter

2019

Journal of Theoretical Biology

121

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Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.

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Section: Self-organisation and Patterningmentioning

confidence: 99%

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Painter

2019

Journal of Theoretical Biology

121

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“…In view of the global attractivity discussed above, we only need to analyze its local stability. By applying some abstract stability results based on analytic semigroup theories (see [5,19]), to get the stability of the spatially homogeneous steady state (u * , v * , w * , z * ), it suffices to prove that the steady state is spectrally stable, i.e., the linearized operator has only eigenvalues with nonnegative real parts (also see [4,31,38]). Linearizing (3) at (u * , v * , w * , z * ) leads to the following system…”

Section: Huanhuan Qiu and Shangjiang Guomentioning

confidence: 99%

Global existence and stability in a two-species chemotaxis system

Qiu¹,

Guo²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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This paper deals with the following two-species chemotaxis system        ut = ∆u − χ 1 ∇ • (u∇v) + µ 1 u(1 − u − a 1 w), x ∈ Ω, t > 0, vt = ∆v − v + h(w), x ∈ Ω, t > 0, wt = ∆w − χ 2 ∇ • (w∇z) + µ 2 w(1 − w − a 2 u), x ∈ Ω, t > 0, zt = ∆z − z + h(u), x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C 1-regular function. The main objectives of this paper are twofold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium (u * , v * , w * , z *) may be globally attractive in the weak competition case (i.e., 0 < a 1 , a 2 < 1), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., a 1 > 1 > a 2 > 0). In the fully strong competition case (i.e. a 1 , a 2 > 1), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous (u * , v * , w * , z *). The matter which species ultimately wins out depends crucially on the starting advantage each species has.

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“…In [37], Wang and Xu proved rigorously the existence of a boundary spike solution of (2). Using a phase plane analysis, Chen, Hao, Wang, Wu, and Zhang established in [8] the existence, uniqueness, and local exponential stability of the spike steady state solution, together with rigorous arbitrary high order internal and boundary layer asymptotic expansions. To obtain more detailed information on the stability of the spike solution, we reinvestigated in [38] the associated eigenvalue problem in a general setting with a systematic method.…”

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confidence: 99%

“…2 We shall prove the evolution limit system (7) in a subsequent paper. In this paper we prove only the limit system (8). For this, we assume for simplicity that T = 0 and w(•, T − ) = v 0 is a non-constant increasing function.…”

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confidence: 99%

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Dynamics of spike in a Keller-Segel's minimal chemotaxis model

Chen¹,

Hao²,

Lai³

et al. 2017

Discrete &Amp; Continuous Dynamical Systems - A

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The dynamics are studied for the Keller-Segel's minimal chemotaxis model τ ut = (ux − kuvx)x, vt = vxx − v + u on a bounded interval with homogeneous Neumann boundary conditions, where τ 0 and k 1 are parameters and the total mass of u is scaled to be one. In general, the dynamics can be divided into three stages: the first stage is very short in which u quickly becomes a delta like function with mass concentrated near the point of global maximum of v; in the second stage, the point of the global maximum of v drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the profile of the solution evolves to a steady state profile. This paper considers a special case in which the relaxation parameter τ is set to be zero, so the first stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.

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Stability of spiky solution of Keller–Segel's minimal chemotaxis model

Cited by 25 publications

References 17 publications

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Global existence and stability in a two-species chemotaxis system

Dynamics of spike in a Keller-Segel's minimal chemotaxis model

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