The stability and dynamics of a spike in the 1D Keller-Segel model

K. Kang, T. Kolokolnikov, M. J. Ward

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

In the limit of a large mass M ≫ 1 and on a finite interval of length 2L, an equilibrium spike solution to the classical Keller-Segel chemotaxis model with a linear chemotactic function is constructed asymptotically. By calculating an asymptotic formula for the translational eigenvalue for M ≫ 1, it is shown that the equilibrium spike solution is unstable to translations of the spike profile. If in addition L ≫ 1, the equilibrium spike is shown to be metastable as a result of an asymptotically exponentially small eigenvalue. For M ≫ 1 and L ≫ 1, an asymptotic ordinary differential equation for the metastable spike motion is derived that shows that the spike drifts exponentially slowly towards one of the boundaries of the domain. For a certain reduced Keller-Segel model, corresponding to a domain of small length, a solution with a spike at each of the two boundaries is constructed. This solution is found to be metastable, and it is shown that there is an exponentially slow exchange of mass between the two spikes that occurs over very long timescales. For arbitrary initial conditions, energy methods are used to show the global existence of solutions. The relationship between this reduced Keller-Segel model and a Burgers-type equation modelling the upward propagation of a flame front in a finite channel is emphasized. Full numerical computations are used to confirm the asymptotic results.

Original languageEnglish
Pages (from-to)140-162
Number of pages23
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Volume72
Issue number2
DOIs
Publication statusPublished - 2007 Apr 1

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Keller-Segel Model
Spike
Reduced Model
Ordinary differential equations
Smallest Eigenvalue
Chemotaxis
Energy Method
Flame
Asymptotic Formula
Numerical Computation
Linear Function
Global Existence
Existence of Solutions
Ordinary differential equation
Time Scales
Initial conditions
Unstable
Propagation
Eigenvalue
Interval

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Cite this

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abstract = "In the limit of a large mass M ≫ 1 and on a finite interval of length 2L, an equilibrium spike solution to the classical Keller-Segel chemotaxis model with a linear chemotactic function is constructed asymptotically. By calculating an asymptotic formula for the translational eigenvalue for M ≫ 1, it is shown that the equilibrium spike solution is unstable to translations of the spike profile. If in addition L ≫ 1, the equilibrium spike is shown to be metastable as a result of an asymptotically exponentially small eigenvalue. For M ≫ 1 and L ≫ 1, an asymptotic ordinary differential equation for the metastable spike motion is derived that shows that the spike drifts exponentially slowly towards one of the boundaries of the domain. For a certain reduced Keller-Segel model, corresponding to a domain of small length, a solution with a spike at each of the two boundaries is constructed. This solution is found to be metastable, and it is shown that there is an exponentially slow exchange of mass between the two spikes that occurs over very long timescales. For arbitrary initial conditions, energy methods are used to show the global existence of solutions. The relationship between this reduced Keller-Segel model and a Burgers-type equation modelling the upward propagation of a flame front in a finite channel is emphasized. Full numerical computations are used to confirm the asymptotic results.",
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The stability and dynamics of a spike in the 1D Keller-Segel model. / Kang, K.; Kolokolnikov, T.; Ward, M. J.

In: IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), Vol. 72, No. 2, 01.04.2007, p. 140-162.

Research output: Contribution to journalArticle

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