### 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 language | English |
---|---|

Pages (from-to) | 140-162 |

Number of pages | 23 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 72 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Apr 1 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*,

*72*(2), 140-162. https://doi.org/10.1093/imamat/hxl028

}

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*, vol. 72, no. 2, pp. 140-162. https://doi.org/10.1093/imamat/hxl028

**The stability and dynamics of a spike in the 1D Keller-Segel model.** / Kang, K.; Kolokolnikov, T.; Ward, M. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The stability and dynamics of a spike in the 1D Keller-Segel model

AU - Kang, K.

AU - Kolokolnikov, T.

AU - Ward, M. J.

PY - 2007/4/1

Y1 - 2007/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34047178454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34047178454&partnerID=8YFLogxK

U2 - 10.1093/imamat/hxl028

DO - 10.1093/imamat/hxl028

M3 - Article

AN - SCOPUS:34047178454

VL - 72

SP - 140

EP - 162

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 2

ER -