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.
|Number of pages||23|
|Journal||IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)|
|Publication status||Published - 2007 Apr 1|
Bibliographical noteFunding Information:
MJW was supported by the Natural Sciences and Engineering Research Council (NSERC, Canada) under grant 81541, TK was partially supported by an NSERC Postgraduate Fellowship and KK was supported by a PIMS Postgraduate Fellowship. MJW is grateful to Prof. R. Carretero (San Diego State University) and Prof. R. Russell (Simon Fraser University) for preliminary numerical experiments suggesting the possibility of metastability in the Keller–Segel model.
All Science Journal Classification (ASJC) codes
- Applied Mathematics