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Chemotactic cellular migration: Smooth and discontinous travelling wave solutions

Landman, Kerry A. , Pettet, Graeme J., & Newgreen, Donald F. (2003) Chemotactic cellular migration: Smooth and discontinous travelling wave solutions. SIAM Journal on Applied Mathematics, 63(5), pp. 1666-1681.

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Abstract

A simple model of chemotactic cell migration gives rise to travelling wave solutions. By varying the cellular growth rate and chemoattractant production rate, travelling waves with both smooth and discontinuous fronts are found using phase plane analysis. The phase plane exhibits a curve of singularities whose position relative to the equilibrium points in the phase plane determines the nature of the heteroclinic orbits, where they exist. Smooth solutions have trajectories connecting the steady states lying to one side of the singular curve. Travelling shock waves arise by connecting trajectories passing through a special point in the singular curve and recrossing the singular curve, by way of a iscontinuity. Hyperbolic partial differential equation theory gives the necessary shock condition. Conditions on the parameter values determine when the solutions are smooth travelling waves versus discontinuous travelling wave solutions. These conditions provide bounds on the travelling wave speeds, corresponding to bounds on the chemotactic velocity or bounds on cellular growth rate. This analysis gives rise to the possibility of representing sharp fronts to waves of invading cells through a simple chemotactic term, without introducing a nonlinear diffusion term. This is more appropriate when cell populations are sufficiently dense.

Impact and interest:

11 citations in Scopus
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12 citations in Web of Science®

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309 since deposited on 10 Jun 2004
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ID Code: 101
Item Type: Journal Article
Additional Information: The contents of this journal can be freely accessed online via the journal’s web page (see hypertext link).
Keywords: migration, chemotaxis, travelling wave, phase plane, shock
DOI: 10.1137/S0036139902404694
ISSN: 1095-712X
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Biological Mathematics (010202)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Copyright Owner: Copyright 2003 Society for Industrial and Applied Mathematics
Copyright Statement: Reproduced in accordance with the copyright policy of the publisher.
Deposited On: 10 Jun 2004
Last Modified: 29 Feb 2012 23:01

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