Tactically-driven nonmonotone travelling waves
Models of cell invasion incorporating directed cell movement up a gradient of an external substance and carrying capacity-limited proliferation give rise to travelling wave solutions. Travelling wave profiles with various shapes, including smooth monotonically decreasing, shock-fronted monotonically decreasing and shock-fronted nonmonotone shapes, have been reported previously in the literature. The existence of tacticallydriven shock-fronted nonmonotone travelling wave solutions is analysed for the first time. We develop a necessary condition for nonmonotone shock-fronted solutions. This condition shows that some of the previously reported shock-fronted nonmonotone solutions are genuine while others are a consequence of numerical error. Our results demonstrate that, for certain conditions, travelling wave solutions can be either smooth and monotone, smooth and nonmonotone or discontinuous and nonmonotone. These different shapes correspond to different invasion speeds. A necessary and sufficient condition for the travelling wave with minimum wave speed to be nonmonotone is presented. Several common forms of the tactic sensitivity function have the potential to satisfy the newly developed condition for nonmonotone shock-fronted solutions developed in this work.
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|Item Type:||Journal Article|
|Keywords:||Travelling wave, Chemotaxis, Haptotaxis, Phase plane, Shock|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000)|
|Divisions:||Current > Institutes > Institute of Health and Biomedical Innovation|
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Current > Schools > School of Psychology & Counselling
|Copyright Owner:||Copyright 2008 Elsevier B V.|
|Copyright Statement:||This is the author’s version of a work that was accepted for publication in Physica D Nonlinear Phenomena. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D Nonlinear Phenomena, [VOL 237, ISSUE 5, (2008)] DOI: 10.1016/j.physd.2007.10.003|
|Deposited On:||21 Sep 2012 10:00|
|Last Modified:||26 Sep 2012 19:18|
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