Existence of travelling wave solutions for a model of tumour invasion

Harley, K., van Heijster, P., Marangell, R., Pettet, G.J., & Wechselberger, M. (2014) Existence of travelling wave solutions for a model of tumour invasion. SIAM Journal Applied Dynamical Systems, 13(1), pp. 366-396.

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Abstract

The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed.

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ID Code: 67124
Item Type: Journal Article
Refereed: Yes
Additional URLs:
Keywords: advection-reaction-diffusion systems, canards, singularly perturbed systems, travelling wave solutions
DOI: 10.1137/130923129
ISSN: 0036-1399
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > PURE MATHEMATICS (010100) > Partial Differential Equations (010110)
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) > APPLIED MATHEMATICS (010200) > Dynamical Systems in Applications (010204)
Divisions: Current > Schools > School of Mathematical Sciences
Copyright Owner: Copyright 2014 Society for Industrial and Applied Mathematics
Deposited On: 10 Feb 2014 22:47
Last Modified: 03 Apr 2014 01:50

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