Novel solutions for a model of wound healing angiogenesis

Harley, K., van Heijster, P., Marangell, R., Pettet, G.J., & Wechselberger, M. (2014) Novel solutions for a model of wound healing angiogenesis. Nonlinearity, 27(12), pp. 2975-3003.

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Abstract

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions.

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ID Code: 78753
Item Type: Journal Article
Refereed: Yes
DOI: 10.1088/0951-7715/27/12/2975
ISSN: 0951-7715
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) > APPLIED MATHEMATICS (010200) > Dynamical Systems in Applications (010204)
Divisions: Current > Institutes > Institute of Health and Biomedical Innovation
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Past > Schools > Mathematical Sciences
Funding:
Copyright Owner: Copyright 2014 Institute of Physics Publishing Ltd.
Deposited On: 18 Nov 2014 23:44
Last Modified: 10 Dec 2015 18:31

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