Open AccessSingular travelling waves in a model for tumour encapsulation
Author(s) -
John R. King,
Judith Pérez-Velázquez,
Helen M. Byrne
Publication year - 2009
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2009.25.195
Subject(s) - ode , traveling wave , singularity , convection , mathematical analysis , reaction–diffusion system , connective tissue , physics , mechanics , classical mechanics , mathematics , biology , genetics
In this paper we study an existing mathematical model of tumour encapsulation comprising two reaction-convection-diffusion equations for tumour-cell and connective-tissue densities. The existence of travelling-wave solutions has previously been shown in certain parameter regimes, corresponding to a connective tissue wave which moves in concert with an advancing front of the tumour cells. We extend these results by constructing novel classes of travelling waves for parameter regimes not previously treated asymptotically; we term these singular because they do not correspond to regular trajectories of the corresponding ODE system. Associated with this singularity is a number of further (inner) asymptotic regions in which the dynamics is not governed by the travelling-wave formulation, but which we also characterise