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February  2009, 25(1): 195-230. doi: 10.3934/dcds.2009.25.195

Singular travelling waves in a model for tumour encapsulation

John R. King 1, , Judith Pérez-Velázquez 1, and  H.M. Byrne 2,

1. 

Centre for Mathematical Medicine, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom, United Kingdom
2. 

Centre for Mathematical Medicine, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD

Received  August 2007 Revised  March 2008 Published  June 2009

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.
Keywords: reaction-convection-diffusion systems., singular travelling waves.
Mathematics Subject Classification: Primary 92C50, 35M10; Secondary 92B05, 34C6.
Citation: John R. King, Judith Pérez-Velázquez, H.M. Byrne. Singular travelling waves in a model for tumour encapsulation. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 195-230. doi: 10.3934/dcds.2009.25.195
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