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Travelling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth

Published online by Cambridge University Press:  24 February 2015

M. BERTSCH ,
D. HILHORST ,
H. IZUHARA ,
M. MIMURA  and
T. WAKASA
M. BERTSCH
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy; IAC-CNR, Rome, Italy email: [email protected]
D. HILHORST
Affiliation:
CNRS, Laboratoire de Mathématique, Analyse Numérique et EDP, Université de Paris-Sud, F-91405 Orsay Cedex, France email: [email protected]
H. IZUHARA
Affiliation:
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1, Nakano, Nakanoku, Tokyo, 164-8525, Japan email: [email protected], [email protected]
M. MIMURA
Affiliation:
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1, Nakano, Nakanoku, Tokyo, 164-8525, Japan email: [email protected], [email protected]
T. WAKASA
Affiliation:
Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyushu, 804-8550, Japan email: [email protected]
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Abstract

We consider a cell growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell populations with contact inhibition. Numerical experiments show that there is a parameter regime where, for a large class of initial data, the large time behaviour of the solutions is described by a segregated travelling wave solution with positive wave speed c. Here, the word segregated expresses the fact that the different types of cells are spatially segregated, and that the single densities are discontinuous at the moving interface which separates the two populations. In this paper, we show that, for each wave speed c > c, there exists an overlapping travelling wave solution, whose profile is continuous and no longer segregated. We also show that, for a large class of initial functions, the overlapping travelling wave solutions cannot represent the large time profile of the solutions of the system of partial differential equations. The structure of the travelling wave solutions strongly resembles that of the scalar Fisher-KPP equation, for which the special role played by the travelling wave solution with minimal speed has been extensively studied.

Keywords

travelling wave solutionscontact inhibition of cellstumour growtha parabolic-hyperbolic systemFisher-KPP equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 3 , June 2015 , pp. 297 - 323
Copyright
Copyright © Cambridge University Press 2015 

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