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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
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy; IAC-CNR, Rome, Italy email: bertsch.michiel@gmail.com
D. HILHORST
Affiliation:
CNRS, Laboratoire de Mathématique, Analyse Numérique et EDP, Université de Paris-Sud, F-91405 Orsay Cedex, France email: danielle.hilhorst@math.u-psud.fr
H. IZUHARA
Affiliation:
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1, Nakano, Nakanoku, Tokyo, 164-8525, Japan email: hiro.izuhara@gmail.com, mimura.masayasu@gmail.com
M. MIMURA
Affiliation:
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1, Nakano, Nakanoku, Tokyo, 164-8525, Japan email: hiro.izuhara@gmail.com, mimura.masayasu@gmail.com
T. WAKASA
Affiliation:
Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyushu, 804-8550, Japan email: wakasa@mns.kyutech.ac.jp

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.

Type
Papers
Copyright
Copyright © Cambridge University Press 2015 

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References

[1]Abercrombie, M. (1970) Contact inhibition in tissue culture. In Vitro 6, 128142.Google Scholar
[2]Bertsch, M., Dal Passo, R. & Mimura, M. (2010) A free boundary problem arising in a simplifies tumour growth model of contact inhibition. Interfaces Free Boundaries 12, 235250.Google Scholar
[3]Bertsch, M., Hilhorst, D., Izuhara, H. & Mimura, M. (2012) A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth. Differ. Equ. Appl. 4, 137157.Google Scholar
[4]Bertsch, M., Hilhorst, D., Izuhara, H., Mimura, M. & Wakasa, T. A limit problem for a nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, In preparation.Google Scholar
[5]Bertsch, M., Mimura, M. & Wakasa, T. (2012) Modeling contact inhibition of growth: Traveling waves. Netw. Heterogeneous Media 8, 131147.Google Scholar
[6]Biró, Z. (2002) Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type. Adv. Nonlinear Stud. 2, 357371.Google Scholar
[7]Chaplain, M., Graziano, L. & Preziosi, L. (2006) Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math. Med. Biol. 23, 197229.CrossRefGoogle ScholarPubMed
[8]Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7, 335369.CrossRefGoogle Scholar
[9]Hartman, Ph. (1964) Ordinary Differential Equations, J. Wiley & Sons, New York.Google Scholar
[10]Kolmogorov, N. S., Petrovsky, N. & Piskunov, I. G. (1937) Étude de l'équation de la diffusion avec croissance de la quantité de matière e son application a un problème biologique, Bull. Univ. État Moscou, Série internationale A 1, 126.Google Scholar
[11]Sherratt, J. A. (2000) Wavefront propagation in a competition equation with a new motility term modeling contact inhibition between cell populations. Proc. R. Soc. Lond. A 456, 23652386.CrossRefGoogle Scholar
[12]Zhu, H., Yuan, W. & Ou, C. (2008) Justification for wavefront propagation in a tumour growth model with contact inhibition. Proc. R. Soc. A 464 (2008), 12571273.Google Scholar