Skip to main content Accessibility help

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

  • M. BERTSCH (a1), D. HILHORST (a2), H. IZUHARA (a3), M. MIMURA (a3) and T. WAKASA (a4)...


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.



Hide All
[1]Abercrombie, M. (1970) Contact inhibition in tissue culture. In Vitro 6, 128142.
[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.
[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.
[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.
[5]Bertsch, M., Mimura, M. & Wakasa, T. (2012) Modeling contact inhibition of growth: Traveling waves. Netw. Heterogeneous Media 8, 131147.
[6]Biró, Z. (2002) Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type. Adv. Nonlinear Stud. 2, 357371.
[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.
[8]Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7, 335369.
[9]Hartman, Ph. (1964) Ordinary Differential Equations, J. Wiley & Sons, New York.
[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.
[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.
[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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed