Modeling contact inhibition of growth: Traveling waves

Michiel Bertsch; Masayasu Mimura; Tohru Wakasa; Michiel Bertsch; Masayasu Mimura; Tohru Wakasa

doi:10.3934/nhm.2013.8.131

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 131-147. doi: 10.3934/nhm.2013.8.131

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Modeling contact inhibition of growth: Traveling waves

1.
Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, University of Rome Tor Vergata, Via dei Taurini 19, 00185 Rome
2.
Meiji Institute of Advanced Mathematical Sciences, Meiji University, 1-1-1, Higashi-mita, Tama-ku, Kawasaki, 214-8571
3.
Department of Basic Sciences, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyuchu, 804-8550

Received: 01 January 2012 Revised: 01 October 2012
Primary: 35R35; Secondary: 35Q92, 92C15, 92C17.

We consider a simplified 1-dimensional PDE-model describing the effect of contact inhibition in growth processes of normal and abnormal cells. Varying the value of a significant parameter, numerical tests suggest two different types of contact inhibition between the cell populations: the two populations move with constant velocity and exhibit spatial segregation, or they stop to move and regions of coexistence are formed. In order to understand the different mechanisms, we prove that there exists a segregated traveling wave solution for a unique wave speed, and we present numerical results on the ``stability" of the segregated waves. We conjecture the existence of a non-segregated standing wave for certain parameter values.
- Degenerate parabolic PDE,
- traveling wave solutions,
- free boundary,
- tumor growth model,
- population dynamics of cells
Citation: Michiel Bertsch, Masayasu Mimura, Tohru Wakasa. Modeling contact inhibition of growth: Traveling waves[J]. Networks and Heterogeneous Media, 2013, 8(1): 131-147. doi: 10.3934/nhm.2013.8.131

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Abstract

We consider a simplified 1-dimensional PDE-model describing the effect of contact inhibition in growth processes of normal and abnormal cells. Varying the value of a significant parameter, numerical tests suggest two different types of contact inhibition between the cell populations: the two populations move with constant velocity and exhibit spatial segregation, or they stop to move and regions of coexistence are formed. In order to understand the different mechanisms, we prove that there exists a segregated traveling wave solution for a unique wave speed, and we present numerical results on the ``stability" of the segregated waves. We conjecture the existence of a non-segregated standing wave for certain parameter values.

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