Spatial pattern of discrete and ultradiscrete Gray-Scott model

Keisuke Matsuya; Mikio Murata

doi:10.3934/dcdsb.2015.20.173

Article Contents

2015, Volume 20, Issue 1: 173-187. Doi: 10.3934/dcdsb.2015.20.173

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Spatial pattern of discrete and ultradiscrete Gray-Scott model

Keisuke Matsuya^1, and
Mikio Murata^2,

1.
Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914
2.
Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho Koganei-shi, Tokyo 184-8588

Received: March 31, 2013

Revised: November 30, 2013

Published: December 2014

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Abstract

Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton. In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations. In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters. The resulting systems give a traveling pulse and a self-replication pattern with appropriate initial data and parameters. The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern. A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern and a self-replication pattern are also constructed.

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Mathematics Subject Classification: Primary: 39A12; Secondary: 35K57.

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References

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