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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 travelling 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, a self-replication
pattern and a chaotic pattern, is also constructed.