Spatial pattern of discrete and ultradiscrete Gray-Scott model

TL;DR

This paper introduces a discretized and ultradiscretized Gray-Scott model that generates diverse spatial patterns, including traveling pulses, self-replication, and fractal structures, linking to cellular automata like Rule 90.

Contribution

It presents a novel discretization and ultradiscretization of the Gray-Scott model, producing complex patterns and connecting to cellular automata.

Findings

The ultradiscrete Gray-Scott model produces Sierpinski gasket patterns.

The models generate traveling pulses and self-replication patterns.

A 2+1D ultradiscrete model creates ring, self-replication, and chaotic patterns.

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.