Emergence of Turing Patterns in a Simple Cellular Automata-Like Model via Exchange of Integer Values between Adjacent Cells

Takeshi Ishida

doi:10.1155/2020/2308074

Abstract

The Turing pattern model is one of the theories used to describe organism formation patterns. Using this model, self-organized patterns emerge due to differences in the concentrations of activators and inhibitors. Here a cellular automata (CA)-like model was constructed wherein the Turing patterns emerged via the exchange of integer values between adjacent cells. In this simple hexagonal grid model, each cell state changed according to information exchanged from the six adjacent cells. The distinguishing characteristic of this model is that it presents a different pattern formation mechanism using only one kind of token, such as a chemical agent that ages via spatial diffusion. Using this CA-like model, various Turing-like patterns (spots or stripes) emerge when changing two of four parameters. This model has the ability to support Turing instability that propagates in the neighborhood space; global patterns are observed to spread from locally limited patterns. This model is not a substitute for a conventional Turing model but rather is a simplified Turing model. Using this model, it is possible to control the formation of multiple robots into such forms as circle groups or dividing a circle group into two groups, for example. In the field of information networks, the presented model could be applied to groups of Internet-of-Things devices to create macroscopic spatial structures to control data traffic.

Highlights

Introduction eTuring pattern model was introduced by Alan Turing in 1952 [1] to account for morphogenesis via interactions between activating and inhibiting factors. is typical self-organization model is described by reaction–diffusion (RD) equations that use different diffusion coefficients for two morphogens that are equivalent to the activating and inhibiting factors. e general RD equations can be written as 휕푢 휕푡 = 푑1∇2푢 + 푓(푢, v), (1) 휕v 휕푡 푑2∇2v + 푔(푢, v), (2)
(b) A hexagonal grid is isotropic, whereas a square grid is not. Since this model includes the process of distributing tokens to adjacent cells, it is simpler to apply when the distances between adjacent cells are equal. is model can be applied to a square grid, but the pattern that is created is not isotropic
(d) Some of the tokens are distributed to adjacent cells. e tokens are created only in black cells but they can move on either black cells or white cells. e residual token ratio is the second parameter, which is the fraction of unchanged tokens in each cell. e parameter is fixed for all cells and all time-steps

Summary

Introduction

Turing pattern model was introduced by Alan Turing in 1952 [1] to account for morphogenesis (a developmental process of patterns found in nature) via interactions between activating and inhibiting factors. Is typical self-organization model is described by reaction–diffusion (RD) equations that use different diffusion coefficients for two morphogens that are equivalent to the activating and inhibiting factors. E general RD equations can be written as 휕푢 휕푡 = 푑1∇2푢 + 푓(푢, v), (1) 휕v 휕푡 푑2∇2v + 푔(푢, v), (2)

Methods

Results

Conclusion