Competing deterministic growth models in two dimensions

TL;DR

This paper studies two-dimensional cellular automata where two competing colors spread from low initial densities, revealing phase transitions and power-law relationships that determine which color dominates or if space remains empty.

Contribution

It introduces a model of competing growth in two dimensions and characterizes phase behavior based on initial densities and dimensional spreading dynamics.

Findings

Three distinct phases depending on initial densities

Power-law relationships determine dominance of colors

Conditions for space remaining empty or one color prevailing

Abstract

We consider three-state cellular automata in two dimensions in which two colored states, blue and red, compete for control of the empty background, starting from low initial densities $p$ and $q$. When the dynamics of both colored types are one-dimensional, the dynamics has three distinct phases, characterized by a power relationship between $p$ and $q$: two in which one of the colors is prevalent, and one when the colored types block each other and leave most of the space forever empty. When one of the colors spread in two dimensions and the other in one dimension, we also establish a power relation between $p$ and $q$ that characterizes which of the two colors eventually controls most of the space.