Implementation of Lenia as a Reaction-Diffusion System

TL;DR

This paper explores the connection between Lenia, a cellular automaton, and reaction-diffusion systems, showing that asymptotic Lenia can be modeled by an RD system but not as a chemical system due to reaction kinetics.

Contribution

It demonstrates that asymptotic Lenia can be described by differential equations and replicated by a reaction-diffusion system, bridging CA and RD models.

Findings

Asymptotic Lenia is independent of time-step ticks.

Asymptotic Lenia can be described by differential equations.

RD Lenia cannot be interpreted as a chemical system.

Abstract

The relationship between reaction-diffusion (RD) systems, characterized by continuous spatiotemporal states, and cellular automata (CA), marked by discrete spatiotemporal states, remains poorly understood. This paper delves into this relationship through an examination of a recently developed CA known as Lenia. We demonstrate that asymptotic Lenia, a variant of Lenia, can be comprehensively described by differential equations, and, unlike the original Lenia, it is independent of time-step ticks. Further, we establish that this formulation is mathematically equivalent to a generalization of the kernel-based Turing model (KT model). Stemming from these insights, we establish that asymptotic Lenia can be replicated by an RD system composed solely of diffusion and spatially local reaction terms, resulting in the simulated asymptotic Lenia based on an RD system, or "RD Lenia". However, our RD Lenia cannot be construed as a chemical system since the reaction term fails to satisfy mass-action kinetics.