Generative complexity of Gray-Scott model

TL;DR

This paper investigates the morphological complexity of the Gray-Scott reaction-diffusion system by quantifying the diversity and entropy of concentration profiles generated under various feeding and removal rates, revealing complex wave and localization patterns.

Contribution

It introduces a novel analysis of the Gray-Scott model's generative complexity using multiple entropy-based metrics and identifies conditions producing highly complex patterns.

Findings

Maximum complexity occurs with wave-fragments and traveling localizations.

Complexity correlates with specific feeding and removal rate parameters.

Gray-Scott patterns resemble phenomena in excitable media and cellular automata.

Abstract

In the Gray-Scott reaction-diffusion system one reactant is constantly fed in the system, another reactant is reproduced by consuming the supplied reactant and also converted to an inert product. The rate of feeding one reactant in the system and the rate of removing another reactant from the system determine configurations of concentration profiles: stripes, spots, waves. We calculate the generative complexity --- a morphological complexity of concentration profiles grown from a point-wise perturbation of the medium --- of the Gray-Scott system for a range of the feeding and removal rates. The morphological complexity is evaluated using Shannon entropy, Simpson diversity, approximation of Lempel-Ziv complexity, and expressivity (Shannon entropy divided by space-filling). We analyse behaviour of the systems with highest values of the generative morphological complexity and show that the Gray-Scott systems expressing highest levels of the complexity are composed of the wave-fragments (similar to wave-fragments in sub-excitable media) and travelling localisations (similar to quasi-dissipative solitons and gliders in Conway's Game of Life).