Non-Archimedean Models of Morphogenesis

TL;DR

This paper explores p-adic reaction-diffusion systems, revealing unique Turing patterns that form clusters with different internal patterns, differing from classical alternating domain patterns, and linking these to large network behaviors.

Contribution

It introduces a non-Archimedean (p-adic) framework for reaction-diffusion systems, establishing instability criteria and characterizing novel clustered Turing patterns.

Findings

Turing patterns in p-adic systems form clusters with diverse internal patterns.

Patterns are not classical alternating domains but consist of multiple clusters.

The patterns resemble those produced by reaction-diffusion on large networks.

Abstract

We study a p-adic reaction-diffusion system and the associated Turing patterns. We establish an instability criteria and show that the Turing patterns are not classical patterns consisting of alternating domains. Instead of this, a Turing pattern consists of several domains (clusters), each of them supporting a different pattern but with the same parameter values. This type of patterns are typically produced by reaction-diffusion equations on large networks.