Turing's diffusive threshold in random reaction-diffusion systems

TL;DR

This paper investigates how increasing the number of species in reaction-diffusion systems affects the diffusive threshold for Turing instabilities, finding that larger systems are more likely to exhibit physically realistic thresholds.

Contribution

It introduces a probabilistic analysis of the diffusive threshold in random reaction-diffusion systems with more than two species, revealing that larger systems tend to have lower, more physical thresholds.

Findings

Diffusive threshold decreases with more species, becoming more physically realistic.

Most instabilities in larger systems cannot be simplified to fewer species.

Probability of lower thresholds increases as the number of species grows.

Abstract

Turing instabilities of reaction-diffusion systems can only arise if the diffusivities of the chemical species are sufficiently different. This threshold is unphysical in most systems with $N=2$ diffusing species, forcing experimental realizations of the instability to rely on fluctuations or additional nondiffusing species. Here we ask whether this diffusive threshold lowers for $N>2$ to allow "true" Turing instabilities. Inspired by May's analysis of the stability of random ecological communities, we analyze the probability distribution of the diffusive threshold in reaction-diffusion systems defined by random matrices describing linearized dynamics near a homogeneous fixed point. In the numerically tractable cases $N\leqslant 6$, we find that the diffusive threshold becomes more likely to be smaller and physical as $N$ increases and that most of these many-species instabilities cannot be described by reduced models with fewer species.