Turing’s diffusive threshold in random reaction-diffusion systems

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 generic systems with N = 2 diffusing species, forcing experimental realizations of the instability to rely on fluctuations or additional non-diffusing 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 threshold for reactiondiffusion systems whose linearized dynamics near a homogeneous fixed point are given by a random matrix. In the numerically tractable cases of N ⩽ 6, we find that the diffusive threshold generically decreases as N increases and that these many-species instabilities generally require all species to be diffusing.