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 reaction-diffusion systems whose linearized dynamics near a homogeneous fixed point are given by a random matrix . In the numerically tractable cases of N\leqslant 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 .
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 .
R
generic
most
R
non-diffusing
nondiffusing
R
threshold for
probability distribution of the diffusive threshold in
R
whose
defined by random matrices describing
D
are given by a random matrix
D
of
R
generically decreases
becomes more likely to be smaller and physical
A
most of
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generally require all species to be diffusing
cannot be described by reduced models with fewer species