Turing’s Diffusive Threshold in Random Reaction-Diffusion Systems

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 $N⩽6$, 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 diffusing species.

Figure 1

Turing’s diffusive threshold for $N=2$. (a) Cartoon of the diffusive threshold and the fine-tuning (FT) problem for $R\approx 1$ and $R\gg 1$. The diffusivity difference required mathematically is unphysical in the hatched region $\mathcal{D}⩽D_2^{*}⩽D_2^{\max }$. (b) Distribution $P(D_2^{*})$, supported on the (scaled) interval $[1,D_2^{\max }(R)]$, estimated for different $R$. (c) Plot of $\mathbb{P}(D_2^{*}<\mathcal{D})$ [shaded areas in panels (a) and (b)] against $R$, revealing the diffusive threshold. Markers: estimates from panel (b); solid line: exact result [28] for $R>\mathcal{D}$ [35].

Figure 2

Results for $N=3$. (a) Contours of $D_3(d_u,d_v)$ in the positive $(d_u,d_v)$ quadrant. (b) Smoothed distribution $P(D_3^{*})$, estimated for different $R$. Inset: same plot, scaled to $[1,D_2^{\max }(R)]$ for comparison to $N=2$ in Fig. 1. (c) $\mathbb{P}(D_N^{*}<\mathcal{D})$ against $R$ for $N\in \{2,3\}$: the diffusive threshold lowers for $N=3$ compared to $N=2$. (d) Proportion ${\varphi }_N(\mathcal{D})$ of random Jacobians that have a physical Turing instability, plotted against $R$, for $N\in \{2,3\}$. Inset: proportion ${\tau }_N$ of random Jacobians that have a (physical or unphysical) Turing instability, averaged over $R$, for $N\in \{2,3\}$ [35].

Figure 3

Results for binary systems with $4⩽N⩽6$. (a) $\mathbb{P}(D_N^{*}<\mathcal{D})$ against $R$ for $4⩽N⩽6$, revealing further lowering of the diffusive threshold compared to $N=3$. (b) Proportion ${\sigma }_N$ of random stable kinetic Jacobians that have a (binary, if $N>3$) Turing instability, averaged over $R$, and plotted against $N$. (c) Proportion ${\varphi }_N(\mathcal{D})$ of random Jacobians that have a physical Turing instability plotted against $R$, for $3⩽N⩽6$. Inset: proportion ${\tau }_N$ of random Jacobians that have a (physical or unphysical) Turing instability, averaged over $R$, for $3⩽N⩽6$ [35].

Figure 4

Slow and fast diffusers in binary Turing instabilities with $3⩽N⩽6$. (a) Proportion $f_N$ of Turing unstable systems with $n⩾2$ fast diffusers plotted against $N$, averaged over $R$. (b) Proportion $r_N$ of systems that remain Turing unstable at $d=0$, plotted against $N$, averaged over $R$. Filled markers: all Turing unstable systems; unfilled markers: physical Turing unstable systems with $D_N^{*}<\mathcal{D}$ [35].