Turing patterns in systems with high-order interactions

TL;DR

This work develops a general framework for Turing pattern formation in systems with high-order interactions modeled on hypergraphs, introducing nonlinear diffusive-like couplings that vanish at homogeneous states. By performing linear stability analysis and exploiting a Master Stability Function approach, the authors show that the interplay between interaction order and topology can either widen or restrict the parameter regions supporting diffusion-driven instabilities, depending on the diffusion coefficients and coupling structure. Analytically tractable cases (natural coupling and regular topologies) are complemented by numerical studies of general topologies, revealing that multi-body diffusion can enable patterns where pairwise interactions fail or suppress previously possible patterns. The results offer design principles for controlling spatial patterns in multispecies or many-body systems and point to extensions toward Turing waves and directed hypergraphs in future work.

Abstract

Turing theory of pattern formation is among the most popular theoretical means to account for the variety of spatio-temporal structures observed in Nature and, for this reason, finds applications in many different fields. While Turing patterns have been thoroughly investigated on continuous support and on networks, only a few attempts have been made towards their characterization in systems with higher-order interactions. In this paper, we propose a way to include group interactions in reaction-diffusion systems, and we study their effects on the formation of Turing patterns. To achieve this goal, we rewrite the problem originally studied by Turing in a general form that accounts for a microscropic description of interactions of any order in the form of a hypergraph, and we prove that the interplay between the different orders of interaction may either enhance or repress the emergence of Turing patterns. Our results shed light on the mechanisms of pattern-formation in systems with many-body interactions and pave the way for further extensions of Turing original framework.

Figures (6)

• Figure 1: Turing instability in networks with nonlinear diffusion. Right panel: Turing instability regions for the Brusselator model with $D_u^{(1)}=0.1$, $D_v^{(1)}=1$ and $\sigma_1=1$. The analytical red curves indicate the instability regions for the linear coupling, while the blue region is for the cubic coupling, as obtained numerically. We can observe that, for lower values of the parameters $(b,c)$ the cubic coupling allows for pattern formation, where the linear one does not, while for greater values we find the opposite situation. This can be visualized through the dispersion relations reported in the left panels: on the bottom left for lower values of the parameter (only the system subject to cubic coupling can go unstable), while on the upper left for greater values of the parameter (only the linear case yields patterns).
• Figure 2: Turing instability with nonlinear diffusion in high-order structures. Brusselator model with $b=3$, $c=3.5$, $D_u^{(1,2)}=0.1$, $D_v^{(1,2)}=2$, $\sigma_{1,2}=1$; the initial perturbation is $\sim 10^{-2}$. On the upper left panel, the dispersion law: the blue line is the continuous one, while the cyan dots indicate the discrete counterpart. On the upper right panel, Turing patterns for the $u$ species. On the bottom panel the hypergraph on which the system has been simulated: $11$ nodes, $11$ links and $3$$2$-hyperedges (i.e., triangles).
• Figure 3: Triangular $2$-lattice of $16$ nodes with periodic boundary conditions; Brusselator model with $b=5.5$, $c=7$, $D_u^{(1)}=1$, $D_v^{(1)}=0.5$, $D_u^{(2)}=0.1$, $D_v^{(2)}=1$, $\sigma_1=0.01$ and $\sigma_2=1$; the initial perturbation is $\sim 10^{-2}$. In the left panel, the dispersion law for the high-order case (blue line and magenta dots) compared with the case where only pairwise interactions are present (purple dashed line). In the right panel, an example of a Turing pattern for the $u$ species.
• Figure 4: Triangular $2$-lattice of $16$ nodes with periodic boundary conditions; Brusselator model with $b=5.5$, $c=7$, $D_u^{(1)}=0.1$, $D_v^{(1)}=1.5$, $D_u^{(2)}=1$, $D_v^{(2)}=0.5$, $\sigma_1=0.7$ and $\sigma_2=0.2$; the initial perturbation is $\sim 10^{-2}$. In the left panel, the dispersion law for the high-order case (blue line and magenta dots) compared with the case where only pairwise interactions are present (purple dashed line). As the high-order dispersion law is always negative, there is no emergence of Turing patterns, as also shown in the right panel for the $u$ species.
• Figure 5: All-to-all 2-hypergraph of $5$ nodes; Brusselator model with $b=5.5$, $c=7$, $D_u^{(1)}=1$, $D_v^{(1)}=0.1$, $D_u^{(2)}=0.07$, $D_v^{(2)}=1$, $\sigma_1=0.01$ and $\sigma_2=1$; the initial perturbation is $\sim 10^{-2}$. On the left panel, the discrete dispersion law; note that the continuous curve is a fictitious dispersion law, but it is a Master Stability Function, as explained in the text; for this reason the continuous curve can be thought as a function of a parameter $\gamma$. On the right panel, an example of a Turing pattern for the $u$ species.