Non-reciprocal interactions enhance heterogeneity

TL;DR

This paper addresses pattern formation in networks of immobile units connected by non-reciprocal long-range interactions. It develops a general mean-field framework using a non-symmetric consensus Laplacian $\mathcal{L}$ and a Master Stability Function to determine when a homogeneous equilibrium becomes unstable due to non-reciprocal coupling, even in the absence of diffusion. The authors derive an instability condition via polynomials $S_1(\xi)$ and $S_2(\xi)$ evaluated on the eigenvalues $\Lambda^{(\alpha)}$ of $\mathcal{L}$ and demonstrate, across multiple canonical models (Brusselator, Mimura–Murray, Volterra, FitzHugh–Nagumo, Stuart–Landau), that non-reciprocity can trigger spatial or temporal patterns by exploiting the complex spectrum. They provide a general sufficient condition for $d\geq 2$ that guarantees pattern formation under non-reciprocal interactions and show how to construct networks with prescribed spectra to realize these instabilities. The work broadens understanding of self-organization beyond diffusion-driven (Turing) mechanisms and offers a versatile approach for modeling heterogeneous patterns in ecological, chemical, and physical systems with non-reciprocal couplings.

Abstract

We study a process of pattern formation for a generic model of species anchored to the nodes of a network where local reactions take place, and that experience non-reciprocal long-range interactions, encoded by the network directed links. By assuming the system to exhibit a stable homogeneous equilibrium whenever only local interactions are considered, we prove that such equilibrium can turn unstable once suitable non-reciprocal long-range interactions are allowed for. Stated differently we propose sufficient conditions allowing for patterns to emerge using a non-symmetric coupling, while initial perturbations about the homogenous equilibrium fade away assuming reciprocal coupling. The instability, precursor of the emerging spatio-temporal patterns, can be traced back, via a linear stability analysis, to the complex spectrum of an interaction non-symmetric Laplace operator. Taken together, our results pave the way for the understanding of the many and heterogeneous patterns of complexity found in ecological, chemical or physical systems composed by interacting parts, once no diffusion takes place.

Figures (10)

• Figure 1: A schematic visual representation of the model \ref{['eq:sysnA']}. Nodes (large white circles) contain two species (blue and red dots). We then focus on the growth rate of the "blue" species in the $i$-th node, that results from two terms; the first one is the local reaction $\mathbf{f}$, represented by the light-blue and light-red curved arrows, involving only "blue" and "red" species indexed by $i$ (green oval). The second contribution arises from the long-range interactions $\mathbf{F}$, represented as violet triangles pointing from each nearby node to the $i$-th one. The average of these terms impacts the growth rate of the "blue" specie in node $i$ (orange oval).
• Figure 2: Instability region and patterns for the Brusselator model. In panel a) we report the region of the complex plane $(\Re \Lambda,\Im \Lambda)$ for which the instability condition is satisfied (grey). For the chosen parameters values ($b = 4.3$ and $c= 5.0$) we can observe that the instability region does not intersect the real axis and thus only non-reciprocal coupling can exhibit complex eigenvalues (white dots) entering into the instability region and thus initiate the pattern as shown in panel b) where we report $u_i(t)$ vs $t$ starting from initial conditions close to the stable equilibrium, $u_*=1$. Any symmetric coupling determines real eigenvalue (black dots in panel a)) that cannot give rise to the instability, as shown in panel c) where we report $u_i(t)$ vs $t$ starting from the same initial conditions used in panel b). The underlying coupling is obtained with a directed Erdős-Rényi network with $n=50$ nodes and a probability for a direct link to exist between two nodes is $p=0.05$.
• Figure 3: Instability region and patterns for the Mimura-Murray model. In panel a) we report the region of the complex plane $(\Re \Lambda,\Im \Lambda)$ for which the instability condition is satisfied (grey). For the chosen parameters values ($a = 35$, $b=15$, $c= 20$ and $d=2/5$) we can observe that the instability region does not intersect the real axis and thus only non-reciprocal coupling can exhibit complex eigenvalues (white dots) entering into the instability region and thus initiate the pattern as shown in panel b) where we report $u_i(t)$ vs $t$ starting from initial conditions close to the stable equilibrium, $u_* \sim 2.28$, $v_*\sim 3.20$. Any symmetric coupling determines real eigenvalue (black dots in panel a)) that cannot give rise to the instability, as shown in panel c) where we report $u_i(t)$ vs $t$ starting from the same initial conditions used in panel b). The underlying coupling is obtained with a directed Erdős-Rényi network with $n=50$ nodes and a probability for a direct link to exist between two nodes is $p=0.05$.
• Figure 4:
• Figure 5: Instability region and patterns for the FitzHugh-Nagumo model. In panel a) we report the region of the complex plane $(\Re \Lambda,\Im \Lambda)$ for which the instability condition is satisfied (grey). For the chosen parameters values ($\alpha=1.5$, $\beta=0$, $\gamma=2.5$ and $\mu = 0.01$) we can observe that the instability region intersects the real axis and thus both reciprocal (black dots) and non-reciprocal coupling (white dots) can exhibit eigenvalues entering into the instability region and thus initiate the pattern as shown in panels b) and c) where we report $u_i(t)$ vs $t$ starting from initial conditions close to the stable equilibrium, $u_*=0$ for the asymmetric coupling and the symmetric one. The underlying coupling is obtained with a directed Erdős-Rényi network with $n=30$ nodes and a probability for a direct link to exist between two nodes is $p=0.1$.