# Generalized patterns from local and non local reactions

## TL;DR

This paper introduces a novel pattern formation mechanism in networked systems with local and non-local reactions, analyzing stability and symmetry breaking without relying on classical Turing conditions.

## Contribution

It provides analytical insights into how homogeneous states become unstable through a new reactive Laplacian spectrum, enabling alternative pattern formation models.

## Key findings

- Homogeneous states can become unstable under external disturbances.
- A new reactive Laplacian spectrum influences stability.
- Patterns can emerge without classical Turing conditions.

## Abstract

A class of systems is considered, where immobile species associated to distinct patches, the nodes of a network, interact both locally and at a long-range, as specified by an (interaction) adjacency matrix. Non local interactions are treated in a mean-field setting which enables the system to reach a homogeneous consensus state, either constant or time dependent. We provide analytical evidence that such homogeneous solution can turn unstable under externally imposed disturbances, following a symmetry breaking mechanism which anticipates the subsequent outbreak of the patterns. The onset of the instability can be traced back, via a linear stability analysis, to a dispersion relation that is shaped by the spectrum of an unconventional reactive Laplacian. The proposed mechanism prescinds from the classical Local Activation and Lateral Inhibition scheme, which sits at the core of the Turing recipe for diffusion driven instabilities. Examples of systems displaying a fixed-point or a limit cycle, in their uncoupled versions, are discussed. Taken together, our results pave the way for alternative mechanisms of pattern formation, opening new possibilities for modeling ecological, chemical and physical interacting systems.