field-field.qmd

---
title: "Field–Field Interactions"
subtitle: "Reaction-diffusion dynamics"
---

## Overview

The field–field interaction governs the evolution of concentration fields through reaction-diffusion PDEs. Four models have been explored:

- [Brusselator](brusselator.qmd): Classic two-component system with cubic autocatalysis (hexagonal + stripes + labyrinthine/vermiform via nonlinear diffusion)
- [Gray-Scott](grayscott.qmd): Autocatalytic system with feed/kill dynamics (radial only)
- [FitzHugh-Nagumo](fhn.qmd): Excitable two-variable system from neuroscience (hexagonal + square)
- [Schnakenberg](schnakenberg.qmd): Minimal activator-inhibitor system (radial only, genuine steady state)

## General Form

$$
\frac{\partial C_i}{\partial t} = D_i \nabla^2 C_i + R_i(C_1, C_2, \ldots)
$$

where $D_i$ is the diffusion coefficient and $R_i$ is the reaction term specific to each model.

## Turing Instability

Pattern formation requires differential diffusion rates and appropriate reaction kinetics. The general condition for Turing instability in a two-component system:

1. **Homogeneous stability**: The uniform steady state must be stable without diffusion
2. **Diffusion-driven instability**: Diffusion destabilizes certain spatial modes

This requires an **activator** (short-range, slower diffusion) and an **inhibitor** (long-range, faster diffusion).

## Implementation

Field dynamics are computed on a discrete mesh using finite differences for the Laplacian. The mesh is coupled to particles through interpolation operators.

```
Mesh (field values) ←→ Particles (positions, velocities)
```