grayscott.qmd

---
title: "Gray-Scott Model"
subtitle: "Radial patterns without hexagonal symmetry breaking"
---

## Field Dynamics

The Gray-Scott model uses autocatalytic reaction with feed/kill dynamics:

$$
\frac{\partial U}{\partial t} = D_u \nabla^2 U - UV^2 + F(1-U)
$$

$$
\frac{\partial V}{\partial t} = D_v \nabla^2 V + UV^2 - (F+k)V
$$

where:

- $U$ is the "feed" species (replenished at rate $F$)
- $V$ is the "autocatalyst" (self-replicating via $UV^2$)
- $F$ is the feed rate
- $k$ is the kill rate

### Pattern Regimes (Pearson 1993)

| Regime | F | k | Pattern |
|--------|---|---|---------|
| α (alpha) | 0.010 | 0.047 | Spots |
| β (beta) | 0.026 | 0.052 | Replicating spots |
| γ (gamma) | 0.035 | 0.060 | Worms/labyrinths |
| δ (delta) | 0.028 | 0.053 | Mitosis |
| ε (epsilon) | 0.018 | 0.055 | Chaotic |
| λ (lambda) | 0.040 | 0.065 | Stripes |

### Parameters (Block 12)

| Symbol | Description | Current | Range |
|--------|-------------|---------|-------|
| $D_u$ | U diffusion | 0.20 | 0.1–0.5 |
| $D_v$ | V diffusion | 0.05 | 0.05–0.2 |
| $F$ | Feed rate | 0.035 | 0.01–0.06 |
| $k$ | Kill rate | 0.063 | 0.04–0.07 |
| time_scale | Dynamics speed | 50 | 1–100 |

::: {.callout-note}
## Block 12 Configuration
Diffusion ratio $D_u/D_v = 4$ maintained. time_scale increased to 50 to match Brusselator dynamics speed. Exploring F-k parameter plane to locate stripe/worm regime.
:::

---

## Particle Coupling

Same diffusiophoretic dynamics as Brusselator, but with U and V fields:

$$
\mathbf{v}_i = M_1 \nabla U\big|_{\mathbf{x}_i} + M_2 \nabla V\big|_{\mathbf{x}_i} + \sqrt{\frac{2}{\text{Pe}}} \boldsymbol{\eta}(t)
$$

::: {.callout-tip}
## Coupling Stabilized (Block 12)
Earlier challenges with negative V fields and NaN explosions have been resolved. The system now produces stable Turing patterns with particles organizing at field boundaries.
:::

---

## Current Status

::: {.callout-warning}
## Key Finding: No Hexagonal Symmetry Breaking
Gray-Scott produces **radial/concentric patterns only**. Even with extended simulation (6000 frames), no hexagonal symmetry breaking occurs. This is a fundamental difference from Brusselator dynamics.
:::

### Key Observations

- Gray-Scott produces **concentric ring patterns**, NOT hexagonal spots
- Qualitatively different from Brusselator — purely radial organization
- Extended simulation does not induce symmetry breaking
- Particles form voronoi-like network at field boundaries
- Moderate clustering (0.4) vs Brusselator's tight clusters (0.6+)

---

## Exploration Gallery

### Iter 9 (7/10)

![C1/C2 fields develop nested ring structures that evolve into multi-lobed flower-like asymmetric patterns. Three particle types stratify --- blue core, orange intermediate, green outer shell. Late frames show patterns breaking into elaborate multi-lobed structures with finer granularity than parent. Bottom panels show branching green/orange clusters dispersing outward with dendritic morphology.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_009.png){.lightbox group="gs"}

### Iter 14 (8/10)

![C1/C2 fields develop elaborate multi-lobed flower/yin-yang patterns with asymmetric branching. Three particle types create tissue-like architecture: blue forms innermost core clusters, orange occupies intermediate zones forming branch arms, green creates outermost boundary layer. Cross-type adhesion sharpens inter-type boundaries into discrete layered domains. Late frames show a distinctive flower/mandala morphology with 3--5 branching arms and clear type stratification.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_014.png){.lightbox group="gs"}