The Gray-Scott model is one of the most studied reaction-diffusion systems, first systematically explored by Pearson in his 1993 Science paper. He mapped the (F, k) parameter plane computationally and identified ~12 qualitatively distinct pattern regimes. The model has since become the canonical testbed for pattern formation, bifurcation theory, and nonlinear dynamics — essentially the hydrogen atom of reaction-diffusion systems.

The equations: du/dt = D_u∇²u − uv² + F(1−u) and dv/dt = D_v∇²v + uv² − (F+k)v. The condition D_u > D_v (substrate diffuses faster than activator) is the essential ingredient for pattern formation — long-range inhibition combined with short-range activation. The feed rate F replenishes U from a reservoir held at u=1; the kill rate k removes V.

The (F, k) plane is crossed by several codimension-1 bifurcation curves (saddle-node, Turing, Hopf). Small changes in F or k can push the system across a bifurcation, producing qualitatively different behavior. Try dragging the F slider while the simulation runs to see live transitions between regimes.

The simulation uses Forward-Time Centered-Space (FTCS) with a 5-point Laplacian, dt=1.0, dx=1.0, D_u=0.2097, D_v=0.105. Periodic boundary conditions wrap the grid like a torus. The stability constraint dt ≤ dx²/(4D_u) ≈ 1.19 is satisfied, so dt=1.0 is marginally stable. The scheme is O(dt, dx²) and trivially parallelizable on the GPU.