🌊 Population Wave — Spatial Spread of Species

Diffusivity D (0.1–3.0) sets how fast individuals disperse by random movement; Growth rate r (0.1–2.0) sets the logistic reproduction rate; and Allee threshold a (0–0.5) sets the minimum density penalty used when the Allee effect toggle is enabled. Both D and r feed directly into the theoretical wave speed shown as 2√(Dr), so moving either slider updates that stat immediately, while Reset reseeds a fresh 8-cell-radius colony at the centre of the 150×150 grid.

Each frame runs four explicit Euler steps of the Fisher-KPP equation. The Laplacian is approximated with the five-point stencil (sum of four neighbours minus four times the centre cell, divided by Δx²=1), the time step is capped at dt = min(0.1, 1/(4D)) to satisfy the 2D diffusion CFL stability condition, and growth uses either ru(1−u) or, with Allee effect on, ru(u−a)(1−u). Boundaries are zero-flux, so density cannot leave the grid.

The simulation tracks the radius where density first drops to 0.5 by scanning outward from the centre, then divides the change in that radius by elapsed simulated time to get a measured speed, updated roughly every half a time unit. The "Theory 2√(Dr)" stat is the closed-form Fisher prediction computed directly from the current D and r sliders. Early in a run the measured speed is below theory while the front steepens from its initial smooth profile; after a short transient the two values converge.

With the Allee toggle off, growth is the standard logistic term ru(1−u), and any positive density eventually invades the whole grid. Switching it on changes growth to ru(u−a)(1−u): once local density falls below the Allee threshold a, that term turns negative, so patches below a shrink instead of growing. Raising a with the slider makes the effective invasion threshold stricter, and a high enough value can stall the front completely or make the colony collapse rather than spread.

Turning it on builds a per-cell diffusivity map where each of the 150×150 cells gets D multiplied by a random factor between 0.5 and 1.5, generated once at reset. During stepping, the simulation uses this local value instead of the uniform D everywhere the toggle is active, so the wave front advances faster through high-diffusivity patches and lags in low-diffusivity ones — producing the same kind of irregular, fingered spread seen when real invasions follow roads, rivers, or forest corridors rather than open, uniform ground.