Invasive Species Front: Reaction-Diffusion Spread Simulator

Invasion Parameters

Diffusion rate D 0.18

Invasive growth rate r_m 1.20

Competitive pressure α 1.30

Landscape State

Native species (resident) Invasive species (spreading) Bare / uncolonised ground

Invaded area

0.0%

Native remaining

100%

Front speed

— cells/gen

Generations

0.0

Theoretical Fisher-KPP speed 2√(D·r_m)

—

How it works

The landscape is a grid of cells, each holding a local density of native species n and invasive species m (0–1, fraction of local carrying capacity). Both grow logistically and compete; only the invader disperses between neighbouring cells:

dm/dt = D·∇²m + r_m·m·(1 − m − b·n)
dn/dt =        r_n·n·(1 − n − α·m)

D  = diffusion rate (how fast the invader spreads)
r_m, r_n = intrinsic growth rates
α  = competitive pressure the invader exerts on natives
b  = fixed resistance natives offer the invader

This is a Fisher-KPP reaction-diffusion system: local logistic growth plus spatial diffusion produces a travelling wave of colonisation with an asymptotic front speed c ≈ 2√(D·r_m), shown live next to the measured speed so you can see how closely a simple two-parameter formula predicts a much more complex, competing, two-species field.

D — raise it to model a fast disperser (windborne seeds, mobile animal); lower it for a slow crawler (a rhizome, a sessile invertebrate).
r_m — how quickly the invader reproduces once established; higher values steepen and speed up the front.
α — how aggressively the invader displaces natives where it is present; high α can hollow out native cover even where the invasion front has barely arrived.
Containment line — drops local diffusivity by 95% along one strip of the grid (a herbicide corridor, a mechanical-removal buffer, a quarantine fence). Whether it actually holds the front depends on whether growth pressure still lets the population "leak" a slow trickle across — a real dynamic behind eradication programmes for species like zebra mussels or Japanese knotweed.

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