Invasive Plant Range Front (2D): Fisher-KPP Travelling Wave, Allee Effect, Wind, Road Corridors and Long-Distance Jumps

Presets

Live readouts

Graphs

Infested area (km2, density u above 0.3) against time

Equivalent radius R = sqrt(area/pi) against time, with the theory R0 + c t

Density profile along the east-west line through the source: the travelling wave now and 2, 4, 6 years ago

Numerical checks

running...

How to use

Pick a preset or move any slider: the model reacts live. The orange-red colour is the plant density, the white line (bright band in 3D) is the front u = 0.3. Click the map to plant a new infestation or to spray herbicide, depending on the click tool. Dashed arcs (birds in 3D) are long-distance jumps: cyan or green means a new focus was founded, grey means the seed fell on occupied or unsuitable ground. Space pauses, R restarts.

What to try: (1) Compare the measured front speed with the theory in the readout, then double r and watch c change by the square root of two. (2) Switch on the Allee law with start radius 8, then 16. (3) Add a 0.5 km/yr wind and read the downwind and upwind speeds. (4) Turn on jumps with a fat tail (alpha 1.1) and watch satellite colonies run ahead of the front. (5) Use the Road corridor preset to see the verge channel the invasion before the river stops it. (6) In Containment, chase the new foci with the herbicide before they merge.

The science

An invasive plant spreads as a wave of occupation. Everything the page shows comes from one partial differential equation for the local density u(x, y, t) of the plant, scaled to the local carrying capacity (u = 1):

∂u/∂t = ∇·( D m(x) ∇u ) − w·∇u + r h(x) g(u)

Here D (km²/yr) is the local dispersal coefficient of seeds and rhizomes, m(x) is a habitat multiplier (a river is a near-wall, a road verge is a corridor), w is a wind drift, r (1/yr) is the intrinsic growth rate and h(x) is habitat quality (0 in water, small on bare rock, 0.42 in shade, 1 in meadow). The growth law g is u(1−u) (Fisher, 1937) or, with a strong Allee effect, u(1−u)(u−a). One grid cell is 0.1 km, the valley is 12.8 km wide, a year is a year.

1. The pulled front. With g = u(1−u) any seed grows, however small. The advancing edge, where u is tiny, is a linearised problem u_t = D u_xx + r u, and the front moves as fast as the leading edge can grow and diffuse at the same time (Fisher; Kolmogorov, Petrovsky and Piskunov, 1937):

c* = 2 √(r D), front width ∼ 1/λ*, λ* = √(r/D)

Because the wave is pulled by this thin edge, the speed approaches c* from below with a logarithmic delay (Bramson, 1983): x_f(t) = c* t − (3/2λ*) ln t + const. The checks below use exactly that corrected speed, not the naive c*. In two dimensions the circular front is slower while it is small, dR/dt ≈ c − D/R.

2. The Allee effect. If small colonies lose (mate or pollinator limitation, seed predation), g = u(1−u)(u−a) is negative below the threshold a. Now the front is pushed by the dense bulk, and for this cubic the speed is known in closed form:

c = √(r D / 2) (1 − 2a), the wave reverses (colony retreats) for a > 1/2

A colony also needs a large enough nucleus: below a critical radius of the order of D/c the edge curvature wins and it dies out. Try the two Allee presets: same law, start radius 16 versus 8 cells.

3. Wind. An advection term shifts the wave: downwind c* + w, upwind c* − w (and the front stalls upwind once w > c*). The numerical test compares (c_down − c_up)/2 with w.

4. Long-distance jumps and stratified diffusion. Real invaders also travel by birds, animals, vehicles, floods and gardeners. Each year the infested area A produces about j A jumps. Their length follows a Pareto (power-law) tail, P(L > l) = (x_m/l)^α, in a random direction (biased downwind when there is wind). A jump only founds a new focus on suitable, empty ground, and with the Allee law a small propagule can fail. The result is stratified diffusion (Shigesada and Kawasaki): satellite colonies grow, merge, and the effective speed exceeds the diffusion speed. For a tail exponent α ≤ 1 there is no finite mean jump and the invasion keeps accelerating.

5. Landscape. The habitat map is generated, not photographed: meadow (h = 1), farmland (0.8), woodland (0.42, D x 0.75), road verge (0.7, D multiplied by your corridor slider), river (h = 0, D x 0.1), bridge (D x 1 across the river) and a rock ridge (h = 0.06). Click the map to plant a new infestation or to spray herbicide (with the residual option the treated soil keeps only 15 percent of its growth rate).

Numerics. Explicit finite volumes on a 128 x 128 grid, face diffusivities by the harmonic mean (flux-conservative), central advection, time step below min(dx²/(4 D m_max), CFL, 0.1/r), zero-flux boundaries, a deterministic random generator (mulberry32) for the jumps. The area, the front line and the profiles use the threshold u = 0.3.

Numerical checks run in the page. (1) A separate 1D solver (r = D = 1, dx = 0.25) against the Bramson-corrected Fisher speed. (2) The same solver with an Allee term against sqrt(rD/2)(1−2a). (3) With wind: the up/down asymmetry against w. (4) This very 2D engine run as a plane front on a uniform strip against the Bramson-corrected c*. (5) Mass balance on the real closed map: the change of ∑u equals the integral of the reaction term, which tests the conservative diffusion stencil.

Simplifications. One plant species with one density field (no age or seed-bank structure), deterministic local dynamics (no demographic noise at the front, which in reality slows pulled fronts), isotropic dispersal apart from wind and the map, a stylised landscape, jumps whose source is the occupied ground, and no climate. Parameter values are illustrative, not fitted to one species. The 3D view is a stylised knotweed-like plant, not a particular real species.

Sources to spot-check (from memory): Fisher 1937 (Ann Eugenics); Kolmogorov, Petrovsky, Piskunov 1937; Skellam 1951 (Biometrika); Bramson 1983 (Mem AMS); Lewis and Kareiva 1993 (Theor Popul Biol); Kot, Lewis, van den Driessche 1996 (Ecology); Shigesada and Kawasaki 1997 (Biological Invasions).