the Fisher-KPP equation
/ FISH-er; K-P-P (Kolmogorov-Petrovsky-Piskunov) /
How does an advantageous gene, or an invading species, spread across a continent? Fisher asked exactly this in 1937, and the equation that bears his name (and that of three Russian mathematicians, Kolmogorov, Petrovsky and Piskunov, who analysed it the same year) is the cleanest answer. It is the reaction-diffusion equation with logistic growth: locally the population grows fast when rare and saturates when crowded, and diffusion carries it into new territory. The result is a wave of advance moving at a definite speed.
The equation is u_t = D u_xx + r u(1 - u), where u(x, t) is the population density scaled so 1 is the carrying capacity. The reaction f(u) = r u(1 - u) has two fixed points: u = 0 (empty, unstable — any small population grows) and u = 1 (full, stable). A travelling-wave solution u(x, t) = U(x - c t) connects these two states: ahead of the front it is near 0, behind it is near 1, and it advances at speed c. Plugging the ansatz in turns the PDE into an ODE for the profile U, and a phase-plane analysis shows fronts exist for every speed c >= c_min = 2 sqrt(r D). The remarkable fact (Kolmogorov-Petrovsky-Piskunov) is that for natural compactly-supported initial data the solution selects exactly the MINIMAL speed c = 2 sqrt(r D) — the slowest admissible front is the one nature picks.
Why is this a landmark? It is the prototype of front propagation 'pulled' by growth at the leading edge, and the 2 sqrt(r D) speed appears all over biology, ecology, combustion and even the spread of ideas. It also teaches a subtle lesson about nonlinear PDEs: a whole family of travelling waves exists mathematically, but the dynamics selects one. The selected speed depends only on the LINEARIZATION at the unstable state (the growth rate r and diffusion D at u = 0), which is why these are called 'pulled' fronts — the front is dragged forward by exponentially small leading-edge population, not pushed from behind.
With D = 1 and r = 1, the minimal wave speed is c = 2 sqrt(1 * 1) = 2. Release a small lump of u near the origin: it spreads and steepens into a front whose position grows like 2 t, so after time t = 100 the wave has reached about x = 200. Doubling the diffusion D to 4 speeds the front only to 2 sqrt(4) = 4 — the speed scales as the square root, not linearly.
Logistic growth + diffusion gives a front advancing at speed 2 sqrt(r D).
A common misconception is that the front 'chooses' its speed by some global rule. In fact the selected speed is fixed by the LINEAR behaviour at the leading edge (the unstable state u = 0); these 'pulled' fronts contrast with 'pushed' fronts (e.g. with bistable reactions) whose speed depends on the full nonlinearity.