FisherKPPtype

FisherKPPtype refers to a class of reaction-diffusion equations of Kolmogorov–Petrovsky–Piskunov (KPP) type, used to model spreading phenomena in space and time.

Standard form: ∂u/∂t = D ∂^2u/∂x^2 + f(u), with diffusion coefficient D > 0 and a reaction term f satisfying f(0) = 0, f'(0) > 0, and f(u) ≤ f'(0) u for u in [0,1], with f(1) typically equal to 0. The nonlinearity is usually monostable, meaning the trivial state u = 0 loses stability and a positive conducting state can invade.

Traveling wave solutions exist: u(x,t) = U(x − ct) connecting equilibria 0 and 1, with a minimal wave speed c* = 2 sqrt(D f'(0)). Waves with speed c ≥ c* exist, and under the KPP condition the fronts are often described as pulled, meaning their speed is determined by growth in the leading edge where u is small.

Applications and extensions: FisherKPPtype models are used to describe spreading of advantageous genes in populations, ecological invasion, disease fronts, and sometimes combustion or flame fronts. Analytical results typically address existence and stability of traveling waves, speed selection, and how heterogeneity in space or higher dimensions modifies wave behavior.

History: The framework originates from the joint work of Fisher (1937) and Kolmogorov, Petrovsky, and Piskunov (1937), who introduced the equation to capture gene spread and population invasion. The Fisher–KPP terminology is often used to denote this monostable reaction-diffusion class.