Fisher–KPP equation
The Fisher–KPP equation combines logistic growth with diffusion and is a classical model of spatial invasion.
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}+ru\left(1-\frac{u}{K}\right).\]Here \(D\) controls spatial spreading, \(r\) controls local growth and \(K\) is carrying capacity.
Travelling fronts
The equation admits travelling-wave solutions connecting the low-density state to the populated state. For the classical equation, travelling fronts exist for speeds
\[c\ge 2\sqrt{rD}.\]Biological meaning
The model can represent the advance of an invading population into previously unoccupied habitat.
Key idea. Fisher–KPP dynamics combine local population growth with movement to produce a propagating invasion front.