Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation

Abstract

A family of travelling wave solutions to the Fisher–KPP equation with speeds c=±5∕6 can be expressed exactly using Weierstraß elliptic functions. The well-known solution for c=5∕6, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ending at the origin. For c=−5∕6, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite z. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a Fisher–Stefan type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed.