Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation. (April 2021) - McCue, Scott W.; El-Hachem, Maud; Simpson, Matthew J.
Abstract: A family of travelling wave solutions to the Fisher–KPP equation with speeds c = ± 5 ∕ 6 can be expressed exactly using Weierstra ß 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.