Convergence and wave-speed limit under fast-transition-rate scaling

Establish rigorous convergence, as the transition-rate scaling parameter tends to zero, of traveling-wave solutions of the fast-transition-rate scaled general go-or-grow reaction–diffusion system for migrating cells u and proliferating cells v (with linear diffusion in u, proliferation term g(u+v)v, and bidirectional transition rates α(u,v) and β(u,v) scaled by 1/ε) to the reduced limiting system defined by the algebraic balance α(u0,v0)u0 = β(u0,v0)v0 coupled to the evolution of the total population; determine the limiting invasion speed of the fast-transition-rate scaled system and ascertain how it relates to the minimum wave speed of the corresponding FKPP-type equation obtained under fast-transition-rate scaling.

Background

The paper introduces a fast-transition-rate scaling for the general go-or-grow model, where phenotype-switching rates are accelerated relative to diffusion and proliferation. Under this singular perturbation, the leading-order behavior is governed by an implicit algebraic constraint linking u and v and a PDE for the total population, which in special cases reduces to an FKPP-type equation.

While formal reductions are derived, the authors highlight an unresolved question regarding rigorous convergence of solutions—specifically traveling waves—as the scaling parameter goes to zero, and how the corresponding invasion speed behaves in the limit compared with the FKPP minimum wave speed.