Boundary-endpoint traveling waves with unequal diffusivities

Determine whether nonzero-speed bounded nonnegative traveling waves exist in the critical three-species competition-diffusion system when the diffusivities are unequal and the wave connects distinct boundary equilibria of the critical simplex.

Background

The paper studies bounded nonnegative traveling-wave profiles (U,V,W) connecting two distinct equilibria A and B on the critical simplex U+V+W=1 for the three-species competition-diffusion system. The authors establish nonexistence when both endpoints are interior points of the simplex, for all wave speeds, and also rule out standing waves for arbitrary endpoints and all-speed waves when the three diffusivities are equal.

The unresolved case is the combination of unequal diffusivities, nonzero wave speed, and boundary endpoints. The paper identifies waves connecting two vertices, such as (1,0,0) and (0,1,0), with the third species forming a localized positive pulse in the interior as a representative configuration. The obstruction is that the two-species phase-plane argument does not extend directly: at an extremum of 1-U-V-W, the relation U'+V'+W'=0 does not determine the sign of the diffusivity-weighted sum U'/d_1+V'/d_2+W'/d_3.