  We address the uniqueness of the nonzero stationary state for a
reaction-diffusion system of Fisher-KPP type that does not satisfy the
comparison principle. Although the uniqueness is false in general, it turns out
to be true under biologically natural assumptions on the parameters. This
Liouville-type result is then used to characterize the wake of traveling waves.
All results are extended to an analogous nonlocal reaction-diffusion equation
that contains as a particular case the cane toads equation with bounded traits.
