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

A Liouville-type result for non-cooperative Fisher--KPP systems and nonlocal equations in cylinders