NonMonotonicity of Traveling Wave Profiles for a Unimodal Recursive System

Recursive systems of unimodal type may exhibit rich propagation dynamics. The shape of traveling wave profiles is one of the unsolved questions in this challenging topic. In this paper, we obtain criteria for the nonmonotonicity of wave profiles in terms of an eigenvalue problem. The proofs rely on establishing several nonlocal Harnack type inequalities, which may be of interest on their own. The existence and uniqueness of traveling waves are also obtained based on these inequalities.