Threshold Traveling Waves for a Grindrod Individual-Clustering Model with Bistable Growth: A Large-Speed Construction

We study threshold traveling waves in a one-dimensional Grindrod individual-clustering model with strong-Allee growth and an elliptically regularized directed velocity. The waves connect the stable positive equilibrium to the unstable Allee threshold, rather than to extinction. Unlike logistic growth, the strong-Allee law makes the directed movement aggregative on a threshold-adjacent density interval, where the formal unregularized effective diffusivity can become negative. For fixed a∈(0,1), δ∈(0,1) and ε,λ>0, we construct a smooth wave at every sufficiently large prescribed speed. The construction combines a phase-fixed scalar inverse with two complementary elliptic resolvent estimates. These distinguish the small Lipschitz constant needed for contraction from the smaller forcing that determines the correction: the coupled density differs from its scalar reference front by O(c−2). We identify the leading correction explicitly, prove strict decrease and confinement between the end states on an expanding interval, and localize any interior minimum lying between those states to within O(c−2) of an endpoint. A threshold-preserving cutoff converges to the original wave with an explicit two-term error bound. Numerical refinement and residual checks support the predicted large-speed scaling and illustrate the cutoff convergence. Global monotonicity, global uniqueness and nonlinear stability remain open.