Global stability conjecture for locally stable steady states in competition models

Consider the nonlocal competition model described above, with two population densities and a locally stable steady state. Global stability conjecture. The locally stable steady state is globally asymptotically stable. This conjecture concerns whether local stability determines the global long-term behavior of the competition system. For the symmetric PDE case, it has been completely resolved, while the corresponding nonsymmetric nonlocal-dispersal setting is the subject of the paper.