Global asymptotic stability conjecture for locally stable steady states in competition models

Consider the competition model described in the paper, in the parameter regime where both semi-trivial steady states exist. A steady state is called locally stable if it is locally asymptotically stable for the model's dynamics, and globally asymptotically stable if every relevant nonnegative solution converges to it as time tends to infinity. Global asymptotic stability conjecture. The locally stable steady state is globally asymptotically stable. This conjecture was proposed for competition models with random diffusion and has been completely resolved when $0<bc\leq 1$; the stated paper investigates global dynamics for symmetric nonlocal dispersals and develops methods applicable to more general dispersal strategies.