Propagation phenomena for a two-species Lotka-Volterra strong competition system with nonlocal dispersal

This paper is concerned with the propagation phenomena for a two-species Lotka-Volterra strong competition system with nonlocal dispersal. We first establish the existence of bistable traveling waves by appealing to the theory of monotone semiflows. Then we use a dynamical systems approach to prove that such bistable traveling waves are asymptotically stable and unique modulo translation. Finally, we study the spreading properties of solutions for a class of initial conditions by the comparison arguments and the method of super- and subsolutions. It is shown that for initial conditions where both species u and v are initially absent from the right half-line x > 0, and the species v dominates the species u around x = -8 initially, if v spreads in absence of u slower than u in absence of v, then solutions of the initial value problem will approach a propagating terrace, which connects the unstable state (0, 0) to the stable state (1, 0), and then the stable state (1, 0) to the other stable state (0, 1).