  We establish spreading properties of the Lotka-Volterra competition-diffusion
system. When the initial data vanish on a right half-line, we derive the exact
spreading speeds and prove the convergence to homogeneous equilibrium states
between successive invasion fronts. Our method is inspired by the geometric
optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis.
Our main result settles an open question raised by Shigesada et al. in 1997,
and shows that one of the species spreads to the right with a nonlocally pulled
front.
