  This is part two of our study on the spreading properties of the
Lotka-Volterra competition-diffusion systems with a stable coexistence state.
We focus on the case when the initial data are exponential decaying. By
establishing a comparison principle for Hamilton-Jacobi equations, we are able
to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin,
Evans and Souganidis. As a result, the exact formulas of spreading speeds and
their dependence on initial data are derived. Our results indicate that
sometimes the spreading speed of the slower species is nonlocally determined.
Connections of our results with the traveling profile due to Tang and Fife, as
well as the more recent spreading result of Girardin and Lam, will be
discussed.
