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

Asymptotic spreading of interacting species with multiple fronts II: Exponentially decaying initial data