Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: The strong-weak type with pushed front

We consider the classical two-species Lotka-Volterra competition-diffusion system in the strong-weak competition case. When the corresponding minimal speed of the traveling waves is nonlinearly selected, we show that the solution of the Cauchy problem uniformly converges to the minimal traveling wave in two different situations, for which the invading speed is locally determined: (i) one species is an invasive one and the other is a native species; (ii) both two species are invasive species.