Linear determinacy on the propagation phenomena of the Lotka-Volterra competition system

In this paper, we mainly consider the speed selection problem for the classical Lotka-Volterra competition system. We propose a sufficient and necessary condition for this long-standing problem, which reveals the essence of linearly selected problem for monostable dynamical system. Moreover, by expanding the common parameter range, our result allows us to obtain a complete observation on the transition phenomenon between linear and nonlinear speed selection. AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35B40 (Asymptotic behavior of solutions).