  In this paper, we consider the speed selection problem of the scalar
reaction-diffusion equations and the two-species Lotka-Volterra competition
systems. We propose a sufficient and necessary criterion for this long-standing
problem of monostable dynamical systems for the first time, from a different
point of view as compared to the classical result on single equations in
[Lucia-Muratov-Novaga, CPAM 2004]. Moreover, our results can further reveal the
essence of the linear determined problem from a new viewpoint on the decay rate
of traveling wave solutions. Our results seem to be the first that provide a
sufficient and necessary criterion for the speed selection of the two-species
Lotka-Volterra competition system.
