Non-monotone traveling waves of the weak competition Lotka-Volterra system

We investigate traveling wave solutions in the two-species reaction-diffusion Lotka–Volterra competition system under weak competition. For the strict weak competition regime $ (b0) $, we construct refined upper and lower solutions combined with the Schauder fixed point theorem to establish the existence of traveling waves for all wave speeds $ s\geq s^*: = \max\{2, 2\sqrt{ad}\} $, and provide verifiable sufficient conditions for the emergence of non-monotone waves. Such conditions for non-monotone waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also holds for the critical speed case $ s = s^* $. In addition, in the critical strong-weak competition case $ (b0) $, we rigorously prove, for the first time, the existence of front-pulse traveling waves.