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 $(b<a<1/c,\,d>0)$, 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-monotonic waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also hold for the critical speed case $s=s^*$. In addition, in the critical weak competition case $(b<a=1/c,\,d>0)$, we rigorously prove, for the first time, the existence of front-pulse traveling waves.