Traveling wave solutions for a competitive reaction–diffusion system and their asymptotics

For a competitive PDE system, we study the existence, uniqueness, and asymptotic behaviors of the traveling wave solutions connecting a monoculture equilibrium to a co-existence equilibrium. We use the method of upper–lower solutions to prove the existence of traveling wave solutions, and investigate the asymptotic behavior of the traveling waves in relation to various interacting parameters of the system. By comparing with the upper solution, we obtain asymptotic description of the solution for large x or t in relation to the interacting parameters, and show the uniqueness of traveling wave solutions connecting the two equilibria with such asymptotic rates. Numerical results are also presented to illustrate the theoretical results.