Existence of Front–Back-Pulse Solutions of a Three-Species Lotka–Volterra Competition–Diffusion System

The existence of nonmonotone traveling wave solutions of the three-species Lotka–Volterra competition diffusion system under strong competition is established. A traveling wave solution can be considered as a heteroclinic orbit of a vector field in $${\mathbb {R}}^6$$ . Under suitable assumptions on parameters of the equations, we apply a bifurcation theory of heteroclinic orbits to show that a three-species traveling wave can bifurcate from two two-species waves which connect to a common equilibrium. The three components of the three-species wave obtained are positive and have the profiles that one is a front, one is a back, and the third component is a pulse between the previous two with a long middle part close to a constant. As applications of our result, we find several explicit regions of parameters of the equations where the bifurcation of three-species traveling waves occur.