Global structure of traveling waves in 3-component competition-diffusion systems

Traveling wave solutions of three-component systems with competition and diffusion are considered. It is assumed that the first and second species move quite solwly relative to the third one. Other parameters in reaction terms are fixed to satisfy the following : When the first species is absent, the second and third ones can coexist stably and when the second one is absent, the first and third ones can also coexist stably. That is, we assume the above systems are bistable. First, we show the existence and stability of traveling wave solutions which connect two stable states. When we colide these stable traveling wave solutions each other on the one-dimensional line, we show that there are two categories depending on parameters : One is blocking phenomena, that is, they are blocked by a stable standing pulse solution. The other is annihilation, that is, they annihilate and then recover the stable state. Next we introduce a time constant to the first and second equations. For these systems with some parameters, we show there exist multiple traveling front solutions and study their stability properties. Finally we study the stability of planar traveling wave solutions on the two-dimensional plane and show that there are also two categories depending on parameters : One is stable, and the other is unstable and their interfaces become complicated.