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

考虑了具有竞争和扩散的三分量系统的行波解。假设第一和第二物种相对于第三物种运动相当缓慢。反应项中的其他参数被固定以满足以下条件：当第一物种不存在时，第二物种和第三物种可以稳定共存，并且当第二物种不存在时，第一物种和第三物种也可以稳定共存。也就是说，我们假设上述系统是连续的。首先，我们证明了连接两个稳定状态的行波解的存在性和稳定性。当这些稳定行波解在一维直线上相互碰撞时，我们发现了两类依赖于参数的情形：一类是阻塞现象，即它们被一个稳定的驻脉冲解阻塞。另一种是湮灭，即它们湮灭然后恢复稳定状态。接下来，我们将时间常数引入第一和第二方程。对于这些具有参数的系统，我们证明了存在多个行波解，并研究了它们的稳定性。最后，我们研究了二维平面上的平面行波解的稳定性，并证明了根据参数的不同，平面行波解也可以分为两类：一类是稳定的，另一类是不稳定的，它们的界面变得复杂。

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.