We study Cauchy problem to scaled Lotka-Volterra model with 2 species :

 

, ,

 

where   and . Moreover, we consider strong competition case, that is, .

The existence and stability of traveling wave solution connecting 2 equilibrium was proved [2] [4]. With assumptions in these paper, we may observe the evolution solution converges in shape and behave like a wave solution as time large enough.

In Muratove's paper [3], he defined a propagation speed for evolution solution to system with gradient type reaction terms and proved it's monotone increasing. Now, we try to find such phenomenon or to proper define propagation speed without such structure. To do so, we do numerical simulations.

However, we have trouble to capture motions of evolution solutions when doing numerical simulations where we consider compact domain with Neumann boundary condition. The evolution solution will become uniform [1]. Moreover, Mottoni proved that it will converge to uniform or function. Inspired by Muratov's paper [3], we can use moving frame to capture traveling wave solution numerically. However, the propagation speed of evolution solution is not defined without gradient type structure.

Our main idea is that, the evolution solution will converge to traveling wave, so the limit of speed is wave speed. If we insist of using moving frame, we could approximate it by wave speed but might lose some information outside of domain while its converging process. To improve the simulation process, we consider time depending frame speed.