Turbulence in diffusion replicator equation

Dynamical behaviors in the diffusion replicator equation of three species are numerically studied. We point out the significant role of the heteroclinic cycle in the equation, and analyze the details of the turbulent solution that appeared in this system. Firstly, the bifurcation diagram for a certain parameter setting is drawn. Then it is shown that the turbulence appears with the supercritical Hopf bifurcation of a stationary uniform solution and it disappears under a subcritical-type bifurcation. Secondly, the statistical property of the turbulence near the supercritical Hopf onset point is analyzed precisely. Further, the correlation lengths and correlation times obey some characteristic scaling laws.