Nonlinear multivariate selfconsistent Fokker-Planck equation for multicomponent reaction-diffusion systems

Mean field approximation is extended to multicomponent stochastic reaction-dif- fusion systems. A multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (Brusselator) with spatially correlated multiplicative noise, is obtained. The evolution of probability density and statistical characteristics of the system in the region of Turing bifurcation are studied. Numerical study of the equation solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple «repumping» of probability density through bimodality.