Langevin approach for intrinsic fluctuations of chemical reactions with Hopf bifurcation

The characteristics of Langevin equations for the intrinsic fluctuations of chemical reactions are investigated via the analyses on the Brusselator model in the parameter domain of spirally stable focus. Two forms of Langevin equations are shown to be equivalent for the results of two statistical measures. The comparisons in the results of two measures between Langevin equations and chemical master equation are given: The difference in stationary probability densities is significant for systems close to the bifurcation point, even the system size is large; the power spectra of Langevin equations with white noise are qualitatively the same as that of chemical master equation, but the discrepancy is found between Langevin equations with colored noises and chemical master equation, if the length of correlation-time is comparable with the correlation-time of a system. As the linearized Langevin equations possess singularity at the supercritical Hopf bifurcation point for the statistical measures, the Langevin equations displace the bifurcation points, and the amount of displacement is a decreasing function of the system size and the length of correlation-time of noises.