Dynamics of the Schlogl models on lattices of low spatial dimension

The steady states and dynamics of the two Schlogl models on one- and two-dimensional lattices are studied using master equation techniques in tandem with simulations. It is found that the classic bistable behavior of model II is modified to monostable behavior at low dimension. An explanation of this modification is proposed in terms of the effective potential that appears in the dynamical equations on considering the significant effect of fluctuation correlations. The behavior can be modeled by replacing the transient average fluctuation correlation by its asymptotic value plus Gaussian white noise and analyzing the resulting effective potential obtained from the Fokker-Planck equation with multiplicative noise. For model I the transcritical bifurcation point is shifted to lower values of the forward rate k(2) of the second step of the reaction scheme and this shift can also be explained via an effective potential as a function of the average asymptotic fluctuation correlation. Further addition of noise to the asymptotic value is irrelevant for this model since the noise term in the corresponding Fokker-Planck equation turns out to be purely additive.